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1
GEOMETRIC INTEGRATORS FOR
CONTINUUM DYNAMICS
by
MATTHEW F DIXON
A thesis presented for the degree of
Doctor of Philosophy of the University of London
and the Diploma of Imperial College
Department of Mathematics
Imperial College
Huxley Building
180 Queens Gate
London SW7 2BZ
United Kingdom
JULY 2007
Copyright
Copyright in text of this thesis rests with the Author. Copies (by any process) either
in full, or of extracts, may be made only in accordance with instructions given by
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Details may be obtained from the Librarian. This page must form part of any such
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instructions may not be made without the permission (in writing) of the Author.
The ownership of any intellectual property rights which may be described in this
thesis is vested in Imperial College, subject to any prior agreement to the contrary, and
may not be made available for use by third parties without the written permission of
the University, which will prescribe the terms and conditions of any such agreement.
Further information on the conditions under which disclosures and exploitation may
take place is available from the Imperial College registry.
ii
Abstract
We develop a unified computational framework for deriving geometric integrators which
implement the conservation laws of Hamiltonian continuum dynamics in the convective
and spatial representations. In an abstract setting, these representations of continuum
dynamics correspond to the body and spatial representations of the rigid body (Holm
et al. 1986) - the latter of which has no spatial dependence. We apply the discrete
Clebsch approach (Cotter & Holm 2006) to the body representation of the rigid body
to give a momentum map through which aMoser-Veselov (MV) integrator is recovered.
The discrete Clebsch approach also gives a momentum map through which a MV in-
tegrator for motion in the spatial representation, with an advected quantity, is defined.
We then apply the discrete Clebsch approach to ellipsoidal and elastic rod continuum
motions. In each case, the discrete Clebsch approach gives momentum maps encoding
discrete conservation laws corresponding to those of the continuous system, which we
verify by numerical experiment.
Practitioners seek to implement geometric integrators for fluids through a unified
discrete framework. We turn to free-Lagrange methods, which compute the spatial
variables using a dynamically generated mesh. We investigate a variational formulation
of the free-Lagrange method for shallow water and show that the semi-discrete shallow
water equations conserve energy and have an associated divergence law. We, however,
obtain an evolution equation for the potential vorticity with a non-zero right hand side
attributed to the discrete curl of the discrete gradient operator. We apply symplectic
integrators to the semi-discrete scheme and present numerical results demonstrating
that the variational free-Lagrange method for rotating shallow water conserves energy
over long-time simulations and exhibits the geostrophic adjustment mechanism.
Finally, in Chapter 5 we describe the implementation of boundary conditions using
geometric integrators for fluid dynamics to address the problem of preserving sym-
plectic structure of a Hamiltonian particle-mesh method (Frank et al. 2002) for shallow
water with spatial velocity free-slip boundary conditions. We formulate the boundary
condition in terms of ghost particles and show by numerical experiment that this ap-
proach gives energy-conserving numerical approximations of rotating shallow water in
a bounded domain.
Acknowledgements
I would like to thank my supervisors, Darryl Holm and Sebastian Reich for their dir-
ection and extensive assistance with understanding the theory of geometric mechanics
and the development of the computational skills needed to produce the numerical res-
ults in this thesis. Thanks is extended to Colin Cotter for providing an abundant source
of ideas, assistance and encouragement. Others that deserve special mention are Mark
Petersen, Matthew Hecht, Beth Wingate, Todd Ringler and Jean-Luc Thiffeault.
I declare that the material presented in this thesis is my own work and any material
which is not my own has been acknowledged.
Signed: Matthew Francis Dixon Date: 4th July, 2007
iv
Contents
Copyright ii
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Extended Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Important related works . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Contributions of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.1 A computational framework . . . . . . . . . . . . . . . . . . . . . 14
1.4.2 Development of new DMV algorithms . . . . . . . . . . . . . . . 15
1.4.3 Geometric integrators for shallow water . . . . . . . . . . . . . . 16
1.5 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Moser-Veselov Integrators for Spatial and Body Representations of
Rigid Body Motions 20
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 The Free Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.1 Discrete velocities . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Discrete Constrained Variational Principle . . . . . . . . . . . . . . . . . 28
2.4 Symmetry Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Clebsch Potentials and Momentum Maps . . . . . . . . . . . . . . . . . 32
2.5.1 Geometric preliminaries . . . . . . . . . . . . . . . . . . . . . . . 32
2.5.2 The Clebsch approach . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5.3 The discrete Clebsch approach . . . . . . . . . . . . . . . . . . . 36
2.5.4 The body representation . . . . . . . . . . . . . . . . . . . . . . . 41
2.5.5 The spatial representation . . . . . . . . . . . . . . . . . . . . . . 46
2.6 Poisson Brackets on Semidirect Products . . . . . . . . . . . . . . . . . . 51
2.7 The Heavy Top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.7.1 The body representation . . . . . . . . . . . . . . . . . . . . . . . 54
2.8 The Coupled Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . 59
v
2.8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.8.2 The body representation . . . . . . . . . . . . . . . . . . . . . . . 60
2.9 The Cayley-Klein Parameterisation of Rigid Body Motion . . . . . . . . 66
2.9.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.9.2 Momentum maps and Hopf fibrations . . . . . . . . . . . . . . . 69
2.10 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.10.1 Body and spatial DMV algorithms for the rigid body . . . . . . . 72
2.10.2 Body DMV algorithm for the heavy top and coupled rigid body 78
2.10.3 Comparison with a Lie-Poisson integrator based on splitting . . . 82
2.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3 Moser-Veselov Integrators for Elastic Body and Rod Motions 88
3.0.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.1 Free Ellipsoidal Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.1.2 Convective and spatial representations of discrete ellipsoidal motion 94
3.2 The Pseudo-Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.2.1 Polar decomposition of discrete pseudo-rigid body motion . . . . 99
3.2.2 Conservation of circulation . . . . . . . . . . . . . . . . . . . . . 101
3.2.3 MV integrators for Mooney-Rivlin materials . . . . . . . . . . . . 103
3.3 Elastic Rod Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.3.1 The discrete Kirchhoff rod analogy . . . . . . . . . . . . . . . . . 105
3.3.2 Time dependent discrete Kirchhoff rod models . . . . . . . . . . 108
3.4 The Geometrically Exact Elastic Rod Model . . . . . . . . . . . . . . . . 112
3.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.4.2 The variational formulation of the geometrically exact elastic rod
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.5 The Discrete Geometrically Exact Elastic Rod Model . . . . . . . . . . . 115
3.6 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.6.1 DMV algorithms for pseudo-rigid bodies and elastic rods . . . . 118
3.6.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.7.1 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.7.2 Proceeding Chapters . . . . . . . . . . . . . . . . . . . . . . . . . 132
4 A Variational Free-Lagrange Method for Shallow Water 134
4.1 The Lagrangian Description of Shallow Water . . . . . . . . . . . . . . . 135
4.2 The Variational Free-Lagrange Method for Shallow Water . . . . . . . . 137
4.2.1 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . 139
vi
4.3 The Variational Free-Lagrange Equations for 1D Shallow Water . . . . . 139
4.3.1 Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.4 A Variational Free-Lagrange method for 2D Shallow Water . . . . . . . 141
4.4.1 Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.5 The Shallow Water Vorticity Equation . . . . . . . . . . . . . . . . . . . 144
4.6 The Semi-Discrete Divergence Form of the Shallow Water Equations . . 148
4.7 A Symplectic Time Stepping Scheme . . . . . . . . . . . . . . . . . . . . 149
4.8 1D Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.8.1 Experiment 1: conservative properties of VFL . . . . . . . . . . . 151
4.8.2 Experiment 2: geostrophic adjustment . . . . . . . . . . . . . . . 154
4.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5 A Hamiltonian Particle Mesh Method for ShallowWater in a Bounded
Domain 161
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.2 The Hamiltonian Particle Mesh Approximation . . . . . . . . . . . . . . 163
5.2.1 Symplectic time stepping . . . . . . . . . . . . . . . . . . . . . . 167
5.3 Layer Depth Smoothing on a Bounded Domain . . . . . . . . . . . . . . 168
5.3.1 A smoothing operator for 1D shallow water . . . . . . . . . . . . 169
5.3.2 A smoothing operator for 2D shallow water . . . . . . . . . . . . 170
5.4 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.4.1 HPM for 1D (non-rotating) shallow water . . . . . . . . . . . . . 172
5.4.2 The 2D (non-rotating) shallow water equations in a channel . . . 173
5.4.3 Rotating shallow water in a channel . . . . . . . . . . . . . . . . 176
5.5 Summary and Further Research . . . . . . . . . . . . . . . . . . . . . . . 178
6 Summary 179
6.1 Contribution of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.1.1 Development of a unified framework . . . . . . . . . . . . . . . . 179
6.1.2 Development of new DMV algorithms . . . . . . . . . . . . . . . 181
6.1.3 Geometric integrators for shallow water . . . . . . . . . . . . . . 182
6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.2.1 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
A Properties of MV Integrators 193
A.1 Body and Spatial Representations in Continuous
and Discrete Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
A.2 MV Integrators for the Cayley-Klein Parameterisation of Rigid Body
Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
vii
A.3 The Spatial DMV Algorithm for the Rigid Body . . . . . . . . . . . . . 196
A.4 A DMV Algorithm for Coupled Rigid Body Motion . . . . . . . . . . . . 196
B The Variational Description of Elastic Body and Rod Models 198
B.1 The Anisotropic Pseudo-Rigid Body . . . . . . . . . . . . . . . . . . . . 198
B.2 The Geometrically Exact Elastic Rod Model . . . . . . . . . . . . . . . . 200
C Additional aspects of the Variational Free-Lagrange Method 204
C.1 Canonical Formulation of the Variational Free-Lagrange Method . . . . 204
C.2 Representation on a Fixed Grid . . . . . . . . . . . . . . . . . . . . . . . 205
viii
Chapter 1
Introduction
1.1 Motivation
Geometric integrators transfer powerful concepts in geometric mechanics to compu-
tational continuum dynamics by preserving properties of the continuous system such
as the geometric structure, symmetries and phase space volume. There are numerous
reasons for using geometric integrators to preserve these properties in computer simu-
lations. Intuitively, one might expect that integrators which preserve properties of the
continuum produce solutions that capture the qualitative features of the continuum dy-
namics too. An increasingly diverse range of continuum models would indeed confirm
this to be the case.
Zero-Helicity flows For a captivating example of this, we refer the reader to the
work of Holm & Kimura (1991) who apply a volume preserving integrator to the in-
tegrable and zero-Helicity Chandrasekhar flow. This flow is used to the study of the
onset of Rayleigh Benard convection in incompressible fluids. The intricate spatially
periodic heteroclinic network of saddle-focus connections which is present analytically
is also exhibited by the numerical solution.
Liquid Crystal Devices LCDs operate on the principle that a suitable applied
field will change the orientation of the liquid crystals, while conserving the pointwise-
norm. Lewis & Nigam (2003) presents a norm-preserving integrator for micromagnetic
applications which have the same constraints. By constraining the geometric integrator
to the sphere, the authors are able to demonstrate a number of versatile and robust
algorithms for modelling micromagnetic applications which are equally applicable to
LCDs.
1
Elastic rods Elastic rods are a core model for studying a wide range of applications
in engineering science and increasingly biomechanics. Idealised rod models are used
to study dynamical properties of rods leading to critical insight into their failure and
fatigue mechanisms under dynamic loads. Although the implications of symmetries
in these models on their dynamics is not fully understood, the preservation of these
symmetries under discretisation has motivated the development of a class of geometric
integrators referred to as Mechanical integrators (see Barth, Leimkuhler & Reich 1999,
Gonzalez & Simo 1996, Marsden, O’Reilly, Wicklin & Zombro 1991). Most crucially,
geometric integrators enable us to understand the role of symmetries and correspond-
ing conservation laws in the dynamics of elastic rods and in the very least provide a
benchmark for more established, classical, numerical methods for rod models in which
numerical dissipation hinders the integrity and efficiency of simulations.
Geophysical fluid dynamics The free evolution of many idealised geophysical flu-
ids are constrained by the existence of conservation laws. Ripa (1981) points out that
these are most easily found in the Lagrangian description because they correspond to
symmetries of Hamilton’s principle. For example a particle relabelling symmetry in
shallow water models gives Ertel’s theorem for potential vorticity conservation. In con-
trast, use of a beta-plane approximation in Hamilton’s principle breaks homogeneity in
the meridional direction and isotropy in the horizontal plane. Consequently a conser-
vation law for Enstrophy is not present as is in an f-plane approximation. Hamilton’s
principle is not only a good starting point for finding conservation laws under various
geophysical approximations but also for deriving new geophysical models (see Salmon
1983, for an introduction to a balanced model now referred to as L1 dynamics). A
discrete variational approach would appear to be the only way for a numerical analyst
to formulate a computational analogue to many of these models without ”exercising
his bias” (Salmon 1983).
Numerical stability Conservation of energy is often regarded as a strong manifest-
ation of unconditional numerical stability (Lewis & Simo 1994). The ever diversifying
set of computational models which crucially rely on the ability to tractably simulate
long-time dynamics, drives the need to develop new computationally competitive geo-
metric integrators. Implementers of global climate models, for example, are challenged
by the need to resolve complex geophysical fluid dynamics over long periods. The be-
nefits of geometric integrators are four-fold. Firstly, by conserving energy, geometric
integrators enable the simulation to be run at larger time-steps offering potential com-
putational savings. To illustrate this point, the reader should refer to the description of
the Stormer Verlets integrated particle shallow water (Frank, Gottwald & Reich 2002,
Frank & Reich 2004) and particle Euler Slice (Dixon & Reich 2004) models.
2
Conservation laws Secondly, geometric integrators for continuum dynamics provide
the capability to selectively resolve the dynamical features which are manifestations of
the conservation law. For example, one can introduce a discrete Bjerknes’ circulation
theorem (an extension of Kelvin’s circulation theorem to include forcing terms) in an
idealised oceanographic model in contrast to the traditional and brute-force approach
of adaptively resolving dynamical features by grid refinement (without an explicit rep-
resentation of a discrete circulation theorem). This paradigm is a direct extension of
geometric integrators for ordinary differential equations, where symplectic integrators
have largely superceeded adaptive generic Runge-Kutta methods as a more efficient nu-
merical approach to simulating Hamiltonian dynamics. Certainly, the inadequacy and
inability of existing global climate prediction models to recover observable features, such
as the western boundary current intensification (Stommel 1948), suggests the need for a
new generation of numerical methods. This generation of numerical methods shall im-
plement a conferred mathematical understanding of the dynamics into computational
models, in turn providing new scientific insight.
Unified computational framework Of course, this idea is far from new, but the
idea of deriving geometric integrators for continuum dynamics in a unified and consist-
ent way has arguably been a more recent pursuit (see Lewis & Simo 1994). Starting with
the priviledged mathematical insight that geometric mechanics has provided, one can
systematically derive variational integrators by discretising Hamilton’s action principle
and using standard methods of variational calculus. The theoretical investigations into
this approach are partially listed later in this Chapter, but the real hurdle is convincing
practitioners to adopt this approach - for numerous good reasons too.
Firstly, there is the need to compound the literature on geometric continuum dy-
namics and geometric integrators to provide an accessible and pedagogical context
for implementors of geometric integrators for continuum dynamics. In particular this
should address how to systematically recover geometric integrators for the various rep-
resentations of continuum dynamics, paying close attention to the classification of geo-
metric structure that is preserved when physical quantities are advected by the flow.
Secondly, there is the need to understand how to cast existing numerical methods
for continuum dynamics into a variational framework to derive geometric variants of
established approaches. The reason for doing this might be to simply improve the
stability of the scheme through application of a symplectic integrator to a Hamiltonian
construction of it, or to transfer concepts in geometric mechanics to the computational
model.
Thirdly, numerous computational aspects remain unaddressed, but the implement-
ation of boundary conditions in a framework for computational dynamics must surely
3
be one of the most fundamental of these. Finally, a much neglected aspect of numer-
ical methods for continuum dynamics is their ability to systematically be verified upon
implementation. Conserved quantities provide additional measures with which to as-
sess the validity of implementation. Moreover, a variational framework systematically
provides the exact form of these discrete conserved quantities.
1.2 Extended Synopsis
Holm, Marsden & Ratiu (1986) present a unified geometric approach for the study of
idealised Hamiltonian continuum models (fluids, plasmas, elasticity, etc.) in the ma-
terial, inverse material, spatial and convective representations. This approach is based
on maps, referred to as momentum maps, which carry the Poisson brackets in one
representation into another. In this thesis, we investigate a corresponding computa-
tional variational framework for the convective and spatial representations of continuum
dynamics in which momentum maps are fundamental to the derivation of geometric
integrators which preserve Lie-Poisson structure1.
Holm et al. (1986) show that the convective and spatial representations of continuum
dynamics correspond to the body and spatial representations of rigid body dynamics.
The rigid body therefore serves as a suitable starting point for developing a unified
computational framework.
The discrete Clebsch approach We begin by applying the discrete Clebsch ap-
proach of Cotter & Holm (2006) to formulate geometric integrators for the body and
spatial representation of rigid body dynamics. The distinguishing feature of the dis-
crete Clebsch approach arises from its construction in a reduced discrete Hamilton’s
action principle. By adding so called Clebsch constraints for the reconstruction of the
dynamics on the full phase space, the discrete Clebsch approach gives the momentum
maps for the cotangent lifted actions to the reduced phase space. These momentum
maps are the conserved quantities associated with the Noether symmetries, see Noether
(1918). Through the momentum map associated with symmetry reduction to the body
representation of the reduced dynamics, the discrete flow on phase space is the same
integrable discretisation of rigid body dynamics discovered by Moser and Veselov in
1991. These integrators are commonly referred to as Moser-Veselov integrators and are
well established for the body representation of the rigid body.
The spatial representation The spatial representation of the dynamics of the rigid
body differs from the body representation by the presence of an advected quantity- the
inertia matrix rotates in the spatial frame. Following Holm et al. (1986), we augment
1All of this terminology is conventional in geometric mechanics (see Marsden & Ratiu 1999)
4
the Lagrangian with the advected quantity and apply the discrete Clebsch again to give
a momentum map through which the MV integrator for the spatial representation is
defined. We will show that the spatial representation of the MV integrator matches the
form of the discrete Euler-Poincare equations derived by Bobenko & Suris (1999a) in a
discrete Euler-Poincare framework. Bobenko & Suris (1999a) showed that the discrete
EP equations for the spatial representation of the dynamics are Lie-Poisson w.r.t. to
the dual of a semi-direct product Lie algebra, a result which we verify for the spatial
MV integrators.
Spatial versus body discrete Moser-Veselov (DMV) algorithms Informally
put, semi-direct product group actions define how advected quantities feedback on the
body dynamics and are fundamental to the Lagrangian description of dynamics with
advected quantities. Given the significance of semi-direct product in continuum dynam-
ics, we show how to modify the DMV algorithm, developed by McLachlan & Zanna
(2005) for the body representation of rigid bodies, to solve the spatial MV integrat-
ors. We provide several numerical experiments to study the comparative conservative
properties of the spatial DMV algorithm with the body DMV algorithm.
Heavy tops and coupled rigid bodies We apply the discrete Clebsch approach to
give MV integrators for the body representation of heavy tops and coupled rigid bodies
respectively as these provide examples of motions with potential and coupling terms
and, in the latter case, can only be solved by adapting the DMV algorithms. These
examples prepare us for more challenging examples in elastic dynamics which feature
potential forces and coupling motions.
Cayley-Klein parameters Before applying the discrete Clebsch approach to the
convective and spatial representations of elastic dynamics, we will show that the dis-
crete Clebsch approach also gives MV integrators for the Cayley-Klein parameterisa-
tion of the rigid body (see Whittaker 1944). MV integrators for rigid body motions
formulated as SO(3) matrices, represent the attitude of the rigid body as a three-stage
rotation by each Euler angle about its corresponding principal axis. One could para-
meterise the rigid body dynamics in a minimal representation in terms of these angles.
This parameterisation is well know to exhibit singularities however (see Leimkuhler &
Reich 2005). Parameterisation of the rotation in terms of four Cayley-Klein parameters
provides a double covering of the rigid body configuration to avoid singularities. These
parameters constitute the group of unit quaternions which is isomorphic to the matrix
group SU(2). Leimkuhler & Reich (2005) point out that it is better to use rotation
5
matrices for parameterisation, despite the introduction of redundant variables together
with constraints, because this leads to a global parameterisation.
We will show that the discrete Clebsch approach can be easily cast in terms of SU(2)
matrices to give a MV integrator for the Cayley-Klein parameterisation of rigid body
motion together with the conserved momentum maps corresponding to symmetries of
the discrete Lagrangian.
Ellipsoidal motion The rigid body example serves only to present the geometric
principles governing our framework. We will further the development of our computa-
tional framework by applying the discrete Clebsch approach to give MV integrators for
the convective and spatial representations of ellipsoidal motion on the group GL(n)+.
In this generalised model, either the shape matrix, describing the shape of the ellipsoid
in an Eulerian frame, or the right Cauchy-Green matrix, describing the shape of the
Eulerian frame in the body, are advected quantities depending on whether the motion
is in the spatial or convective representation. In each case, the MV integrators define
a co-adjoint action on the dual of a semi-direct product Lie-algebra and preserve the
Lie-Poisson bracket defined on the dual of this Lie-algebra.
The Pseudo-rigid body In the absence of a generalised DMV algorithm for MV
integrators on GL(n)+, we will apply a polar decomposition to a special case of ellips-
oidal motion on GL(3)+, referred to as isotropic pseudo-rigid body motion, in which
the body is initially spherical. Applying the discrete Clebsch approach gives MV in-
tegrators for the polar components of pseudo-rigid body motion. These components
describe the orientation, stretching and internal circulation of the rigid body, where the
internal circulation is coupled to the orientation through a Coriolis term. The adapted
DMV algorithm for solving these coupled polar component MV integrators is based on
the DMV algorithm which we develop for the coupled rigid body.
Conservation Laws The polar decomposed isotropic pseudo-rigid body is invariant
under left and right actions of SO(3). The discrete Clebsch approach gives the left and
right conserved momentum maps corresponding to these Noether symmetries. These
momentum maps are the angular momentum and vorticity of the internal circulation.
We will show that the conservation of vorticity is the discrete Kelvin circulation law
for ellipsoidal motions. This law corresponds to a relabelling symmetry and is central
to the geometric description of idealised continuum dynamics.
Kirchhoff elastic Rods So far, we have only described the application of our frame-
work to the spatial and convective representations of discrete time finite dimensional
motions, yet, continuum motion is infinite dimensional. One of the simplest examples
6
of such motions is that of the Kirchhoff rod which is defined by a continuum of or-
thonormal frames describing the orientation of an inextensible rod. It is well known
that the static configuration of the Kirchhoff rod is in one-to-one correspondence with
the motion of the Lagrange top, by a result known as the Kirchhoff kinetic analogy.
We will show that the analogy only holds in the discrete case, i.e. when there are
a finite number of orthonormal frames and time steps, if the spatial discretisation of
the rod corresponds to the temporal discretisation of the Lagrange top. This analogy
extends to the correspondence between dynamical Kirchhoff rods and sequences of
elastically coupled Lagrange tops. It follows that the discrete rod model then exhibits
a discrete compatibility equation in the same form as the discrete auxiliary equation for
the relative orientation matrix in the discrete coupled rigid body model.
Geometrically exact elastic rod models Following Krishnaprasad, Marsden &
Simo (1988), we will apply our computational framework to the convective represent-
ation of a fully non-linear rod model which bends, twists, shears and stretches. This
elastic rod model is referred to by Krishnaprasad et al. (1988) as a geometrically exact
elastic rod model. This model represents the motion of the rod as an elastic coupling
of its line of centroids and orthonormal frames. We will show that the discrete Clebsch
approach gives the conserved momentum map for the total spatial angular momentum
and use numerical experiments to show that the DMV algorithm is conservative.
Geometric integrators for Hamiltonian fluids Implementation of geometric in-
tegrators seeks to transfer powerful concepts in the convective and spatial representa-
tions of Hamiltonian fluid dynamics using a unified framework. The configuration space
for fluids is the infinite dimensional group of diffeomorphisms. Symmetry reduction by
this group action to the convective and spatial representations is challenging in the
discrete case.
This thesis considers geometric integrators for Hamiltonian fluids which preserve the
canonical symplectic structure of a finite dimensional system of fluid particles and are
expressed in terms of the material velocity and pressure field. The challenging problem
of developing a unified discrete framework for fluids based on the finite dimensional
representation of the group of diffeomorphisms developed by Zhong & Scovel (1994)
and Zeitlin (2004) is not considered here.
Variational free-Lagrange method We will consider Hamiltonian shallow water
as a model for furthering application of our framework. This model describes the
motion of an incompressible inviscid layer of fluid, under the influence of gravity and
Coriolis forces. Following the method described by Augenbaum (1984), we spatially
discretise the shallow water action principle for a free-Lagrange method. This method
7
represents the material velocity at the position of point particles and the layer depth
on a mesh, dynamically generated from the position of the particles, referred to as
a Voronoi diagram. The use of a Voronoi diagram to represent the layer depth is a
computationally tractable and stable approach.
Stationarity of the discrete action principle gives a semi-discrete method for shallow
water which conserves energy. We also formulate a semi-discrete divergence conserva-
tion law and an evolution equation for the potential vorticity with a non-zero right
hand side. We will finally apply symplectic integrators to the semi-discrete scheme
and present numerical results demonstrating that the VFL method for rotating shallow
water exhibits no secular drift in energy over long-time simulations and the geostrophic
adjustment mechanism of rotating shallow water in a f-plane.
Boundary conditions Finally, we implement boundary conditions using geometric
integrators for Hamiltonian fluid dynamics to address the problem of preserving sym-
plectic structure of a particle-mesh formulation of shallow water with spatial velocity
free-slip boundary conditions. For convenience, we extend the Hamiltonian particle-
mesh method for shallow water in a periodic domain, developed by Frank, Gottwald
& Reich (2002), to bounded domains. We formulate the boundary condition in terms
of ghost particles and show by numerical experiment that this approach gives energy
conservative numerical approximations of rotating shallow water in a bounded domain.
1.3 Literature Survey
Holm et al. (1986) presented the Hamiltonian structure of continuum mechanics in the
material, inverse material, spatial and convective representations.
Body and Spatial Representations of Rigid Body Motions Their work iden-
tified the body and spatial representations of rigid body motions as prototypes for the
respective convective and spatial representations of continuum dynamics. Their com-
parison of the spatial and convective representations also put the Hamiltonian treat-
ments of elasticity by Holm & Kupershmidt (1983) and by Marsden, Ratiu & Weinstein
(1984) into a unified framework.
The convective representation The convective and also the inverse material (aug-
mented Eulerian) representations offer alternative descriptions of continuum models.
The motivation for the convective representation of the continuum arose from a num-
ber of sources in the 1980s, including the study of relativistic adiabatic fluids by Holm
8
(1985), stability analysis of the coupled rigid body-beam and plate models of Krish-
naprasad & Marsden (1987) and the geometrically exact rod and plate models of Krish-
naprasad et al. (1988).
Semi-direct products Holm, Marsden & Ratiu (1998) derived the Euler-Poincare
(EP) formulation of the Eulerian fluid equations for an ideal fluid by applying sym-
plectic reduction to Hamilton’s principle for fluids. Legendre-transforming the EP
theory recovered the semidirect-product Lie-Poisson Hamiltonian theory that had been
discovered and applied earlier for nonlinear stability analysis by Holm, Marsden, Ratiu
& Weinstein (1998). A key step in the analysis of nonlinear stability of fluid equilibria
relies on the existence of Casimirs – quantities whose Lie-Poisson bracket vanishes with
all Eulerian (spatial) fluid variables because of right-invariance of the Eulerian variables
under reparameterisation of the Lagrangian labels. Because their Poisson brackets with
the Hamiltonian vanish, the Casimirs are conserved quantities.
Circulation theorems Fluid mechanics literature widely refers to the reparamet-
erisation of labels as fluid parcel relabelling and attributes the existence of the Kelvin
circulation theorem for ideal flow to the application of Noether’s theorem for the particle
relabelling symmetry group. Holm, Marsden & Ratiu (1998) showed that when advec-
ted quantities are present, a corollary of the EP framework is a geometric form of the
Kelvin circulation theorem referred to as the Kelvin Noether theorem. In this frame-
work, Holm et al. (1986) and Holm, Marsden & Ratiu (1998) further revealed the utility
of simple finite dimensional examples, such as the heavy top, by demonstrating that
they also exhibit a Kelvin Noether theorem. This theorem together with the EP equa-
tions form an essential ingredient in the geometric description of idealised continuum
dynamics.
Variational integrators Geometric numerical methods seek to transfer these power-
ful concepts in geometric mechanics to computational models. The pioneering work of
Moser & Veselov (1991) revealed integrable classical mechanical systems which have in-
tegrable discrete time counterparts. They considered the free rigid body as one example
and derived a discrete analogue to the Euler-Arnold equations for rigid body motion in
the body description. These integrators, referred to by McLachlan & Scovel (1995) as
Moser-Veselov (MV) integrators, conserve the rigid body energy to an arbitrary order
of the time step size and angular momentum to numerical round off.
Moser’s and Veselov’s key step was to form a discrete Hamilton’s action principle
9
and then derive variational integrators by deriving the discrete Euler-Lagrange equa-
tions. Although the number of contributions that later followed this approach are too
extensive to list here, the reader may follow some important aspects of its development
in Bobenko & Suris (1999a), Marsden, Pekarsky & Shkoller (1999), Marsden & West
(2001), McLachlan & Scovel (1995), Wendlandt & Marsden (1997), Leok, Marsden &
Weinstein (2004) who provide a differential geometric foundation for variational integ-
rators applied to mechanical systems. The number of numerical studies supporting the
theory appears less extensive, however.
Explicit integrators Moser-Veselov integrators are solved using an explicit algorithm,
referred to by McLachlan & Scovel (1995) as a DMV algorithm. Cardoso & Leite (2001)
cast the expression for the discrete angular momentum of Moser’s and Veselov’s rigid
body into a matrix Ricatti equation and solved it by Schur decomposing the Hamilto-
nian matrix. With the exception of the Schur decomposition, this DMV algorithm is
explicit. McLachlan & Zanna (2005) provide a more detailed description of this DMV
algorithm and demonstrate how to avoid the costly computation of the Schur form by
using an explicit spectral decomposition of the Hamiltonian instead. The Hamiltonian
can be decomposed in this way whenever its characteristic polynomial can be solved
analytically. The simple models considered herein do not require this optimisation step.
Pseudo-rigid bodies The pseudo-rigid body is a useful prototype for the geometric
description of homogeneous elasticity. Sousa-Dias (2002) applied the EP theory to
the polar decomposed pseudo-rigid body and showed that the motion of the isotropic
pseudo-rigid body is described by two coupled Lax equations for the angular momentum
and vorticity and a second order differential equation on the set of diagonal matrices
with positive determinant.
The geometrically exact rod model Krishnaprasad et al. (1988) applied the ma-
terial, spatial and convective description of continuum mechanics by Holm et al. (1986)
to the study of the Hamiltonian structure of non-linear elasticity. Krishnaprasad et al.
(1988) considered the Poisson structure and the reduced Poisson (Lie-Poisson) struc-
ture of non-linear elastic media before formulating the spatial, material and convective
representations of a geometrically exact rod and plate model. This rod model is a
generalisation of Antman’s director approach for the classical Kirchoff-Love elasticity
model, described in Antman (1995), to allow for finite extension and shear. This is
most conveniently achieved when the kinematics of the rod are described in terms of
orthogonal rotating frames(or orthogonal directors whose origins are fixed along the
rod centroid. Then, the configuration of the rod is described entirely by the attitude
10
of the rotating frames and the rod centroid position.
Simo’s rod model Simo & Vu-Quoc (1986, 1988) formulated a discrete geometrically
exact rod model. Their approach was novel in that it avoided making approximations
which resulted in artificial numerical features such as damping of shear and compres-
sion waves. Their approach combined the Rodrigues formula to compute infinitesimal
rotations and the Newmark variational integrator to compute the displacement of a
finite element discretisation of the rod. Their approach is not structure preserving,
however, nor is it explicit.
Other discrete rod models Symplectic methods for the discrete Kirchhoff rod have
been developed to enforce the rod inextensibility constraint. We note the Impetus-
striction method (Dichmann & Maddocks 1996) and holonomically constrained rod
model (Barth et al. 1999), both of which address the difficult numerical problem of
enforcing inextensibility in Kirchhoff rod simulations. Neither of these approaches
formulate the rod motion in the reduced representation and do not, therefore, provide
expressions for the conserved momentum maps corresponding to rotational symmetries.
Variational free-Lagrange methods Augenbaum (1984) derived a variational free-
Lagrange (VFL) method from a semi-discrete Hamilton’s action principle. The novel
feature of this method is the use of a Voronoi diagram to construct the layer depth
over. The method represents the material velocity field as tangent vectors of particle
positions and the layer depth as piecewise constant functions defined over Voronoi cells
of fixed mass moving with the (single) particle that each cell contains. The cell vertices
are defined at the mid-points between neighbouring particles. A variety of cell polygons
can then be constructed from the particle positions in this way.
Augenbaum’s work did not establish the conservative properties of this method nor
did it detail the use of symplectic time integrators to solve the Hamiltonian ordinary
differential equations describing Voronoi-cell trajectories on phase space
The VFL approach is conceptually similar to the particle moving-grid method
(Nishiguchi & Yabe 1982) which was developed to eliminate the presence of spurious
modes exhibited by the particle-in-cell method developed by Harlow (1964). The ori-
ginal particle-in-cell method represents the fluid mass in fixed cells and tracks particle
cell boundary interaction. An interesting feature of the moving grid variant is that
few particles are needed to resolve large pressure gradients. The concept of discretising
mass into moving mass packets was first developed by Pasta & Ulam (1959) and more
11
recently formed the basis of the energy conserving finite mass method (Gauger et al.
2000). Unlike these approaches, however, the VFL method does not exhibit mesh-
tangling because a Voronoi diagram is generated at each time step (see Augenbaum
1984) rather than moving a mesh of fixed connectivity .
Mesh tangling The use of the Voronoi diagram is not the only means of avoiding
mesh-tangling. Harlen et al. (1995) present a split Eulerian-Lagrangian scheme for
viscoelastic fluids that retains the nodes as material points and reconnects them to
produce the optimal Delaunay triangulation. This offers an advantage over the free-
Lagrange method in that the discrete fluid equations can be solved using standard finite
elements. This avoids one complication with the Voronoi diagram -the property that
the number of sides of a cell may vary over each time interval.
Petera & Nassehi (1996) presents a Galerkin-Lagrange finite element approximation
of a shallow water model for tidal flow which are challenged by fluctuations between dry
and wetted regions. Their basis for node adjustment is a physical one. The nodes of the
mesh are only moved if the neighbourhood of the node changes state between wetted
and dried. This approach avoids excessive mesh distortions but requires a smoothing
operator to eliminate numerical oscillations introduced by succesive modifications of
the mesh in very shallow regions.
Other variational methods Buneman (1982) describes a two-dimensional Eulerian
model based upon the Clebsch representation of Hamilton’s principle. The stability
of the approach appears to be contingent on the invariance of the discrete Hamilton’s
principle under particle relabelling. Salmon (1983) presents a numerical analogue of a
variational shallow water blob model. The curious feature of the design is the flexibility
to allow the blobs to subdivide which appears to be motivated by its application to
the study of the ocean’s main thermocline. A relevant outcome of his work is the
observation that the potential vorticity conservation is difficult to enforce numerically.
Variational formulations of smooth particle hydrodynamics (SPH) methods have
also been developed (see Bonet & Rodriguez-Paz 2005, and the references by the same
first author therein). The authors use the variational principle to derive spatially
dependent smoothing lengths - a limiting feature in conventional SPH methods is the
use of a fixed smoothing length. In particular, there appears to be a notable advantage
in using a variational SPH for very compressible fluids.
Poisson bracket methods An alternative approach to discretising Hamilton’s ac-
tion principle is to formulate numerical schemes which satisfy a Poisson-bracket. By
12
doing so, this approach captures symmetries of the continuum motion and correspond-
ing conservation laws. Strauss & Longcope (1998) construct an adaptive unstructured
zero-residual Galerkin finite element approximation of the vorticity stream function
formulation of the incompressible 2D Magnetohydrodynamical equations. This ap-
proximation exhibits the same anti-symmetry property of the Poisson bracket, proving
critical to magnetic flux and energy conservation. The authors do not, however, at-
tempt the more difficult problem of formulating approximations which also satisfy an
additional property of the Poisson bracket, namely, the Jacobi identity. This has been
the pursuit of Salmon (2004) in the context of shallow water leading to a scheme which
conserves energy and potential enstrophy. The expressions are somewhat complex,
however, perhaps to the detriment of flexibility of the approach.
Boundary Conditions Cotter, Frank & Reich (2004), Frank, Gottwald & Reich
(2002) and Frank & Reich (2004) have successfully applied a geometric numerical
method, referred to as the Hamiltonian Particle-Mesh (HPM) method, to numerous
shallow water models on the torus and the sphere, but did not consider boundary con-
ditions. Several approaches to implementing boundary conditions for smooth particle
hydrodynamics (SPH) methods (Monaghan 2002) with particle approximations of the
Euler equations are discussed by Vila (1999), namely, (i) image or ghost particles (ii)
boundary particles and forces, and (iii) semi-analytic techniques. We note a potential
extension of this approach to variational SPH methods (see Bonet & Rodriguez-Paz
2005). The image particle approach is, however, successfully used by the Hamiltonian
finite mass method of Klinger, Leinen & Yserentant (2005) and we shall consider the
implementation of the image particle approach in the HPM method.
1.3.1 Important related works
Simo’s computational framework Juan Carlos Simo investigated a unified ap-
proach for deriving mechanical integrators on Lie groups for continuum dynamics, as
described in Lewis & Simo (1994). Mechanical integrators are geometric integrators
which conserve any two of energy, symplectic structure or momentum, but not all three.
Simo showed that, given any Hamiltonian flow on a Lie group, the integrated con-
strained canonical Hamiltonian equations on phase space are equivalent to the Co-
adjoint orbits on the reduced Poisson manifold through the momentum map. This
result is the principle governing Simo’s framework, in which momentum preserving
integrators on Lie groups are derived from a discrete time update of the momentum
map.
Simo then used an algorithmic exponential to construct multiplicative integrators
on canonical phase space, which in the case when the group action is SO(3), take the
13
form
Λk+1 = Λkcay(Θ), Λk ∈ SO(3), Θ ∈ R3
Pk+1 = cay(−Θ)Pk, Pk ∈ T∗ΛkSO(3),
(1.1)
where cay : R3 → SO(3) denotes the Cayley transform and Θ is arbitrary at this stage
but can be constrained so that the discrete flow on the phase space is a mechanical
integrator.
1.4 Contributions of this Thesis
The results and partial results that are presented in this thesis are the culmination of
key independent ideas and developments which are now stated.
1.4.1 A computational framework
This thesis pursues the development of a unified computational framework for deriving
geometric integrators for the convective and spatial representation of continuum dy-
namics. This computational framework transfers powerful concepts given by the unified
framework of Holm et al. (1986), for the convective and spatial Hamiltonian continuum
dynamics, to computational continuum dynamics. Specifically,
1 Holm et al. (1986) show that the group action for passing between the represent-
ations generates an infinitesimally equivariant momentum map which carry the
Poisson brackets in one representation to those of the other. Using the discrete
Clebsch approach (Cotter & Holm 2006), we give the corresponding (diagonal)
group actions for passing between the representations and their momentum maps
from the cotangent bundle to the dual of the Lie algebra of the group.
2 Holm et al. (1986) show that the equations of continuum motion with advected
quantities are coadjoint orbits for the action of a semi-direct product Lie-algebra
on the dual of a semi-direct product Lie algebra. These orbits are symplectic foli-
ations of the Poisson manifold P defined by the augmented cotangent bundle. We
show that the discrete Clebsch approach gives discrete equations of motion with
advected quantities which define co-adjoint orbits for the action of the semi-direct
product group on the dual of its Lie-algebra. These orbits are also symplectic
foliations of P . The co-adjoint actions preserves the ± Lie-Poisson brackets on
the dual of this semi-direct product Lie algebra.
14
3 Holm et al. (1986) show through various examples, that these momentum maps
encode fundamental conservation laws of Hamiltonian continuum dynamics. These
conserved momentummaps are generated from the Noether symmetries for passing
to the spatial and convective representations, the latter of which is referred to
as a particle relabelling symmetry. We show that the discrete Clebsch approach
gives
I conserved momentum maps for the polar decomposed pseudo-rigid body
which are the conserved spatial angular momentum and the discrete Kelvin
circulation theorem generated by the respective rotational and material re-
labelling symmetries.
II conserved momentum maps for the geometrically exact elastic rods which are
the total spatial angular momentum generated by the rotational symmetries.
For the latter case, there is a fundamental difference between the form of the
momentum maps derived from the continuum framework and our computational
framework, however. Our computational framework represents the continuum
as a finite dimensional system of particles. The conserved momentum map then
takes the form of a discrete sum over all particle labels and only recovers the form
of the conserved momentum map for the continuum, in the continuum limit of
the particle system.
4 Our computational framework gives a prototype MV integrator for the convective
and spatial representations of compressible fluids.
Metric tensors are central to the theory of continuum mechanics. Holm et al.
(1986) consider a compressible fluid flow, in which the passage to the convective
and spatial representation is by reduction under the group of diffeomorphisms,
and show how the metric tensor and densities respectively transform in the differ-
ent representations. Analogously, we consider ellipsoidal motion and demonstrate
how our framework transforms the metric tensor (the Cauchy-Green matrix) and
shape matrices under reduction by the group GL(n,R)+ to the convective and
spatial representations. The discrete Clebsch approach gives MV integrators for
these representations in which the metric and shape matrix are respectively the
advected quantities.
1.4.2 Development of new DMV algorithms
This thesis shows that, under a forward in-time finite difference approximation of the
continuous Clebsch constrained action principle for the body representation of the rigid
body, that the discrete Clebsch approach recovers the Moser-Veselov integrator. MV
15
integrators are computed using the explicit DMV algorithm developed by McLachlan
& Zanna (2005) for solving the associated matrix Ricatti equation. In parallel with the
development of our computational framework, as described above, we give new DMV
algorithms to solve for the MV integrators and verify their conservative properties by
numerical experiment.
• Rigid body motions in the spatial representation We modify the DMV algorithm,
for the body representation of rigid bodies, to solve the spatial MV integrat-
ors. We then provide several numerical experiments to study the comparative
conservative properties of the spatial DMV algorithm with the DMV algorithm.
The results are largely conclusive and show that, in general, the spatial angular
momentum error profiles differ by a factor of 103.
• Rigidly Coupled motions We develop a DMV algorithm for solving the coupled
matrix Ricatti equation. This equation arises from coupled rigid body motion,
examples of which include the free rigid body motions of the coupled rigid body
and the circulatory and rotational polar components of the pseudo-rigid body
motion. We implement a model of a Mooney-Rivlin (Mooney & Rivlin 1977)
type pseudo-rigid body to describe the stretching and rotational components of
the motion and show that the DMV algorithm conserves angular momentum and
vorticity (relative to the Lagrangian frame) to an order of 10−15 and the energy
error exhibits no secular drift with a mean to the order of 10−3 (over 104 time
steps).
• Elastically coupled motions We solve a system of elastically coupled MV integrat-
ors for the elastically coupled director motions of the geometrically exact elastic
rod. The DMV algorithm for an elastically coupled rigid body differs from that
of the rigidly coupled rigid body. In the former case, the coupling is only through
the source term and not, as in the latter case, through the Coriolis term and re-
quires minor modification. The discrete Clebsch approach also gives a variational
integrator for the material representation of the rod centroid positions which is
equivalent to a Stormer-Verlet symplectic integrator. The DMV algorithm for a
rod of 50 sections conserves total spatial angular momentum to an order of 10−8,
linear momentum to an order of 10−11 and energy levels exhibit no secular drift
with a mean error to the order of 10−2 (after 104 time steps).
1.4.3 Geometric integrators for shallow water
In the absence of a general theory for the finite dimensional representation of the
group of diffeomorphisms, this thesis considers geometric integrators for shallow water
which preserve the canonical symplectic structure of a finite dimensional system of
16
fluid particles and are expressed in terms of the material velocity and the Eulerian
layer depth.
• Variational free-Lagrange method A variational free-Lagrange method for rotat-
ing shallow water with bottom topography is presented. We will establish the
conservative properties of the semi-discrete shallow water equations and derive
the semi-discrete shallow water divergence conservation law and potential vorti-
city evolution equations. The semi-discrete divergence equation is given by the
extrema of the discrete action principle. The semi-discrete potential vorticity
equation, however, is formed through a choice of a discrete curl operator which
is not resolved from the extrema of the action principle. This suggests the need
for an additional constraint in the action principle to constrain the form of the
discrete curl operator so that a semi-discrete potential vorticity conservation law
is also exhibited by the VFL method. Numerical results are also presented which
show that the VFL method for 1D rotating shallow water conserves energy to
an order of the integrator and exhibits the geostrophic adjustment mechanism of
rotating shallow water.
• Boundary conditions in the HPM method We extend the HPMmethod to bounded
rotating shallow water flows by implicity introducing ghost or image particles. We
demonstrate that the HPM approximation of rotating shallow water in a bounded
domain conserves mass, exhibits no secular drift in the energy level and remains
stable over long-time simulations. We also simulate the motion of a vortex pair
in a channel of rotating shallow water and show the motion of the vortex pair as
it reaches the channel wall.
1.5 Overview
Chapters 2 and 3 both start with largely abstract details and end with specific examples
and details of numerical experiments.
Chapter 2 We begin with a review of the geometric description of the free rigid body
and the necessary preliminaries of geometric mechanics. We derive Moser-Veselov in-
tegrators for body and spatial representations of rigid body motions using a discrete
variational, referred to as the discrete Clebsch approach (Cotter & Holm 2006). The
continuous time Clebsch approach provides a systematic means of deriving the Euler
Poincare equations from a symmetry reduced Hamilton’s action principle and the mo-
mentum maps (see Marsden & Ratiu 1999) associated with this symmetry reduction.
Through examples of a top and a coupled rigid body we develop a geometric de-
scription of the discrete time motion in both representations and then compare these
17
descriptions with their well-known continuous counterparts. Where appropriate, we
verify that the momentum maps of the integrators are conserved and use numerical ex-
periment primarily to (i) validate the conservative properties of the DMV algorithms,
(ii) compute the Casimirs and (iii) in the case of the rigid body, compare the numer-
ical with the analytic solution. We assess the comparative performance of the DMV
algorithm with the explicit Lie-Poisson integrator of McLachlan (1993) which is based
on a splitting of the rigid body Hamiltonian.
We also demonstrate the utility of the discrete Clebsch approach by deriving a MV
integrator for motions on SU(2) which corresponds to the Cayley-Klein parameterisa-
tion of discrete time Lagrange top motion. This parameterisation avoids the notorious
problem of Gimble-lock which occurs under Euler angle parameterisations.
Chapter 3 further applies the discrete Clebsch approach to derive Moser-Veselov
integrators for the convective and spatial representations of ellipsoidal motion. In Sec-
tion 3.1.1, we then derive a MV integrator for the polar decomposed pseudo-rigid body
motion on GL(3)+. We show that the discrete Clebsch approach gives two conserved
momentum maps representing the angular momentum and encoding a Kelvin circula-
tion theorem, given in Section 3.2.2. We also describe a new algorithm for solving the
coupled MV integrators for the internal circulatory and rotational motions.
We then develop a discrete form of the Kirchhoff rod analogy theorem which states
that the motion of a Lagrange top in discrete time is in one-to-one correspondence
with the frames of an isotropic symmetric inextensible elastic rod at equilibrium. By
extending this analogy further, we show that the dynamical discrete rod motion is in
one-to-one correspond with an elastically coupled series of Lagrange tops. This rod
motion exhibits a discrete compatibility equation of the form of the discrete auxiliary
equation for the relative orientation matrix in the coupled rigid body model. We then
derive a MV integrator for an extensible (and shearable) elastic rod model, referred to
by Krishnaprasad et al. (1988) as a geometrically exact elastic rod model. Numerical
results show the conservative properties of the DMV algorithms for solving the MV
integrators for the pseudo-rigid body and geometrically exact elastic rod model.
Chapter 4 We turn to rather more implementational aspects of variational integ-
rators for canonical Hamiltonian fluid dynamics. We consider a rotating shallow water
model since it is ubiquitous in geophysical fluid dynamics and extend geometric numer-
ical methods to represent the layer-depth in a Voronoi diagram. The use of a Voronoi
diagram is a key step in developing variational moving-mesh based numerical methods
which avoid the difficulty associated with tangling. We start by deriving a variational
free-Lagrange method for rotating shallow water with bottom topography and then
validate conservative properties by analysis and numerical experiment. The method
18
generates a Voronoi diagram from the position of particles at each time step. Numer-
ical experiments of 1D rotating shallow water demonstrate conservation of energy to
the order of the integrator. This approach is well suited to long-time simulations on
the sphere and seeks application in global climate models.
Chapter 5 We finally address the formulation of boundary conditions in geomet-
ric integrators for rotating shallow water by extending the Hamiltonian particle mesh
(HPM) method to bounded shallow water models. By treating planar boundaries as a
particle symmetry, we show how image or ghost particles may be implicitly introduced
into the HPM method. We also derive a finite element approximation of the Helm-
holtz operator which is used to dispersively regularise the layer depth and thus relaxes
the CFL stability constraint. A Neumann boundary condition on the regularised layer
depth is imposed naturally so that the resulting Helmholtz matrix is symmetric.
Appendix A gives a series of tables comparing the MV integrators with the continu-
ous Euler-Poincare equations for various rigid body motions, MV integrators for the
Cayley-Klein parameterisation of the rigid body formulated as SU(2) matrices and qua-
ternions. Appendix A also gives the DMV algorithms for solving MV integrators for the
spatial representation of the rigid body and the coupled rigid body. Appendix B gives
the Euler-Poincare description of the anisotropic pseudo-rigid body and the Lagrange-
Poincare description of the geometrically exact elastic rod. Finally, Appendix C gives
a summary of the Hamiltonian aspects of the variational free-Lagrange method and a
useful modelling step, referred to rezoning for computing the material variables over a
fixed mesh.
19
Chapter 2
Moser-Veselov Integrators for
Spatial and Body
Representations of Rigid Body
Motions
Synopsis In this review Chapter, we consider the problem of formulating Moser-
Veselov (MV) integrators for the body and spatial representations of rigid body motions
as a prototype for a unified variational framework for the convective and spatial repres-
entations of continuum dynamics. We apply the discrete Clebsch approach of Cotter &
Holm (2006) to (i) derive conserved momentum maps corresponding to symmetries of
the discrete Lagrangian and then (ii) discrete Euler equations for discrete rigid body
motions. The body representation of these discrete Euler equations match the integ-
rable discretisation of rigid body motion discovered by Moser & Veselov (1991). For
this reason, McLachlan & Zanna (2005) refer to these discrete Euler equations as a
Moser-Veselov (MV) integrator. The discrete Euler equations are also equivalent to
the discrete Euler-Poincare equations later given by Bobenko & Suris (1999a). For
consistency with McLachlan & Zanna (2005) , we will refer, however, to these discrete
Euler equations as MV integrators as this is more natural terminology for discussing
the computational aspects of these equations.
In the spatial representation, the framework gives a spatial variant of the MV
integrator which has an additional equation for the advection of the inertia matrix.
This integrator, again, matches the discrete Euler-Poincare equations with an advected
quantity given by Bobenko & Suris (1999a).
Using the discrete Clebsch approach, we use examples of heavy tops and coupled
rigid bodies to derive body MV integrators for motion driven by a potential and coupled
20
motion and give their corresponding DMV algorithms. We also observe that a MV
integrator for the Cayley-Klein parameterised rigid body follows from our framework.
In Section 2.10, we demonstrate the conservative properties of the DMV algorithms
for the heavy top and coupled rigid body by numerical experiment. This Chapter
reviews the preliminary theory and computations necessary to address MV integrators
and DMV algorithms for pseudo-rigid body and elastic rod motions in Chapter 3.
2.1 Introduction
Holm et al. (1986) showed that the body and spatial representations of rigid body mo-
tion correspond, respectively, to the convective and spatial representations of continuum
dynamics. The convective representation expresses the kinematics of the continuum in
terms of the convective velocity and the spatial or, equivalently, Eulerian representa-
tion expresses the kinematics in terms of the spatial velocity. In order to define these
velocities, we shall briefly review the terminology and notations of geometric continuum
dynamics given by Holm et al. (1986).
Continuum dynamics preliminaries The configuration of a material point (or
label) ` ∈ B, where B is the continuum reference space, is a diffeomorphic map g ∈
Diff(B) (a smooth invertible map with a smooth inverse) to a spatial point in the
continuum container (taken to be the Euclidean three space R3)
x(`) = g ∙ `. (2.1)
The motion of a material point ` is a time dependent curve g(t) ∈ Diff(B) defining
a trajectory of the material point in the container
x(t, `) = g(t) ∙ `. (2.2)
Definition 2.1.0.1 (The spatial velocity). The spatial velocity is the time derivative
of the motion evaluated at a fixed spatial point and takes the right invariant form
u(x, t) = g(t)g−1(t) ∙ x. (2.3)
Conversely, the configuration of a (fixed) spatial point x ∈ R3 is the inverse map
g−1 ∈ Diff(R3) to a material point ` in the continuum reference space
` = g−1 ∙ x. (2.4)
The motion of a (fixed) spatial point x, is a time dependent curve g−1(t) ∈ Diff(R3)
defining a trajectory of the spatial point in the reference space
21
`(t) = g−1(t) ∙ x. (2.5)
Definition 2.1.0.2 (The convective velocity). The convective velocity is the time de-
rivative of the motion of a spatial point, evaluated at a fixed material coordinate and
takes the left invariant form
V(`, t) = −g−1(t)g(t) ∙ `. (2.6)
We will review the correspondence between the spatial and convective representa-
tions of continuum mechanics and the spatial and body representations of rigid body
dynamics (Holm et al. 1986) once we have revisited the standard description of rigid
body dynamics in the next Section.
MV integrators With a view to developing a unified computational framework for
both spatial and convective representations, we will apply the discrete Clebsch approach
(Cotter & Holm 2006), developed for continuum dynamics, to derive the conserved
momentum maps associated with the Noether symmetries of the discrete Lagrangian for
body and spatial representations of rigid body dynamics in discrete time. Through these
momentum maps, we will show that the discrete Clebsch approach yields a known class
of variational integrators which were discovered by Moser & Veselov (1991). Following
McLachlan & Scovel (1995), these integrators have become commonly referred to as
Moser-Veselov (MV) integrators.
Body MV integrators It is well known that the body representation of these in-
tegrators has two distinguishing features:
1 Firstly, the time discretisation preserves the integrability of the body represent-
ation of the continuous rigid body motion. Integrable motions exhibit the same
number of invariants as their number of degrees of freedom (Moser & Veselov
1991). Motion in discrete time is consequently on the intersection of the level
sets of the same invariants and gives dynamics which are consistent with the con-
tinuous motion. We discuss the geometric properties of the MV integrator at the
end of Section 2.10.
2 Secondly, discrete Moser-Veselov integrators are solved using explicit algorithms
and hence avoid iterative computations with conditional convergence criteria. We
will review the performance of these algorithms by numerical experiments, the
details of which are provided in Section 2.10.
22
Spatial MV integrators
1 We will show that the application of the discrete Clebsch approach in the spatial
representation gives a spatial MV integrator with an (auxiliary) equation for the
advection of the Inertia matrix. These equations are the same as the discrete
Euler-Poincare equations with an advected quantity derived by Bobenko & Suris
(1999a). Bobenko & Suris (1999a) show these discrete Euler-Poincare equations
are Lie-Poisson w.r.t. to the dual of a semi-direct product Lie algebra, a result,
which we confirm for the spatial MV integrators for the rigid body.
2 The spatial DMV algorithms for solving the spatial MV integrators are also ex-
plicit. We give an algorithm for solving the spatial MV integrator in Section
A.3 of the Appendix and provide several numerical experiments to compare the
conservative properties of the spatial with the standard body DMV algorithm in
Section 2.10.
Caveat 2.1.0.3. Preservation of the Lie-Poisson structure on the dual of the semi-
direct product Lie algebra and demonstration of the conservative properties of the al-
gorithm by numerical experiment render the spatial MV integrator for the rigid body
a suitable prototype for the geometric integration of Hamiltonian continuum dynamics
with advected quantities.
Heavy tops and coupled rigid bodies We apply the discrete Clebsch approach
to give MV integrators for the body representation of both rigid body motions in a
potential field and coupled rigid body motions, referred to as heavy tops and coupled
rigid bodies respectively. We also consider the application of the DMV algorithms to
these integrators. These examples prepare us for the modelling of more challenging
models in Chapter 3 which exhibit potential and motion coupling terms.
Cayley-Klein parameters With a view to developing a unified computational vari-
ational framework, we will show that the discrete Clebsch approach also gives MV
integrators for the Cayley-Klein parameterisation of the rigid body. This Chapter
largely reviews MV integrators for rigid body motions formulated as SO(3) matrices.
These matrices represent the attitude of the rigid body as a three-stage rotation by
each Euler angle about its corresponding principal axis. Parameterisation of the rigid
body dynamics in terms of these angles and their momenta gives rigid body equations
which exhibit singularities (see Leimkuhler & Reich 2005). These singularities cause
Gimble-lock, a loss of degree of freedom of motion when one of the three stages of
rotation maps a principal axis to the previous position of another principal axis. The
combination of Euler-angles resulting in a singularity must be excluded and for this
reason, the Euler-angle parameterisation is referred to as non-global.
23
It is preferable to represent the rotation in terms of four Cayley-Klein parameters,
which form a global parameterisation (see Whittaker 1944). The Cayley-Klein para-
meters constitute the group of unit quaternions and are isomorphic to the matrix group
SU(2) consisting of all 2-by-2 unitary matrices with unit determinant. The discrete
Clebsch approach can be easily cast in terms of SU(2) matrices to give a MV integrator
for the Cayley-Klein parameterisation of rigid body motion together with the conserved
momentum maps corresponding to symmetries of the discrete Lagrangian. In doing so,
the framework gives a three-way correspondence between the equations of motion and
momentum maps formulated as matrices of SO(3), SU(2) and quaternions. The cor-
respondence between the first two can be observed by comparing Tables (A.1, pg. 193)
and (A.2, pg. 195) of Appendix A. Table A.2 compares the SU(2) and quaternionic for-
mulation of the MV integrator for the Cayley-Klein parameterised rigid body equations
and the conserved momentum maps.
Numerical experiments Finally, Section 2.10 presents numerical experiments which
demonstrate the conservative properties, computational efficiency and accuracy of the
numerical solutions of the body and spatial representations of the SO(3) formulated
MV integrators. We now begin by recalling the geometric description of the free rigid
body given by Marsden & Ratiu (1999).
2.2 The Free Rigid Body
In this Section, we review the geometric description and notation for the free rigid
body given in (Marsden & Ratiu 1999, Chapter 15) as a discrete time problem in
both the body and the spatial representations. Much of these follow the notations and
conventions reviewed in the previous Section and given in Holm et al. (1986).
Consider a free rigid body as a solid body, occupying a reference configuration
B ⊂ R3, which is free to move in R3 by rotations about a fixed point. Material points
` ∈ B are position vectors whose components, relative to a fixed orthonormal basis
(E1,E2,E3) in B, are the material coordinates. For the rigid body, a configuration of B
is a C1, invertible and orientation preserving map ψ : B → R3 from material points to
spatial points in R3. The spatial points are position vectors whose components, relative
to (e1, e2, e3), the right-handed orthonormal basis of R3 are spatial coordinates. This
basis is commonly referred to as the spatial (or Eulerian) frame.
Adapting the terminology for the discrete time case, we define a discrete motion
ψk := ψ(tk) of B as a family of discrete time dependent configurations of B. The discrete
motion gives a k parameterised sequence of spatial points representing the position of
a material point at time tk
xk = ψk ∙ `, k ∈ Z+. (2.7)
24
The discrete motion satisfies ψ0 ∙ ` = `. Marsden & Ratiu (1999) explain how the last
property together with rigidity of the body and continuity of the motion imply that the
configuration of B may be identified with the matrix SO(3) and the k parameterised
sequence of spatial points is given by
xk = Λk`, Λk ∈ G = SO(3), (2.8)
where, for notational convenience, Λk := Λ(tk). Λ is commonly referred to as the
attitude of the body.
The body coordinates of a material position vector are its components relative to a
time-dependent basis (ξ1, ξ2, ξ3)(tk) which is defined by
ξi(tk) = ΛkEi, i := 1→ 3, (2.9)
and hence is attached to the rigid body that rotates about the origin. This basis is
commonly referred to as the body frame.
The spatial position vector relative to the spatial frame is equal to the material position
vector relative to the body frame
xk = x(tk)i︸ ︷︷ ︸spatial
ei = Λk `i︸︷︷︸material
Ei = `i︸︷︷︸body
ξi(tk). (2.10)
Remark 2.2.0.4 (The continuous angular velocities). We recall from the geometric
description of the continuous rigid body given by Marsden et al. (1999), that the body
angular velocity Ω and spatial angular velocity ω are respectively given by
Ω = ΛTt Λt, ω = ΛtΛT , Λ ∈ SO(3). (2.11)
The body angular velocity is left invariant and corresponds (up to a minus sign) to the
convective velocity of continuum dynamics given in equation (2.6). The spatial angular
velocity is right invariant and corresponds to the spatial velocity of continuum dynamics
given in equation (2.3).
2.2.1 Discrete velocities
Moser & Veselov (1991) introduce the notion of a discrete velocity. In order to define
this, we first recall from the definition of the motion that the position of material points
in the container at time tk is given by
xk = Λk`. (2.12)
25
Conversely, the position of spatial points in the reference space at time tk is given by
`k = ΛTk x. (2.13)
Substituting equation (2.13) into equation (2.12) evaluated at time tk+1, gives the
following relation between spatial points at consecutive times
xk+1 = ωk+1xk, (2.14)
where ωk+1 = Λk+1ΛTk is referred to by Moser & Veselov (1991) as the discrete spatial
angular velocity.
Similarly, substituting equation (2.12) into equation (2.13) evaluated at time tk+1,
gives the following relation between material points at consecutive times
`k+1 = ΩTk+1`k, (2.15)
where Ωk+1 = ΛTkΛk+1 is referred to by Moser & Veselov (1991) as the discrete body
angular velocity. The two velocities are related to each other by a rotation
Ωk+1 = ΛTk ωk+1Λk. (2.16)
Order of the discrete velocities
The reason why Moser & Veselov (1991) refers to Ωk+1 and ωk+1 as discrete velocities is
because they approximate their continuous velocity counterparts to O(Δt2). To show
this, we revisit once more the geometric description of the (continuous) free rigid body
given by Marsden & Ratiu (1999), which describes the dynamics of the rigid body
on velocity phase space in terms of the body angular velocity by the reconstruction
formula
Λ(t) = Λ(t)Ω. (2.17)
The solution to this equation is given by a t-parameterised curve on G = SO(3)
Λ(t) = Λ0exp{Ωt}. (2.18)
Expressing this equation as an incremental solution
Λ(t+Δt) = Λ(t)exp{ΩΔt}, (2.19)
26
and then taking the step described by Lewis & Nigam (2003) of replacing the exponen-
tial function with the Cayley transform
cay : so(3)→ SO(3) , cay(Ω) = (Id +Ω
2)(Id −
Ω
2)−1, (2.20)
where Id denotes the identity matrix, gives
Λ(t+Δt) = Λ(t)(Id +ΩΔt
2)(Id − Ω
Δt
2)−1. (2.21)
Keeping only the first two terms of the binomial expansion of the denominator and
neglecting the remaining O(Δt2) term gives
Λ(t+Δt) = Λ(t)(Id +ΩΔt). (2.22)
Comparing this equation evaluated at time t = kΔt with the definition of Ωk+1 and
recalling the definition Λk := Λ(tk) gives the O(Δt2) approximation of Ω
Ωk+1 = Id +ΩΔt, (2.23)
or, equivalently, the finite difference approximation
Ω ≈ΛTkΔt(Λk+1 − Λk), (2.24)
which satisfies the definition of Ωk+1, given in equation (2.16), when substituted into
equation (2.23). By analogy with the reconstruction formula given in equation (2.17)
we refer to the equation
Λk+1 = ΛkΩk+1, (2.25)
as the discrete reconstruction formula. This formula reconstructs the dynamics on
G×G from Ωk+1.
This explanation holds for the spatial representation also. We note that the (con-
tinuous) body and spatial angular velocities are right and left invariant respectively, so
too are the corresponding body and spatial discrete angular velocities.
With the spatial and body discrete angular velocities defined, we now consider the
geometric mechanics of rigid body motion in discrete time by following Moser & Veselov
(1991).
27
2.3 Discrete Constrained Variational Principle
Moser & Veselov (1991) consider a functional
Sd =∑
k
L(Λk,Λk+1), (2.26)
where the discrete Lagrangian L : G×G→ R is a smooth function defined as
Lk := L(Λk,Λk+1) = Tr(ΛkI0ΛTk+1)︸ ︷︷ ︸
Kinetic energy
, (2.27)
in which I0 is a positive definite, symmetric and constant matrix referred to, in the
context of rigid body mechanics, as the inertia matrix.
Remark 2.3.0.1. The discrete Lagrangian is the first order finite difference approxim-
ation of the continuous Lagrangian Tr2
(ΛI0Λ
t). We show this by substituting Λ(tk) ≈
(Λk+1−Λk)h into the continuous Lagrangian to give
Tr
h2((Λk+1 − Λk) I0
(ΛTk+1 − Λ
Tk
))=Tr
h2(Λk+1I0Λ
Tk+1 + ΛkI0Λ
Tk − Λk+1I0Λ
Tk − ΛkI0Λ
Tk+1
)
=Tr
h2(I0 − Λk+1I0Λ
Tk
)
= −Tr
h2(ΛkI0Λ
Tk+1
)+Tr(I0)
h2,
(2.28)
where we have made use of the properties that (i) the trace operator is both invariant
under cyclic permutations Tr(AB) = Tr(BA) and transposition Tr(AB) = Tr(BTAT )
and (ii) orthogonality of the attitude matrix ΛTkΛk = Id. The last termTr(I0)h2is ignored
because it is constant in the body frame.
With the use of the following definition given by Wendlandt & Marsden (1997)
(which is stated in a general form for any Lie group G), we may state the invariance
properties of this Lagrangian.
Definition 2.3.0.2 (Diagonal action (Wendlandt & Marsden 1997)). The (left) diag-
onal action of G on G×G is defined as Ψ : G× (G×G)→ G×G | Ψ(f, (g, h)) =
f ∙ (g, h) = (fg, fh). Denote by Ψf , for all f ∈ G, the continuous linear transformation
Ψf : G×G→ G×G given by (g, h)→ Ψ(f, (g, h)).
The discrete time Lagrangian in equation (2.27) is invariant under the (left) di-
agonal action of Ψg. For this reason the group action of G on itself is referred to
as a symmetry of the discrete Lagrangian. In the continuous case, it is well known
28
that this symmetry gives a new reduced Lagrangian by a process referred to as sym-
metry reduction (Marsden & Ratiu 1999). The reduced Lagrangian for the rigid body
is expressed solely in terms of the body angular velocity (rather than velocity phase
space variables) and by Euler-Poincare reduction (Marsden & Ratiu 1999), gives the
Euler-Poincare equations defining the reduced dynamics.
It is natural to question the analogous process of symmetry reduction in the discrete
case, if only to obtain sufficient intuition to distinguish this process between the discrete
and continuous cases. We begin in the next Section by using recent work by Leok,
Marsden and Weinstein Leok et al. (2004) to help us visualise the otherwise largely
abstract notions described in the remainder of this Chapter.
2.4 Symmetry Reduction
Differential geometric aside Leok, Marsden & Weinstein (2004) show that a prin-
cipal G-Bundle furnishes the geometric description of the symmetry reduction of the
discrete Lagrangian to the body representation. A principal G-bundle is a bundle
(Q,S, π) where G acts freely on a bundle space B by left translation and is isomorphic
to (Q,Q/G, πQ/G). For the rigid body discrete Lagrangian, the bundle space Q = G×G,
the shape space is G ' G × G/G and the projection π : Q → S is isomorphic to the
natural projection πQ/G : Q → Q/G. At time tk, the natural projection is given by
the invariant action Ψ(ΛTk , (Λk,Λk+1)). The bundle space and natural projection of
the principal G-bundle furnishing the geometric description of the symmetry reduction
of the rigid body Lagrangian to the body representation are illustrated in Figure 2.1
below.
Body representation Following Moser & Veselov (1991), we reduce the discrete
Lagrangian defined on G×G to the body reduced Lagrangian l : G→ R defined on G
and given in body variables by
l(Ωk+1) = Tr(Ωk+1I0) (2.29)
The lowercase l denotes that the Lagrangian has been reduced from G×G to G.
Remark 2.4.0.3. This Lagrangian can be also obtained from the reduced continuous
Lagrangian by substituting the finite difference approximation Ω(tk) ≈ΛTkh (Λk+1 − Λk)
into the continuous time Lagrangian Tr2
(ΩI0Ω
T). As shown in the previous Section,
this finite difference approximation is consistent with a second order approximation of
the Cayley transform of ΩΔt which provides an algorithmically convenient form of the
exponential function (Lewis & Nigam 2003).
29
Ωk+1
e
G=SO(3)
G=SO(3)
ΨΛTk
ΨΛTk
Λk
Λk+1
Figure 2.1: This Figure shows the principal G-Bundle (Q,S, π) furnishing the description of symmetryreduction of the discrete Lagrangian for the rigid body at time tk (Leok et al. 2004). This bundle consists ofa bundle space Q = G × G, a shape space S = G ' G × G/G (not shown on the Figure) and a projectionπ : Q → S which is isomorphic to the natural projection πQ/G = Q → G × G/G. At time tk, this naturalprojection is defined by the diagonal action of ΛTk on (Λk,Λk+1) and is illustrated by the two curved arrows.
The constrained coordinate formulation Still following Moser & Veselov (1991),
we take one further step and embed G in the linear space of real matrices V (the
symmetric part of which is denoted V ) and use holonomic constraints in the form of
matrix Lagrange multipliers Θk+1 to constrain the family of curves satisfying δSd = 0
to G. This formulation is explained in Wendlandt & Marsden (1997) in which it is
referred to as the constrained coordinate formulation. An alternative formulation (which
is shown by these authors to be equivalent), is referred to as the generalised coordinate
formulation. This formulation uses a coordinate chart to extremise the discrete action
principle directly on any G. We do not pursue this alternative approach here but
instead consider the holonomically constrained Lagrangian lc : V → R defined in body
variables as
lc(Ωk+1) = Tr(Ωk+1I0)− Tr(Θk+1(Ωk+1Ω
Tk+1 − Id)
)(2.30)
30
in which the trace operator gives the pairing between elements of V and V ∗, and Θk+1
is a symmetric matrix Lagrange multiplier enforcing the orthogonality constraint on
Ωk+1. Note that Id denotes the identity matrix and the superscript c on lc denotes that
the reduced discrete Lagrangian is holonomically constrained. We shall now derive the
reduced discrete Lagrangian in the spatial representation.
Spatial representation The spatial representation distinguishes itself from the body
representation by not only expressing the dynamics in terms of discrete spatial angular
velocities but also by having an additional dynamical variable, the inertia matrix. The
inertia matrix, which is fixed in the body frame, rotates relative to the spatial frame
of the body. Holm et al. (1986) show that the rotation of the inertia matrix, as the
rigid body moves relative to the spatial frame, provides an example of a much more
general physical process in continuum dynamics, referred to as advection. Our purpose
here, is to show how the approach taken by Moser & Veselov (1991) can be extended
to include advected quantities. The spatial representation of the rigid body provides a
convenient example to explain this.
To transform to spatial variables, we follow Holm et al. (1986) and define a Lag-
rangian on the augmented space,
LI0 : G×G→ R, (2.31)
given by (Λk,Λk+1)→ L(Λk,Λk+1, I0) where
L(Λk,Λk+1, I0) = Tr(ΛkI0ΛTk+1). (2.32)
The notation LI0 denotes that I0 ∈ V ∗ becomes a dynamical variable in the spatial
representation. To discuss the symmetries of this Lagrangian, we must first define how
G acts on V ∗. Let φLg (I0) be the left group translation on V∗ given by the map
φLg : V∗ → V ∗ | φLg (I0) = g ∙ I0 = gI0g
T . (2.33)
The corresponding right group translation is given by the map
φRg∗ : V ∗ → V ∗ | φR(I0) = I0 ∙ g = g
−1I0g−T , (2.34)
which is equal to the left translation g−1 ∙I0. We use the definition of these translations
to describe how G acts on the augmented state space G×G× V ∗.
Definition 2.4.0.4 (Augmented diagonal action). The (right) augmented diagonal
31
action of G on G × G × V ∗ is defined as Ψ : (G × G × V ∗) × G → G × G ×
V ∗ | Ψ((g, h, a), f) = (g, h, a) ∙ f = (gf, hf, f−1af−T ). The continuous linear trans-
formation Ψf : G×G× V ∗ → G×G× V ∗ is given by (g, h, a)→ Ψ((g, h, a), f).
The right translation Ψg leaves the discrete time Lagrangian LI invariant
LI0∙g(Λkg,Λk+1g) = LI0(Λk,Λk+1). (2.35)
We observe this property from the form of LI0
LI0∙g(Λkg,Λk+1g) = Tr(Λkgg−1I0g
−T gTΛTk+1). (2.36)
Symmetry reduction of the Lagrangian by the augmented diagonal action gives the
reduced Lagrangian lI : G→ R given in spatial variables as
lIk(ωk+1) = Tr(ωk+1Ik). (2.37)
Analogous to the body reduced Lagrangian, we shall take one further step and define
a holonomically constrained Lagrangian lcI : V → R given by
lcIk(ωk+1) = Tr(ωk+1Ik)− Tr(Θk+1(ω
Tk+1ωk+1 − Id)
). (2.38)
This equation defines the spatial representation of the discrete Lagrangian considered
by Moser & Veselov (1991) which is extended to include the advected inertia matrix.
In order to transfer concepts in continuous geometric mechanics to the discrete
formulation, we need to review how the body and spatial representations of the discrete
Lagrangians give corresponding equations of motion and conservation laws using the
discrete Clebsch approach (Cotter & Holm 2006).
2.5 Clebsch Potentials and Momentum Maps
We shall give a brief introduction to an approach for obtaining encodings of the mo-
mentum conservation laws in continuous time, referred to as the Clebsch approach. In
order to do so, we must define some standard terminology of geometric mechanics. This
is best done in the context of an example.
2.5.1 Geometric preliminaries
We revisit the classical continuous time rigid body once again, only this time we will
review the construction of vector fields on the configuration space Q = G, where G =
SO(3).
32
The trajectories of the rigid body are described by the left translation of G on itself
Φg : G → G, for all g ∈ G, given by h → Φ(g, h) = gh. Recall that the motion of
the rigid body Λ(t) defines a continuous t-parameterised path with the property that
Λ(0) = Id and Λ(0) = ζ, ζ ∈ g = so(3). The infinitesimal generator of Φg in the
direction of ζ is the vector field ζG on G defined by
ζG(g) =d
dt
∣∣t=0ΦΛ(t)(g). (2.39)
The path Λ(t) = exp(tζ) is an integral curve of ζG(g) and by definition
Λ(t) = ζG(Λ(t)) = ζΛ(t), (2.40)
where Λ(t) ∈ TΛ(t)G. The Lagrangian for this motion is the smooth function
L : TG→ R, (2.41)
that takes a form for which Φg is a symmetry transformation of the Lagrangian- its
tangent lift on TG leaves L invariant. Analogously, Φg is a symmetry of the Hamiltonian
H : T ∗G→ R if the cotangent lift of Φg on T ∗G leaves H invariant. For example, when
the symmetry transformation is given by the left action of G, the cotangent lift of this
translation Φ′g([Λ, P ]) = (gΛ, g−TP ) and Φ′g(Λ) is a symmetry of H if H ◦ Φg(z) =
H(z), z ∈ T ∗G. The cotangent bundle is an example of a more general manifold
referred to as a Poisson manifold (P, {, }), equipped with a Poisson bracket. It is
conventional to simply denote the Poisson manifold as P.
The action Φ : G × P → P is referred to as canonical if Φg is a Poisson map for
every g ∈ G, that is
{F1 ◦ Φg, F2 ◦ Φg} = {F1, F2} ◦ Φg, (2.42)
for every smooth scalar function F1, F2 ∈ F(P).
Recall that the infinitesimal generator of Φ on G in the direction of ζ is the vector
field ζG on G given by equation (2.40). The corresponding infinitesimal generator for
the action of Φ on P in the direction of ζ is the vector field ζP on P defined by
ζP =d
dt
∣∣t=0Φg(t)(z). (2.43)
A Hamiltonian vector field on P is the vector field XH defined by
XH [F ] = {F,H}, (2.44)
for all smooth scalar functions F ∈ F(P). It follows from the property of the Poisson
bracket that XH [H] = 0 meaning that the Hamiltonian vector field conserves the
33
Hamiltonian. We are interested in the case when ζP is globally Hamiltonian, i.e. there
is some global Hamiltonian, a linear map Jζ : P → F(P), such that XJζ = ζP. We
will now clarify why we are interested in a global Hamiltonian of this form through the
following definition.
Definition 2.5.1.1 (Momentum Map (Marsden & Ratiu 1999)). Let G act canonically
on a Poisson manifold P. A momentum map for this action is a map J : P→ g∗ such
that the map Jζ : P→ F(P) : Jζ(z) = 〈J(z), ζ〉 satisfies
XJζ = ζP, ∀ζ. (2.45)
Momentum maps are conserved quantities when they correspond to symmetries.
The following fundamental theorem of geometric mechanics formalises this statement.
Theorem 2.5.1.2 (Noether’s theorem (Marsden & Ratiu 1999)). The Hamiltonian
version of Noether’s theorem states that if Φ acts canonically on P with a momentum
map J such that H is G invariant, then J is conserved by the flow of XH .
Theorem 2.5.1.3 (Momentum maps for lifted actions (Marsden & Ratiu 1999)). If
ζP is a vector field on P = T∗G given by the cotangent lift of the infinitesimal action of
G to T ∗G, then ζP is Hamiltonian with an infinitesimally equivariant momentum map
J : P→ g∗ given by
〈J([Λ, P ]), ζ〉 = 〈P, ζG(Λ)〉. (2.46)
The brackets on the right hand-side denote the pairing of T ∗ΛG with TΛG and on
the left hand-side, the pairing of g∗ with g.
By the theorem of Canonical Momentum maps (Marsden & Ratiu 1999), it fol-
lows that all infinitesimally equivariant momentum maps are Poisson. This property
is fundamental to the development of geometric integrators for (reduced) continuum
dynamics, since the existence of a momentum map in a discrete geometric framework
implies preservation of Poisson structure in the reduced phase space g∗.
2.5.2 The Clebsch approach
The naming of this approach is misleading since Clebsch did not pioneer this approach
as the name suggests. In fact, there a complex history behind this approach which
we do not attempt to detail here but rather refer the reader to Seliger & Whitham
(1968). The most crucial step appears to have originated from the difficulties that Lin
(1963) encountered when formulating a variational principle for continuum mechanics
in Eulerian variables because of the non-canonical form of Hamilton’s principle. Lin
(1963) was able to derive the equations of isentropic fluid flow by using the Clebsch
34
representation (Clebsch 1859) in which the Eulerian velocity is, in the most basic case,
expressed in terms of Clebsch potentials :
u = ∇χ+ λ∇μ. (2.47)
So by expressing the velocity field in a general form, Lin (1963) was able to recover the
equations of motion for the Clebsch potentials from a Hamilton’s principle. The con-
nection was also made between the potential representation of Electromagnetic fields
and the variational principle for Maxwell’s equations. This connection was soon exten-
ded to a wider range of idealised flows by numerous authors (see, for example, Seliger
& Whitham 1968, Bretherton 1970). Seliger & Whitham (1968) pointed out that Lin’s
approach remained somewhat of a mystery as a mathematical device. Moreover, at
that time, no general methodology existed for deriving equations of continuum motion
in the Eulerian representation from the canonical Hamilton’s principle, expressed in
the material representation.
Marsden & Weinstein (1974) addressed this by explicating a geometric approach for
deriving Euler-Poincare equations of motion using abstract Clebsch variables, the theory
of symmetry reduction and canonical maps. In the context of fluid mechanics, the
Clebsch variables take the form of potentials but in other contexts may not necessarily
do so. Cendra & Marsden (1987) refer to this abstract Clebsch setting as the Clebsch
approach for deriving variational principles for symmetry reduced motions and illustrate
it with the example of the free rigid body.
Example: free rigid body motion Given the left invariant Lagrangian `(Ω) for
the body representation of the free rigid body, Cendra & Marsden (1987) define a new
Lagrangian, which in our notation, is given by ˜ : T (G × V × V∗) → R, for a linear
vector space V ⊂ Rn×n. For the rigid body, n = 3 and G = SO(3) and ˜ takes the form
˜(A,Λ, Λ, P, P ) = `(Ω) + 〈P, Λ− ΩG(Λ)〉, (2.48)
where Ω = AT A, P and Λ are the Clebsch variables, ΩG(Λ) is the infinitesimal generator
of the action of G on Λ in the direction of Ω and the equation paired with P , is the Lin
constraint (we will refer to this constraint from hereon as the Clebsch constraint only
for simplicity of terminology). Note that the˜ symbol over ` denotes the addition of
a Clebsch constraint. In this context, the Lagrange multiplier P denotes the canonical
rigid body momentum. Ω is a solution to the Euler equations if (Ω,Λ, P ) is a critical
point of Hamilton’s action principle
S =
∫ t2
t1
`′dt. (2.49)
35
By the standard procedure of calculus of variations, we obtain the Clebsch representa-
tion for the body representation of the free rigid body
Ωˆ1 =(ΛTP )23 − (ΛTP )32
2I1, Ωˆ2 =
(ΛTP )31 − (ΛTP )132I2
, Ωˆ3 =(ΛTP )12 − (ΛTP )21
2I3,
(2.50)
where Ij are the principal moments of inertia.
Example: perfect incompressible fluid motion To connect this example with
the standard notion of Clebsch potentials, consider the Hamilton’s action principle for
Euler’s equations for a perfect incompressible fluid which takes the form
S =
∫ t2
t1
∫C1
2||u(`, t)||2 + 〈P (`, t), ∂ta(`, t) + u(`, t) ∙ ∇a(`, t)〉d
3`dt, (2.51)
where a and P are the passively advected Clebsch potentials. A special case which eases
comparison between the forms of the Clebsch constraint for perfect fluids and the rigid
body is when a(`, t) = X(`, t) is a fluid parcel position and P (`, t) is the corresponding
conjugate momenta.
Stationarity of S gives the familiar form of the Clebsch representation of the Eu-
lerian velocity in terms of these Clebsch potentials
ui = −∂`ia(`, t) ∙P(`, t) + ∂`iφ, (2.52)
where φ is a Lagrange multiplier paired with an incompressibility condition. Equation
2.52 is the perfect fluid analogue of the rigid body form of the Clebsch potential rep-
resentation given in equation 2.50. So, although the form of the Clebsch representation
is context dependent, the procedure for its derivation is systematic and amenable to
formulation of a discrete analogue, the discrete Clebsch approach of Cotter & Holm
(2006).
2.5.3 The discrete Clebsch approach
The discrete Clebsch approach (Cotter & Holm 2006), which we will now briefly in-
troduce, provides a systematic means of deriving momentum maps for cotangent lifted
actions in the discrete Lagrangian framework. Consider first the left reduced continuous
Lagrangian on g defined as l : g→ R. A Clebsch (Lin) constrained reduced Lagrangian
l, is given by
l = l(Ω) + 〈P, Λ− ΛΩ〉︸ ︷︷ ︸Clebsch constraint
, (2.53)
36
in which 〈∙, ∙〉 denotes the pairing of elements in T ∗ΛG with TΛG and Ω is the body
angular velocity.
Just as the discrete reduced body Lagrangian given by equation (2.29) follows from
a finite difference approximation of l, so too does the discrete form of l. To show this,
we approximate the reconstruction formula for describing the dynamics on velocity
phase space from the body angular velocity Ω
Λ(t) = Λ(t)Ω, (2.54)
with finite differences to give
Λ(tk)− Λ(tk)Ω =1
h
((Λk+1 − Λk)− ΛkΛ
Tk (Λk+1 − Λk)
)+O(h2)
=1
h(Λk+1 − ΛkΩk+1) +O(h
2).
(2.55)
This equation evaluates to the discrete reconstruction formula for the body represent-
ation of the reduced rigid body dynamics in discrete time given in equation (2.25).
We then formulate this discrete reconstruction formula in constrained coordinates by
embedding G in V, the linear space of all 3-by-3 real matrices as described in Wend-
landt & Marsden (1997). The Clebsch constraint is reparameterised by h to give
Pk := hP (tk) ∈ T ∗ΛkV. This constraint represents the (reparameterised) canonical mo-
mentum of the rigid body at time tk. The Clebsch constrained body reduced discrete
Lagrangian thus takes the form
lk+1 = lc(Ωk+1) + 〈Pk+1,Λk+1 − ΛkΩk+1〉, (2.56)
where the superscript k on lk is used to denote that the Lagrangian is evaluated at
time tk.
Stationarity of the discrete action principle can be expressed in terms of variations
in the dynamical variables
δSd =∑
k
δlk+1 =∑
k
〈∇Λk lk+1, δΛk〉
+ 〈∇Λk lk+1, δΛk+1〉
+ 〈∇Pk+1 lk+1, δPk+1〉
+ 〈∇Ωk+1 lk+1, δΩk+1〉 = 0,
(2.57)
where ∇v lk : V → V∗ denotes the gradient of lk with respect to the linear matrix v ∈ V ,
and is defined in Bobenko & Suris (1999a) by the formula
37
〈∇v lk, w〉 =
d
dε
∣∣ε=0
lk(v + εw), ∀w ∈ V . (2.58)
The term in equation (2.57) paired with δΛk is the discrete Euler-Lagrange equation
〈∇Λk lk+1 +∇Λk l
k, δΛk〉 = 0, (2.59)
which evaluates to
〈−Pk+1ΩTk+1 + Pk, δΛk〉 = 0, (2.60)
giving the discrete flow of Pk as
Pk+1 = PkΩk+1. (2.61)
In the same way, the stationary discrete action principle also recovers the discrete
reconstruction formula
Λk+1 = ΛkΩk+1, (2.62)
from the expression paired with variations Pk+1.
Symplectic flow on the cotangent bundle Equations (2.61) and (2.62) define
the (discrete) flow map on zk := [Λk, Pk] as a cotangent lifted right translation (see
Marsden & Ratiu 1999) ΦΩk+1(Λk) of the form zk+1 = Φ′Δt(zk) := Φ
′Ωk+1(zk) :=
(ΛkΩk+1, PkΩ−Tk+1).
The discrete flow on the cotangent bundle given by equations (2.76) preserves the
symplectic two-form dPk ∧ dΛk.
Proof.
dPk+1 ∧ dΛk+1 = d(PkΩk+1) ∧ Ωk+1dΛk
= −Ωk+1dΛk ∧ dPkΩk+1
= −dΛk ∧ dPkΩk+1ΩTk+1
= dPk ∧ dΛk.
(2.63)
The discrete flow on the cotangent bundle preserves the symplectic two-form. Re-
ferring to Hairer et al. (2002) we point out that this property is equivalent to the
statement that the flow is Poisson w.r.t. the canonical Poisson bracket {, } on the
cotangent bundle meaning that
38
{F1 ◦ φ, F2 ◦ φ}(zk) = {F1, F2}(zk+1). (2.64)
This is a convenient point at which to define the fibre derivative for the discrete
Lagrangian. This definition will be used to prove the proceeding discrete Noether’s
theorem given by Wendlandt & Marsden (1997) in generalised coordinates.
Definition 2.5.3.1 (Discrete Time Fibre Derivative (Wendlandt & Marsden 1997)).
The fibre derivative for the discrete (unreduced) Lagrangian on G×G is a smooth map
of the form
FL : G×G→ T ∗G , (g1, g0) 7→ (g0,∇g0L(g0, g1)) , (2.65)
where ∇gL : G→ T ∗G denotes the gradient of L with respect to the group g ∈ G, and
is defined in Bobenko & Suris (1999a) by the formula
〈∇gL, g〉 =d
dε
∣∣ε=0
L(g(ε)). (2.66)
We shall now state the discrete Noether’s theorem which is adapted from Wendlandt
& Marsden (1997) for the discrete Clebsch approach.
Theorem 2.5.3.2 (Discrete Noether’s Theorem). If the diagonal lift of the left action
Φ to the diagonal action Ψ, is a symmetry of the discrete Lagrangian L : G × G →
R, then the corresponding symplectic flow Φ′Δt (corresponding to the discrete Euler-
Lagrange equations) on the cotangent bundle preserves the (left) momentum map JLk+1◦
Φ′Δt = JLk , where J
Lk+1 := J
L([Λk, Pk]).
Proof. We follow the general proof, given in (Wendlandt & Marsden 1997, Section
3.3), for any Lie group g ∈ G. The proof is most conveniently formulated in generalised
coordinates. Invariance of Lk := L(Λk,Λk+1) under the diagonal (left) action Ψ implies
invariance of the discrete action principle
d
dε
∣∣ε=0
Sd =∑
k
d
dε
∣∣ε=0
L (exp(Ωε)Λk, exp(Ωε)Λk+1) =∑
k
d
dε
∣∣ε=0
Lk = 0. (2.67)
Using the chain rule we express the derivative of Lk w.r.t. to ε in terms of derivatives
of Lk in Λk and Λk+1 giving
d
dε
∣∣ε=0Lk = 〈∇ΛkL
k, ζG(Λk)〉+ 〈∇Λk+1Lk, ζG(Λk+1)〉 = 0. (2.68)
Substitution of this expression into the discrete action principle gives, after a shift of
the time parameterisation index, the discrete Euler-Lagrange equations paired with
39
ζG(Λk+1) which take the form
〈∇Λk+1Lk+1 +∇Λk+1L
k, ζG(Λk+1)〉 = 0. (2.69)
Subtracting the expression in equation (2.68) from this last equation gives
〈∇Λk+1Lk+1, ζG(Λk+1)〉 = 〈∇ΛkL
k, ζG(Λk)〉. (2.70)
Substituting the definition of Pk = ∇ΛkLk, from the definition of the fibre derivative
given above for the discrete Lagrangian Lk, and the definition for the infinitesimal
generator for the left action of G on itself into the above expression gives
〈Pk+1, ζΛk+1)〉 = 〈Pk, ζΛk〉. (2.71)
Substituting the discrete reconstruction formula Λk+1 = ΛkΩk+1 from equation
(2.25) into the above expression gives
〈Pk+1 − PkΩ−Tk+1, ζG(Λk)〉 = 0. (2.72)
The image of the fibre derivatives given by the discrete Euler-Lagrange equations is
the discrete symplectic flow (see lemma 2.5.3 on pg. 38 for a proof that this flow is
symplectic) on T ∗G given by
Λk+1 = ΛkΩk+1,
Pk+1 = PkΩ−Tk+1.
(2.73)
Equation (2.71) gives the statement of conservation of the infinitesimally equivariant
left momentum map
〈skew(Pk+1ΛTk+1)− skew(PkΛ
Tk ), ζ〉 = 0. (2.74)
So the symplectic flow on the cotangent bundle corresponding to the left invariant
discrete Lagrangian conserves the right infinitesimally equivariant momentum map.
We will now use the discrete Clebsch approach in the body and spatial representa-
tions to derive momentum maps and the equations of motions.
40
2.5.4 The body representation
The Clebsch constrained reduced discrete Lagrangian given in equation (2.56) takes
the form
lk+1 = Tr (I0Ωk+1) +Tr
2
(P Tk+1(Λk+1 − ΛkΩk+1)
)− Tr
(Θk+1(Ωk+1Ω
Tk+1 − Id)
),
(2.75)
where the second term is the Clebsch constraint for the discrete auxiliary equation and
the last term is the holonomic constraint on Ωk+1. Recall that the stationary discrete
action principle gives the discrete symplectic flow on the cotangent bundle
Pk+1 = PkΩk+1,
Λk+1 = Λk Ωk+1.(2.76)
The momentum maps gives the corresponding preserved geometric structure on the
reduced phase space of the rigid body so(3)∗.
Derivation of momentum maps The derivative ∇Ωk+1 l, paired with the variation
δΩk+1 in the discrete action principle given by equation (2.57) is a Clebsch relation
which evaluates to
I0 −1
2ΛTk Pk+1 = Θk+1Ωk+1. (2.77)
We use the symmetry property of Θk+1 and the expression for Pk+1 given by equa-
tion (2.76) to give
I0ΩTk+1 −
1
2ΛTk Pk = Ωk+1I0 −
1
2P Tk Λk. (2.78)
This equation can be written as
JRk+1 = skew(ΛTk Pk), (2.79)
which by the theorem of momentum maps for lifted actions (see 2.5.1.3) satisfies the
definition of a right momentum map
〈JRk+1, ζ〉 = 〈skew(ΛTk Pk), ζ〉
= 〈Pk,Λkζ〉
= 〈Pk, ζG(Λk)〉.
(2.80)
Holm, Marsden & Ratiu (1998) show that the momentum map for cotangent lifted
actions takes the form
41
JRk+1 = Pk � Λk, (2.81)
where the bilinear operator � : V∗ × V → g∗ is defined by the pairing
〈Pk � Λk, ζ〉 = 〈Pk, ζG(Λk)〉, (2.82)
where T ∗ΛG ⊂ V . The diamond operator, as notation, thus makes explicit the property
that equation (2.81) is a momentum map for cotangent lifted actions.
The spatial angular momentum is conserved The discrete Noether’s theorem
states that the left momentum map JLk+1 is preserved by the symplectic flow on the
cotangent bundle corresponding to the left invariant form of the discrete Euler-Lagrange
equations. The spatial angular momentum is the left momentum map
mk+1 = ΛkMk+1ΛTk = Λkskew(Λ
Tk Pk)Λ
Tk = skew(PkΛ
Tk ) = J
Lk+1. (2.83)
Substitution of equations (2.76) defining the discrete symplectic flow on the cotangent
bundle T ∗G into the right momentum map given in equation (2.81) gives the recursion
relation on the cotangent bundle
ΛTk Pk = ΩTkΛ
Tk−1Pk−1Ωk. (2.84)
The right momentum map projects the skew-symmetric component of this equation
onto so(3)∗, giving the discrete Euler equation for the body representation of rigid
body motion
Mk+1 = Ad∗ΩkMk, Mk+1 = I0Ω
Tk+1 − Ωk+1I0, (2.85)
where the body angular momentum Mk+1 ∈ so(3)∗ is defined as
Mk+1 := 2skew(∇Ωk+1 lkΩT ), (2.86)
in which ∇g lk : G → T ∗G denotes the gradient of lk with respect to the group G =
SO(3), and is given by equation (2.66).
Equation (2.85) defines a Moser-Veselov integrator on the dual of the Lie algebra
so(3)∗ as a co-adjoint action of G. The co-adjoint action is defined by the pairing
(Marsden & Ratiu 1999)
〈Ad∗gα, ζ〉 = 〈α,Adgζ〉, (2.87)
42
where ζ ∈ g, α ∈ g∗and Ad : G × g → g : Adgζ = gζg−1. Substituting the form of
Adg into this pairing
〈gTαg−T , ζ〉 = 〈α, gζg−1〉, (2.88)
implying that Ad∗g takes the form Ad∗gα = gTαg−T .
Notational remark 2.5.4.1. Moser & Veselov (1991) define the Moser-Veselov in-
tegrator in slightly different notation. In their notation, Mk = ωTk J − Jωk, where
ωk := XTk Xk−1 is the discrete body angular velocity, Xk is the attitude of the body
and J is the fixed inertia matrix. ωk is not to be mistaken with our definition of
the discrete spatial angular velocity. A comparison of Mk, given by equation (2.85),
with the definition in Moser & Veselov (1991) gives the relation between the different
notations for the discrete body angular momentum Ωk = −ωTk . Table A.1 of Appendix
A provides a summary comparing our notation with Moser & Veselov (1991). Our
notation is chosen to be as consistent as possible with the convention of continuous
geometric mechanics (see Marsden & Ratiu 1999), for the purpose of developing a uni-
fied discrete framework for transferring concepts in continuous geometric mechanics to
computational models. In particular, the use of Ωk and ωk to denote the respective body
and spatial angular discrete velocities is necessary for exposition of a discrete unified
computational framework. We also find it convenient to define Ωk and ωk so that the
discrete time reconstruction formulae
Λk = Λk−1Ωk Λk = ωkΛk−1, (2.89)
take a similar form to the continuous reconstruction formulae
Λ = ΛΩ Λ = ωΛ. (2.90)
Remark 2.5.4.2 (Relation to Bobenko’s and Suris’s discrete Euler-Poincare equa-
tions). Bobenko & Suris (1999a) give, in our notation, the discrete Euler-Poincare
equation on G as
Mk+1 = Ad∗ΩkMk, (2.91)
where the body angular momentum Mk+1 = dΩk+1 lk is expressed in terms of left Lie
derivatives of lk w.r.t. to Ωk+1, and dgl : G→ g∗ denotes the (left) Lie derivative of L
w.r.t. to g and is given in Bobenko & Suris (1999a) by the formula
〈dgl, ζ〉 =d
dε
∣∣ε=0
l(exp(εζ)g), ∀ζ ∈ g. (2.92)
The relation between the gradient ∇gl and the Lie derivative of l w.r.t. to g is given in
43
Bobenko & Suris (1999a) by
〈T ∗Rg∇gl, ζ〉 = 〈dgl, ζ〉, ∀ζ, (2.93)
where T ∗Rg is the cotangent lift of right translations in the group given by the pairing
〈T ∗RgP,A〉 = 〈P, TRgA〉, P ∈ T ∗gG,A ∈ TgG, (2.94)
and TRgA is the tangent lift of the right translation by g on A given by TRgA = Ag.
Using the expression for the tangent lift of the right translation of g, equation (2.94)
evaluates to
〈PgT , A〉 = 〈P,Ag〉. (2.95)
It follows from the definition of T ∗Rg and equation (2.93) that
〈dΩk+1 lk, ζ〉 = 〈∇Ωk+1 l
kΩTk+1, ζ〉 (2.96)
and the discrete EP equation in equation (2.91) is therefore (up to a factor of 2) the
Moser-Veselov integrator (2.85) that we recovered by application of the discrete Clebsch
approach.
We now review the properties of MV integrators.
Moser-Veselov integrators
Poisson integrators MV integrators for the body representation of the rigid body
belong to a class of Poisson structure preserving integrators referred to as Poisson
integrators. Hairer et al. (2002) give the following definition of a Poisson integrator.
Definition 2.5.4.3 (Poisson Integrator (Hairer et al. 2002)). A discrete flow map φΔt
defined on a Poisson manifold (M, {, }) is Poisson with respect to the Poisson bracket
{, } if
{F1, F2} ◦ φΔt = {F1 ◦ φΔt, F2 ◦ φΔt}, (2.97)
where F1 and F2 are scalar functions. If ΦΔt defines a path on the level sets of Casimir
functions C (which Poisson commute with any function F , {C,F}=0) of this Pois-
son bracket, then ΦΔt is referred to as a Poisson integrator. (Lie-Poisson integrators
preserve the Lie-Poisson structure {, } on the dual of the Lie algebra).
Definition 2.5.4.4 (The so(3)∗ Lie-Poisson bracket for the rigid body (Moser &
Veselov 1991)). The so(3)∗(' R3) Lie-Poisson bracket for the body representation of
the (continuous) rigid body is given by
{F1, F2}(M) = −Tr(∇TMC[∇MF1,∇MF2]), (2.98)
44
where F1 and F2 are scalar functions, ∇MF1 denotes the skew-symmetric matrix of
partial derivatives ∂F1/∂Mij, C =||M ||222 is the Casimir of this bracket and M is the
body angular momentum of the rigid body in so(3)∗.
Definition 2.5.4.5 (Body MV flow map (Moser & Veselov 1991)). Equation (2.85)
defines a discrete flow Mk+1 = ΦΔt(Mk) where ΦΔt is the discrete flow map taking the
form of the co-adjoint action
ΦΔt(Mk) = ΦΩk(Mk) := Ad∗ΩkMk. (2.99)
ΦΔt defines a Lie-Poisson integrator with respect to the so(3)∗ Lie-Poisson bracket
for the rigid body (Moser & Veselov 1991).
Proof. We verify that M ′ = Φg(M) is Poisson w.r.t. to the Lie-Poisson bracket given
by equation (2.100). From the definition of the so(3)∗ Poisson bracket for the rigid
body
{F1 ◦ Φg, F2 ◦ Φg}(M) = −Tr(∇TMC[g∇M ′F1g
T , g∇M ′F2gT ])
= −Tr(g(∇TM ′C
)gT [g∇M ′F1g
T , g∇M ′F2gT ])
= −Tr((∇TM ′C
)[∇M ′F1,∇M ′F2]) = {F1, F2}(M
′),
(2.100)
which proves that Φg is Poisson.
ΦΔt preserves the Casimir ||M ||2 of the rigid body bracket and it follows that ΦΔtsatisfies the definition of a Lie-Poisson integrator.
Remark 2.5.4.6. Rigid body Lie-Poisson integrators hence define a discrete flow on
the intersection of the momentum sphere (the Casimir of this bracket) and approximate
energy ellipsoids (approximated to within an order of Δt).
DMV algorithm Cardoso & Leite (2001) cast equation (2.85) into a discrete matrix
Ricatti equation and solve the eigenvalue problem using the Hamiltonian for this Ricatti
equation. McLachlan & Zanna (2005) provide a detailed description of the explicit
algorithm for implementing these steps and propose an optimised DMV algorithm. This
optimised algorithm avoids the use of a Schur factorisation to compute the eigenvalues
and eigenvectors of the Hamiltonian Ricatti equation. The numerical experiments
presented in Section 2.10 are performed using the unoptimised version of the DMV
algorithm, however.
45
2.5.5 The spatial representation
We now derive the equations of motion in the spatial representation by applying the
discrete Clebsch approach. Recall that the motivation for deriving these equations in
the spatial representation is to formulate a prototype MV integrator for the spatial
representation of continuum dynamics, in which quantities are advected. Following
Holm et al. (1986), who show that the spatial representation of the rigid body (in which
the inertia matrix is advected) corresponds to the spatial representation of continuum
dynamics, we pursue a MV integrator for the spatial representation of the rigid body.
In spatial variables, Ik and ωk := ΛkΛTk−1, the Clebsch constrained discrete Lagrangian
lIk is given by adding Clebsch constraints to the reduced discrete Lagrangian given in
equation (2.38)
lIk := Tr (Ikωk+1) +Tr
2(P Tk+1(Λk+1 − ωk+1Λk)
︸ ︷︷ ︸Clebsch constraint for Λk
+Tr
2(Jk+1(Ik+1 − ωk+1Ikω
Tk+1))
︸ ︷︷ ︸Clebsch constraint for Ik
−Tr(Θk+1(ωk+1ω
Tk+1 − Id)
).
(2.101)
Remark 2.5.5.1. The description of the Clebsch constraint for the evolution of Λk
is analogous to the description given in Section 2.5.3. The Clebsch constraint for the
evolution of Ik follows from expressing Ik = ΛkI0ΛTk as a recursion in Ik.
Stationarity of the Clebsch constrained discrete action principle
δSd =∑
k
lIk = 0, (2.102)
gives, as terms paired with variations in δΛk and δIk, the respective equations
[∇Λk lIk(Λk−1,Λk)] + [∇Λk lIk+1(Λk,Λk+1)] = 0,
[∇Ik lIk(Ik−1, Ik)] + [∇Ik lIk+1(Ik, Ik+1)] = 0.(2.103)
Evaluation of the derivatives of lIk in each equation w.r.t. Λk and Ik gives
[Pk/2] + [−ωTk+1Pk+1/2] = 0,
[Jk/2] + [∇Ik lIk − ωTk+1Jk+1ωk+1/2] = 0.
(2.104)
where the matrix derivative ∇Ik lIk evaluates to ∇Ik lIk = sym(ωk+1). Subsequent re-
arrangement of these equations gives,
46
Pk+1 = ω−Tk+1Pk,
Jk+1 = ωk+1(2∇Ik lIk + Jk)ωTk+1.
(2.105)
These two equations together with the discrete reconstruction formula for Λk and the
discrete auxiliary equation for Ik define the discrete flow on the augmented cotangent
bundle T ∗(G× V ∗).
Definition 2.5.5.2 (The discrete flow on the augmented cotangent bundle). The
discrete flow Φ′Δt : T∗(G × V ∗) → T ∗(G × V ∗) on the augmented cotangent bundle
zk ∈ T ∗(G× V ∗) is a smooth map zk+1 = Φ′Δt(zk) given by
Λk+1 = ωk+1Λk,
Pk+1 = ω−Tk+1Pk,
Ik+1 = φωk+1(Ik),
Jk+1 = φω−Tk+1(2∇Ik lIk + Jk).
(2.106)
The Clebsch relation, the expression paired with δωk+1, gives
(Ik − P
Tk+1Λ
Tk /2− Jk+1ωk+1Ik
)ωTk+1 = Θk+1. (2.107)
Using the symmetry property of the holonomic constraint Θk+1, the above equation
gives the map
mk+1 = skew(Pk+1ΛTk+1) + [Jk+1, Ik+1], (2.108)
where the spatial angular momentum mk+1 ∈ so(3)∗ is defined as
mk+1 := IkωTk+1 − ωk+1Ik. (2.109)
Using this definition of mk+1, we show that the spatial angular momentum is related
to the body angular momentum Mk+1 by the transformation
mk+1 = ΛkI0ΛTkΛkΛ
Tk+1 − Λk+1Λ
TkΛkI0Λ
Tk
= Λk(I0ΛTk+1Λk − Λ
TkΛk+1I0)Λ
Tk
= Λk(I0ΩTk+1 − Ωk+1I0)Λ
Tk
= ΛkMk+1ΛTk .
(2.110)
We shall now show that the map given by equation (2.108) is a left momentum map.
47
Lemma 2.5.5.3. The map mk : T∗(G× V ∗)→ g∗ given by
mk = skew(PkΛTk ) + [Jk, Ik] (2.111)
is an infinitesimally equivariant left momentum map of the general form
JL = Λ � P + J � I, (2.112)
for cotangent lifted left actions of ζ ∈ g on the augmented cotangent bundle T ∗(G×V ∗).
Proof. We first state the form of the left action of ζ on G×V ∗ and then use the pairing
between the augmented cotangent and tangent space to verify the form of the map
given by equation (2.111).
Consider the case when Λ is an element of any finite dimensional Lie group G. The
left translation on G× V ∗, for all g ∈ G, is given by the map
φg : G× V∗ → G× V ∗ : φg(Λ, I0) := (gΛ, gI0g
T ). (2.113)
The infinitesimal generator of φg on G × V ∗ is the left action of the Lie algebra ζ on
G× V ∗ given by
ζG×V ∗(Λ, I) := (ζΛ, ζI + IζT ), (2.114)
where I = gI0gT . The pairing between elements of the augmented cotangent space
T ∗(Λ,I)(G× V∗) and the augmented tangent space T(Λ,I)(G× V
∗) is defined as
〈(P, J), (ζΛ, ζI + IζT )〉 = 〈P, ζΛ〉+ 〈J, ζI + IζT 〉, (2.115)
where the first term on the right hand side pairs elements of T ∗ΛG and TΛG and the
second term pairs elements of the symmetric matrices V and TvV ' V ∗. Expressing
the last equation as a pairing of elements of g and g∗ gives
〈PΛT + 2JI, ζ〉 = 〈(P, J), (ζΛ, ζI + IζT )〉. (2.116)
By the theorem of momentum maps for lifted actions (Marsden & Ratiu 1999) given,
for convenience, on page 34, the expression on g∗ is an infinitesimally equivariant left
momentum map
JL = Λ � P + J � I. (2.117)
The bilinear operator � pairing elements in the first term of this expression is defined
as � : V ×V∗ → g∗, where T ∗ΛG ⊂ V∗, G ⊂ V and in the second term, � : V × V ∗ → g∗.
48
Rotations For the case when g ∈ G = SO(3) and g = so(3)∗, equation (2.116) takes
the form
〈skew(PΛT ) + [J, I], ζ〉 = 〈(P, J), (ζΛ, [ζ, I])〉, (2.118)
and setting Λ = Λk, P = Pk, I = Ik and J = Jk in this expression finishes the proof.
Discrete equations of motion Substituting the discrete flow equations on the aug-
mented co-tangent bundle into the left momentum map gives the spatial representation
of the discrete EP equations with an advected parameter
mk+1 = skew(Pk+1ΛTk+1) + [Jk+1, Ik+1]
= Ad∗ωTk+1
(skew(PkΛ
Tk ) + [Jk + 2∇Ik lIk , Ik]
)
= Ad∗ωTk+1(mk + 2[∇Ik lIk , Ik]) ,
(2.119)
which, together with the discrete auxiliary equation for Ik give the spatial MV integ-
rator
Ad∗ωk+1mk+1 = mk + 2∇Ik lIk � Ik,
Ik+1 = φωk+1(Ik),(2.120)
where the bilinear operator � is defined as � : V × V ∗ → so(3)∗.
These equations define the spatial representation of the MV integrator and are dis-
tinguished from the body representation by a discrete auxiliary equation for Ik and
an additional term 2∇Ik lIk(ωk+1) � Ik contributing to the expression for mk in so(3)∗.
These equations match the discrete Euler-Poincare equations for the spatial represent-
ation of the rigid body given by Bobenko & Suris (1999a) (up to a factor of 2 in front
of the derivative of lIk).
Using the invariance property of ∇Ik lIk(ωk+1) under transformations
∇Ik lIk = φ∗ωTk+1(∇Ik lIk) = ωk+1∇Ik lIkω
Tk+1, (2.121)
where φ∗g : V → V denotes the left translation on V by g, for all g ∈ G, defined by the
pairing between elements of V and V ∗
〈φ∗g(v), I〉 = 〈v, φg(I)〉, (2.122)
49
the spatial MV integrator, given by equations (2.120), may be written as
mk+1 = Ad∗ωTk+1
mk + 2∇Ik lIk � φωk+1Ik,
Ik+1 = φωk+1(Ik).(2.123)
The relation of the spatial and body MV integrators to their continuous time coun-
terparts are summarised in Figure 2.5.5.
Ad∗Λ−1(m, I)
M = ad∗ΩM←−−→ m = ad∗ωm+∇I lI � I
Ad∗Λ(M, I0)
↓ Ω ≈ΛTkh (Λk+1 − Λk) ω ≈ (Λk+1 − Λk)
ΛTkh ↓
Ad∗(Λk)−1(mk, Ik)
Mk+1 = Ad∗ΩkMk
←−−→ mk+1 = Ad
∗ωTk+1(mk + 2∇Ik lIk � Ik)
Ad∗(Λk)(Mk, I0)
Figure 2.2: The relation between the continuous body and spatial representations andthe body and spatial representations of the rigid body dynamics. Bobenko & Suris(1999a) show that the spatial representation of the discrete Euler Poincare equationfor the rigid body is also integrable.
Conservation of spatial angular momentum Before reviewing the geometric
properties of the spatial MV equations, we firstly verify, through the momentum map
given by equation (2.111), that the spatial angular momentum is conserved by the
discrete flow on the augmented cotangent bundle.
Lemma 2.5.5.4 (Conservation of spatial angular momentum). The spatial MV integ-
rator given by equations (2.120) conserves the spatial angular momentum mk.
Proof. Evaluating the expression for ∇Ik lIk and using the form of the bilinear operator
a � b = [a, b] gives
2∇Ik lIk(ωk+1) � Ik = [ωk+1 + ωTk+1, Ik]
= ωTk+1Ik − Ikωk+1 − (IkωTk+1 − ω
Tk+1Ik)
= Ad∗ωk+1(IkωTk+1 − ωk+1Ik)−mk+1
= Ad∗ωk+1mk+1 −mk+1.
(2.124)
50
Substituting this relation into equation (2.120) gives
Ad∗ωk+1mk+1 = mk +Ad∗ωk+1
mk+1 −mk+1, (2.125)
which implies the statement of conservation of spatial angular momentum mk+1 =
mk.
In the next Section, we will show that the spatial MV integrator is a co-adjoint
action in the dual of a semi-direct product Lie-algebra and preserves the Lie-Poisson
structure on this Lie-algebra.
2.6 Poisson Brackets on Semidirect Products
We will show that the spatial representation of the MV integrator is Lie-Poisson w.r.t.
to the same Lie-Poisson bracket on the dual of the semi-direct product algebra. This
result was independently shown by Bobenko & Suris (1999a) for the right reduced
discrete EP equations. We begin by introducing the following terminology.
Definition 2.6.0.5 (Lie-Poisson bracket on s∗ (Holm et al. 1986)). The (±) Lie-
Poisson brackets [s∗ = g n V ]± on the dual of the semi-direct product algebra s∗ are
given by (Holm et al. 1986, equation (3.11L), pg 59) as
{F1, F2}±(m, I) = ±〈m, [∇mF1,∇mF2]〉 ± 〈I,∇mF1 ∙ ∇IF2 −∇mF2 ∙ ∇IF1〉, (2.126)
where F1 and F2 are scalar functions defined on s∗ and ∇mF1 ∙ ∇IF2 denotes the left
action of ∇mF1 ∈ g on ∇IF2 ∈ V given by the pairing
〈ζ ∙ v, I〉 = 〈v,LζI〉. (2.127)
Lζv denotes the Lie derivative of ζ ∈ g on I ∈ V ∗, the vector field on V ∗ defined by the
left action of ζ on V ∗
Lζv = ζT v + vζ. (2.128)
Definition 2.6.0.6 (CoAdjoint actions of semi-direct products (Holm et al. 1986)).
The co-adjoint action of the semi-direct product (g, v) ∈ S = G n V on (M, I) ∈ s∗ =
g∗ n V ∗ is given by
Ad∗(g,v)−1(m, I) = (Ad∗gTm+ v � φg(I), φg(I)), (2.129)
where g acts on m by the co-adjoint action and on I by the left action. The bilinear
operator is defined as � : V × V ∗ → g∗.
51
Definition 2.6.0.7 (Spatial MV flow map). Equation (2.120) defines a discrete flow
(mk+1, Ik+1) = ΦΔt(mk, Ik) on s∗ = g∗nV ∗, where ΦΔt is the discrete flow map taking
the form of the co-adjoint action of (ωk+1, 2∇Ik lIk) ∈ Gn V on (Mk, Ik) ∈ s∗ given by
ΦΔt(mk, Ik) = Ad∗(ωk+1,2∇Ik lIk )
−1(mk, Ik) = (Ad∗ωTk+1
mk+2∇Ik lIk �φωk+1(Ik), φωk+1(Ik)).
(2.130)
ΦΔt defines a Lie-Poisson integrator with respect to the (±) Lie-Poisson brackets
[s∗]± on the dual of the semi-direct product Lie algebra s∗.
Proof. Recall that a map Φ is Poisson w.r.t. to [s∗]± if
{F1 ◦ Φ, F2 ◦ Φ}(m, I) = {F1, F2}(Φ(m, I)). (2.131)
We first note how derivatives of a function F on s∗ transform under the map (m′, I ′) =
Φ(m, I) := Ad∗(g,v)(m, I) by application of the chain rule
∇mF = AdgT∇m′F, ∇IF = φ∗g(∇I′F − [v,∇m′F ]). (2.132)
Using these expressions, the left hand side of equation of (2.131) evaluates to
{F1 ◦ Φ, F2 ◦ Φ}(m, I) = ±〈m,AdgT [∇′mF1,∇
′mF2]〉
+±〈I,AdgT (∇m′F1) ∙(φ∗g(∇I′F2 +∇m′F2 ∙ v)
)
−AdgT (∇m′F2) ∙(φ∗g(∇I′F1 +∇m′F1 ∙ v)
)〉,
(2.133)
On expressing m in terms of m′ and I in terms of I ′, the first bracket becomes
〈Ad∗g−T (m′ − v � I ′), AdgT [∇
′mF1,∇
′mF2]〉 = 〈m
′ − v � I ′, [∇′mF1,∇′mF2]〉. (2.134)
where from the definition of the diamond operator
〈v � I ′, [∇′mF1,∇′mF2]〉 = 〈I
′, [∇′mF1,∇′mF2] ∙ v〉. (2.135)
The second bracket becomes
〈φg−1(I′), AdgT (∇m′F1) ∙φ
∗g(∇I′F2+∇m′F2 ∙v)−AdgT (∇m′F2) ∙φ
∗g(∇I′F1+∇m′F1 ∙v)〉,
(2.136)
which simplifies to
〈I ′,∇m′F1 ∙ (∇I′F2 +∇m′F2 ∙ v)−∇m′F2 ∙ (∇I′F1 +∇m′F1 ∙ v)〉. (2.137)
52
This bracket rearranges to the form
〈I ′, [∇m′F1,∇m′F2] ∙ v〉, (2.138)
which cancels with the bracket given in equation (2.135). The discrete flow map there-
fore satisfies equation (2.131).
Holm et al. (1986) show that the Casimir functions on the Poisson manifold [s∗]± are
the set of functions invariant under the co-adjoint action of the Lie group S = GnV ∗ for
any finite dimensional Lie group G. It follows from the definition of ΦΔt as a co-adjoint
action of S, that the discrete flow defines a Poisson integrator.
Remark 2.6.0.8. For the case when G = SO(3), it is trivial to show that co-adjoint
actions of S preserve the Casimirs, the family of functions of ||m|| and ||I||.
Numerical experiments In Section 2.10, we implement the spatial MV integrator
using an adaptation of the DMV algorithm given by McLachlan & Zanna (2005). This
spatial variant of the DMV algorithm, referred to as the spatial DMV algorithm is
provided in Appendix A.3. We then compare the conservative properties of the spatial
DMV algorithm with the body DMV algorithm. These numerical experiments provide
insight into how the presence of an advected quantity affects the conservative properties
of the DMV algorithm.
In the next Section, we apply the discrete Clebsch approach to formulate the body
representations for two further examples of rigid body motion. The first example is
the heavy top and shows how body MV integrators are formulated using the discrete
Clebsch approach when there is a potential field. The second example is a coupled
rigid body, and shows how MV integrators are coupled. The formulation of coupled
MV integrators gives insight into how to apply MV integrators to elastic rod models,
as explicated in the next Chapter.
Each of these examples exhibit an advected quantity in the body representation.
The spatial representation provides little further insight into how to formulate MV
integrators for continuum dynamics with advected quantities and we do not belabour
its formulation.
For convenience, a summary of the main terms describing the MV integrator in the
body representation for these examples is given in a series of Tables in Appendix A.1.
Alongside each term in the Tables, we also provide its continuous (time) counterpart.
53
2.7 The Heavy Top
In this Section, we extend the review of the spatial and body representations of the rigid
body, presented thus far, to the heavy top. We consider the kinematics and symmetries
of the heavy top, described in the body frame by the orientation Γk of the vertical axis
z and the body angular velocity (see Figure 2.3). In the spatial representation, the
position of center of mass χk rotates relative to the point of support of the top and,
as previously described for the spatial representation of the free rigid body, the inertia
matrix also rotates.
2.7.1 The body representation
Figure 2.3: The heavy top as viewed in the body frame. The top is attached to the spatial frame at anarbitrary point (in this diagram this point is the origin). The motion of the top is composed of two components,precession and spinning. The unit vector z, representing the direction of gravity, rotates about each axis of theheavy top with body angular velocity Ω. Spatial angular momentum is only conserved for motions purely aboutE3, however. The body frame also spins about its centreline axis but this is only observable in the spatial frame.The vector χ0 from the point of support to the centre of mass of the top (c.o.m.), which lies on the centrelineof the top, remains fixed in the body frame.
The heavy top differs from the rigid body in the body representation by the presence
of an advected quantity, the body unit vector Γk, representing the direction of the unit
54
gravity vector z in the body frame given by
Γk = ΛTk z. (2.139)
The discrete auxiliary equation for Γk is given by substitution of the rearranged defin-
ition of Γk
z = ΛkΓk, (2.140)
into the definition of Γk+1 to give the discrete auxiliary equation for Γk of the form
Γk+1 = ΛTk+1z = Λ
Tk+1ΛkΓk = Ω
Tk+1Γk. (2.141)
The holonomically constrained discrete Lagrangian lcΓk : V → R for heavy top
motion in the body representation is defined as
lcΓk(Ωk+1) = Tr (I0Ωk+1) +mgh2〈Γk, χ0〉
− Tr(Θk+1(Ωk+1Ω
Tk+1 − Id)
),
(2.142)
where the first term is the kinetic energy, the second term is the gravitational potential,
the last term is the holonomic constraint restricting Ωk+1 to G = SO(3) and h is
the fixed time interval. The corresponding Clebsch constrained discrete Lagrangian is
defined as
lΓk = lcΓk+Tr
2
(P Tk+1(Λk+1 − ΛkΩk+1)
)+ 〈Jk+1,Γk+1 − Ω
Tk+1Γk〉, (2.143)
where the second and third terms are the Clebsch constraints for the discrete recon-
struction formula and the discrete auxiliary equation for Γk.
The stationary discrete action principle gives the following expressions paired with
Λk and Γk respectively
∇Λk lΓk−1(Λk−1,Λk) +∇Λk lΓk(Λk,Λk+1) = 0,
∇Γk lΓk−1(Γk−1,Γk) +∇Γk lΓk(Γk,Γk+1) = 0.(2.144)
Evaluation of the derivatives of lΓk
Pk/2− ΩTk+1Pk+1/2 = 0,
mgh2χ0 + Jk − Ωk+1Jk+1 = 0,(2.145)
where the second equation is derived by expressing the vector pairing as matrix pairings,
55
a ∙ b = Tr(abT ), and abT is a tensor product a⊗ b, gives
〈Jk+1,Γk+1 − ΩTk+1Γk〉 = Tr
(Jk+1(Γ
Tk+1 − Γ
TkΩk+1)
). (2.146)
Subsequent rearrangement of equations (2.145) gives,
Pk+1 = PkΩk+1,
Jk+1 = ΩTk+1 (Jk +∇Γk lΓk)︸ ︷︷ ︸
Gk
, (2.147)
where the derivative ∇Γk lΓk evaluates to ∇Γk lΓk = mgh2χ0.
The Clebsch relation, following from the discrete action principle, takes the form
I0 − ΛTk Pk+1/2− ΓkJ
Tk+1 = Θk+1Ωk+1. (2.148)
Using the symmetry property of the Lagrange multiplier Θk+1 and the recursions given
by equations (2.147) gives
I0ΩTk+1 − Λ
Tk Pk/2− ΓkG
Tk = Ωk+1I0 − P
Tk Λk/2−GkΓ
Tk . (2.149)
Using the tensor relation involving the hap map ˆ
abT − baT = [bˆ,aˆ], (2.150)
we write equation 2.149 as the map
Mk+1 = JRk+1 = skew(Λ
Tk Pk) + [Gkˆ,Γkˆ], (2.151)
where Mk+1 := I0ΩTk+1 − Ωk+1I0.
Recall from the theorem of momentum maps for tangent lifted actions (see 2.5.1.3
on pg. 34) that the right infinitesimally equivariant momentum map for left actions on
G× R3 is given by
〈JRk+1, ζ〉 = 〈Pk, ζG(Λk)〉+ 〈Gkˆ,−[ζ,Γkˆ]〉, (2.152)
in which JRk+1 is of the form
JRk+1 = Pk � Λk +Gk � Γk, (2.153)
where Gk = Jk +∇Γk lΓk , the � operator in the first term is the same form as for the
free rigid body and the � operator in the second term � : R3 ×R3 → so(3)∗ is given by
a � b = (a× b)ˆ = [aˆ, bˆ].
56
Substitution of the two expressions for Pk and Jk given by equation (2.147) into
equation (2.153) for the right momentum map gives
Mk+1 = ΩTkΛ
Tk−1Pk−1Ωk + [(Jk +∇Γk lΓk)ˆ,Γkˆ]
= ΩTk (Mk − [Gk−1ˆ,Γk−1ˆ])Ωk + [(Jk +∇Γk lΓk)ˆ,Γkˆ]
= Ad∗ΩTkMk − [Jkˆ,Γkˆ] + [(Jk +∇Γk lΓk)ˆ,Γkˆ]
= Ad∗ΩTkMk + [∇Γk lΓkˆ,Γkˆ],
(2.154)
which is the definition of the body representation of the MV integrator for the heavy
top on s∗ = so∗(3)nR3
Mk+1 = Ad∗ΩTkMk +∇Γk lΓk � Γk,
Γk = ΩkΓk−1,(2.155)
with the body angular momentum defined as
Mk+1 :=2
h2skew
((∇Ωk+1 lΓk)Ω
Tk+1
)= I0Ω
Tk+1 − Ωk+1I0. (2.156)
It follows from the previous Section, that the MV integrator for the heavy top, given by
equation (2.155), defines a co-adjoint action of S = SO(3)n R3 on s∗ = (so(3)× R3)∗
of the form
Ad∗(Ωk,∇Γk lΓk )−1(Mk,Γk) = (Ad
∗ΩTkMk +∇Γk lΓk � φΩk(Γk−1), φΩk(Γk−1)), (2.157)
and is Lie-Poisson w.r.t. to the ± Lie-Poisson brackets on s∗ given by
{F1, F2}±(Mk,Γk) = −1
2Tr(±MT
k [∇MkF1,∇MkF2])
± Γk ∙ (Φ(∇MkF1)∇ΓkF2 − Φ(∇MkF2)∇ΓkF1) .(2.158)
In Section 2.10, we verify the conservative properties of these equations by numerical
experiment.
This example demonstrates the formulation of a MV integrator for advected quant-
ities arising from the potential energy. In the next example, we shall consider the
formulation of MV integrators for coupled rigid body motion. The coupled rigid body
57
serves as a simple example of more complex multi-body systems considered in the next
Chapter.
58
2.8 The Coupled Rigid Body
Figure 2.4: The coupled rigid body as viewed in the frame of body 1. Each body is attached from its centreof mass (c.o.m.) to an ideal spherical joint. In the R3 reduced configuration space, the origin of the spatialframe is the centre of mass (C.O.M.) of the coupled rigid body. In the frame of body 1, the motion is composedof two components, precession and spinning. The vector Λd02, representing the position of the centre of mass ofbody 2 in the frame of body 1, rotates about the origin with relative orientation Λ = ΛT1 Λ2. φ and ψ denotethe angles between the body axes E1 and the vertical and θ denotes the angle between the body attachmentsat the joint. Each body also spins about its axes, but only the spin of body 2 is observable.
2.8.1 Preliminaries
Following Patrick (1989), we outline the Lagrangian description of the free motion of
two rigid bodies coupled with an ideal spherical joint, placed in a container whose origin
is fixed at the center of mass (c.o.m.) of the coupled body (as shown in Figure 2.4)
and Λ1,Λ2 ∈ G = SO(3) denote the attitude of body 1 and 2 respectively. The basic
configuration space, under the assumption that the centre of mass of the coupled body
is stationary, is C = G×G.
Patrick (1989) denotes the total mass as m = m1 +m2, the position of the center
of mass of each body as di and the attitude of body 2 in the frame of body 1, referred
to by Grossman, Krishnaprasad & Marsden (1988) as the relative orientation matrix,
as Λ = ΛT1 Λ2. ε =m1m2m denotes the reduced mass.
59
Patrick (1989) considers the Lagrangian L : TC → R which takes the form of the
kinetic energy of the coupled rigid body
L =2∑
i=1
1
2Tr(ΛiI1Λ
Ti )+
ε
2||Λidi|| − ε〈Λ1d1, Λ2d2〉.
(2.159)
which (except for the case when g1 = g2) is not invariant under tangent lifted left actions
of (g1, g2) ∈ G ×G on TC on account of an addition term gT1 g2 ∈ SO(3) appearing in
the coupling component of the kinetic energy, which for the case when g1 = Λ1 and
g2 = Λ2 is the relative orientation matrix.
Following Holm et al. (1986), we consider the Lagrangian
LΛ0 : TC→ R, (2.160)
which is given by the map Λ→ L(Λi, Λi,Λ), where L is defined on the augmented space
TC× SO(3). The variable Λ0 = Id denotes that Λ becomes a dynamical parameter in
the reduced configuration.
Under the tangent lift of the linear translation Φ(g1,g2) of g1 and g2 on SO(3)3
defined as
Φ(g1,g2)(Λ1,Λ2,Λ0) := Φg1 ◦ Φg2 := (g1Λ1, g2Λ2, g1Λ
0gT2 ), (2.161)
the Lagrangian LΛ0 becomes
LΛ(g1Λ1, g1Λ1; g2Λ2, g2Λ2) =2∑
i=1
1
2Tr(giΛiI1Λ
Ti g
Ti ) +
ε
2||giΛidi||
− εdT1 ΛT1 g
T1 Λg2Λ2d2,
(2.162)
where Λ = g1Λ0gT2 .
2.8.2 The body representation
To express the kinetic energy of the coupled rigid body as a quadratic form in the body
angular velocities, Grossman et al. (1988) introduce a modified inertia matrix, which
we denote as Ii (Grossman et al. (1988) denote this as J), defined as Ii = Ii + ε(dˆi)2,
where (∙) denotes the hat map ˆ : R3 → so(3).
Grossman et al. (1988) give the following reduced Lagrangian lΛ : so(3)×so(3)→ R
for the body representation given as the quadratic form
60
lΛ =1
2〈Ω,JΩ〉, (2.163)
where Ω := [Ω1,Ω2]. Ωi is the left invariant body angular velocity of body i and J is
the symmetric positive definite metric which Grossman et al. (1988) define as
J =
(I1 εA
εAT I2
)
where A := d1ˆTΛd2ˆ.
Expanding the quadratic form in equation (2.163) gives
lΛ =1
2
2∑
i=1
Tr(ΩTi IiΩi)− εTr(ΩT1AΩ2), (2.165)
where the first term is the kinetic energy of each free rigid body motion and the second
term is the kinetic energy of the rigid coupling motion. The modified inertia matrices
Ii and the position vectors di are fixed in the body frame but the relative orientation
matrix Λ is an advected quantity.
The discrete coupled rigid body
Repeating the procedure described for deriving the discrete Lagrangian for the body
representation of the free rigid body, we substitute the finite difference approximation
of the body angular velocities Ωi(tk) ≈ 1hΛ
iT
k (Λik+1 − Λ
ik) into the Lagrangian defined
in equation (2.165) to give the body representation of the holonomically constrained
discrete Lagrangian lcg : V2 → R given by
lcΛk =2∑
i=1
Tr(IiΩ
k+1i
)+ εTr
((Id − Ω
k+1T
1 )Ak(Id − Ωk+12 )
)
− Tr(Θk+1i (Ωk+1i Ωk+1
T
i − Id)),
(2.166)
where the first term is the approximation of the kinetic energy of the free motion of
each rigid body and the second term is the approximation of the coupling component
of the kinetic energy1. The third term is the holonomic constraint for restricting each
Ωk+1i to SO(3).
We obtain the discrete auxiliary equation for Λk by formulating the definition of
1The discrete Lagrangian has been factored by − 1h2which explains the change in sign of the coupling
component under discretisation of the Lagrangian.
61
Λk = ΛkT
1 Λk2 as a recursion
Λk+1 = Λk+1T
1 Λk1ΛkΛk
T
2 Λk+12 = Ωk+1
T
1 ΛkΩk+12 . (2.167)
The corresponding Clebsch constrained discrete Lagrangian is given by
lΛk = lcΛk +
2∑
i=1
Tr
2
(P k+1
T
i (Λk+1i − ΛkiΩk+1i )
)+ Tr
(Jk+1
T
(Λk+1 − Ωk+1T
1 ΛkΩk+12 )),
(2.168)
where the second and third terms are the Clebsch constraints for the discrete recon-
struction formula and the relative orientation matrix Λk ∈ V .
Under stationarity of the discrete action principle, the discrete Clebsch approach
gives the following two discrete Euler-Lagrange equations, paired with variations in Λkiand Λk respectively
∇Λki lΛk−1(Λk−1i ,Λki ) +∇Λki lΛk(Λ
ki ,Λ
k+1i ) = 0, i ∈ {1, 2},
∇Λk lΛk−1(Λk−1,Λk) +∇Λk lΛk(Λ
k,Λk+1) = 0.(2.169)
Evaluation of the derivatives of lΛk w.r.t. Λki and subsequent rearrangement gives,
P k+1i = P ki Ωk+1i . (2.170)
The derivative of lΛk w.r.t. Λk evaluates to
−Jk +Ωk+11 Jk+1Ωk+1T
2 +∇Λk lΛk = 0, (2.171)
where the derivative ∇Λk lΛk evaluates to
∇Λk lΛk = εd1ˆ(Id − Ωk+11 )(Id − Ω
k+1T
2 )d2ˆT . (2.172)
Equation (2.173) rearranges to
Jk+1 = Ωk+1T
1 (∇Λk lΛk + Jk)
︸ ︷︷ ︸Gk
Ωk+12 . (2.173)
The two Clebsch relations paired with δΩk+1i evaluate respectively to
Ck+11 − ΛkT
1 P k+11 /2− ΛkΩk+12 Jk+1T
= Θk+11 Ωk+11 ,
Ck+12 − ΛkT
2 P k+12 /2− ΛkT
Ωk+11 Jk+1 = Θk+12 Ωk+12 ,(2.174)
62
where Ck+1i := ∇Ωk+1ilΛk take the form
Ck+11 = I1 + εAk(Ωk+12 − Id),
Ck+12 = I2 + εAkT (Ωk+11 − Id).
(2.175)
The recursions for P ki and Jk given by equations (2.173) and (2.170), together with the
symmetry property of the Lagrange multipliers Θk+1i , yield the maps:
Mk+11 := Ck+11 Ωk+1
T
1 − Ωk+11 Ck+1T
1 = skew(ΛkT
1 P k1 + ΛkGk
T
),
Mk+12 := Ck+12 Ωk+1
T
2 − Ωk+12 Ck+1T
2 = skew(ΛkT
2 P k2 + ΛkTGk),
(2.176)
where Gk := ∇Λk lΛk + Jk.
From the theorem for momentum maps of lifted actions, the momentum maps for
the respective cotangent lifted left actions of ζ1 and ζ2 on P = T ∗C× SO(3) defined as
JRi : P → g∗ = so(3)∗, i ∈ {1, 2}, (2.177)
are infinitesimally equivariant momentum maps given by
〈JR1 , ζ〉 = 〈Pk1 , ζG(Λ
k1)〉+ 〈G
k,Λkζ〉,
〈JR2 , ζ〉 = 〈Pk2 , ζG(Λ
k2)〉+ 〈G
k,−ζΛk〉,(2.178)
and take the form
JR1 = Pk1 � Λ
k1 +G
k � Λk,
JR2 = Pk2 � Λ
k2 +G
k � Λk,(2.179)
where the second term is paired with the bilinear operator � : V∗×V . These expressions
evaluate to
JR1 = skew(ΛkT
1 P k1 + ΛkGk
T
),
JR2 = skew(ΛkT
2 P k2 + ΛkTGk),
(2.180)
verifying that equations (2.176) give the momentum maps associated with the reduction
to body representation, by the actions Φg1 and Φg2 defined in equation (2.161).
Conservation of spatial angular momentum The following calculation verifies
that the total spatial angular momentum is conserved by the discrete flow on the
augmented cotangent bundle through the momentum map given by equation (2.179).
63
The total spatial angular momentum is
mk+1 = mk+11 +mk+1
2 =2∑
i=1
ΛkiMk+1i Λk
T
i
=2∑
i=1
skew(ΛkiΛkT
i P ki ΛkT
i + Λk1Λ
kGkT
ΛkT
1 + Λk2Λ
kTGkΛkT
2 )
=2∑
i=1
skew(P k−1i ΩkiΩkT
i Λk−1Ti + Λk2G
kTΛkT
1 + Λk1G
kΛkT
2 )
=2∑
i=1
skew(P k−1i Λk−1T
i + Λk−12 Gk−1T
Λk−1T
1 + Λk−11 Gk−1Λk−1T
2 )
= mk1 +m
k2 = m
k.
(2.181)
Substituting the expressions for the discrete flow on the augmented cotangent bundle,
given by the discrete auxiliary equation, the discrete construction formulae and equa-
tions (2.170) and (2.173), into the right momentum map given in equation (2.179)
gives
Mk+11 = Ad∗
Ωk1(skew(Λk−1
T
1 P k−11 )) + skew(Λk(JkT
+∇TΛk lΛk)),
= Ad∗Ωk1
(Mk1 − skew(Λ
k−1Gk−1T
))+ skew(Λk(Jk
T
+∇TΛk lΛk)),
= Ad∗Ωk1
(Mk1 − skew(Λ
k−1Ωk2JkTΩk
T
1 ))+ skew(Λk(Jk
T
+∇TΛk lΛk)),
= Ad∗Ωk1Mk1 − skew(Λ
kJkT
) + skew(Λk(JkT
+∇TΛk lΛk)),
= Ad∗Ωk1Mk1 + skew(Λ
k∇TΛk lΛk),
(2.182)
and by an identical procedure
Mk+12 = Ad∗
Ωk2Mk2 + skew(Λ
kT∇Λk lΛk), (2.183)
which defines the body representation of the MV integrator for the coupled rigid body
defined on so(3)∗ × so(3)∗ × SO(3)
Mk+1i = Ad∗
ΩkiMki +∇Λk lΛk � Λ
k, i ∈ {1, 2}, (2.184)
together with the discrete auxiliary equation for the relative orientation matrix Λk in
64
the frame of body 1
Λk = φ(Ωk1 ,Ωk2)(Λk−1) = Ωk
T
1 Λk−1Ωk2. (2.185)
The MV integrator for the coupled rigid body is co-adjoint action of the Lie group
S = SO(3)2 × V∗ on the dual of the semi-direct product Lie algebra s∗ = so(3)∗ ×
so(3)∗ × SO(3) given by
Ad∗(Ωk1 ,Ω
k2 ,∇Λk lΛk )
−1(Mk1 ,M
k2 ,Λ
k) =
2∑
i=1
Ad∗ΩkiTM
ki +∇Λk lΛk � φ(Ωk1 ,Ωk2)(Λ
k−1), φ(Ωk1 ,Ωk2)(Λk−1).
(2.186)
A DMV algorithm for coupled rigid body motion The form of the momenta
given by equations (2.176) can be cast into a coupled matrix Ricatti equation
M1 =M′1 + J(Ω2)Ω
T1 − Ω1J(Ω2)
T ,
M2 =M′2 + J(Ω1)Ω
T2 − Ω2J(Ω1)
T ,(2.187)
where M ′i is an uncoupled term of the form M ′i = IiΩTi − ΩiIi and J(Ωi) is a function
of Ωi which takes the form
J(Ω1) = εAkT (Ωk+11 − Id),
J(Ω2) = εAk(Ωk+12 − Id).
(2.188)
Numerical experiments We provide an algorithm in Appendix A.4 for solving
equations (2.187) for the discrete angular velocities Ωk+11 and Ωk+12 . Preliminary nu-
merical experiments showing the conservative properties of the MV integrator for the
coupled rigid body are presented in Section 2.10. Grossman et al. (1988) give the Lie-
Poisson bracket for the coupled rigid body on so(3)∗×so(3)∗×SO(3). This bracket has
a Casimir of the form C = ||M1+ΛM2ΛT ||2 which we verify by numerical experiment.
Tables 5 and 6 of Appendix A summarise the main results of this Section.
We now consider how to adapt the discrete Clebsch approach to give the Cayley-
Klein parameterisation of the rigid body.
65
2.9 The Cayley-Klein Parameterisation of Rigid BodyMo-
tion
2.9.1 Background
It is well known that the parameterisation using Euler-angles of rigid body motion is
not global. This has important practical consequences, most familiar of which is the
problem of Gimble-lock. The Cayley-Klein parameterisation of rigid body motion is,
however, not only singularity free but equal to simple combinations of the components
of the quaternions. We shall therefore briefly review some relevant definitions before
deriving Moser-Veselov integrators for Cayley-Klein parameterisations of the rigid body.
Following Kosenko (1998), we consider the map from the quaternions to SU(2),
whose elements consist of all 2 × 2 unimodular unitary matrices,
h : q 7→1
|q|
(q0 − ıq3 −q2 − ıq1q2 + ıq1 q0 + ıq3
)
(2.189)
is a diffeomorphism on S3. When the quaternions have unit norm, they form a multi-
plicative group Q and h defines a group isomorphism h : Q → SU(2).
The matrix elements of SU(2) are the components of a quaternion q = q01+ q1i1+
q2i2+ q3i3 and |q|2 = 1 which, by definition, describes a rotation θ about a unit vector
[k1, k2, k3] given by qj = kjsin(θ2
), j := 1→ 3 and q0 = cos
(θ2
).
Elements of the Lie algebra su(2), all traceless skew-Hermitian matrices, may be
identified with the pure quaternions (R3,×) by the isomorphism
x 7→ x =1
2ıσjxj , (2.190)
whereσj2ı are the Pauli spin matrices which form the basis of su(2). In this basis, the
natural pairing between elements of x ∈ su(2) and its dual y ∈ su(2)∗ can be written
as
〈x, y〉 = −2Tr(xy) = x ∙ y. (2.191)
This pairing defines the map ˇ which identifies elements of su(2)∗ with R3. The dual of
su(2) thus also consists of all traceless skew-Hermitian matrices.
The MV integrators are now adapted from the Euler-angle to the Cayley-Klein
parameterisation by identifying the configuration space with SU(2) instead of SO(3).
Accordingly, the discrete body angular ”velocity” becomes
Ωk+1 = Λ†kΛk+1, (2.192)
where † denotes the adjoint (conjugate transpose) of the matrix and the Lagrangian is
66
then a function SU(2)× SU(2)→ R given by
L = Tr(Ω†k+1J(Ωk+1)
). (2.193)
The operator J takes the form J(Ωk+1) = Id − (1 − Ii)Ωk+1Tr2(E†iΩk+1
). So for
the special case when the principal moments of inertia I1 = I2 = 1 and I3 6= 1, J takes
the form J(Ωk+1) = Id − (1 − I3)Ωk+1Tr2(E†3Ωk+1
). Recall that this is a rigid body
motion with an axis of symmetry about the body axis E3. The discrete Lagrangian
is invariant under the left action of SU(2) and, when two of the principal moments of
inertia are equal, is also right invariant under the subgroup consisting of all matrices
of the form
g =
(α ıβ
ıβ α
)
, α, β ∈ R. (2.194)
This subgroup defines rotations θ about the E3 axis, the axis of symmetry of the
rigid body. This rotation is equivalent to the quaternion q01+ 0ı1 + 0ı2 + q3ı3, where
q0 = cos(θ/2) and q3 = sin(θ/2) which is mapped to a double covering of SO(3) of the
form
g =
q20 − q23 −2q0q3 0
2q0q3 q20 − q23 0
0 0 q20 + q23
. (2.195)
Now let a Hermitian matrix Uk+1 = Ωk+1Ω†k+1 and denote the real and imaginary
parts of a complex matrix as <{} and ={}. By analogy with equation (2.75), the
Clebsch constrained discrete Lagrangian lk+1 (where˜denotes the addition of Clebsch
constraints) for the body representation of the rigid body becomes
lk+1 = Tr(Ω†k+1J(Ωk+1)
)−Tr
2
(P†k+1(Λk+1 − ΛkΩk+1)
)
− Tr(<{Θ†k+1}(<{Uk+1} − Id)
)+ Tr
(={Θ†k+1}={Uk+1}
).
(2.196)
The last two terms in the definition of lk+1 are simply the real and imaginary parts
of the unitary constraint when Θk+1 is Hermitian. To show this, first split Θk+1 and
Uk+1 into its real and imaginary components
67
Tr(Θ†k+1(Uk+1 − Id)
)= Tr
(<{Θ†k+1}(<{Uk+1} − Id)
)+ ıT r
(<{Θ†k+1}={Uk+1}
)
+ ıT r(={Θ†k+1}(<{Uk+1} − Id)
)− Tr
(={Θ†k+1}={Uk+1}
).
(2.197)
Since any Hermitian matrix U has the property, <{U} = <{U}T and ={U} = −={U}T ,
it follows by the property of the trace operator that when the <{Θ†k+1} is symmetric
and ={Θ†k+1} is anti-symmetric, the unitary constraint on the left hand side of equa-
tion (2.197) is equivalent to the constraint in the discrete Lagrangian when Θk+1 is a
Hermitian matrix. We therefore use the unitary constraint with a Hermitian Lagrange
multiplier Θk+1 from hereon. The derivation of the Euler-Lagrange equations is then
a trivial generalisation of the procedure outlined in equation 2.5 to complex matrices.
The Clebsch relation paired with the variation δΩk+1 in the discrete stationary
action principle gives the expression
JRk+1 = skew(Λ†kPk), (2.198)
where skew now denotes the skew-Hermitian component. This expression satisfies the
definition of an equivariant right momentum map for cotangent lifted left actions of
SU(2)
〈JRk+1, ζ〉 = 〈Pk, ζGΛk〉
= 〈Pk � Λk, ζ〉,(2.199)
where the bilinear operator � is given by � : V∗ × V → su(2)∗. The image of this
map is the body angular momentum Mk+1 := Ak+1Ω†k+1 − Ωk+1A
†k+1 ∈ su(2)
∗, where
Ak+1 is defined as Ak+1 = ∇Ωk+1 lk+1, which from the definition of J takes the form
Id − IiTr(Ω†k+1Ei
)Ei.
Substitution of the following discrete flow equations on the cotangent bundle T ∗SU(2)
Pk+1 = PkΩk+1,
Λk+1 = ΛkΩk+1,(2.200)
into the right momentum map gives the discrete basic EP equation for the co-adjoint
orbits in su(2)∗ describing rigid body motion in the body representation
Mk+1 = Ad∗Ω†kMk. (2.201)
68
2.9.2 Momentum maps and Hopf fibrations
We now consider the form of the right momentum map for cotangent lifted right actions
of ζ ∈ su(2) on T∗G. Substitution of the equations for the discrete flow on the cotangent
bundle into the right momentum map gives
Λ†k+1Pk+1 = Ω†k+1Λ
†kPkΩk+1, (2.202)
which leaves the determinant of Λ†kPk invariant. When scaled by the constant1|P0|, this
product is a special unitary matrix. The value of this constant is obtained by equating
the expression for the body angular momentum with the momentum map to give
Pk = Λk+1Ak+1,1
|Ak+1|Ak+1 ∈ SU(2). (2.203)
where |Pk| = |Ak+1| = |A0|. The pre-image of the momentum map is therefore iso-
morphic to S3 with radius√|A0|. In the case of a rigid body with distinct principal
moments of inertia, the radius is√1 + I2i Ω
2i (t0), i := 1 → 3. The momentum map is
the Hopf fibration (Marsden & Ratiu 1999)
JRk+1 : S3 := {z ∈ C2| |z|2 = 1 + I2i Ω
2i (t0)} −→ S2 := {x ∈ R3| |x|2 =
3∑
i=1
M2i (t0)}.
(2.204)
This result is consistent with the continuous time geometric description of the rigid
body for which the momentum map for the SU(2) action on C2, the Cayley-Klein
parameters and the family of Hopf fibrations on concentric three-spheres in C2 are all
equivalent. We refer the reader to Table A.2 in Appendix A which compares the main
expressions for the MV integrators in terms of SU(2) matrices and quaternions.
2.10 Numerical Experiments
This Section presents results demonstrating the conservative properties, computa-
tional efficiency and accuracy of the rigid body, heavy top and coupled rigid body
integrators.
The components of the body momentum are compared with the analytic solution
for the rigid body only, and the Matlab Ode45 integration of the Euler-Arnold ordinary
differential equations and their variants for the heavy top and coupled rigid body. The
tolerance of the Ode45 routine is set to 10−15. The components of the quaternions and
69
the body angular momentum computed by the DMV algorithm for the Cayley-Klein
parameterisation of the symmetric rigid body is also presented.
The time step for all numerical experiments is Δt = 0.1. Although the Figure
captions give details of each experiment, we point out a few general features here.
Rigid body experiments
• Firstly, the choice of initial parameters in each experiment avoids intersection
of the body momenta with fixed points. It is well known that the co-adjoint
orbits of the classical rigid body with distinct moments of inertia have saddle
points at (0,±π, 0) (which are connected by four heteroclinic orbits) and centers
at (±π, 0, 0) and (0, 0,±π). We find that the numerical solution does not become
unstable, however, provided the time step is no larger than approximately 0.5.
• Figures 2.5 and 2.6 show that there is a good agreement between the numerical
results and the analytic solution for the rigid body and confirm conservation of
spatial angular momentum and the Casimirs ||M ||22 and (det(I), ||I||) for the body
and spatial representations respectively.
• The numerical round-off error in each representation depends upon the principle
moments of inertia. Figure 2.7 shows the comparative errors in the spatial angular
momentum and energy after 104 time steps for the case when I1 = I2 > I3. The
error in the (i) spatial angular momentum is O(10−8) and O(10−11) and (ii) energy
is O(10−7) and O(10−10) for the respective body and spatial representations.
Figure 2.8 shows the comparative errors in the spatial angular momentum and
energy after 104 time steps for the case when I1 > I2 > I3. The errors in the
(i) spatial angular momentum are O(10−14) and O(10−11) and (ii) energy are
O(10−13) and O(10−10) for the respective body and spatial algorithms.
• Each of these graphs also include the results of the explicit Runge-Kutta method
of Dormand & Prince (1980) (implemented in the Matlab ode45 routine). These
graphs do not indicate comparative performance of the DMV algorithms with
the Ode45 method. In each experiment, the ode45 solver was run at the smallest
time step possible (tol=10−15) purely to provide a quantitative benchmark for
the DMV algorithm.
• The comparative computational performance of the DMV algorithm with an ex-
plicit Lie-Poisson integrator based on splitting of the rigid body Hamiltonian
(McLachlan 1993) is shown in Section 2.10.3. Although this Lie-Poisson method
70
is not derived in a unified discrete framework it does provide a performance bench-
mark for the DMV algorithm of McLachlan & Zanna (2005). Both methods are
explicit and conserve the Hamiltonian to O(Δt)2 but the error in the Casimir and
spatial angular momentum conservation and computational efficiency differ. This
is because the DMV algorithm is formulated as matrices and uses a Schur fac-
torisation to solve the matrix Ricatti equation. The explicit Lie-Poisson method
based on splitting is formulated in vectors and performs only matrix vector multi-
plications. McLachlan (1993) present an optimised DMV algorithm, which avoids
the Schur factorisation, but we do not pursue this here.
Figures 2.9 and 2.10 show the respective components of the quaternions and the
body angular momentum computed by the DMV algorithm for the Cayley-Klein
parameterised rigid body motion. The group of quaternions forms a double cov-
ering of SO(3) and consequently the components of the quaternions are observed
to vary over two time scales. Only the first two components of the body angular
momentum vary periodically because the moments of inertia I1 = I2.
Heavy top and coupled rigid body experiments
• Figure 2.11 shows the components of the body angular momentum of heavy top
motion and the error in the Casimir 〈M,Γ〉 of the heavy top Lie-Poisson bracket.
We observe that whenever the first and second components of the body angular
momentum are non-zero, heavy top motion breaks the S1 symmetry about the
vertical axis. The experiment confirms that the Casimir 〈M,Γ〉 is conserved.
• We perform two coupled rigid body experiments in which two identical bodies are
subject to the same initial conditions (i) but are initially at right angles to each
other and (ii) are initially aligned with each other. In the first experiment, shown
in Figure 2.12, we observe non-periodic behaviour in the body angular momentum
components caused by exchanges of momentum between the two bodies. The
second component of the momentum changes the most, ranging from −1 to 0.8.
In the second experiment, shown in Figure 2.13, we recover a rigid body motion
similar to that shown in Figure 2.5, except that M3 varies.
71
2.10.1 Body and spatial DMV algorithms for the rigid body
0 100 200 300 400 500 600 700 800 900 1000-0.02
0
0.02
M1-M
* 1
time steps
0 100 200 300 400 500 600 700 800 900 1000-0.02
0
0.02
time steps
M2-M
* 2
0 100 200 300 400 500 600 700 800 900 1000-2
0
2x 10
-9
time steps
M3-M
* 3
0 100 200 300 400 500 600 700 800 900 10000
2x 10
-4
time steps
||M-M
* ||2 2
0 100 200 300 400 500 600 700 800 900 1000-2
0
2x 10
-12
time steps
Spatial DMVBody DMVOde45
|| Ik ||
2 - || I
0 ||
2det(I
k)-det(I
0)
Figure 2.5: This Figure compares numerical simulations of a rigid body over 1000 timesteps for the case when the principal moments of inertia I1 = I2 > I3 (I1 = 2, I2 =2, I3 = 1). The top three graphs show the error in each component of the body angularmomentum of rigid body motion (M∗ denotes the analytic momentum). The bottomtwo graphs show the error in the Casimirs ||M ||22 and ||I||2, det(I) of rigid body flow inthe respective body and spatial representations. The graph labelled (i) ”body DMV”is the solution computed by the body DMV algorithm, (ii) ”spatial DMV” is the bodyframe translated solution computed by the spatial DMV algorithm (which computes theangular momentum and moment of inertia in the spatial representation) (iii) ”ode45” isan explicit Runge-Kutta (4,5) integrated solution of the Euler-Arnold equations usingthe Matlab routine, ode45 and (iv) ”analytic” is the analytical solution. The initialconditions for this simulation are the initial body angular momentum components givenas M1(0) = 0.1, M2(0) = 0, M3(0) = 1. The top three graphs show that the DMVmomentum matches the analytical solution which describes the rolling of a cone ofconstant angle in the body on a second cone of constant angle fixed in space (Marsden &Ratiu 1999). The 2nd from bottom graph shows that the body DMV algorithm preciselycomputes the Casimir ||M ||22 (Marsden & Ratiu 1999) suggesting preservation of therigid body Lie-Poisson structure and consequently that the DMV angular momentumremains on the sphere. The bottom graph shows that the Casimirs ||I||2 and det(I)of the spatial DMV algorithm are correctly computed suggesting that the Lie-Poissonstructure on the dual of the semi-direct product Lie algebra is also preserved.
72
0 100 200 300 400 500 600 700 800 900 1000-0.01
0
0.01
M1-M
* 1
time steps
0 100 200 300 400 500 600 700 800 900 1000-0.02
0
0.02
time steps
M2-M
* 2
0 100 200 300 400 500 600 700 800 900 1000-5
0
5x 10
-3
time steps
M3-M
* 3
0 100 200 300 400 500 600 700 800 900 10000
2x 10
-4
time steps
||M-M
* ||2 2
0 100 200 300 400 500 600 700 800 900 1000-5
0
5x 10
-13
time steps
Spatial DMVBody DMVOde45
|| Ik ||
2 - || I
0 ||
2det(I
k)-det(I
0)
Figure 2.6: This Figure compares numerical simulations of a rigid body over 1000time steps for the case when the principal moments of inertia I1 > I2 > I3 (I1 =3.5, I2 = 2.5, I3 = 2). The top three graphs each the error in each component of thebody angular momentum of rigid body motion (M∗ denotes the analytic momentum).The bottom two show the error in the Casimirs ||M ||22 and ||I||2, det(I) of rigid bodyflow in the respective body and spatial representations. The initial conditions forthis simulation are the initial body angular momentum components given as M1(0) =−0.5, M2(0) = 0, M3(0) = 1. The top three graphs show that the DMV momentummatches the analytical solution describing the intersection of energy ellipsoids withco-adjoint orbits of SO(3) which are two-spheres Marsden & Ratiu (1999). Note thatalthough our choice of simulation parameters avoids the flow intersecting either of thetwo saddle points at (0,±||M ||2, 0) or centers at (±||M ||2, 0, 0) and (0, 0,±||M ||2), thesolution does not become unstable provided the time step Δt < 0.5.
73
0 2000 4000 6000 8000 1000010
-20
10-15
10-10
10-5
time steps
|E-E
* |/E*
0 2000 4000 6000 8000 1000010
-20
10-15
10-10
10-5
time steps
||m-m
0|| 2
Spatial DMV Body DMV Ode45
Figure 2.7: This Figure compares the energy and spatial angular momentum error innumerical simulations of rigid body motion over 104 time steps for the case when theprincipal moments of inertia I1 = I2 > I3 (I1 = 2, I2 = 2, I3 = 1). The top graphshows the relative energy error growth in the solutions computed by the DMV algorithmand the ode45 integrator and the bottom graph shows the error in the approximatedspatial angular momentum. The initial conditions for this simulation are the initialbody angular momentum components given as M1(0) = 0.1, M2(0) = 0, M3(0) = 1.The graphs show that the error in the energy and spatial angular momentum computedby the body DMV algorithm is higher than the error computed by the spatial DMValgorithm. The bottom graph also shows that the spatial DMV algorithm conservesspatial angular momentum to numerical round off.
74
0 2000 4000 6000 8000 1000010
-20
10-15
10-10
10-5
time steps
|E-E
* |/E*
0 2000 4000 6000 8000 1000010
-16
10-14
10-12
10-10
time steps
||m-m
0|| 2
Spatial DMVBody DMVOde45
Figure 2.8: This Figure compares the energy and spatial angular momentum error innumerical simulations of rigid body motion over 104 time steps for the case when theprincipal moments of inertia I1 > I2 > I3 (I1 = 3.5, I2 = 2.5, I3 = 2). The top graphshows the relative energy error growth in the solutions computed by the DMV algorithmand the ode45 integrator and the bottom graph shows the error in the approximatedspatial angular momentum. The initial conditions for this simulation are the initialbody angular momentum components given as M1(0) = −0.5, M2(0) = 0, M3(0) = 1.In contrast with the previous Figure, the graphs show that the error in the energy andspatial angular momentum computed by the spatial DMV algorithm is higher than theerror computed by the body DMV algorithm. The bottom graph also shows that thebody DMV algorithm conserves spatial angular momentum to numerical round off.
75
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1
0
1
time steps
q 0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1
0
1
time steps
q 1
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1
0
1
time steps
q 2
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1
0
1
time steps
q 3
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1e-11
0
1e-11
time steps
|q|-
12
DMV
Figure 2.9: This Figure shows the components of the quaternions computed by theDMV algorithm for the Cayley-Klein parameterised rigid body motion whose principalmoments of inertia are I1 = I2 = 1 and I3 = 0.5. The group of quaternions providesa double covering of SO(3) and each component is observed to vary over two timescales. The initial conditions for this simulation over 104 time steps are M1(0) =−0.5, M2(0) = 0.1, M3(0) = 0.5.
76
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1
0
1
time steps
M1
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1
0
1
time steps
M2
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1e-15
0
1e-15
time steps
M -
13
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1e-11
0
1e-11
time steps||M|||
-||M
(0)|
|2
222
DMV
Figure 2.10: This Figure shows the components of the angular momentum computedby the DMV algorithm for the Cayley-Klein parameterised rigid body whose principalmoments of inertia are I1 = I2 = 1 and I3 = 0.5 and initial conditions are the same asfor the previous Figure. This Figure verifies that (i) the first two components of thebody angular momentum are periodic, (ii) the third component is conserved and (iii)the norm of the momentum is a Casimir.
77
2.10.2 Body DMV algorithm for the heavy top and coupled rigid
body
0 100 200 300 400 500 600 700 800 900 1000-0.2
0
0.2
M1
0 100 200 300 400 500 600 700 800 900 1000-0.2
0
0.2
M2
0 100 200 300 400 500 600 700 800 900 1000-5
0
5
10x 10
-7
M3-1
0 100 200 300 400 500 600 700 800 900 1000-2
0
2
4x 10
-4
time steps
<M
,Γ>
- <
M0,Γ
0>
Body DMVOde45
Figure 2.11: This Figure compares numerical simulations of the body representationof the heavy top over 1000 time steps for the case when the principal moments ofinertia I1 = I2 > I3 (I1 = 2, I2 = 2, I3 = 1). The top three graphs each show acomponent of the body angular momentum of heavy top motion and the bottom graphshows the error in the body DMV and ode45 computed Casimir 〈M,Γ〉 of the heavy topLie-Poisson bracket. The initial conditions for this simulation are the initial (i) bodyangular momentum components and (ii) position of the vertical axis in the body framegiven respectively as M1(0) = 0.1, M2(0) = 0, M3(0) = 1 and Γ = [0, 0, 1]. Wheneverthe first and second components of the body angular momentum are non-zero, heavytop motion breaks the S1 symmetry about the vertical axis. The bottom graph showsthat there is no secular growth in 〈M,Γ〉.
78
0 200 400 600 800 1000-2
0
2
time steps
M1
0 200 400 600 800 1000-1
0
1
time steps
M2
0 200 400 600 800 10000
1
2
time steps
M3
Body 1 Body 2
Figure 2.12: This Figure compares numerical simulations of the coupled rigid body, asseen in the frame of body 1, over 1000 time steps for the case when the two identicalrigid bodies are initially positioned at right angles to each other. The top three graphseach show a component of the body angular momentum of body 1 and 2. The initialconditions for this simulation are the initial (i) body angular momentum components(ii) orientation of the bodies relative to their E3 axes and (iii) angle between themechanical attachments at the ball and socket joint given respectively as M2(0) =M1(0) = [0.5, 0, 1], φ(0) = ψ(0) and θ(0) = π
2 . The principal moments of inertia ofthe two identical rigid bodies are I1 = I2 > I3 (I1 = 2, I2 = 2, I3 = 1). The graphsshow that the components of body angular momentum are not periodic and extremevalues are different from those of the single (uncoupled) rigid body shown in Figure2.5, indicating transfer of angular momentum between the two bodies.
79
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-0.02
-0.01
0
0.01
0.02
M1-M
1ode4
5
time steps
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-0.01
-0.005
0
0.005
0.01
time steps
M2-M
2ode4
5
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-2
-1
0
1
2x 10
-3
time steps
M3-M
3ode4
5
Figure 2.13: This Figure compares numerical simulations of the coupled rigid body, asseen in the frame of body 1, over 10000 time steps for the case when the two identicalrigid bodies are initially aligned with each other. The three graphs each show thedifference of each component of the body angular momentum of body 1 between theDMV algorithm and ode45 computations. The initial conditions for this simulationare the initial (i) body angular momentum components (ii) orientation of the bodiesrelative to their E3 axes and (iii) angle between the mechanical attachments at theball and socket joint given respectively as M2(0) = M1(0) = [0.5, 0, 1], φ(0) = ψ(0)and θ(0) = 0. The principal moments of inertia of the two identical rigid bodies areI1 = I2 > I3 (I1 = 2, I2 = 2, I3 = 1).
80
0 200 400 600 800 1000-5
0
5x 10
-12
time steps
||M||2 2
0 200 400 600 800 1000-1
0
1x 10
-3
time steps
Ene
rgy
erro
r
0 200 400 600 800 1000-2
0
2x 10
-11
time steps
||m-m
0|| 2
Figure 2.14: This Figure shows the error in computation of the Casimirs and conservedspatial angular momentum of the coupled rigid body motion, as viewed in the frameof body 1, for the case when both identical bodies are initially positioned at rightangles to each other. The top graph shows the absolute error in DMV and ode45computations of the C.R.B. Casimir ||M ||22 = ||M1+ΛM2Λ
T ||22 (Grossman et al. 1988).The middle and bottom graphs show the comparative error in the energy and spatialangular momentum of the coupled rigid body. The initial conditions for this simulationare the initial (i) body angular momentum components (ii) orientation of the bodiesrelative to their E3 axes and (iii) angle between the mechanical attachments at theball and socket joint given respectively as M2(0) = M1(0) = [0.5, 0, 1], φ(0) = ψ(0)and θ(0) = π
2 . The principal moments of inertia of the two identical rigid bodies areI1 = I2 > I3 (I1 = 2, I2 = 2, I3 = 1).
81
2.10.3 Comparison with a Lie-Poisson integrator based on splitting
Figure 2.15: This Figure shows the comparative error in the Hamiltonian as computedby the (body) DMV algorithm for the rigid body and the splitting integrator, over arange of time steps Δt. The splitting integrator is an explicit Lie-Poisson integratorbased on a splitting of the rigid body Hamiltonian (McLachlan 1993). The Figureshows that both methods are second-order accurate in Δt - their slope coincides withthe slope of the triangle, which represents an O(Δt2) scaling law.
82
Figure 2.16: This Figure shows the comparative cpu time of the (body) DMV algorithmfor the rigid body against the splitting integrator, over a range of total number ofsimulations N . The splitting integrator is an explicit Lie-Poisson algorithm based on asplitting of the rigid body Hamiltonian (McLachlan 1993). Both methods are explicit,but the DMV algorithm is not as efficient for two reasons; firsty the DMV algorithmcomputes the momentum and angular velocities in matrix form where as the splittingintegrator computes the momenta as vectors. Secondly, the DMV algorithm uses aSchur factorisation, which is a computationally expensive step. McLachlan & Zanna(2005) show that this step can be replaced with an explicit expression for computingthe eigenvalues of the Hamiltonian associated with the matrix Ricatti equation, but wedo not implement this here.
83
0 2000 4000 6000 8000 10000-18
-16
-14
-12
-10
-8
-6
-4
-2
0
2x 10
-13
time steps
||M||-
||M0||
DMVSplitting
0 2000 4000 6000 8000 10000-20
-15
-10
-5
0
5x 10
-13
time steps
||m-m
0||
DMVSplitting
Figure 2.17: These graphs show the comparative errors in the rigid body Casimir andspatial angular momentum as computed by the (body) DMV algorithm and the splittingmethod of McLachlan (1993). The time step size is 0.01, the moments of inertia areI1 = 1, I2 = 2, I3 = 3 and the initial conditions are M0 = [−0.5, 0, 1]. The reason whythe error growth computed by the DMV algorithm is larger can be attributed to boththe roundoff error from matrix operations and error in the Schur factorisation.
84
2.11 Summary
We pursue the discrete Clebsch approach by Cotter & Holm (2006) as the basis of a
unified computational framework for deriving geometric integrators for the convective
and spatial description of continuum dynamics. Holm et al. (1986) showed that the
spatial and body representations of rigid body motion correspond to the spatial and
convective representations of continuum dynamics. Under a finite difference approxim-
ation of the continuous Clebsch constrained action principle, we show that the discrete
Clebsch approach gives the same integrator derived by Moser & Veselov (1991) for the
body representation of the free rigid body. We also show that the discrete Clebsch
approach gives a corresponding integrator, referred to as a spatial Moser-Veselov (MV)
integrator, for the spatial description of the rigid body, and takes the form of the dis-
crete Euler-Poincare equations for the spatial representation of the rigid body derived
by Bobenko & Suris (1999a). The discrete EP equations are known to correspond to
the Lie-Poisson equations on the dual of a semi-direct Lie-algebra.
In pursuit of a unified framework, we then apply the discrete Clebsch approach
to give MV integrators for the heavy top and the coupled rigid body. These examples
demonstrate how the discrete Clebsch approach is applied for potential and coupled mo-
tions respectively. Furthering this pursuit still, we also show how the discrete Clebsch
approach gives MV integrators for (the singularity free) Cayley-Klein parameterised
rigid body motion. Finally, we present numerical results which show a comparative
study of the conservative properties of the body and spatial DMV algorithms for the
rigid body and the conservative properties of the body DMV algorithm for the heavy
top and coupled rigid body motions. This Chapter provides the necessary geometric
preliminaries and assessment of the DMV algorithms for extension to elastic bodies, a
subject which we pursue in the next Chapter.
The discrete Clebsch approach The discrete Clebsch approach (Cotter & Holm
2006) gives a (Hamiltonian) discrete Noether’s theorem which states the condition for
the discrete Euler-Lagrange equations to conserve the momentum map. We derived
the momentum maps corresponding to the symmetry reductions of the discrete (time)
Lagrangians and show that the form of the right momentum map is the same as its
continuous form. Through this momentum map, the discrete time body representation
of the dynamics on the cotangent bundle gives the Moser and Veselov integrator. The
key point, here, is that the discrete Clebsch approach only recovers the MV integrator if
the discretisation of the body angular velocity in the reconstruction formula is the same
as that used to derive Moser and Veselov’s discrete Lagrangian from the continuous
Lagrangian.
85
Lie-Poisson structure This Chapter has pursued the relationship between vari-
ational integrators, derived in the discrete Clebsch framework, and existing studies of
discrete integrable rigid body systems. In the body representation, we recover a dis-
crete integrable analogue of the Euler-Arnold equations, first discovered by Moser &
Veselov (1991). In the spatial representation we obtain discrete EP equations with
an advected parameter. These correspond to the Lie-Poisson equations on the dual
space of a semidirect product Lie algebra discovered by Bobenko & Suris (1999a). This
discovery provides a discrete extension to the work by Holm, Marsden & Ratiu (1998)
who developed the theory of EP entirely within a Lagrangian framework so that the
EP equations with advection always correspond to Lie-Poisson Hamiltonian systems on
the dual of a semi-direct product Lie-algebra. Consequently, the DMV equations have
a family of Casimirs associated with the Lie-Poisson bracket for these systems, two of
which we confirmed by numerical simulation of the spatial representation of the rigid
body.
Heavy tops We applied the discrete Clebsch approach to the body representation
of the heavy top to give a MV integrator with a discrete auxiliary equation for the
advection of the gravity vector. This integrator is Lie-Poisson w.r.t. to the dual of the
semi-direct product Lie-algebra s∗ = (so(3)×R3)∗. The MV integrator is solved using
the standard DMV algorithm proposed by McLachlan & Zanna (2005).
Coupled rigid bodies Application of the discrete Clebsch approach to the coupled
rigid body model of Grossman et al. (1988) gives a coupled MV integrator for each
body and an auxiliary equation for the relative orientation matrix. We will refer to
this equation in Chapter 3 as it turns out to be the same as the compatibility equation
for fully discrete elastic rod models. The MV integrator conserves total spatial angular
momentum and is Lie-Poisson w.r.t. to the dual of the semi-direct product Lie-algebra
s∗ = (so(3)×so(3))∗×SO(3). We derive the DMV algorithm for this coupled integrator
from the coupled matrix Ricatti equation. The DMV algorithm for coupled motion is
used in Chapter 3 to solve for the isotropic pseudo-rigid body.
Cayley-Klein parameterisation By simply replacing SO(3) with the matrix group
SU(2) in the discrete action principle, the discrete Clebsch approach gives a DMV al-
gorithm for the Cayley-Klein parameterisation of rigid body motion. It is well known
that this parametrisation is singularity-free in contrast with Euler-angle parameterisa-
tion. The DMV algorithm is explicit when the motion has at least one axis of symmetry,
i.e. is a Lagrange top. Unlike other integrators for the Cayley-Klein parameterised rigid
body (see Hairer et al. 2002, Leimkuhler & Reich 2005, Wendlandt & Marsden 1997)
our approach is unique in that it is systematically derived in a framework for body and
86
spatial representations (although we do not give the spatial representation here). Our
framework provides additional geometric insight into the DMV algorithm. Specifically,
we show that the momentum map for the symmetry reduction by SU(2) is a Hopf
fibration (as it is in the continuous case).
The discrete Clebsch approach gives the correspondence between the quaternionic
formulation and the (standard) SO(3) formulation of DMV integrators. The question
of comparing variational discretisations of rigid body motion formulated in quaternions
with SO(3) formulated MV integrators was posed in Wendlandt & Marsden (1997). We
propose a comparative numerical study of the quaternionic formulation of MV integ-
rators with alternative quaternionic discretisations , such as those given in Wendlandt
& Marsden (1997), as the subject of future research.
Numerical experiments We provide results from several numerical experiments to
demonstrate the conservative properties, computational efficiency and accuracy of the
DMV algorithms for all the rigid body motions considered in this Chapter.
Appendix Appendix A give Tables comparing the MV integrators with the continu-
ous form of the rigid body, heavy top and coupled rigid body. These Tables show that
the discrete and continuous versions of the equations of motion and the momentum
maps from the cotangent bundle to the dual of the Lie algebras are remarkably similar.
Appendix A also provides a Table summarising the SU(2) DMV algorithm for the body
representation of the rigid body in Cayley-Klein parameters and the corresponding for-
mulation in quaternions.
Chapter 3 The rigid body motions considered in this Chapter served only as ex-
amples to (i) explain the geometric principles guiding the development of a unified
computational framework for continuum dynamics and (ii) demonstrate the conser-
vative properties of the DMV algorithms. We continue in the next Chapter by first
generalising the application of the discrete Clebsch approach to give MV integrators
for the convective and spatial representations of ellipsoidal motions and then applying
the approach to more challenging problems in elastic body dynamics.
87
Chapter 3
Moser-Veselov Integrators for
Elastic Body and Rod Motions
Synopsis This Chapter develops a unified computational framework for the convect-
ive and spatial representations of elastic body motion, polar decomposed pseudo-rigid
body motion and geometrically exact elastic rods. In this Chapter, we apply the dis-
crete Clebsch approach of Cotter & Holm (2006) to give Moser-Veselov integrators for
two examples of elastic motions.
We firstly present MV integrators for the spatial and convective representations of free
ellipsoidal motions on GL(n,R)+ (denoted GL(n)+ from hereon) before restricting our
attention to a pseudo-rigid body.
Polar decomposition of the pseudo-rigid body motion gives a MV integrator which (i)
exhibits a discrete momentum and Kelvin circulation law and (ii) is solved using the
DMV algorithm for the coupled rigid body motions given in Appendix A.4, based on
the DMV algorithm for the rigid body given by McLachlan & Zanna (2005). Numerical
simulations of a Mooney-Rivlin material (Mooney & Rivlin 1977) conserve vorticity to
10−15 and exhibit no secular drift in the energy levels.
We then develop MV integrators for continuum elastic rods by using a discrete
Kirchhoff kinetic analogy to apply the MV integrator for the Lagrange top to the static
inextensible rod. Extending this analogy to the time dependent case gives a discrete
compatibility equation for the dynamic elastic rod model which takes the same form as
the discrete auxiliary equation for the relative orientation matrix, given by the discrete
formulation of the coupled rigid body model of Chapter 2. We go on to formulate a
MV integrator for an extensible and shearable elastic rod, which Krishnaprasad et al.
(1988) refer to as a geometrically exact elastic rod. Numerical simulations of this model,
with 50 rod sections, conserve spatial angular momentum to an order of 10−8, linear
momentum to an order of 10−11 and exhibit no secular drift in the energy error whose
88
mean is the order of 10−2 (over 104 time steps).
The main theorems and equations of motion of this Chapter are stated in the Table
below.
Result Equation page
MV integrators for ellipsoidal motion on GL(n)+ (3.27), (3.35) pg. 96 & pg. 98A discrete Kelvin circulation theorem for the pseudo-rigid body (3.53) pg. 102MV integrators for the polar decomposed pseudo-rigid body (3.60) pg. 104A discrete Kirchhoff rod analogy (3.3.1.2) pg. 106MV integrators for the geometrically exact rod (3.106) pg. 117
Table 3.1: A summary of the main theorems and equations of motion in this Chapter.
3.0.1 Overview
In the late 1980’s and early 1990’s Juan Carlos Simo proposed a general approach for
deriving integrators of Hamiltonian continuum systems on manifolds from the Hamilto-
nian on phase space Simo & Wong (1991), Simo et al. (1992), Lewis & Simo (1994),
Simo & Tarnow (1994). This general approach prescribes algorithms which exactly
preserve any two of energy, momentum or symplectic structure. We develop an altern-
ative general approach for deriving continuum integrators from a discrete variational
principle using the discrete Clebsch approach of Cotter & Holm (2006). The purpose
of this is to
• generalise the derivation of Moser-Veselov integrators and the momentum maps
corresponding to cotangent lifted actions of gl(n) for the convective and spatial
representations of ellipsoidal motion on GL(n)+. In this generalised model, either
the shape matrix or the right Cauchy-Green matrix is an advected quantity de-
pending on whether the motion is the spatial or body representation. Just as
we saw in the previous Chapter for the spatial representation of the rigid body
(in which the inertia matrix is an advected quantity), convective and spatial MV
integrators for motion with advected quantities define a co-adjoint action on the
dual of a semi-direct product Lie-algebra and are Poisson w.r.t. to the corres-
ponding Lie-Poisson bracket.
• formulate a MV integrator for the three dimensional isotropic pseudo-rigid body.
This model arises as a special case of ellipsoidal motion when the configuration
is GL(3)+ and the reference configuration is a sphere (the shape matrix is the
identity matrix). Using polar decomposition, we derive MV integrators for the
polar components of pseudo-rigid body motion in Section 3.2.1. These MV in-
tegrators provide a discrete analogue of the Euler-Poincare equations for polar
89
decomposed pseudo-rigid body motions derived by Sousa-Dias (2002). We also
give the left and right momentum maps corresponding to cotangent lifted actions
of so(3) on the cotangent bundle. These maps correspond to conservation of
angular momentum and vorticity.
Relation to fluid dynamics The analogy with fluid dynamics becomes appar-
ent from the latter conserved quantity. The invariance of the Lagrangian under
the action of S is referred to in the context of fluid dynamics as the ”particle re-
labelling symmetry” (see Ripa 1981, Salmon 1982). There, invariance of the vari-
ational principle for idealised fluids under continuous relabelling of fluid particle
labels by the action of the group of diffeomorphisms is associated with conser-
vation of fluid vorticity. Of course, the symmetry transformations in fluids are
both spatially and time dependent. This aside, the analogy with fluids can be
extended - the momentum map corresponding to conserved vorticity is shown to
give a discrete Kelvin circulation theorem. We shall return to the discussion of
particle relabelling symmetries in the next Chapter.
Finally, we implement a model of a Mooney-Rivlin material to describe the
stretching and rotational components of the motion and show that the discrete
Moser-Veselov conserves angular momentum and vorticity (relative to the Lag-
rangian frame) to an order of 10−15 and exhibit no secular drift in the energy
error whose mean is the order of 10−3 (over 104 time steps).
• derive discrete elastic rod models. The key step is the use of a discrete variant of
the Kirchhoff kinetic analogy in the discrete action principle for the discrete Lag-
range top. The discrete Clebsch approach then gives inextensible and extensible,
shearable discrete rod models in Sections 3.3.1 and 3.5 respectively. The latter
model is referred to as a geometrically exact elastic rod model. We implement
a geometrically exact elastic rod (with periodic boundary conditions) using an
explicit discrete Moser-Veselov integrator to compute the orientation of the dir-
ectors and a Stormer-Verlet symplectic integrator to compute the position of the
rod centroid. The discrete Moser-Veselov integrator for 50 rod sections conserves
total spatial angular momentum to an order of 10−8 over 104 time steps and
exhibits no secular drift in energy error whose mean is the order of 10−2. This
establishes a new and explicit geometric integrator for the geometrically exact
rod model.
90
We begin by reviewing the preliminaries of continuous ellipsoidal motion given by
Sousa-Dias (2002) and then formulate the necessary terminology to derive MV integ-
rators for the convective and spatial representations of ellipsoidal motion. Then in
Section 3.2, we consider the formulation of a MV integrator for the polar decomposed
motion of a Pseudo-rigid body and give the discrete conservation laws corresponding
to symmetries of the discrete Lagrangian.
In Section 3.3, we consider the problem of extending the application of MV integ-
rators to elastic rods. We first review Goriely & Nizette (1999), Chouaıeb (2003) for
the preliminaries of elastic rod models. Following the approach of Bobenko & Suris
(1999b), who consider general discrete time Lagrangian mechanics on Lie groups, we
systematically extend the Moser-Veselov integrators for rigid bodies to inextensible
elastic rod models using the discrete Clebsch approach.
We restrict our consideration of the MV integrator for the heavy top, presented
in Chapter 2, to a Lagrange top (I1 = I2). The MV integrator for the Lagrange top
relates to the equilibrium configuration of a static symmetric inextensible rod through a
discrete Kirchhoff kinetic analogy. We then extend this analogy and show that discrete
time motions of the symmetric inextensible rod correspond to discrete time motions of
elastically coupled Lagrange tops.
By relaxing the inextensibility constraint and extending the configuration to include
the position of the rod centroid, we arrive at the discrete geometrically exact elastic
rod model proposed by Krishnaprasad et al. (1988). Sections 3.4 and 3.5 describe
the derivation and discretisation of this model. Finally numerical simulations of the
isotropic pseudo-rigid body and geometrically exact elastic rod model are presented in
Section 3.6.
3.1 Free Ellipsoidal Motion
We follow Sousa-Dias (2002), who consider the geometric description of a free pseudo-
rigid body as a deformable motion in which the configuration space is identified with
the group of all invertible matrices with positive determinant GL(3)+. The free pseudo-
rigid body is a particular case of free ellipsoidal motion on GL(n)+.
We extend MV integrators to this motion by first generalising some relevant results
of Chapter 2 to ellipsoidal motions. For simplicity, it is assumed that the body is free
and there are no external forces. Decomposition of the pseudo-rigid body motion not
only gives insight into the geometry of deformation but is the basis of our proposed
computationally tractable extension of the DMV algorithm for rigid bodies.
We now generalise the body and spatial MV integrators for rigid body motion,
presented in the last Chapter, to the corresponding convective and spatial MV in-
tegrators for compressible ellipsoidal motion, in which the Lie group GL(n)+ is the
91
configuration space. The theory of elasticity describes how homogeneous compressible
elastic bodies deform in terms of ellipsoidal motions.
Ellipsoidal motion has two key properties useful for the development of a unified
computational framework. Firstly, it is the most general form of MV integrators for spa-
tially independent motion on finite dimensional Lie groups (except for the restriction to
real matrices with positive determinant) and secondly, as we will shortly briefly discuss,
the convective and spatial representations of ellipsoidal motion are a useful prototype
for the development of geometric integrators for the corresponding representations of
Hamiltonian compressible fluids.
We shall begin by describing the preliminaries of ellipsoidal motion leading to two
theorems stating the form of the convective and spatial MV integrators for ellipsoidal
motion.
3.1.1 Preliminaries
Consider the ellipsoidal motion in a container C (taken to be the Euclidean three space
R3) of a material point in a body B given by
x(t, `) = Q(t) ∙ `, (3.1)
where Q(t) ∈ GL(n)+ and x(0, `) = `. The kinetic energy of this motion, in the
material representation, is given by
L =1
2
∫
Cρ(x)x ∙ C0xd
nx =1
2
∫
Bρ(`)Q ∙ `, C0Q ∙ `d
n` =1
2Tr(C0QI0Q
T), (3.2)
where C0 ∈ V is a symmetric positive definite Riemannian metric on C defining the
shape of the container and I0 ∈ V ∗ : (I0)ab =∫B ρ(`)`
a`bdn` is a symmetric positive
definite Riemannian metric on B defining the shape of the body.
A Riemannian metric pairs two tangent vectors v,w ∈ Rn producing a real number.
Once a local basis is chosen, the metric takes the form of a constant matrix. The
simplest example is the dot product vigijwj , which is defined by the Riemannian metric
gij . In Rn, the metric is represented as the identity matrix gi,j = (Id)i,j .
The case I0 = Id and C0 = Id respectively describes a spherical reference config-
uration in Rn. The deformation gradient ∂xi/∂`j = Qij(t) is a function only of time.
Following Holm et al. (1986), we define the symmetric positive definite Cauchy-Green
metric as the product of the deformation gradients
Ct = QTC0Q. (3.3)
92
Before proceeding further, we define the notation for how g acts on each metric.
The group G acts on V by the right translation
φ∗g : V → V : φ∗g(C) = gTCgT , (3.4)
and recall that G acts on V ∗ by the left translation
φg : V → V : φg(I) = gIgT . (3.5)
We now express the material representation of the kinetic energy given by equation
(3.2) in the convective and spatial form through the following symmetry actions of
g ∈ GL(n)+ defined respectively as
Lφ∗g(C0)(g−1Q) := L(g−1Q, gTC0g) =
1
2Tr(g−1QI0Q
T g−T gTC0g), (3.6)
and
Lφg(I0)(Qg−1) := L(Qg−1, gI0g
T ) =1
2Tr(Qg−1gI0g
T g−T QTC0). (3.7)
The convective and spatial representations of the Lagrangian follow when g takes
the value Q, giving the reduced Lagrangian with advected parameters in the body
representation lC : g→ R of the form
lCt(Γ) = l(Γ, Ct) =1
2Tr(ΓI0Γ
TCt), (3.8)
where g = gl(n) and Γ is (minus) the left Q invariant convective velocity
Γ := Q−1Q. (3.9)
In the spatial representation, the Lagrangian lI : g→ R takes the form
lIt(γ) = l(Γ, It) =1
2Tr(γItγ
TC0), (3.10)
which is expressed in terms of the right Q invariant spatial velocity
γ := QQ−1. (3.11)
Remark 3.1.1.1. Note how the metrics become advected quantities in either represent-
ation. In the convective representation, the metric Ct (commonly referred to as the right
Cauchy-Green strain tensor), defined on C, is advected, whereas in the spatial repres-
entation, the body shape metric It, defined on B is instead advected. This has a natural
interpretation. In the spatial representation, the body deforms relative to the container,
whereas in the convective representation, the container deforms relative to the reference
93
space. The latter interpretation provides an interpretation of the Cauchy-Green metric
as defining the shape of the container in the reference space. The Cauchy-Green metric
is therefore the analogue of the body shape metric, which defines the shape of the body
in the container.
3.1.2 Convective and spatial representations of discrete ellipsoidal
motion
kinematics We state the discrete time kinematics by generalising the notation and
terminology presented in Chapter 2 for the rigid body. The discrete motion Qk =
Q(tk) ∈ GL(n)+ is a k parameterised sequence of configurations giving the sequence of
spatial points in the container
xk(`) = Qk ∙ `. (3.12)
The discrete convective velocity (up to a minus sign) and discrete spatial velocity are
respectively defined by the recursion relations
`k+1 = Γ−1k+1`k,
xk+1 = γk+1xk,(3.13)
Discrete reconstruction formulae As shown in Chapter 2, substitution of the
finite difference approximations
hΓ(tk) ≈ Λ−1k (Λk+1 − Λk),
hγ(tk) ≈ (Λk+1 − Λk)Λ−1k ,
(3.14)
into the convective and spatial continuous reconstruction formulae respectively given
by
Q = QΓ,
Q = γQ,(3.15)
and using the definition of the discrete velocities gives the convective and spatial rep-
resentations of the discrete reconstruction formula
94
Qk+1 = QkΓk+1,
Qk+1 = γk+1Qk.(3.16)
The finite difference approximation of the convective reduced Lagrangian, given
by equation (3.8), defines (up to a factor of h2) the Ck augmented discrete material
Lagrangian LCk : G×G→ R
LCk(Id, Q−1k Qk+1) :=
1
2Tr(Q−1k (Qk+1 −Qk)I0(Qk+1 −Qk)
TQ−Tk Ck
), (3.17)
where the Cauchy-Green metric Ck is given by Ck = QTkC0Qk. Expressing this equation
in terms of Γk+1, gives the Ck augmented discrete convective Lagrangian lCk : G→ R
lCk(Γk+1) :=1
2Tr((Γk+1 − Id)I0(Γ
Tk+1 − Id)Ck
). (3.18)
Analogously, the finite difference approximation of the spatial reduced Lagrangian,
given by equation (3.10), defines (up to a factor of h2) the Ik augmented discrete
material Lagrangian LIk : G×G→ R
LIk(Id, Qk+1Q−1k ) :=
1
2Tr((Qk+1 −Qk)Q
−1k IkQ
−Tk (Qk+1 −Qk)
TC0
), (3.19)
where the body shape metric Ik is given by Ik = QkI0QTk .
Expressing this equation in terms of γk+1, gives the Ik augmented discrete spatial
Lagrangian lIk : G→ R
lIk(γk+1) :=1
2Tr((γk+1 − Id)Ik(γ
Tk+1 − Id)C0
). (3.20)
In each representation, substitution of the rearranged expressions C0 = Q−Tk CkQTk
and I0 = Q−1k IkQ
−Tk into
Ck+1 = QTk+1C0Qk+1, (3.21)
and
Ik+1 = Qk+1I0QTk+1, (3.22)
gives the discrete auxiliary equations for Ck
Ck+1 = φ∗Γk+1(Ck) = Γ
Tk+1CkΓk+1, (3.23)
95
and for Ik
Ik+1 = φγk+1(Ik) = γk+1IkγTk+1. (3.24)
We seek the left and right momentum maps corresponding to the left and right
augmented diagonal actions of g ∈ G = GL(n)+ on G×G×V ∗ by applying the discrete
Clebsch approach. Recall, from Chapter 2, that we first add Clebsch constraints for
the discrete reconstruction formula and the discrete auxiliary equation to the discrete
Lagrangians lCk and lIk defined in equations (3.8) and (3.10) giving
lCk(Γk+1) = lCk(Γk+1)+ 〈Pk+1, Qk+1−QkΓk+1〉+ 〈Jk+1, Ck+1−ΓTk+1CkΓk+1〉, (3.25)
and
lIk(γk+1) = lIk(γk+1) + 〈Pk+1, Qk+1 − γk+1Qk〉+ 〈Jk+1, Ik+1 − γk+1IkγTk+1〉. (3.26)
Convective representation
Theorem 3.1.2.1 (Convective MV integrators for ellipsoidal motion on GL(n)+). The
MV integrator for the convective representation of (free) ellipsoidal motion on GL(n)+
is given by
Ad∗Γ−1k
Mk+1 =Mk + 2Ck−1 � ∇Ck−1 lCk−1 ,
Ck = φ∗Γk(Ck−1),
(3.27)
where lCk−1 is the convective discrete Lagrangian (evaluated at time tk−1) given by
equation (3.18) and φ∗g is defined by the right action of g on the space of symmetric
matrices V .
Proof. In the convective representation, stationarity of the discrete action principle
δSd =∑
k δlCk = 0 gives the following expressions paired with variations in the dy-
namical variables
〈Pk − Pk+1ΓTk+1, δQk〉 = 0↔ Pk+1 = PkΓ
−Tk+1,
〈Jk − Γk+1Jk+1ΓTk+1 +∇Ck lCk , δCk〉 = 0⇔ Jk+1 = Γ
−1k+1 (Jk +∇Ck lCk) Γ
−Tk+1,
〈∇Γk+1 lCk −QTk Pk+1 − 2CkΓk+1Jk+1, δΓk+1〉 = 0.
(3.28)
We substitute the first two expressions for Pk+1 and Jk+1 into the last to give
96
∇Γk+1 lCk −QTk PkΓ
−Tk+1 − 2CkΓk+1Γ
−1k+1 (Jk +∇Ck lCk) Γ
−Tk+1 = 0. (3.29)
Right multiplication of this expression by ΓTk+1 gives
∇Γk+1 lCkΓTk+1 −Q
Tk Pk − 2CkJk − 2Ck∇Ck l(Ck) = 0. (3.30)
Finally substituting the expressions for the derivatives
Mk+1 := ∇Γk+1 lCk = Ck(Γk+1−Id)I0 and ∇Ck lCk =1
2(Γk+1−Id)I0(Γ
Tk+1−Id), (3.31)
into equation (3.30) simplifies to the expression for the map Mk+1 : T∗(G × V ) → g∗
given by
Mk+1 = QTk Pk + 2CkJk. (3.32)
It follows by the theorem of momentummaps for lifted actions (see theorem 2.5.1.3, pg. 34),
that the map Mk+1 = QTk Pk+2CkJk is an infinitesimally equivariant right momentum
map JR for cotangent lifted left actions of ζ ∈ g on T ∗(G× V ) given by the pairing
〈JR, ζ〉 = 〈Pk, ζG(Qk)〉+ 〈Jk, ζTCk + Ckζ〉. (3.33)
JR takes the form JR = Q � P + C � J , where the bilinear operator � in the first term
is defined as � : V × V∗ → g∗ and in the second term as � : V × V ∗ → g∗, where V ∈ V
is a symmetric matrix.
Substituting the discrete flow equations (3.28), the discrete auxiliary equation (3.23)
and the discrete reconstruction formula (3.16) into the right momentum map given by
equation (3.32) gives
Mk+1 = Ad∗Γk
(QTk−1Pk−1
)+ 2Ck−1
(Jk−1 +∇Ck−1 lCk−1
)
= Ad∗Γk (Mk − 2Ck−1Jk−1) + 2Ck−1(Jk−1 +∇Ck−1 lCk−1
= Ad∗Γk(Mk + 2Ck−1∇Ck−1 lCk−1
).
(3.34)
This expression together with the discrete auxiliary equation give the form of the con-
vective MV integrator in the theorem in which the pairing between elements of V and
V ∗ has been replaced by the operator �.
97
Spatial representation
Theorem 3.1.2.2 (Spatial MV integrators for ellipsoidal motion on GL(n)+). The
MV integrator for the spatial representation of (free) ellipsoidal motion is given by
Ad∗γkmk+1 = mk + 2∇Ik−1 lIk−1 � Ik−1,
Ik = φγk(Ik−1),(3.35)
where lIk−1 is the spatial discrete Lagrangian (evaluated at time tk−1) given by equation
(3.20) and φg is defined by the left action of g on the space of symmetric matrices V∗.
Proof. In the spatial representation, stationarity of the discrete action principle δSd =∑
k δlIk = 0 gives the following expressions paired with variations in the dynamical
variables
〈Pk − γTk+1Pk+1, δQk〉 = 0↔ Pk+1 = Γ
−Tk+1Pk,
〈Jk − γTk+1Jk+1γk+1 +∇Ik lIk , δIk〉 = 0⇔ Jk+1 = γ
−Tk+1 (Jk +∇Ik lIk) γ
−1k+1,
〈∇γk+1 lIk − Pk+1QTk − 2Jk+1γk+1Ik, δγk+1〉 = 0.
(3.36)
We substitute the first two expressions for Pk+1 and Jk+1 into the last expression to
give
∇γk+1 lIk − γ−Tk+1PkQ
Tk − 2γ
−Tk+1 (Jk +∇Ik lIk) Ik = 0. (3.37)
Left multiplication of this expression by γTk+1 gives
γTk+1∇γk+1 lIk − PkQTk − 2 (Jk +∇Ik lIkIk) = 0. (3.38)
Finally substituting the expressions for the derivatives
mk+1 := ∇γk+1 lIk = (γk+1 − Id)Ik and ∇Ik lIk =1
2(γTk+1 − Id)(γk+1 − Id), (3.39)
into equation (3.38) simplifies to the expression for the map mk+1 : T∗(G× V ∗) → g∗
given by
mk+1 = PkQTk + 2JkIk, (3.40)
which is an infinitesimally equivariant left momentum map for cotangent lifted right
actions of ζ ∈ g on T ∗(G × V ∗) (see the right momentum map for the convective
representation for further details).
Substituting the discrete flow equations (3.36), the discrete auxiliary equation (3.24)
98
and the discrete reconstruction formula (3.16) into the left momentum map given by
equation (3.40) gives
mk+1 = Ad∗γ−1k
(Pk−1Q
Tk−1
)+ 2
(Jk−1 +∇Ik−1 lIk−1Ik−1
)
= Ad∗γ−1k(mk − 2Jk−1Ik−1) + 2
(Jk−1 +∇Ik−1 lIk−1Ik−1
)
= Ad∗γ−1k
(mk + 2∇Ik−1 lIk−1Ik−1
).
(3.41)
This expression together with the discrete auxiliary equation give the form of the spatial
MV integrator in the theorem.
DMV algorithms The convective and spatial MV integrators for ellipsoidal motion
can not be solved using the DMV algorithm proposed by McLachlan & Zanna (2005).
Rather than develop a new algorithm, we shall instead restrict our discussion to a spe-
cific homogeneous elasticity model for which McLachlan and Zanna’s DMV algorithm
can be adapted.
3.2 The Pseudo-Rigid Body
We restrict our consideration of MV integrators for ellipsoidal motions on G = GL(3)+.
Following Sousa-Dias (2002), we consider a discrete Lagrangian on the polar decom-
posed configuration space and begin by reviewing the Lagrangian description of the po-
lar decomposed isotropic pseudo-rigid body. This is the simplest case when the reference
configuration is spherical. To stimulate future research, we provide the Euler-Poincare
description of the polar decomposed anisotropic pseudo rigid body in Appendix B.
3.2.1 Polar decomposition of discrete pseudo-rigid body motion
Definition 3.2.1.1 (Polar Decomposition (Sousa-Dias 2002)). The polar decomposition
of any n × n invertible matrix with positive determinant Q ∈ GL(n)+ is Q = RTDS,
where R,S ∈ SO(n)+ and D ∈ D(n) where D(n) is the space of all diagonal matrices
with positive determinant.
Remark 3.2.1.2 (Uniqueness of ”Polar” decomposition). This decomposition is re-
ferred to by Sousa-Dias (2002) as ”bi-polar” decomposition so as to avoid conflict with
the standard definition of polar decomposition as Q = RTU , where R is an orthogonal
matrix and U is a symmetric positive definite matrix (Ciarlet 1988). Further diag-
onalisation of U gives an equivalent decomposition as the definition above. For this
reason, our definition of polar decomposition is only unique when the corresponding
99
diagonalisation of U is unique. Diagonalisation of symmetric positive definite matrices
is unique, up to permutations of the eigenvalues, if the eigenvalues of U are distinct
(see, for example, Horn & Johnson 1991).
Consider a Lagrangian defined on TGL(3)+ of the form
L =Tr
2(QQT ), (3.42)
which is invariant under the left and right actions of SO(3). We polar decompose the
velocity to give
Q = RT (−ΩD + D +Dω)S, (3.43)
where Ω := RRT and ω := SST are the respective right invariant angular velocity and
internal circulation angular velocity. Reduction of L by the left and right actions of
SO(3) gives the reduced Lagrangian l : so(3)2 × TD(3)→ R given by
l =Tr
2
(−Ω2D2 − ω2D2 + 2ΩD + D2
). (3.44)
For the purposes of forming a discrete Kelvin circulation theorem, it is simplest to
neglect the stretching term D here, although we will include stretching in the derivation
of MV integrators for pseudo-rigid bodies in Section 3.2.3.
Discrete motion Consider the discrete action principle for the isotropic (Q = RTS ∈
SO(3)) free pseudo-rigid body
Sd =∑
k
Tr(ΩTk+1ωk+1
)−Tr
2
(P Tk+1(Qk+1 − uk+1Qk)
)
− Tr(ΘRk+1(Ωk+1Ω
Tk+1 − Id)
)− Tr
(ΘSk+1(ωk+1ω
Tk+1 − Id)
),
(3.45)
which is formed by finite difference discretisation of the terms in Ω and ω, where the
discrete velocities are Ωk+1 := Rk+1RTk and ωk+1 = Sk+1S
Tk . Variations in Qk and Pk
give the update equations
Pk+1 = uk+1Pk,
Qk+1 = uk+1Qk,(3.46)
where uk+1 := Qk+1QTk . These equations preserve the symplectic two-form dQk ∧ dPk.
Variations in Ωk+1 give the expression for the momentum map Mk+1 : T∗SO(3) →
100
so(3)∗ of the form
Mk+1 = ωk+1ΩTk+1 − Ωk+1ω
Tk+1 = Sk+1(Q
Tk Pk − P
Tk Qk)S
Tk+1, (3.47)
in which the update equation for Pk+1 and the polar decomposed expression for uk+1
has been used. This expression defines the relation Pk = Qk+1, which is consistent with
the definition of the discrete kinetic energy 〈Qk+1, Qk〉. The angular momentum in the
Lagrangian coordinate frame takes the form
RTk+1Mk+1Rk+1 = PkQTk −QkP
Tk . (3.48)
Similarly, variations in ωk+1 give the expression for the momentum map
Nk+1 = Ωk+1ωTk+1 − ωk+1Ω
Tk+1 = ωk+1Rk(QkP
Tk − PkQ
Tk )R
Tk ω
Tk+1, (3.49)
which is the vorticity in the Lagrangian coordinate frame
STk+1Nk+1Sk+1 = PTk Qk −Q
Tk Pk. (3.50)
An identical proof to that given in Section 2.5.5 of Chapter 2 confirms the con-
servation of mk+1 = RTk+1Mk+1Rk+1 = mk and nk+1 = STk+1Nk+1Sk+1 = nk. The
former conserved quantity is the angular momentum and the latter is the vorticity in
the Lagrangian coordinate frame.
3.2.2 Conservation of circulation
Before stating a discrete Kelvin circulation theorem, it is useful to recall the continuous
Kelvin circulation theorem for the Pseudo-rigid body. In order to do this, we shall
briefly introduce some new notation and terminology.
Background
Recall that the spatial coordinate x of a label in a pseudo-rigid body ` is given by the
motion x = Q(t)`, Q ∈ GL(3)+. It follows from the definition of this motion that the
(right Q invariant) spatial velocity takes the form u = QQ−1x. Kelvin’s circulation
theorem states that the circulation (differential) one-form u ∙ dx around a closed loop
moving with velocity u is conserved. This can be more concisely expressed using the
exterior derivative
d
dtd(u ∙ dx) = 0, along u =
dx
dt. (3.51)
101
Substituting the definition of u into the differential two-form gives
d(QQ−1x ∙ dx) = (QQ−1)ijdxj ∧ dxi
=1
2(Q−T QT − QQ−1)ijdx
i ∧ dxj
=1
2(Q−T QT − QQ−1)ijd(Qia`
a) ∧ d(Qjb`b)
=1
2(Q−T QT − QQ−1)ijQjbQiad`
a ∧ d`b
=1
2(Q−Tim QTmj − QilQ
−1lj )QiaQjbd`
a ∧ d`b
=1
2(δmaQ
TmjQjb − Q
TliQiaδlb)d`
a ∧ d`b
=1
2(QTQ−QT Q)abd`
a ∧ d`b,
(3.52)
where the steps in the second and fourth lines follow from the respective skew-symmetry
and bi-linearity of the differential two-form. The Kelvin circulation theorem is therefore
equivalent to the statement that QTQ−QT Q is a constant of motion along u = dxdt . This
constant of motion is the vorticity of the internal material, relative to the Lagrangian
frame. The vorticity is a conserved momentum map corresponding to invariance of the
continuous Lagrangian under the tangent lifted right action of S on the tangent bundle
TQ. This symmetry is commonly referred to as a ”particle relabelling symmetry”.
Equally, the invariance of the Lagrangian under the tangent lifted right action of R
on the tangent bundle corresponds to another conserved momentum map, the angular
momentum of the pseudo-rigid body.
Discrete Kelvin circulation theorem
Definition 3.2.2.1. It follows from the definition of the forward map that the (right Q
invariant) discrete spatial velocity takes the form uk+1 := Qk+1Q−1k such that xk+1 =
uk+1xk. Consider the circulation (differential) one-form xk+1 ∙dxk around a closed loop
with position xk+1 at time tk+1.
We shall now state the discrete Kelvin circulation theorem and show, analogously
to the continuous case, that it is equivalent to conservation of discrete vorticity.
Theorem 3.2.2.2 (Discrete time Kelvin circulation). The change in the exterior de-
rivative of the circulation one-form about a closed loop c(xk+1) is
Δtd(xk+1 ∙ dxk) = 0 along xk+1 = ukxk. (3.53)
102
Proof. Substituting the reconstruction formula xk+1 = uk+1xk into the differential two-
form gives
d(uk+1xk ∙ dxk) = (Qk+1Q−1k )ijdx
jk ∧ dx
ik
=1
2(Q−Tk QTk+1 −Qk+1Q
−1k )ijdx
ik ∧ dx
jk
=1
2(Q−Tk QTk+1 −Qk+1Q
−1k )ijd((Qk)ia`
a) ∧ d((Qk)jb`b)
=1
2(Q−Tk QTk+1 −Qk+1Q
−1k )ij(Qk)ia(Qk)jbd`
a ∧ d`b
=1
2
((Qk)
−Tim (Qk+1)
Tmj − (Qk+1)il(Qk)
−1lj
)(Qk)ia(Qk)jbd`
a ∧ d`b
=1
2
(δma(Qk+1)
Tmj(Qk)jb − (Qk+1)
Tli(Qk)iaδlb
)d`a ∧ d`b
=1
2(QTk+1Qk −Q
TkQk+1)abd`
a ∧ d`b,
(3.54)
where the steps in the second and fourth lines follow from the respective skew-symmetry
and bi-linearity of the differential two-form dxk ∧ dxk. It follows that the discrete
time motion of an isotropic Pseudo-rigid body satisfies a discrete Kelvin circulation
theorem because the discrete vorticity STk+1Nk+1Sk+1 is a momentum map which, upon
substitution of Pk = Qk+1 into its definition in equation (3.50), gives the constant
expression QTk+1Qk − QTkQk+1. Conservation of discrete vorticity corresponds to the
invariance of the discrete Lagrangian under right actions of Sk.
3.2.3 MV integrators for Mooney-Rivlin materials
The MV integrator, with the stretching component Dk, is now derived to include an
elastic potential term. Following Sousa-Dias (2002), we choose to model a rubbery
material classified as a Mooney-Rivlin material. We shall revisit the Lagrangian for
the free polar decomposed pseudo-rigid body in equation (3.44) and now include the
stretching components in the finite difference of ˙RTDS. This approximation gives (up
to a factor of −h2) the holonomically constrained discrete Lagrangian lck : (V×D)2 → R
given by
lck =Tr
2
((4(Ωk+1 + ωk+1)− 6Id)D
2k − 2Ωk+1Dkωk+1Dk − (Dk+1 −Dk)
2)
−Tr
2
(ΘSk+1(ωk+1ω
Tk+1 − Id)
)−Tr
2
(ΘRk+1(Ωk+1Ω
Tk+1 − Id)
),
(3.55)
where (Dk+1 −Dk)/h is referred to as the discrete stretching velocity (in the material
representation) and the last two terms are the holonomic constraints on ωk+1 and Ωk+1
103
respectively.
The Clebsch constrained discrete Lagrangian takes the form
lk = lck + Tr
(PR
T
k+1(Rk+1 − Ωk+1Rk))+ Tr
(PS
T
k+1(Sk+1 − ωk+1Sk)), (3.56)
where the second and third terms are constraints for the update of the pseudo-rigid
body orientation and internal circulation respectively.
Application of the discrete Clebsch approach gives the following Clebsch relations,
paired with δΩk+1 and δωk+1 respectively,
Dk(2Id − ωTk+1)Dk − P
Rk+1R
Tk = Θ
Rk+1Ωk+1,
Dk(2Id − ΩTk+1)Dk − P
Sk+1S
Tk = Θ
Sk+1ωk+1,
(3.57)
and from the symmetry of ΘRk+1 and ΘSk+1, equation (3.57) gives the left momentum
maps for the cotangent lifted actions of SO(3) on T ∗(SO(3)× SO(3)×D)
Mk+1 := J(ωk+1)ΩTk+1 − Ωk+1J
T (ωk+1) = PRk+1R
Tk+1,
Nk+1 := J(Ωk+1)ωTk+1 − ωk+1J
T (Ωk+1) = PSk+1S
Tk+1.
(3.58)
Substituting the discrete flow equations
PRk+1 = Ωk+1PRk ,
PSk+1 = ωk+1PSk ,
(3.59)
into the momentum maps in equation (3.58) gives the MV integrators for the rotational
and internal circulatory momenta
Ad∗Ωk+1Mk+1 =Mk,
Ad∗ωk+1Nk+1 = Nk.(3.60)
Stretching component We add a potential energy term of the form −h2W (Dk) to
the discrete Lagrangian in equation (3.56). The function W (Dk) is given by
W (Dk) = aI1(D2k) + bI2(D
2k) + c|Dk|
2 − dLog(|Dk|), a, b, c, d > 0, (3.61)
104
and characterises a Mooney-Rivlin material (see Sousa-Dias 2002). Note that Ii are
the principal invariants of D2k whose expressions are given in Sousa-Dias (2002). The
discrete Euler-Lagrange equations in Dk define an additive update of the diagonal
matrix Dk given by
Dk+1 = −4πD(−Ωk+1 + 2Id − ωk+1)Dk
+ πD(Ωk+1Dkωk+1 + ωk+1DkΩk+1)−Dk−1 + h2∇DkW (Dk),
(3.62)
where πD denotes projection of the principal diagonal of the matrix. Note that the
elements of Dk are not guaranteed to stay positive. To alleviate this feature, we ensure
that the last term in the potential energy term is sufficiently large to prevent the
determinant of Dk from vanishing during simulation. The simulations of the polar
components of the motion of a Mooney-Rivlin material are provided in Section 3.6.
We shall now consider the formulation of MV integrators for elastic rods and begin
by reviewing the description of the Kirchhoff rod given by Goriely & Nizette (1999).
3.3 Elastic Rod Preliminaries
3.3.1 The discrete Kirchhoff rod analogy
Consider a discrete ribbon, a space curve {φ(Si)}i∈{0,1,...,N} parameterised by arc-length
Si := iΔS, with three smooth orthonormal unit vector-fields, d2(Si) which is orthogonal
to the curve, d3(Si) is aligned with the unit tangent vector to the curve d3(Si) =
t(Si) = φ′(Si) and d1(Si) = d2 × d3. The triad {d1,d2,d3} is related to the Frenet
basis {n,b, t} at each Si by an orthogonal transformation about t(Si). For this reason,
the triad is referred to as an adapted Frenet frame. The Euler angle of rotation about
t(Si) is referred to as the twist angle. The ribbon simplifies to the Bernoulli’s elastica
when the triad coincides with the Frenet basis. When the principal moments of inertia
about the axes d1 and d2 are equal, the ribbon has equal principal bending stiffnesses
and is referred to as a symmetric ribbon.
The definition of the ribbon will now be used to define the discrete elastic rod
(also known as Kirchhoff’s elastica). We refer the reader to Antman (1995), Goriely
& Nizette (1999) for a detailed description of the kinematics of the continuous elastic
rod. The elastic rod is described as symmetric when its ribbon is symmetric.
Definition 3.3.1.1 (Discrete Elastic Rod). In the absence of external torques, it follows
from the definition of a continuum Kirchhoff rod, given by Bobenko & Suris (1999a),
105
Kirchhoff rod Lagrange top
Discrete angular strain at Si Ωi Discrete body angular velocity at time tk ΩkStiffness matrix C0 Inertia matrix I0Rod tension p0 Position of centre of mass χ0
Tangent vector at Si ti Orientation of gravity vector at time tk Γk
Table 3.2: This Table shows the correspondence between the terms used to describethe discrete Lagrange top and those to describe the discrete Kirchhoff rod.
that a discrete elastic rod is an arc-length Si parameterised ribbon of fixed length L
minimising the functional
Fd =∑
i
li(f(Ω)i), (3.63)
where the reduced Lagrangian l : G→ R is given by
li =∑
i
〈f(hΩ)i, C0f(hΩ)i〉︸ ︷︷ ︸elastic potential
− 〈p0, ti〉︸ ︷︷ ︸inextensibility constraint
, (3.64)
in which Si ∈ {ih, i := 0 → N}, f : so(3) → SO(3) : f(hΩ)i denotes a
transformation of hΩ(Si) into G at position Si, where Ω(Si) is the angular strain at this
position and when the discrete set {Si}i is periodic, h = L/N . p0 is the uniform tension
in the rod which enforces the inextensibility constraint φ(SN )−φ(S1) =∑
i t(Si). C0 is
a spatially independent diagonal matrix comprised of the principal bending and torsional
stiffnesses of the homogeneous rod.
As a result of the following theorem, which is a discrete variant of the Kirchhoff
kinetic analogy given by Bobenko & Suris (1999a), it will be shown that this functional
coincides with the discrete action principle for the body representation of the Lagrange
top under a particular choice of f(hΩ).
Theorem 3.3.1.2 (Discrete Kirchhoff kinetic analogy). The N frames of the arclength
Si parameterised (homogeneous) discrete symmetric elastic rod are in 1-to-1 corres-
pondence with the discrete time motion of the Lagrange top over N time intervals if
f(hΩ)i := Ωki+1 − Id, where the discrete angular strain is given by Ω
ki+1 = Λ
kT
i Λki+1.
Proof. Substitution of f(hΩ)i := ΛkT
i (Λki+1 − Λ
ki ) into the Lagrangian for the discrete
106
rod given by equation (3.64) gives
li =∑
i
Tr((ΛkTi+1 − Λ
kTi )Λ
kiC0Λ
kT
i (Λki+1 − Λ
ki ))− 〈p0, ti〉
=∑
i
Tr((ΛkTi+1Λ
ki − Id)C0(Λ
kT
i Λki+1 − Id)
)− 〈p0, ti〉
=∑
i
Tr(Ωki+1C0)− 〈p0, ti〉.
(3.65)
Now define Pd := {Si, C0, ti,p0}i:=1→N as a subset of variables and constants form-
ing the discrete elastic rod functional Fd and Qd := {tk, I0,mgΓk, χ0}k:=1→N as the
subset of variables and constants forming the discrete Lagrange top action principle
Sd. Forming the discrete elastic rod functional Fd from Qd, instead, gives the reduced
discrete action principle Sd for the Lagrange top given in the form of the first two terms
of equation (2.142) on pg. 55 in Chapter 2.
Replacing the terms in Qd with the those in Pd in the definition of the MV integrator
for the Lagrange top gives
Ni+1 = Ad∗ΩTiNi + p0 � ti,
ti = Ωiti−1,(3.66)
where the dual to the angular strain Ni is given by Ni :=2h2skew
((∇Ωi+1 l)
T Ωi+1
)and
ti := t(Si) is the unit tangent vector to the curve at Si.
Lie-Poisson structure
The Lie-Poisson structure preserved by the discrete elastic rod equations is in one-to-one
correspondence with the structure preserved by the discrete Lagrange top equations,
which are a specialisation of the discrete heavy top equations (2.155) for the case when
the moments of inertia are I1 = I2. It follows from Section 2.7 and the discrete Kirchhoff
rod analogy, that the Lie-Poisson brackets preserved by the discrete elastic rod take
the form
{F1, F2}±(Ni, ti) = −1
2Tr(±NT
i [∇NiF1,∇NiF2])
± ti ∙ (Φ(∇NiF1)∇tiF2 − Φ(∇NiF2)∇tiF1) ,(3.67)
in which the norms of the dual of the angular strain ||Ni|| and the extensibility con-
straints ||ti|| are the Casimirs. Preservation of the Lie-Poisson structure leaves the
107
rod length as an invariant of the discrete flow given by equation (3.66). We shall now
briefly consider the discrete Lagrangian for the discrete time motion of a symmetric
dynamical Kirchhoff rod before turning to the more difficult problem of formulating a
discrete geometrically exact rod.
3.3.2 Time dependent discrete Kirchhoff rod models
We form the discrete Lagrangian for the discrete time motion of the (inextensible)
Kirchhoff rod model by adding a discrete kinetic energy term, for rigid body rotations
of the set of n orthonormal frames, to the discrete Lagrangian given in equation (3.64)
to give
lk =n∑
i=1
Tr(Ωk+1T
i I0)︸ ︷︷ ︸kinetic energy
+ Tr(ΩkT
i+1C0)︸ ︷︷ ︸elastic potential energy
− 〈p0, tki 〉︸ ︷︷ ︸
extensibility constraint
. (3.68)
The essential feature of this model is the relation between the discrete angular
velocities Ωki and the discrete angular strains Ωki which is expressed as a discrete com-
patibility equation.
Lemma 3.3.2.1 (Discrete compatibility equation). The discrete dynamical Kirchhoff
rod exhibits the discrete compatibility equation
Ωk+1i+1 = Ωk+1i+1 Ω
ki+1Ω
k+1T
i . (3.69)
Proof. Temporal recursion of Λki followed by spatial recursion of Λk+1i takes the form
Λk+1i+1 = Λk+1i Ωk+1i+1 = Λ
kiΩ
k+1i Ωk+1i+1 . (3.70)
Conversely, spatial recursion of Λki followed by temporal recursion of Λki+1 gives
Λk+1i+1 = Λki+1Ω
k+1i+1 = Λ
ki Ω
ki+1Ω
k+1i+1 . (3.71)
Equating these two expressions and eliminating Λki gives the discrete compatibility
equation
Ωk+1i+1 = Ωk+1i+1 Ω
ki+1Ω
k+1T
i . (3.72)
Remark 3.3.2.2. The discrete compatibility equation takes the form of the discrete
auxiliary equation for the relative orientation matrix in the discrete coupled rigid body
model given in equation (2.167).
108
Correspondence with the elastically coupled Lagrange top It follows from the
discrete Kirchhoff kinetic analogy that the discrete time motion (over n intervals) of
N frames of the discrete convective representation of the elastic rod are in 1-to-1 cor-
respondence with the discrete time motion (over N intervals) of n elastically coupled
Lagrange tops. From the definition of the discrete Lagrangian for the Lagrange top
given in equation (2.142) and the form of the elastic potential in equation (3.68), the
reduced Lagrangian for n elastically coupled Lagrange tops is given by
lk =n∑
i=1
Tr(Ωk+1T
i I0)︸ ︷︷ ︸kinetic energy
+ Tr(ΩkT
i+1C0)︸ ︷︷ ︸elastic potential energy
− mg〈χ0i ,Γki 〉︸ ︷︷ ︸
gravitational potential energy
. (3.73)
Replacing {tk, I0, C0, χ0i ,Γki } by {Si, C0, I0,p
k0, t
ki } in this discrete Lagrangian gives
the discrete Lagrangian for the discrete dynamical Kirchhoff rod given by equation
(3.68).
The discrete equations of motion
The discrete equations of motion for the dynamical inextensible Kirchhoff rod model
are now derived from the discrete Lagrangian
lk =∑
i
〈Ωk+1i , I0〉+ 〈Ωki+1, C0〉 − 〈p
k0, t
ki 〉. (3.74)
Appending the Clebsch constraints to this Lagrangian gives
lk = lk +∑
i
〈P k+1i
2,Λk+1i − ΛkiΩ
k+1i 〉+ 〈
P ki+12
,Λki+1 − Λki Ω
ki+1〉
+ 〈Jk+1i , tk+1i − Ωk+1iT tki 〉+ 〈J
ki+1, t
ki+1 − Ω
ki+1
T tki 〉
− 〈θk+1i ,Ωk+1i Ωk+1iT − Id〉 − 〈θ
ki+1, Ω
ki+1Ω
ki+1
T − Id〉.
(3.75)
Taking variations in Ωk+1i and Ωki+1 and right multiplying by the transpose of these
matrices respectively gives the Clebsch relations:
I0Ωk+1i
T − ΛkiTP k+1i Ωk+1i
T /2− tki ⊗ Jk+1i Ωk+1i
T = θk+1i , (3.76)
C0Ωki+1
T − ΛkiT P ki+1Ω
ki+1
T /2− tki ⊗ Jki+1Ω
ki+1
T = θki+1. (3.77)
Variations in Λki and tki respectively give
109
P ki − Pk+1i Ωk+1i
T + P ki − Pki+1Ω
ki+1
T = 0,
Jki + Jki +∇tki ltki − Ω
k+1i Jk+1i − Ωki+1J
ki+1 = 0.
(3.78)
Adding equations 3.76 and 3.77 and defining Mk+1i = I0Ω
k+1i
T − Ωk+1i I0 and Nki+1 =
C0Ωki+1
T − Ωki+1C0 gives
Mk+1i +Nk
i+1 + skew((P ki + P
ki )TΛki
)+ 2(Jki + J
ki ) � t
ki + 2∇tki ltki � t
ki = 0. (3.79)
It is convenient to rewrite equation 3.76 as
ΩkiTMk
i Ωki = Ω
kiT(skew(Λk−1i
TP ki ΩkiT ) + 2skew(tk−1i ⊗ JkiΩ
kiT ))Ωki
= Λki � Pki + 2t
ki � J
ki .
(3.80)
Similarly, we can write equation 3.77 as
ΩkiTNk
i Ωki = Ω
kiT(skew(Λki−1
T P ki ΩkiT ) + 2skew(tki−1 ⊗ J
ki Ω
kiT ))Ωki
= Λki � Pki + 2t
ki � J
ki .
(3.81)
Finally, substituting equations 3.80 and 3.81 into equation 3.79 gives the discrete equa-
tions of motion for the dynamical Kirchhoff rod model
Mk+1i +Nk
i+1 −AdΩkiMki −AdΩki
Nki + 2∇tki ltki � t
ki = 0. (3.82)
Note that the only unknowns in this equation at each time step are the body angu-
lar momenta for each node Mk+1i . We also note the symmetry in the equation under
switching Λki+1 with Λk+1i in the definition of Mk
i , Nki ,Ω
ki and Ω
ki . This symmetry of
course only appears because of the choice of discretisation of Ω and Ω. The inextens-
ibility constraint also appears as a forcing term in the above equation of motion.
DMV algorithm In principle, the above equation of motion can be recast as a N
coupled system of matrix Ricatti equations and solved for each Λk+1i with a DMV
algorithm for each matrix Ricatti equation. Once all Λk+1i are known, one can then
compute each Nk+1i from its definition. The presence of the forcing term arising from
the inextensibility constraint renders these equations stiff and a challenge to robustly
solve using the standard form of the DMV algorithm, however. Specifically, we find
that the eigenvalues of the Hamiltonian matrix associated with each of the N matrix
110
Ricatti equations exceed stability constraints. Further research should address how to
regularise the dynamical Kirchhoff rod equations to guarantee stability.
Extension to the geometrically exact elastic rod The discrete Kirchhoff rod
model describes rods which deform by bending and twisting but preserve the length. It
is precisely the imposition of the length constraint which stiffens the system of Matrix
Ricatti equations and renders them challenging to solve. For this reason we consider
an alternative class of elastic rods which deform by shearing and stretching. These are
the class of rod motions which Krishnaprasad et al. (1988) and Simo & Vu-Quoc (1986,
1988) considered leading to the derivation of the geometrically exact elastic rod model.
The purpose of the next Section is to present a MV integrator for this model which can
be solved using a DMV algorithm.
We proceed by reviewing the terminology and notation used by (Krishnaprasad et al.
1988) to describe the geometrically exact rod model. In contrast to the Hamiltonian
approach taken by Krishnaprasad et al. (1988), we formulate the variational descrip-
tion of this rod model. We verify, in Appendix B.2, that the variational description
gives the equations of motion corresponding to the continuum Lie-Poisson equations
given by Krishnaprasad et al. (1988). These equations of motion are referred to, more
generally, as ’Lagrange-Poincare’(LP) equations (see Holm 2002). In general, LP equa-
tions describe partially reduced variational dynamics associated with Lagrangians with
more than one group symmetry. For the geometrically exact rod, the SO(3) reduced
dynamics given by the LP equations are only partially reduced. These equations can
be further reduced by the action of the group of diffeomorphisms Diff(R3), but this
is not a stage which we pursue here because it is only relevant to the continuum rod
model and our interest is pursuing a MV integrator for the discrete rod.
In Section 3.5, we give the MV integrator for the convective angular and linear
momentum equations corresponding to the LP equations in the continuum rod model.
These equations describe the motion of a set of particles, whose position is the rod
centroid at arc-length Sα and is given in coordinates relative to the director frame
at this position. These equations also describe the update of each director frame in
the convective representation. A symplectic integrator is then used to compute the
particle trajectories and the MV integrator, the attitude of the directors positioned on
each particle. Numerical results are presented in Section 3.6 which demonstrate the
conservative properties of the spatial angular momentum, linear momentum (in the
material representation) and energy.
111
3.4 The Geometrically Exact Elastic Rod Model
We begin by reviewing the formulation of the continuum geometrically exact rod model
given by Krishnaprasad et al. (1988).
3.4.1 Preliminaries
Krishnaprasad et al. (1988) consider a rod whose placement in the container C at time
t is defined to be the set
Pt := {x ∈ R3| x = φ(S, t) + ζ(S)αdα(S, t), S ∈ B, (ζ
1(S), ζ2(S)) ∈ A(S), ζ3(S) = 0},
(3.83)
where the family of diffeomorphic maps
φ : B → C, (3.84)
are volume and orientation preserving embeddings of the reference configuration B =
[0, L] in the container. A(S) ⊂ R2 is a (disk-like) cross-sectional area of the rod at a
point in the reference configuration S ∈ [0, L] along a S-parameterised curve, represent-
ing the line of centroids of the rod in its natural (unstressed) reference configuration.
φt(S) = φ(S, t) is the motion of the line of rod centroids given by
φ : [0, L]× R+ → C : φ(S, t0) = S. (3.85)
The special (restricted) theory of Cosserat rods describes the orientation of the
cross-section in the container by the set of director fields {dα(S, t)}α:=1→3, whose third
director d3(S, t) is the normal vector to the cross-sectional area. The directors are
subject to length constraints for all times t and S ∈ [0, L].
||dα(S, t)|| = 1, α = 1, 2; d1(S, t) ∙ d2(S, t) = 0, (3.86)
in addition to the shear limiting constraint
d3(S, t) ∙ φ′(S, t) > 0, d3(S, t) := d1(S, t)× d2(S, t) 6= 0, (3.87)
where ′ = ∂∂S and the set of directors {dα(S, t)}α:=1→3 is a rotating orthonormal frame
whose origin is fixed at the point of the reference line of centroids parameterised by S.
There exists a unique orthogonal transformation
Λt : B → SO(3) : dα(S, t) = Λt(S)Eα(S, t), α := 1→ 3, (3.88)
112
from the inertial frame {Eα}α:=1→3 (the basis of R3) to the set of directors. Thus
{Λt(S)}S∈[0,L] is a S-parameterised curve in SO(3) at each time t.
Adopting the approach taken in Krishnaprasad et al. (1988), which is to regard
the transformations as the state variables rather than the directors themselves, the
configuration space of the rod is the product of smooth maps
C := {Φ = (φ,Λ) : B → R3 × SO(3)}, (3.89)
where the rod motion Φt(S) = Φ(S, t) is a submanifold of SO(3) × R3, which defines
the configuration of the rod at time t. In order to determine the symmetries of the
Lagrangian for this model (which we define in Section 3.4.2), we shall first define the
following SO(3) actions which leave the Lagrangian for this rod invariant.
Lie Group actions We begin by defining the left and right actions of SO(3) on the
rod configuration C for some B ∈ SO(3), A : B → SO(3) which represents the orient-
ation of a director and w ∈Diff(C) which represents the position of the rod centroid
with reference to an arc-length S ∈ B.
ΨL : SO(3)× C → C : ΨL(B, (A(S), w(S))) = B ∙ (A(S), w(S)) := (BA(S), Bw(S)),
(3.90)
ΨR : C × SO(3)→ C : ΨR(B, (A(S), w(S))) = (A(S), w(S)) ∙B := (A(S)B,w(S)),
(3.91)
where the right action of B on w(S) is trivial. Following Krishnaprasad et al. (1988),
we now recall the convective and spatial representations of the velocities and strain
measures arising from symmetry reduction of the tangent lift of the respective group
actions ΨL and ΨR on the tangent bundle TC = {(Φ, Φ) : B → TSO(3)× TC}.
Velocities Associated with the motion of the rod is the material velocity field VΦtdefined by
VΦt(S) := Φt(S) :=(φt(S), Λt(S)
). (3.92)
The spatial velocity of the rod motion is the time derivative of Φt(S) at fixed x ∈ C
uΦt(x) := Φt(S) |x := (uφt(x), ωt(x)) . (3.93)
The pull-back of uΦt by Φt is the convective velocity of the rod motion
VΦt(S) = Φ∗tuΦt(x) = (Φ
∗tuφt ,Φ
∗tωt) = (φ
∗tΛ∗tuφt(x),Λ
∗tωt(x)) := (Vφt(S),Ωt(S)).
(3.94)
113
The definition of the linear and angular velocities in each description is given in
Table 3.3.
component material spatial convective
Linear: Uφt = φt ∈ TφtC, uφt = φt ∙ φ−1t ∈ Tφ0C, Vφ = −Λ
Tt˙(φ−1t ) ◦ φt ∈ Tφ−10
B
Angular: Λt ∈ TΛtSO(3), ωt = ΛtΛTt ∈ so(3), Ωt = Λ
Tt Λt ∈ so(3)
Table 3.3: The material, spatial and convective representations of the velocities.
Strain measures The rod motion describes elastic deformations which result in axial
and torsional strains. The one dimensional deformation gradient of the rod is
Φ′t(S, t) := (φ′t(S, t),Λ
′t(S, t)), (3.95)
where φ′t and Λ′t are the linear (axial) and angular (torsional) strains. The different
representations of these strains are presented in Table 3.4.
Material Convective Spatial
Linear strain measure φ′t − d3 Γ := −ΛTt φ−1t (φ
′t − d3) γ := (φ′t − d3)
′ ∙ φ−1tAngular strain measure Λ′t Ω := ΛTt Λ
′t ω := Λ′tΛ
Tt
Table 3.4: The material, convective and spatial representations of the strain measures.
With all the terms defined, we now describe the variational description of the geo-
metrically exact rod in the next Section.
3.4.2 The variational formulation of the geometrically exact elastic
rod model
Approach Following Krishnaprasad et al. (1988) we consider the material description
of the kinetic energy of the rod which is
K(t) =
∫
BdS
ρ0
2A|φt(S)|
2 +ρ0
2Tr(Λt(S)I0Λt(S)
T ), (3.96)
where ρ0 is the density of the homogeneous rod. The kinetic energy is invariant under
the tangent lift of the actions ΨL and ΨR, the latter of which is the particle relabelling
symmetry. The material description of the potential energy of the rod is the stored
energy
V (t) =
∫
BdS〈(φ′t − d3), C
φ0 (φ
′t − d3)〉+ Tr
(Λ′tC
Λ0 Λ′Tt
). (3.97)
114
It is assumed that along the rod centroid the (i) cross-sectional area, (ii) stiffness
and shear modulus and (iii) elasticity tensor C and moments of inertia are constant.
It is also assumed that the matrix of bending and torsional stiffnesses CΛ0 , the matrix
of shear stiffnesses Cφ0 and the inertia tensor I0 are diagonal and constant in a rotating
frame of directors.
Symmetries Note that invariance of V (t) under the left action of SO(3) is only ob-
tained by imposing that Cφ is isotropic, i.e. the shear stiffnesses in the directions of
each of the directors are the same. Otherwise, rotation of a rod cross-section, para-
meterised by S, breaks the symmetry unless d3(S, t) is aligned with the tangent vector
at φt(S) (this corresponds to the case when there is no rod shearing). V (t) is also
invariant under the right action of SO(3) in the restrictive case when CΛ is isotropic,
i.e., the bending and torsional stiffnesses are the same.
Boundary conditions For simplicity, we shall only consider periodic boundary con-
ditions, i.e. both φt(0) = φt(L) and Λt(0) = Λt(L). With these boundary conditions,
the rod model dynamics are invariant under linear translations and rotations of the rod
in R3.
The Convective Lagrangian The lifted left action of g = ΛTt ∈ SO(3) on the
tangent bundle gives the reduced lagrangian density `(ΛTt φt,ΛTt φt,Λ
Tt φ′t; Ω, Ω) in the
convective representation given by
` =
∫
BdS
ρ0
2A|V|2 +
ρ0
2Tr(ΩI0Ω
T )− Tr(ΩCΛ0 ΩT )− 〈Γ, Cφ0 Γ〉, (3.98)
where V := ΛTt φt and Γ := ΛTt (φ
′t − d3) are the linear material velocity and axial
material strain measure in the rotating frame. All other dynamical variables (in both
the convective and spatial representation) are given in Tables 3.3 and 3.4.
The application of the Clebsch approach (see Holm & Kupershmidt 1983) to derive
the LP equations for the SO(3) reduced rod motion is outlined in Appendix and we
shall not consider the continuum rod further. We shall now consider the application
of the discrete Clebsch approach of Cotter & Holm (2006) to give a MV integrator for
the geometrically exact rod.
3.5 The Discrete Geometrically Exact Elastic Rod Model
Consider the discrete Lagrangian for an extensible and shearable rod in the convective
representation derived by finite difference approximations of the continuous convective
Lagrangian given by equation (3.98)
115
`n+1 =∑
α
ρ02A|Vnα |
2
︸ ︷︷ ︸kin. energy of centroid
+ρ02Tr(I0Ω
n+1α
T )︸ ︷︷ ︸kin. energy of frames
− Tr(CΛ0 Ωnα+1
T )︸ ︷︷ ︸twisting pot. energy
− 〈Γnα, Cφ0 Γ
nα〉︸ ︷︷ ︸
shearing pot. energy
,
(3.99)
where the notation is defined in Table 3.5. The second and third term of this discrete
Lagrangian is similar are also present in the discrete Lagrangian for the (inextensible)
time dependent Kirchhoff rod given by equation (3.68). The subscript α denotes a
particle label and the superscript n denotes the nth time increment.
Recall from Chapter 2, that in the constrained coordinate formulation we holonom-
ically constrain Ωn+1α and Ωnα+1 to give the discrete constrained Lagrangian
`n+1c = `n+1 − {∑
α
Tr(Θn+1α (Ωn+1α Ωn+1α
T − Id))+ Tr
(Θnα+1(Ω
nα+1Ω
nα+1
T − Id))}.
(3.100)
The Clebsch constrained Lagrangian density is
ˆn+1 = `n+1c + 〈Pn+1α ,Λn+1α − ΛnαΩn+1α 〉+ 〈J n+1α ,Λnα+1 − Λ
nαΩ
nα+1〉
+ 〈Pn+1α , φn+1α − φnα − hΛnαV
n+1α 〉+ 〈Jnα+1, φ
nα+1 − φ
nα − hΛ
nα(Γ
nα+1 + E3)〉,
(3.101)
where the first and second terms are the Clebsch constraints for the temporal and
spatial update of the rod frame orientations Λnα and the third and fourth terms are the
Clebsch constraints for the temporal and spatial update of the rod centroid positions
φnα.
From the stationary action principle, one gets after rearranging the derived expres-
sions for the functional derivative of ˆ paired with δΛnα and δφnα,
Pn+1α =(Pnα − J
nα + J
nα+1Ω
nα+1
T + hY n+1α+1
)Ωn+1α ,
Pnα − Pn+1α = Jnα+1 − J
nα .
(3.102)
Linear material velocity Vnα :=ΛTαn
h (φn+1α − φnα)
Linear material strain measure Γnα :=ΛTαn
h (φnα+1 − φ
nα − h(d3)
nα)
Convective angular velocity Ωn+1α := ΛTαnΛn+1α
Convective angular strain measure Ωnα+1 := ΛTαnΛnα+1
Table 3.5: This Table gives the expressions for the discrete linear velocities and strainmeasures.
116
where Y n+1α+1 = P
n+1α ⊗ Vn+1α + Jnα+1 ⊗ (Γ
nα+1 + E3).
The left momentum map for the left SO(3) symmetry reduced discrete Lagrangian
density is given by
Jn+1α = 2skew(ΛnαTPnα)− 2skew(Λ
nαTJ nα ) + h skew
(mn+1α ⊗ Vn+1α
)
︸ ︷︷ ︸=0
+ hskew(nnα+1 ⊗ (Γ
nα+1 + E3)
),
(3.103)
where the linear angular momentum mn+1α := ∇Vn+1α`n+1 and nnα+1 := ∇Γnα+1`
n+1. The
image of this momentum map is the difference of mn+1α := I0Ω
n+1α
T − Ωn+1α I0 and
nnα+1 := Cα0 Ω
nα+1
T − Ωnα+1Cα0 .
Substituting the expressions in equation (3.102) for Pnα and Jnα into the above equa-
tion gives the explicit discrete Lagrange-Poincare equation for the convective angular
momentum
mn+1α = Ωnα
TmnαΩ
nα + Ω
nαTnnαΩ
nα − n
nα+1 − hskew
(nnα+1 ⊗ (Γ
nα+1 + E3)
). (3.104)
The discrete Clebsch relations also give the following expressions in the rotating
frame for the material linear momentum and dual of the material axial strain measure
mn+1α = hΛnαTPn+1α , nnα+1 = hΛ
nαTJnα+1. (3.105)
In summary the discrete linear and angular momentum equations are respectively
mn+1α = ΩnαTmnα − n
nα+1 + Ω
nαTnnα,
mn+1α = Ad∗ΩnαT
mnα +Ad
∗ΩnαT n
nα − n
nα+1 − hskew
(∇Γnα+1`
n+1 ⊗ (Γnα+1 + E3)).
(3.106)
The discrete linear momentum equation is just the discrete Euler-Lagrange equation
∇φnα`nα(φ
nα, φ
n+1α , φnα+1) +∇φnα`
n−1α (φn−1α , φnα, φ
n−1α+1) +∇φnα`
n−1α−1(φ
nα−1, φ
n+1α−1, φ
nα) = 0,
(3.107)
where `nα denotes the αth term of the discrete Lagrangian given in equation (3.99).
These are the discrete equations of motion which are used to generate the numerical
results in Section 3.6 demonstrating the conservative properties of the DMV algorithm
for this rod model. Just as in the continuous case, the last term of the discrete angular
momentum couples the particle positions with the orientation of the directors and
117
represents shearing motion. Future research should establish whether these discrete
flow maps are Poisson w.r.t. the Lie-Poisson bracket for the geometrically exact rod
model given by Krishnaprasad et al. (1988).
We shall now consider an alternative extension of the MV integrator for rigid body
motion, presented in Chapter 2, to pseudo-rigid body motion. In the next Section we
present the DMV algorithms for solving the MV integrators for the pseudo-rigid body
and geometrically exact rod and assess their conservative properties through numerical
experiments.
3.6 Numerical Experiments
3.6.1 DMV algorithms for pseudo-rigid bodies and elastic rods
With the exception of the pseudo-rigid body model, the algorithms used to generate
the numerical results, presented in this Section, are explicit and therefore not subject
to conditions which may affect convergence properties. Details of the iterative step
used to compute the pseudo-rigid body motion are provided in this Section.
The pseudo-rigid body
The discrete internal circulation velocity and rotational velocities of the pseudo-rigid
body are coupled through a Coriolis term. Consequently, the MV integrator for the
pseudo-rigid body casts to a coupled matrix Ricatti equation of the form
Mk+1 =M′k+1 − J(ωk+1)Ω
Tk+1 +Ωk+1J
T (ωk+1),
Nk+1 = N′k+1 − J(Ωk+1)ω
Tk+1 + ωk+1J
T (Ωk+1),(3.108)
where J(Xk+1) := DkXTk+1Dk and M
′k+1 and N
′k+1 are uncoupled matrix Ricatti equa-
tions of the form
M ′k+1 := JkΩTk+1 − Ωk+1Jk,
N ′k+1 := JkωTk+1 − ω
Tk+1Jk,
(3.109)
where Jk = 2D2k.
We solve for the velocity components Ωk+1 and ωk+1 by applying the DMV al-
gorithm for the coupled rigid body motion given in Appendix A.4, to equation (3.108).
Following step 2 of this algorithm, we split equation (3.108) into separate matrix Ric-
atti equations by making the substitution J(Xk+1) ≈ J(Xk) and solve for the velocities
satisfying the split matrix Ricatti equations
118
Ωk+1 = J−1k (−M
′k+1 + S
Rk+1),
ωk+1 = J−1k (−N
′k+1 + S
Sk+1),
(3.110)
where SR and SS are symmetric matrices with expressions determined from the split
matrix Ricatti equations. Following step 3, we recompute M ′k+1 amd N′k+1 using
J(Xk+1) and repeat these steps until Ωk+1 and ωk+1 converge to a specified tolerance
in the matrix two-norm.
In the numerical simulations of a Mooney-Rivlin material, shortly described, only a
few iterations of the DMV algorithm for coupled rigid motions are required on average
for the velocities to converge in the matrix two-norm to an order of 10−15, one order
above the precision of the machine.
Stability of the algorithm Experiments suggest that the DMV algorithm breaks
down for a large initial stretching velocity (Dk+1 − Dk)/h. More precisely, for a suf-
ficiently large stretching velocity, the real parts of the eigenvalues of (−M ′k+1 + SR)T
and (−N ′k+1 + SS)T become negative and no longer satisfy the stability constraint
determined by Cardoso & Leite (2001).
The update of the stretching matrix is explicit but not multiplicative and does there-
fore not ensure that the determinant of Dk remains positive. Consequently, the para-
meterisation of the potential energy is chosen to prevent the determinant vanishing
in numerical simulations. Geometric integrators which preserve positivity in the nu-
merical solution of matrix Riccati equations have been investigated by Dieci & Eirola
(1994) and this model might provide an excellent application of their work.
The geometrically exact rod model
Simo’s computational approach Simo’s computational approach for the geomet-
rically exact rod given by Krishnaprasad et al. (1988) uses an approximation of the
Cayley transform to compute the rotations and the finite element method to spatially
discretise the weakform of the momentum balance equations and the constitutive re-
lations. Nodal incremental rotations and vorticities are computed at each node and
the orientation and position of each frame is computed by interpolation of the nodal
values. This approach, however, requires a Newton-Raphson method to solve for the
new configuration. Simo proposed the use of quaternions to avoid the singularity ex-
hibited by Euler parameterisation and used Spurrier’s algorithm to efficiently compute
the quaternions from an orthonormal matrix.
119
Spurrier’s algorithm provides a singularity free extraction of a Quaternion from a
direction cosine matrix M which is precise for all rotation angles. This algorithm uses
only the largest component of the quaternion (which is greater than or equal to a half)
to compute square roots and the divisor in the computation of the other components.
This step avoids computation of negative square root arguments or division by zero,
both due to numerical imprecision. Essentially, the algorithm determines the largest
of tr(M), Mi,i, i ∈ {1 . . . 3} and computes the components of the quaternions using
different expressions depending upon whether tr(M) is the largest. These expressions
are specified in Spurrier (1977).
Our approach distinguishes itself from Simo’s in two ways
• Our discrete rod model is entirely variational, that is, the discrete model is de-
rived from a discrete Hamilton’s action principle. The configuration update is
performed by two mechanical integrators, a symplectic integrator for the update
of the arc-length parameterised set of rod centroid positions and a Lie-Poisson
integrator for the orientation of the corresponding set of directors. Consequently
energy levels do not drift and angular momentum is conserved to numerical round
off.
• The configuration update is explicit. A Stormer-Verlet second order symplectic
integrator computes the rod centroid positions and a DMV algorithm computes
the orientation of the set of directors.
Details of numerical experiments which demonstrate the conservative properties are
now provided.
3.6.2 Numerical results
Numerical experiments of the pseudo-rigid body and convective representation of the
geometrically exact rod model are performed to demonstrate the conservative properties
of the DMV algorithms and describe the elastic body dynamics. We summarise the
main observations that can be made from the Figures provided in this Section.
The pseudo-rigid body
• The time update of the eigenvalues of the pseudo-rigid body are shown in Figure
3.1. The motion of the Mooney-Rivlin material is observed to pulse with a fre-
quency which increases with the values of the Mooney-Rivlin parameters b = 10
and c = 10. These parameters increase the stiffness of the body. We observe
the determinant of the rigid body configuration varies between approximately 0.8
and 1, thus characterising this motion as quasi-incompressible.
120
• The components of the angular momenta and vorticity relative to the Lagrangian
coordinate frame are shown in Figure 3.2 to be periodic because the Coriolis
coupling between the angular velocity and vorticity is relatively small. The an-
gular momentum and vorticity relative to the Lagrangian coordinate frame are
also shown to remain on the sphere.
• Figure 3.3 shows that the DMV algorithm for the pseudo-rigid body exhibits no
secular drift in the energy error and conserves angular momentum and vorticity
to numerical round-off.
121
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000.5
1
1.5
λ 1
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000.5
1
1.5
λ 2
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000.5
1
1.5
λ 3
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000.5
1
1.5
|λ|
time steps
Figure 3.1: This Figure shows the eigenvalues of numerical simulations of a Pseudo-rigidbody motion over 104 time steps at a time step Δt = 0.05 for the case when the initialeigenvalues are (d1 = 1, d2 = 0.8, d3 = 1/0.8) and the Mooney-Rivlin parametersare a = 0.1, b = 10, c = 10, d = 50. The top three graphs show, respectively, thefirst, second and third eigenvalues of the body which describe the shape of the ellipsealong the principal axis. The bottom graph shows the determinant of the stretchingmatrix which evaluates to the product of the eigenvalues. The initial conditions forthis simulation are the initial body angular momentum and vorticity components, andstretching velocity given as M1(0) = 0.02, M2(0) = 10
−4, M3(0) = 0.0152, N1(0) =−0.022, N2(0) = −10−4, N3(0) = −0.0172 and d1(0) = 2 × 10−4, d2(0) = 1.6 ×10−4, d3(0) = 2.5× 10−4. The tolerance for the iterative step is 10−10. The first threegraphs show that each eigenvalue evolves in pulses which are approximately of the sameorder of magnitude. The bottom graph shows that the body is nearly incompressible,which is characterised by the large values of the parameters b and c.
122
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-0.04
-0.02
0
0.02M
1N
1
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-5
0
5x 10
-3
M2
N2
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-0.05
0
0.05M
3N
3
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100006
7
8x 10
-4
time steps
||M||||N||
Figure 3.2: This Figure shows the components of the body angular momentum andvorticity in numerical simulations of pseudo-rigid body motion over 104 time steps ata time step Δt = 0.05 for the case when the initial eigenvalues are (d1 = 1, d2 =0.8, d3 = 1/0.8) and the Mooney-Rivlin parameters are a = 0.1, b = 10, c = 10, d = 50.The top three graphs show, respectively, the first, second and third component of thebody angular momentumM and vorticity N . The bottom graph shows the norm of thebody angular momentum and vorticity. The initial conditions for this simulation arethe initial body angular momentum and vorticity components, and stretching velocitygiven as M1(0) = 0.02, M2(0) = 10
−4, M3(0) = 0.0152, N1(0) = −0.022, N2(0) =−10−4, N3(0) = −0.0172 and d1(0) = 2×10−4, d2(0) = 1.6×10−4, d3(0) = 2.5×10−4.The tolerance for the iterative step is 10−10. The first three graphs show that eachcomponent of M and N is periodic with different periods. The bottom graph showsthat the body angular momentum and vorticity remain on the sphere.
123
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-3
-2.5
-2
-1.5
-1
-0.5
0
0.5x 10
-3
E-E
0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
2
4
6
8x 10
-15
||m-m
0||
time steps
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-4
-2
0
2
4
6
8
10x 10
-15
||n-n
0||
time steps
Figure 3.3: This Figure shows the error in the energy, and angular momentum and vor-ticity, relative to the Lagrangian coordinate frame, in numerical simulations of pseudo-rigid body motion over 104 time steps at a time step Δt = 0.05 for the case whenthe initial eigenvalues are (d1 = 1, d2 = 0.8, d3 = 1/0.8) and the Mooney-Rivlinparameters are a = 0.1, b = 10, c = 10, d = 50. The top graph shows the energyerror in the solutions computed by the DMV algorithm and the bottom graph showsthe error in the angular momentum and vorticity relative to the Lagrangian coordinateframe. The initial conditions for this simulation are the initial body angular momentumand vorticity components, and stretching velocity given as M1(0) = 0.02, M2(0) =10−4, M3(0) = 0.0152, N1(0) = −0.022, N2(0) = −10−4, N3(0) = −0.0172 andd1(0) = 2× 10−4, d2(0) = 1.6× 10−4, d3(0) = 2.5× 10−4. The tolerance for the iterat-ive step is 10−10. The first graph shows that the mean energy error does not drift andthe bottom two graphs show that the mean angular momentum and vorticity errors,relative to the Lagrangian coordinate frame, are conserved to numerical round-off.
124
The geometrically exact elastic rod
The experimental parameters for simulation of a geometrically exact elastic rod are
the material properties: the principal moments of inertia {I1 = 2, I2 = 2, I3 = 1},
the respective shear and principal bending stiffnesses along axis t1 and t2 are 0.2 and
[2×10−3, 3×10−3] and the respective axial and torsional stiffnesses are 0.2 and 4×10−3.
The rod centroid is initially perturbed so that φi(t0) = a0{sin(π2Si), 0, sin(π2Si)}, where
the amplitude a0 = 0.1, the arc-length parameterisation Si = i LN−1 , i := 1 → N and
the orientation of the N directors at position Si is chosen so that the N − 1 rod rigid
sections are aligned with the tangent vector of φi(t0). The following Figures show the
conservative properties of the rod motion and provide a qualitative description of the
rod motion and are summarised below
• Figure 3.4 shows that the mean error in total spatial angular momentum for 50
rod sections is of the order of 10−8 after 104 steps. Comparison with the numerical
results for spatial angular momentum conservation of the DMV algorithm for the
rigid body and coupled rigid body, presented in Chapter 2, suggest that the linear
elastic coupling term contributes a marginal error to the momentum.
• Figure 3.5 shows that the mean error in total spatial linear momentum is of the
order of 10−11 and that this error also grows linearly with the number of rod
sections.
• Figure 3.6 shows that the rod model (with 50 sections) exhibits no secular drift
in the mean energy error.
125
0 2000 4000 6000 8000 100000
0.5
1
1.5
2
2.5x 10
-8
Time steps
Tot
al S
patia
l Ang
ular
Mom
entu
m E
rror
5 sections25 sections50 sections
Figure 3.4: This Figure shows the total spatial angular momentum error of the rodover 104 time steps of size Δt = 0.1 for rods with 5, 25 and 50 frames. The error scaleslinearly with the number of frames.
126
0 2000 4000 6000 8000 100000
0.5
1
1.5
2
2.5x 10
-11
Time steps
Line
ar M
omen
tum
Err
or
12 sections25 sections50 sections100 sections
Figure 3.5: This Figure shows the total material linear momentum error of the rodover 104 time steps of size Δt = 0.1 for rods with 12, 25, 50 and 100 frames. In eachcase, the mean total linear momentum error is to numerical round-off. The numericalround-off error scales linearly with the number of frames.
127
0 2 4 6 8 10
x 104
-0.016
-0.014
-0.012
-0.01
-0.008
-0.006
-0.004
-0.002
0
Time steps
Ene
rgy
Err
or
Figure 3.6: This Figure shows the energy error of the rod over 104 time steps of sizeΔt = 0.1. There are 50 directors in this model and the initial conditions for thissimulation are given above. The graph shows that the mean energy error is an orderof 10−2 and does not drift.
128
3.7 Summary
In this Chapter, we applied the discrete Clebsch approach of Cotter & Holm (2006) to
derive MV integrators for the convective and spatial representations of free ellipsoidal
models before specialising to a pseudo-rigid body. We also applied the discrete Clebsch
approach to give MV integrators for elastic rod models such as the geometrically exact
elastic rod model.
MV integrators for elastic motions In each case, a critical step is performed in the
formulation of the discrete action principle to give DMV algorithms for these models.
For the pseudo-rigid body, we polar decompose the configuration space GL(3)+ and
derive a MV integrator for the polar components of the rigid body and a corresponding
coupled DMV algorithm. For the elastic rod, a discrete variant of the Kirchhoff kinetic
analogy is used to formulate the discrete Lagrangian from the MV integrator for the
Lagrange top, considered in Chapter 2. The analogy gives a compatibility equation
which takes the form of the discrete auxiliary equation for the relative orientation
matrix given by the MV integrator for the coupled rigid body model, also considered
in Chapter 2. For each model, we show how the discrete Clebsch approach gives
momentum maps which correspond to conservation laws. These conservation laws
are verified by numerical experiment.
Pseudo-rigid Body We present theorems giving MV integrators for the spatial and
convective representations of free ellipsoidal motion on GL(n)+. These integrators
seek a new algorithm to implement them. We then take the critical step of polar
decomposing the motion on GL(3)+ to give MV integrators and DMV algorithms for
the polar components of the pseudo- rigid body corresponding to internal circulation,
stretching and rotation.
We show how to apply the DMV algorithm for coupled rigid body motions to
solve for the internal circulatory and rotational components of the pseudo-rigid body
motion. This is made possible through the observation that the the expressions defining
the momentum and vorticity contain a Coriolis term, coupling the discrete internal
circulatory and rotational velocities, which appears in the same form in the discrete
coupled rigid body model. We cast the expressions for the momentum and vorticity
as a coupled matrix Ricatti equation and solve this using the iterative DMV algorithm
given in Appendix A.4.
We also derive the momentum maps for left and right symmetry reductions which
correspond to conservation of spatial angular momentum and circulation respectively.
This derivation culminates in a discrete Kelvin circulation theorem. Numerical ex-
periments of a Mooney-Rivlin material show that the energy levels exhibit no secular
129
drift and angular momentum and vorticity is conserved to numerical round-off. These
conservative numerical models provide a new and efficient approach to the numerical
modelling of homogeneous elastic materials which exhibit exact conservation laws such
as angular momentum and circulation.
Numerical simulations of the DMV algorithm for the pseudo-rigid body show that it
is only stable for suitable parameterisations of the elastic potential energy term. This
is because the variational integrator for the stretching motion is not constrained to
preserve the strict positivity of the determinant of the diagonal matrix of eigenvalues.
The derivation of a determinant preserving integrator remains a subject for future
research.
Elastic Rod For the elastic rod, we present a discrete variant of the Kirchhoff kinetic
analogy and use it to define the spatial discretisation of the Lagrangian necessary for
the equilibrium configuration of a discrete inextensible elastic rod to be in one-to-
one correspondence with the discrete time motion of the Lagrange top presented in
Chapter 2. We then state the Lie-Poisson structure on the dual of the semi-direct
product Lie algebra which is preserved by the MV integrator. We observe that the
rod inextensibility constraint is intrinsically enforced by preserving the Lie-Poisson
structure.
We then add a discrete kinetic energy term to the discrete action principle and
derive a dynamical inextensible rod model. The critical feature of this model is the
existence of a discrete compatibility equation. We observe that it takes the form of the
discrete auxiliary equation for the relative orientation matrix in the discrete coupled
rigid body model.
Geometrically exact rod model We then extend this discrete model to the (ex-
tensible and shearable) geometrically exact elastic rod of Krishnaprasad et al. (1988),
presented in Section 3.5. In this model, the configuration is split into the position of the
rod centroid and the orientation of the directors. An elastic potential then couples these
two motions. A MV integrator computes the rigid body motions of the rod sections
and an explicit variational integrator computes the particle position of the rod sections
in the body frames at each particle position. Both integrators are explicit. Numerical
simulations of a geometrically exact rod with 50 rod sections conserve spatial angular
momentum to an order of 10−8, linear angular momentum to an order of 10−11 and
exhibit no secular drift in the energy error whose mean is an order of 10−2 (after 104
time steps).
130
3.7.1 Future research
MV integrators for the pseudo-rigid body The stretching component of the dis-
crete pseudo-rigid body motion is currently updated using a discrete Euler-Lagrange
equation. This update does not constrain the stretching matrices to the space of di-
agonal matrices with positive determinant. Further research should consider the for-
mulation of a multiplicative update procedure for the stretching motion. The MV
integrator should also be extended for an anisotropic polar decomposed pseudo-rigid
body in which the shape matrix is advected by both rotations and stretching. In order
to solve this integrator, the DMV algorithm must be extended to solve for stretching
motion.
MV integrators for the geometrically exact rod Future research should address
the extent to which the MV integrator for the geometrically exact rod model is Poisson
with respect to the Lie-Poisson bracket for the continuous SO(3) reduced rod motion
given by Krishnaprasad et al. (1988). The MV integrator for this rod model should
also be adapted to model the director orientations using quaternions, rather than Euler
angles, using the approach described in Chapter 2. The latter is motivated by the need
to model supercoiling and twisting motions of elastic materials such as DNA and other
polymer chains, for which Euler angle parameterisation is not suitable.
The stability of the DMV algorithm for this rod model is conditional upon the
choice of parameterisations of the elastic potential energy term. This problem should
be addressed in two stages. Firstly, the stability of the DMV algorithm for the rigid
body problem should be analysed to determine the bounds on the eigenvalues of the
Hamiltonian for the associated matrix Ricatti equation. Secondly, these bounds should
be expressed in terms of the parameterisations of the elastic energy potential in order
to assess the suitability of the DMV algorithm to rod models. Boundary conditions
should also be investigated.
Compressible fluid dynamics The unified computational framework seeks applic-
ation to the convective and spatial descriptions of fluid dynamics. The convective and
spatial representations of (compressible) ellipsoidal motion provide a basic prototype
for ideal compressible fluid dynamics for reasons which we now state.
• Spatial : Holm et al. (1986) consider a compressible fluid and show that the pas-
sage to the spatial representation is by reduction under the group of diffeomorph-
isms. This group acts on the density by pull-back, but acts trivially on the
metric-tensor on Eulerian space, forming the kinetic energy. Analogously, the
(compressible) ellipsoid reduces to the spatial representation by the right action
131
of GL(n)+. This group acts on the shape matrix by conjugation and trivially on
the right Cauchy-Green Matrix (the metric-tensor on Eulerian space).
• Convective: Conversely, Holm et al. (1986) show that the passage to the con-
vective representation of compressible fluids is by reduction under the group of
spatial diffeomorphisms. This group acts trivially on the density and on the
metric-tensor on Eulerian space by pull-back, forming the kinetic energy. Ana-
logously, the (compressible) ellipsoid reduces to the convective representation by
the left action of GL(n)+. This group acts trivially on the shape matrix and on
the right Cauchy-Green Matrix by conjugation.
In order to extend the MV integrators for ellipsoidal motion to compressible fluids,
we conjecture that two developments are needed, (i) the finite dimensional represent-
ation of the group action of diffeomorphisms on G, where G is a finite dimensional
representation of the group of diffeomorphisms, and (ii) the finite dimensional repres-
entation of the group action of diffeomorphisms on V ∗ by pull-back. Although Zhong &
Scovel (1994), Zeitlin (2004) have pursued the use of a finite dimensional representation
of the group of diffeomorphisms for modelling fluids, it remains an open question as to
how this applies to MV integrators.
3.7.2 Proceeding Chapters
We place the remainder of the thesis in the context of the material explicated thus far on
the derivation of Lie-Poisson integrators in a unified computational framework. Recall
from Chapter 2, that this unified computational framework applies the discrete Clebsch
approach to the Lie symmetry reduced discrete variational principle to give momentum
maps which transfer the canonical Poisson structure to the Lie-Poisson structure on the
(reduced) manifold. This manifold is the dual of the corresponding finite dimensional
Lie algebra. The MV integrators define co-adjoint orbits on the Lie-Poisson manifold
which are non-canonical symplectic foliations.
In Chapters 4 and 5, we restrict our attention to the formulation of geometric in-
tegrators which preserve the canonical symplectic structure on the full (unreduced)
phase space of Hamiltonian particle shallow water models. We will show in the next
Chapter that computational models with favourable long-time conservative properties
can be derived from the material representation of the semi-discrete variational prin-
ciple. Extremising the semi-discrete action principle gives semi-discrete Euler-Lagrange
equations which preserve the canonical symplectic form of the momentum phase space
of the particles. Preliminary numerical results suggest that application of explicit
(canonical) symplectic integrators to these equations gives a viable computational ap-
proach for long time simulations of shallow water. Chapter 5 goes on to demonstrate
132
the preservation of the Hamiltonian structure of particle shallow water models in the
presence of boundary conditions.
We reiterate the need to substantiate the development of a unified computational
framework for computational continuum dynamics by extending the results of this
Chapter. This development entails formulation of a symmetry reduced discrete vari-
ational principle and use of the discrete Clebsch approach to derive momentum maps
transferring the canonical Poisson structure of the particle equations to the reduced
particle phase space in the convective and spatial representations. It remains the sub-
ject of further research to derive integrators which preserve the non-canonical Poisson
structure on this reduced phase space and, by extending the results of this Chapter,
show that they exhibit additional conservation laws (associated with the symmetries of
the discrete variational principle) such as a Kelvin circulation theorem.
133
Chapter 4
A Variational Free-Lagrange
Method for Shallow Water
Synopsis This Chapter derives and investigates a semi-discrete approximation, re-
ferred to as a variational free-Lagrange (VFL) method, for the rotating shallow water
equations. This method is a variational formulation of the free-Lagrange method (see
Fritts 1985). The free-Lagrange method uses a Voronoi diagram to represent the layer-
depth field and is intrinsically locally mass conservative.
The discretisation of the variational principle for shallow water with the free-
Lagrange data structure forms the critical step in the derivation of the VFL shallow
water equations, expressed in terms of the Voronoi cell averaged layer depth and the ma-
terial particle velocities. This Chapter shows that these equations conserve energy and
formulates the corresponding semi-discrete divergence and vorticity equations based on
a discrete divergence and curl operator.
The form of the semi-discrete vorticity equation reveals that the potential vorticity
is not conserved because the discrete curl of the grad operator does not vanish. This
suggests the need for a constraint on the curl operator in the discrete action principle.
This Chapter closes with the presentation of numerical results showing the conservative
properties of the 1D VFL rotating shallow water equations over long time intervals.
Overview We shall begin with a Lagrangian description of the continuum rotating 2D
shallow water equations with varying bottom topography, as derived from a variational
principle. We then define the discretisation of the velocity and layer depth fields and
derive the semi-discrete 1D Euler-Lagrange equations of motion from a semi-discrete
Hamilton’s action principle. We prove that energy is conserved by the semi-discrete
continuity and momentum equations. We then show how this approach can be extended
134
for any two-dimensional Voronoi diagram to yield a weak form of the semi-discrete
Euler-Lagrange equations for shallow water. Using these equations, we derive the
semi-discrete vorticity and divergence equations and show that the potential vorticity
is only conserved if the discrete curl operator is constructed so that the discrete curl of
the gradient operator vanishes.
We give details of the symplectic integrator which we use for numerical simulation
of the 1D rotating shallow water equations on a periodic domain and close the Chapter
with numerical results of the conservative properties of the VFL method. Appendix
C gives the corresponding canonical formulation of the VFL method and outlines a
procedure, referred to as rezoning for approximating the spatial velocity and layer
depth over a fixed grid.
4.1 The Lagrangian Description of Shallow Water
Following Salmon (1983), we recall the Lagrangian description of a rotating fluid, for-
mulated as a continuum of fluid particles. Let each particle be positioned labelled by
some ` ∈ L ⊂ R2 meaning that the label is the initial position of the particle. The
position of a particle in the container R2 is given by a diffeomorphism X = X(`, t)
where
X : L × R+ → R2.
The fluid layer depth over the container h(X(`, t), t) is pulled-back to a fixed time-
independent function on label space h0(`) by the determinant of the Jacobian |J | =∣∣Xi
,a
∣∣ = h0(`)
h(X(`,t),t) , where the short-hand notation Xi,a =
∂Xi
∂`a .
The shallow water equations in a rotational frame are derived from the stationary
state of the action principle
S =
∫ t2
t1
dt
∫da h0{
1
2|X|2 +R(X) ∙ X−
g
2(h0|J |
−1 + 2b)}, (4.1)
where da = d`ad`b and the rotation vector R is given by curl R = 2ω(X), where ω is
the angular velocity of the rotational frame relative to an inertial frame.
The variational derivative of S,
δS =
∫ t2
t1
dt
∫da h0{(X
i +Ri)δXi +gh0
2|J |−2δ|J |} = 0, (4.2)
where the potential has been derived by integrating over the depth of fluid for some
bottom topography b(x) = b
135
∫dA
∫ h+b
b
gzdz =
∫dA
g
2((h+ b)2 − b2) =
∫dA
gh
2(h+ 2b)
=
∫da |J |
g
2h0|J |
−1(h+ 2b)
=
∫da h0
g
2(h+ 2b),
(4.3)
where dA = dX1dX2. Integrating the first term in the stationary action principle by
parts and expressing δ|J | as a function of δX i gives
δS =
∫ t2
t1
dt
∫da h0[−X
i +Rj,iXj − gh0m
i]δX i = 0, (4.4)
where m1 = (|J |−2X2,b),a − (|J |−2X2,a),b and m
2 = (|J |−2X1,a),b − (|J |−2X1,b),a and it is
assumed that δX i(`, t2) = δXi(`, t1) = 0 for some arbitrary times t1 and t2.
The Euler-Lagrange equations for flow on velocity phase space are, after use of the
chain rule h,a = h,iXi,a in the expressions for m
i,
U = −g∇Xh− f0k×U,
X = U,(4.5)
where we have made the f-plane approximation, ∇ × R = f0k, k is the unit vector
in the direction of gravity and f0 is the Coriolis parameter which is given by f0 =
2|ω|. Under the assumption of hyper-regularity of the Lagrangian, the Hamiltonian
may be defined by the Legendre transformation. One may then derive the canonical
Hamiltonian equations of motion which define the flow on the momentum phase space,
dual to the velocity phase space.
In Section 4.3, we semi-discretise the Hamilton’s action principle for shallow water
with a free-Lagrange data structure, derive the semi-discrete Euler-Lagrange equations
and show that these are equivalent to the canonical Hamiltonian particle equations.
We refer to these semi-discrete Euler-Lagrange equations as the VFL equations since
they preserve the variational structure of the flow field and use the data structure of
the free-Lagrange method.
Discrete variational approach in fluids The concept of discretising Hamilton’s
principle with Lagrangian particles in the context of idealised fluid dynamics can be
attributed to Salmon (1983) and Buneman (1982). Salmon (1983) points out that not
136
only is the approach succinct as a beginning point for approximations, but accommod-
ates moving disconnecting fluid boundaries and gives conservation laws corresponding
to symmetries of the Hamiltonian. The subsequent derivations of the discrete equa-
tions of motion parallel his presentation. Bonet & Rodriguez-Paz (2005) give a similar
presentation leading to variational SPH methods for hydrodynamics. The wider scope
of their presentation gives useful details for future generalisation of the approach sub-
sequently described.
We begin with a description of the VFL method and show how the mean layer
depth field is reconstructed at the end of each time step to ensure that the mass in
each moving cell remains constant.
4.2 The Variational Free-Lagrange Method for Shallow
Water
We present the VFL method in a similar notation to that used in the Hamiltonian
particle mesh method (Frank et al. 2002, Frank & Reich 2004), considered in Chapter
5. We follow the standard methodology for deriving particle methods. Firstly the label
space is discretised into N2 particles which are labelled by α. The particle coordinates
in velocity phase space TQ ⊆ R4 are
(XT ,UT ) = ([X1, . . . ,XN2 ]T , [U1, . . . ,UN2 ]
T ),
where Xα = (Xα, Yα)T .
Voronoi Diagram We refer to the set of particle positions X = {X1, . . . ,XN2} as
the set of sites.
Definition 4.2.0.1 (Voronoi Cell). A Voronoi cell Vα with site Xα is a polygon con-
taining the set of points Xβ closer to Xα than to any other site.
A hexagonal Voronoi cell is shown in Figure 4.1 together with the specification of
a local index for referring to neighbouring particles.
The Voronoi diagram is the set of all closed Voronoi cells indexed by α. In its con-
struction, we assume that particle positions are distinct. Note that this assumption
does not prohibit the formation of shocks - characteristics may still collide as they are
not the particle trajectories (see, for example, Whitham 1974, for an explanation of
shock formation). We now describe how the conservation of cell mass law gives the
shallow water layer depth field.
137
Aα
Xβ1
Xα
Xβ2
Xβ3
Xβ4
Xβ5
Xβ6
Δβ1Xα
nβ5
Figure 4.1: A hexagonal (ne = 6) Voronoi cell containing the particle with label indexα. By construction of the cell, each line connecting particle α with its neighbours isa perpendicular bi-sector of a cell edge. A local index i := 1 → ne is used to refer tothe neighbouring particles βi. Since the ne cells edges are in one to one correspondencewith the neighbouring particles, cell edges are also indexed by βi. Each cell edge is oflength Δlβiα .
138
4.2.1 Mass conservation
The statement of conservation of mass of cell α
Dmα
Dt=
D
Dt
∫
Vα
h(x, t)dA = 0, (4.6)
reduces to the form, DDtmα =DDt
(hα(t)Aα(t)
), if the layer depth h(x, t) is chosen to be
piecewise constant over each Voronoi cell, h(x, t) = hα(t), x ∈ Vα. This conservation
law gives an explicit expression to compute the cell averaged layer depth which, in finite
time and using finite difference notation, takes the form
hn+1α =AnαAn+1α
hnα. (4.7)
We now semi-discretise the shallow water variational principle with the ’free-Lagrange
data structure’. This data structure is defined by a particle representation of the ma-
terial velocity field and a Voronoi diagram for the layer depth. We first begin with the
1D non-rotational model.
4.3 The Variational Free-Lagrange Equations for 1D Shal-
low Water
We begin by considering the semi-discrete material description of Hamilton’s action
principle for 1D (non-rotating) shallow water, expressed using the free-Lagrange data
structure,
L =1
2
∑
α
mαU2α −
g
2
∑
α
mαhα, (4.8)
where the first term is the kinetic energy over the particles and the second term is
the potential energy over the Voronoi cells. Substituting the expression for the cell-
averaged layer thickness hα =mαΔXα, where Aα = ΔXα =
12(Xα+1 − Xα−1), into the
above action principle and taking variations gives
∂L
∂Xα=g
2
(m2α−1(ΔXα−1)
−2 −m2α+1(ΔXα+1)−2) , (4.9)
andD
Dt
∂L
∂Xα
= mαUα. (4.10)
The semi-discrete Euler-Lagrange equations are therefore
139
Uα = −g
2mα
(
mα+1hα+1ΔXα+1
−mα−1hα−1ΔXα−1
)
. (4.11)
The right hand side of the above equations defines the discrete gradient operator
grad(hα) :=1
2mα
(
mα+1hα+1ΔXα+1
−mα−1hα−1ΔXα−1
)
. (4.12)
The semi-discrete Euler-Lagrange equation and the semi-discrete continuity equation
D
Dthα = mα
D
Dt(ΔXα)
−1, (4.13)
are referred to as the VFL equations for 1D shallow water.
4.3.1 Energy Conservation
We now show that the semi-discrete shallow water equations conserve energy.
Definition 4.3.1.1 (Hamiltonian for the VFL method). The Hamiltonian for the VFL
method is given by
H =1
2
∑
α
mαU2α +
g
2
∑
α
mαhα, (4.14)
where the first term is the kinetic energy, formed from particle velocities, and the second
term is the potential energy formed from the cell averaged layer depth.
Lemma 4.3.1.2. The Hamiltonian for the VFL method is conserved along particle
trajectoriesD
DtH = 0. (4.15)
Proof 4.3.1.3.
D
DtH =
∑
α
mαUαD
DtUα +
g
2
∑
α
mαD
Dthα
=∑
α
[−gmαUαgrad(hα) +g
2m2α
D
Dt(ΔXα)
−1]
=∑
α
−g
2Uα[mα+1
hα+1
ΔXα+1−mα−1
hα−1
ΔXα−1+mα−1
hα−1
ΔXα−1−mα+1
hα+1
ΔXα+1] = 0.
(4.16)
We now consider the derivation of the 2D VFL rotating shallow water equations.
140
4.4 A Variational Free-Lagrange method for 2D Shallow
Water
The semi-discrete material description of the Hamilton’s action principle, is expressed
using the free-Lagrange data structure
Definition 4.4.0.4 (The Semi-Discrete Hamilton’s Action Principle for Shallow Wa-
ter).
Sd =1
2
∫ tb
ta
dt∑
α
mα
(|Uα(t)|
2 + 2Rα ∙Uα
)− g
∑
α
mα
(h(Xα, t) + 2bα). (4.17)
Following Augenbaum (1984), we express the potential energy V in terms of the cell
Jacobian ˉ|J |α(t) :=h(Xα,t0)h(x,t) , x ∈ Aα to give
V =g
2
∑
α
mα(h(Xα, t0) ˉ|J |−1α (t) + 2bα), (4.18)
Stationarity of the discrete action principle gives
∂L
∂Xα= mαXα =: Pα, (4.19)
and
1
Aα
∫
Aα
∂L∂Xα
dA =1
Aα
∫
Aα
∂L∂|J |α
∂|J |α∂Xα
dA =1
Aα
∑
i
[∂L∂|J ||J |
]βi
α
dnβiα . (4.20)
We formulate the weak form of the Euler-Lagrange particle equations, given by
1
Aα
∫
Aα
{d
dt
∂L
∂Xα−∑
i
[∂L∂|J ||J |
]βi
α
dnβiα }dA = 0, (4.21)
which is satisfied by
d
dt
∂L
∂Xα=∑
i
[∂L∂|J ||J |
]βi
α
dnβiα , (4.22)
where the operator
[∙]βiα =1
2(∙α + ∙βi), (4.23)
expresses the value of a scalar quantity at the βthi edge of cell α as the mean of that
quantity over cell α and cell βi. Evaluating the derivatives inside the brackets [∙]βiα gives
[∂L∂|J ||J |
]βi
α
= −g[mh]βiα, (4.24)
141
giving the final expression for the semi-discrete Euler-Lagrange particle equations
Uα = −ggrad(hα)− f0k×Uα,
Xα = Uα,(4.25)
where
grad(hα) := −1
mα
∑
i
[mh]βiαdnβiα . (4.26)
and dnβiα := nβiα Δl
βiα . The semi-discrete continuity equation takes the form
D
Dthα = −hα(t0)
Aα(t0)
A2α(t)
D
DtAα(t)
= −hα1
Aα
∫
Aα
∇ ∙UdA.(4.27)
From the divergence theorem, the semi-discrete continuity equation can be rewritten
as1
Aα
∫
Aα
∇ ∙UdA =1
Aα
∮
∂Aα
n ∙Udl =1
Aα
ne∑
i=1
nβiα ∙∮
∂Aβiα
Udl, (4.28)
where ∂Aα denotes the boundary of the polygonal Voronoi cell, and ∂Aβiα denotes the
face shared with the neighbouring cell βi with unit normal vector nβiα .
Following Ringler & Randall (2002) we introduce the approximation
1
Aα
ne∑
i=1
nβiα ∙∫
∂Aβiα
Udl ≈1
Aα
ne∑
i=1
dnβiα ∙Uβiα := div(Uα), (4.29)
which gives the definition of the discrete divergence operator div(∙α) over cell α where
dnβiα := nβiα Δl
βiα . (4.30)
Recall that Δlβiα is the length of the side indexed by βi of cell α. The semi-discrete
Euler-Lagrange equation and the continuity equation
D
Dthα = −hαdiv(Uα), (4.31)
formulate the VFL method for rotating shallow water on a f-plane with bottom topo-
graphy.
142
4.4.1 Energy Conservation
We now show that the VFL 2D rotating shallow water equations conserve energy.
Consider the material derivative of the Hamiltonian given by
D
DtH =
∑
α
mαUα ∙D
DtUα + gmα
D
Dthα
=∑
α
−gmαUα ∙ grad(hα)− f0k×Uα − gmαhαdiv(Uα)
=∑
α
gUα ∙∑
i
[mh]βiαdnβiα − f0k×Uα − gmαhα
∑
i
[U]βiα ∙ dnβiα
=∑
α
gUα ∙∑
i
mβi hβidnβiα − f0k×Uα − gmαhα
∑
i
Uβi ∙ dnβiα
=∑
α
Uα ∙
(
g∑
i
mβi hβidnβiα − f0k×Uα − g
∑
i
mβi hβidnβiα
)
= 0,
(4.32)
where the term in [∙]βiα is simplified by the property that∑
i dnβiα = 0 and the last line
is obtained by shifting the indices of the terms of the potential energy.
Remark 4.4.1.1 (Canonical VFL equations on momentum phase space). An alternat-
ive approach to proving energy conservation is outlined in Appendix C. This approach
constructs the canonical VFL equations on momentum phase space and verifies that
they are Hamiltonian. The approach relies on the existence of a smooth and invertible
Legendre transformation.
Remark 4.4.1.2 (Particle relabelling symmetry). The discrete Lagrangian is not in-
variant under continuous transformations of the labels - there is no particle relabelling
symmetry (see Ripa 1981, Salmon 1982). Padhye & Morrison (1996) succinctly sum-
marise the implications of the particle relabelling symmetry for hydrodynamics. The
symmetry gives Ertel’s theorem of conservation of potential vorticity which in turn
recovers the known connection with Kelvin’s circulation theorem along surfaces of con-
stant entropy. The relation between Ertel’s theorem and Hamilton’s action principle
was made earlier by Salmon (1982). Moreover, he pointed out that the Hamilton’s
principle provides a means of unifying all forms of Ertel’s theorem of hydrodynamics.
The discrete Lagrangian is, however, invariant under permutation of the indices.
This symmetry is deemed by Serrano, Espanol & Zuniga (2005) to give an associated
discrete Kelvin circulation theorem. We point out that this theorem is only approximate,
however, and is in fact exhibited by any convergent scheme.
143
4.5 The Shallow Water Vorticity Equation
By analogy with the continuum shallow water theory, we derive the semi-discrete po-
tential vorticity and divergence equations exhibited by the VFL method and show
that the latter quantity is conserved. The following derivation of the shallow water
vorticity equation requires the definition of the discrete curl, div and grad operators.
Semi-discretisation of Hamilton’s action principle with the free-Lagrange data structure
gives the form of the grad operator. Following Ringler & Randall (2002), we formulate
the semi-discrete continuity equation, given by equation (4.31), in terms of the div
operator which is restated below for convenience. Further following Ringler & Randall
(2002), we define the normal component of the discrete curl operator by analogy with
the derivation of the discrete divergence operator given by equation (4.34). The Voro-
noi diagram is assumed to rest in the X-Y plane so that the vertical unit vector k is
normal to the plane. From Stoke’s theorem
1
Aα
∫
Aα
k ∙ ∇ ×UdA =1
Aα
∮
∂Aα
τ ∙Udl =1
Aα
ne∑
i=1
τβiα ∙∮
∂Aβiα
Udl, (4.33)
which approximates to
1
Aα
ne∑
i=1
τβiα ∙∫
∂Aβiα
Udl ≈1
Aα
ne∑
i=1
dτβiα ∙ (U)βiα := k ∙ curl(Uα), (4.34)
where
dτβiα := τβiα Δl
βiα . (4.35)
Recall that Δlβiα is the length of the side indexed by βi of cell α. To summarise, the
discrete div and curl operators take the form
div (∙α) :=1
Aα
ne∑
i=1
dnβiα ∙(∙βiα),
k ∙ curl (∙α) :=1
Aα
ne∑
i=1
dτβiα ∙(∙βiα).
(4.36)
The vertical component of the discrete curl of the semi-discrete Euler-Lagrange
equation over cell α takes the form
k ∙ curl(Uα) = −gk ∙ curl(grad(h(Xα, t))
)− f0k ∙ curl(k×Uα). (4.37)
144
From the definition of curl, the left hand side of equation (4.37)
k ∙ curl(Uα)
=1
Aα
∑
i
Uβiα ∙ dτ
βiα
=D
Dt
(1
Aα
∑
i
Uβiα ∙ dτβiα
)
+1
A2α
D
Dt(Aα)
∑
i
Uβiα ∙ dτβiα −
1
Aα
∑
i
Uβiα ∙D
Dtdτβiα
=D
Dt
(1
Aα
∑
i
Uβiα ∙ dτβiα
)
+1
A2α
D
Dt(Aα)
∑
i
Uβiα ∙ dτβiα +
1
Aα
∑
i
Uβiα ∙ΔUβiα
︸ ︷︷ ︸=0
=
(D
Dt+1
Aα
D
DtAα
)1
Aα
∑
i
Uβiα ∙ dτ
βiα
=
(D
Dt+1
Aα
D
DtAα
)1
Aα
∑
i
Uβiα ∙ dτ
βiα
=
(D
Dt+ div(Uα)
)1
Aα
∑
i
Uβiα ∙ dτ
βiα
=D
Dt
(k ∙ curl(Uα)
)+ div(Uα)k ∙ curl(Uα),
(4.38)
where
D
Dtdτβiα =
D
Dt
(τβiα Δl
βiα
)= Δ
D
Dt(τ l)βiα = ΔU
βiα := U
βi+1α − Uβi−1α . (4.39)
Remark 4.5.0.3. The vanishing under-braced term in the fourth line follows from the
definition of ΔUβiα and the property that the terms cancel around a closed loop
∑
i
Uβiα
(Uβi+1α −Uβi−1α
)= 0. (4.40)
Note that in the continuum limit, the expression takes the form
1
2A
∮
A
d|U|2 = 0. (4.41)
Substituting the definition of the relative vorticity ζα = k ∙ curl(Uα) into the last
line of equation (4.38) simplifies it to
k ∙ curlDUα
Dt=
D
Dtζα + ζαdiv(Uα). (4.42)
Substituting this expression in the semi-discrete vorticity equation in equation (4.37)
145
givesD
Dtζα − ζαdiv(Uα) + f0k ∙ curl
(U⊥)= −gεα, (4.43)
where the term εα := k ∙ curl(grad(hα)
)6= 0 arises from the property that the discrete
curl of the discrete grad operator does not vanish
k ∙ curl(grad(hα)) =1
Aα
∑
i
(grad(hα)
)βiα∙ dτβiα
= −1
mαAα
∑
i
∑
j
[mh]βjαdn
βjα
βi
α
∙ dτβiα
= −1
mαAα
∑
i
∑
j
[mh]βjαnβjα Δl
βjα
βi
α
∙ τβiα Δlβiα = εα 6= 0.
(4.44)
The semi-discrete vorticity equation simplifies to
D
Dt(ζα + f0) + (ζα + f0) div(Uα) = −gεα, (4.45)
where we have used the property of the discrete curl and div operators
k ∙ curl(U⊥α ) =1
Aα
ne∑
i=1
dτβiα ∙(U⊥α
):=1
Aα
ne∑
i=1
dτβiα ∙(Uα × k
)
=1
Aα
ne∑
i=1
dnβiα ∙ (Uα) = div(Uα).
(4.46)
Remark 4.5.0.4 (Discrete curl operator). εα is an error term in the semi-discrete
vorticity equations which arises because the discrete curl of the discrete gradient operator
does not vanish. Certainly, the absence of a discrete potential vorticity law is consistent
with the remark made earlier on particle relabelling symmetry. For further insight, recall
that the discrete curl operator, unlike the discrete gradient operator is not implied from
the discrete Euler-Lagrange equations, but is chosen from a discrete approximation of
the integral identity for curl given by Stoke’s law. The discrete curl operator is therefore
formulated outside of the variational framework. To remedy the property of the discrete
curl operator, we suggest adding a constraint of the form
〈Jβiα ,1
mαAα
∑
i
∑
j
[mh]βjαdn
βjα
βi
α
∙ dτβiα 〉, (4.47)
146
where Jβiα is a Lagrange multiplier. The constrained discrete semi-discrete action prin-
ciple would then give a Mimetic differencing scheme (see Lipnikov et al. 2006, Margolin
et al. 2002). Mimetic differencing schemes are based on discrete operators which pre-
serve critical properties of the original continuous differential operators. Conservation
laws, solution symmetries and relationships between differential operators are just some
examples of such properties. They have been applied to a wide class of problems includ-
ing continuum mechanical models.
In these models, Mimetic differencing schemes have taken the form of arbitrary-
Lagrangian-Eulerian schemes defined on irregular unstructured meshes, rather than on
Voronoi diagrams. The approach is not variational by construction either. These reas-
ons render it more difficult to infer the form of the discrete operators required to make
the VFL method Mimetic. Crucially, we must establish whether existing discrete differ-
ential operators on unstructured meshes, which satisfy curl(grad(∙)) = 0, can be formu-
lated on a Voronoi diagram before attempting to cast this approach into a variational
framework.
Potential vorticity We now derive the semi-discrete potential vorticity equation
from the semi-discrete vorticity equation and the semi-discrete mass continuity equation
D
Dthα = −hαdiv(Uα), (4.48)
over the cell Vα.
Following Salmon (1998), we substitute the expression for the discrete divergence
of the particle velocity Uα into the semi-discrete vorticity equation, given by equation
(4.45) to give the statement of conservation of semi-discrete potential vorticity over cell
α
D
Dt(ζα + f0) +
(ζα + f0hα
)D
Dthα =
D
Dt
(ζα + f0hα
)
= −gεα. (4.49)
This equation states that potential vorticity is only conserved if the discrete curl of the
discrete gradient of the cell averaged layer depth vanishes. Enforcement of this property
would then put the VFL method on a par with the energy and potential vorticity
conserving Poisson-bracket method developed by Salmon (2004). This approach, based
on direct discretisation of the Poisson-bracket may give insight into the form and other
properties of the gradient and curl operators required for the VFL method to conserve
potential vorticity. Curiously, in his earlier work, Salmon (1983) suggests expressing
the rotational velocity in the discrete action principle as a functional of particle labels
and the invariant potential vorticity on particles.
147
4.6 The Semi-Discrete Divergence Form of the Shallow
Water Equations
We now repeat the previous steps for deriving the semi-discrete potential vorticity equa-
tion, only this time, taking the discrete divergence of the semi-discrete Euler-Lagrange
equations. The discrete divergence of the semi-discrete Euler-Lagrange equation over
cell α takes the form
div(Uα) = −gdiv(grad(h(Xα, t))
)− f0div(k×Uα). (4.50)
From the definition of div, the left hand side of equation (4.50)
div(Uα) =1
Aα
∑
i
Uβiα ∙ dn
βiα
=D
Dt
(1
Aα
∑
i
Uβiα ∙ dn
βiα
)
+1
A2α
D
Dt(Aα)
∑
i
Uβiα ∙ dnβiα −
1
Aα
∑
i
Uβiα ∙D
Dtdnβiα
=D
Dt
(1
Aα
∑
i
Uβiα ∙ dn
βiα
)
+1
A2α
D
Dt(Aα)
∑
i
Uβiα ∙ dnβiα −
1
Aα
∑
i
Uβiα ∙Δ(U⊥)βiα
︸ ︷︷ ︸:=Γα 6=0
=
(D
Dt+1
Aα
D
DtAα
)1
Aα
∑
i
Uβiα ∙ dn
βiα − Γα
=
(D
Dt+1
Aα
D
DtAα
)1
Aα
∑
i
Uβiα ∙ dn
βiα − Γα
=
(D
Dt+ div(Uα)
)1
Aα
∑
i
Uβiα ∙ dnβiα − Γα
=D
Dtdiv(Uα) + div(Uα)
2 − Γα.
(4.51)
Denoting the divergence of Uα as δα := div(Uα), the last line of the above equation
becomes
div(D
DtUα) =
D
Dtδα + δ
2α − Γα. (4.52)
So the semi-discrete divergence equation over cell α is
D
Dtδα + δ
2α − Γα + gdiv(grad(hα))− f0div(U
⊥α ) = 0. (4.53)
Remark 4.6.0.5. We compare the semi-discrete shallow water divergence equation
with the continuous form of this equation given by
148
div
(D
DtU
)
= ∂tδ + div(U ∙ ∇U)
= ∂tδ + ∂X(U∂XU + V ∂Y U) + ∂Y (U∂XV + V ∂Y V )
= ∂tδ + (∂XU)2 + (∂Y V )
2 + 2∂Y U∂XV + U∂2XU + V ∂
2Y V + V ∂XY U + U∂XY V
= ∂tδ + (∂XU)2 + (∂Y V )
2 + 2∂Y U∂XV +U ∙ ∇δ
=D
Dtδ + δ2 + 2(∂XV ∂Y U − ∂XU∂Y V )
=D
Dtδ + δ2 − 2det(∇U).
(4.54)
The first two terms correspond to those in the semi-discrete divergence equation
(4.52) and the last term corresponds to Γα which evaluates to1Aα
∮∂AαU ∙ d(U⊥) in the
continuum limit.
4.7 A Symplectic Time Stepping Scheme
The Stormer-Verlet method is used to integrate the 1D VFL shallow water equations of
motion. Similar procedures are described in Dixon & Reich (2004), Frank, Gottwald &
Reich (2002), Frank & Reich (2004). This integrator is a 2nd order explicit partitioned
Runge-Kutta method of the form
Pn+ 1
2α = Pnα −
Δt
2VX(X
nα),
Xn+1α = Xn
α +ΔtPn+ 1
2α
mα,
Pn+1α = Pn+ 1
2α −
Δt
2VX(X
n+1α ),
(4.55)
which preserves the symplectic two-form
ω =N2∑
α=1
dXα ∧ dPα. (4.56)
Computational considerations The gradient of the potential in the above time
stepping scheme is computed on a Voronoi diagram which is newly generated at the
beginning of each time step. Each Voronoi cell may have a variable number of sides and
change shape over each time step. Keeping track of the cell vertices is computationally
149
complex and challenges the scalability of the approach to higher dimensions. Harlen,
Rallison & Szabo (1995) outline an alternative approach which retains the nodes as
material points and reconnects them in the way that optimally triangulates the mesh.
For this shallow water model, this takes the form of a Delauney triangulation which is
dual to the Voronoi diagram. This triangulation can be generated at a relatively low
computational cost.
150
4.8 1D Numerical Experiments
This Section describes two numerical experiments of the 1D VFL rotating shallow water
equations on a periodic domain. The purpose of the first experiment is to investigate
the conservative properties of the VFL method for rotating 1D shallow water. In the
second experiment, we consider the problem of geostrophic adjustment of shallow water
and show that the VFL method produces results which are consistent with the theory
of geostrophic adjustment established by Rossby (1938).
This theory describes the physical mechanism by which perturbed rotating shallow
water recovers to a geostrophically balanced state. Layer depth perturbations h′ of the
scale L′ << LD, where LD =√gH0/f0 is the Rossby deformation radius for shallow
water in a f-plane, result in gravity waves which propagate energy and momentum
away from the source leaving behind a geostrophically balanced flow. This mechanism
is consistent with the advection law oF potential vorticity conservation.
The experiments are initialised by perturbing the Voronoi cell masses with a Gaus-
sian perturbation and the velocity field is initially zero.
4.8.1 Experiment 1: conservative properties of VFL
Unless otherwise stated, we use 128 cells, where each cell is initially of the same size.
The parameters of the simulations are provided in Table 4.1 below.
Parameter Value
Number of cells N 128Time step 0.01Domain length 2πInitial conditions mα(t0) = 1 + 0.1exp(−800(Xα(t0)− L/2)2/L2)
Uα(t0) = 0Vα(t0) = 0
f0 2πg 4π2
Table 4.1: This Table lists the simulation parameters for experiment 1, which investig-ates the conservative properties of the VFL method for 1D rotating shallow water.
Results The following Figures show the energy, potential (and total) vorticity and
potential enstrophy (square of PV) errors. Note that thesse quantities are summed
over all particles. Figure 4.2 shows the scaling of the energy, potential vorticity and
total vorticity with the size of the time step. The error in the energy scales with the
151
order of integrator, O(Δt2), where as the potential and total vorticity do not scale
with this order. This suggests that the VFL method introduces error in the discrete
vorticity. Figure 4.3 shows (from top to bottom) the relative energy, potential vorticity
and potential enstrophy error over 106 time steps, with a time step of 0.01. Each error
does not drift which suggests that the VFL method is long-time stable and conservative.
1 0 -1 -2 -3 -4 -5 -6 -7 -8-25
-20
-15
-10
-5
0
log 2(e
rror
)
log2(Δ t)
Energy errorPotential vorticity error Total vorticity error
Figure 4.2: This Figure shows the scaling laws of the mean energy error, potentialvorticity and total vorticity with the size of the time step over 106 time steps. Theenergy error scales to an order Δt2 which is consistent with the order of the StormerVerlet integrator. Potential vorticity (PV) and total vorticity (TV) error do not scalewith the order of the integrator suggesting that the order of the error of the PV andTV are not governed by the integrator but by the VFL method itself. The simulationparameters are given in Table 4.1.
152
0 1 2 3 4 5 6 7 8 9 10
x 105
-6
-4
-2
0
2
4
6
8x 10
-4 Energy Error
t
E-E
0
0 1 2 3 4 5 6 7 8 9 10
x 105
-1.5
-1
-0.5
0
0.5
1
1.5x 10
-4
|| q
|| -||
q 0||
t
Potential Vorticity Error
0 1 2 3 4 5 6 7 8 9 10
x 105
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2x 10
-4
|| q2 ||
-||q
02 ||
t
Potential Enstrophy Error
Figure 4.3: These graphs show (from top to bottom) the relative energy, potentialvorticity and potential enstrophy error over 106 time steps. Each error does not driftwhich suggests that VFL method is long-time stable and conservative. The simulationparameters are given in Table 4.1.
153
4.8.2 Experiment 2: geostrophic adjustment
This experiment uses the VFL method to simulate the mechanism of geostrophic adjust-
ment of shallow water in a plane in which the layer depth and velocity are independent
of the y-direction (meridional direction)
U = −g∇Xh+ f0V,
V = −f0U,
X = U,
(4.57)
Given a geostrophically balanced layer depth H0, meridional velocity V 0 = gf0∇Xh
and horizontal velocity U0 = 0, we perturb the layer depth h = H0 + h′ by a smooth,
domain centred, gaussian function h′ with support L′ less than the Rossby deformation
radius LD. The dynamics, governed by equations (4.57), are observed to exhibit gravity
waves which propagate energy and momentum away from the source leaving behind
geostrophically balanced flow.
We verify in this numerical experiment that this mechanism only occurs if the scale
of layer depth perturbation is smaller than the Rossby deformation radius by performing
two simulations, one in which the scale of perturbation is larger and one is which it is
smaller than the Rossby deformation radius. The parameters for this simulations are
provided in Table 4.2.
Parameter Value
Number of cells N 512Time step 0.01Domain length 2π
Initial conditions mα(t0) = 1 +∑2
i=1 0.01exp(−βi(Xα(t0)− L/2)2/L2)Uα(t0) = 0Vα(t0) = −
gf0grad(H0α)
f0 2πg 4π2
H0 1LD 1
Table 4.2: This Table lists the simulation parameters for experiment 2 which modelsgeostrophic adjustment when β1 = 10(L
′ << LD) and β2 = 1000(L′ >> LD). H
0α de-
notes the discrete steady state profile, from which the meridional velocity is initialised.
Computational issues
154
Results Figures 4.4 and 4.5 compare two different rotating shallow water regimes
distinguished by the scale of perturbation of the layer depth. In the first case, the layer
depth is perturbed on the scale of the Rossby radius LR = 1 and in the second, on a
scale smaller than the Rossby radius. In the latter case, the sequence of layer depth
graphs (shown at increasing simulation times) in Figure 4.5 show the gravity waves
which propagate from the source. The bottom profile of each of these graphs in this
Figure also shows that a geostrophically balanced region is recovered in the region of
the source.
0 1 2 3 4 5 6 7-0.01
0
0.01time: 0.27 days
x
h-h
0
0 1 2 3 4 5 6 71
1.005
x
h
0 1 2 3 4 5 6 7-0.02
0
0.02
x
u
ui
0 1 2 3 4 5 6 7-0.05
0
0.05
x
v
0 1 2 3 4 5 6 7-0.05
0
0.05
x
v+ g
/f 0 hx
Figure 4.4: These graphs show a snapshot of the shallow water taken at time t = 0.27”days” (1 ”day” is a rotational unit) for which the initial scale of perturbation L′ >>LD, where LD is the Rossby deformation radius. The top two graphs show the layerdepth perturbation and layer depth respectively. The third and fourth graphs showthe horizontal and meridional velocities. The bottom graph shows the difference of themeridional velocity with the layer depth gradient. Note that the source region is notrestored to a geostrophically balanced state because geostrophic adjustment does notoccur. The simulation parameters for this experiment are given in Table 4.2 in whichβ = 1000.
Figure 4.6 show the corresponding energy and potential vorticity errors over 2 ro-
tation units (i.e. 2 days). We firstly observe that the profiles exhibit high frequency
oscillations arising from gravity waves but do not exhibit drift. There are also jumps in
the profiles with an approximate period of 0.3 days. These jumps occur when the grav-
ity waves collide (recall that the domain is periodic). Figure 4.7 shows how the discrete
approximation of the layer depth, over the central region of the domain, converges to
155
0 1 2 3 4 5 6 71
h
0 1 2 3 4 5 6 7-0.05
0
0.05
u
0 1 2 3 4 5 6 7-0.05
0
0.05v
0 1 2 3 4 5 6 7-0.2
0
0.2
v- g
/f0 h
x
0 1 2 3 4 5 6 76
6.5
x
PV
hh
0
0 1 2 3 4 5 6 70.95
1
1.05
h
0 1 2 3 4 5 6 7-0.05
0
0.05
u
0 1 2 3 4 5 6 7-0.1
0
0.1
v
0 1 2 3 4 5 6 7-0.2
0
0.2
v- g
/f0 h
x
0 1 2 3 4 5 6 76
6.5
x
PV
hh
0
Figure 4.5: These graphs show snapshots of the shallow water taken at times t ={0.2, 0.5} ”days” (1 ”day” is a rotational unit) for which the initial scale of perturba-tion L′ << LD, where LD is the Rossby deformation radius. The top graph show thelayer depth perturbation and layer depth respectively. The third and fourth graphsshow the horizontal and meridional velocities. The bottom graph shows the differ-ence of the meridional velocity with the layer depth gradient. This difference is zerowhen the flow is geostrophically balanced. Note that the source region does restoreto a geostrophically balanced state because gravity waves propagate away energy andmomentum to restore the balance. The simulation parameters for this experiment aregiven in Table 4.1, in which β = 50.
the steady state after 0.4 days for various number of grid points. The outer regions of
the domain exhibit gravity waves which have propagated away from the source.
156
0 0.5 1 1.5 2-3
-2
-1
0
1
2
3x 10
-6
time (days)
(E-E
0)
0 0.5 1 1.5 2-1
-0.5
0
0.5
1
1.5x 10
-5
time (days)
(PV
-PV
0)/P
V0
Figure 4.6: These graphs show the energy (top) and potential vorticity (bottom) errorprofiles of shallow water over 2 ”days” (1 ”day” is a rotational unit) for which theinitial scale of perturbation is L′ << LD, where LD is the Rossby deformation radius.Both profiles exhibit high frequency oscillations arising from gravity waves but do notexhibit drift. There are also jumps in the profiles with an approximate period of 0.3days. These jumps occur when the gravity waves collide (recall that the domain isperiodic). The simulation parameters for this experiment are given in Table 4.2 inwhich β = 50.
157
0 1 2 3 4 5 6 70.998
1
1.002
1.004
1.006
1.008
1.01
1.012
1.014
x
h
1282565121024
Figure 4.7: This graph shows the layer depth at 0.4 days for various number of gridpoints. In the central region of the domain, the layer depth has reached a steadystate (geostrophic balance). The graph shows how the discrete approximation of thelayer depth converges to the steady state. The outer regions of the domain exhibitgravity waves which have propagated away from the source. This outer region is notin geostrophic balance after 0.4 days.
158
4.9 Summary
This Chapter derives and investigates a variational integrator for the rotating shallow
water equations which uses a Voronoi diagram to represent the layer-depth. This
method, referred to as the Variational free-Lagrange (VFL) method, locally conserves
mass and conserves energy to the order of the symplectic integrator used to approximate
the semi-discrete Euler-Lagrange equations. This Chapter culminates in three main
outcomes.
Firstly, we investigate the properties of the semi-discrete Euler-Lagrange shallow
water equations. We form discrete gradient, divergence and curl operators and verify
that the discrete divergence of the discrete gradient operator is zero. We verify that
the semi-discrete 2D Euler-Lagrange rotating shallow water equations conserve energy.
Secondly, we form the semi-discrete shallow water divergence and potential vorticity
equations which describe the evolution of the respective divergence and potential vorti-
city of the semi-discrete shallow water equations. We find that conservation of potential
vorticity depends upon the property that the discrete curl of the discrete gradient is
zero. This suggests that an additional constraint is needed in the discrete action prin-
ciple in order to derive a discrete curl operator with the correct property. The addition
of the constraint would, however, destroy the symplecticity of the VFL method.
Thirdly, we integrate the semi-discrete Euler-Lagrange equations using a Stormer-
Verlet symplectic integrator. Augenbaum did not appreciate the importance of pre-
serving symplectic integrator on long-time energy conservation properties and con-
sequently did not present numerical results, as we do, which demonstrate the energy
error scaling and energy and potential vorticity conservative properties of the VFL
method over long time intervals. We also verify that the VFL method exhibits the
mechanism of geostrophic adjustment. These results are only for 1D rotating shal-
low water and further numerical experiments are required to verify the conservative
properties of the VFL method in 2D with bottom topography.
Further research Serrano et al. (2005) consider the application of the free-Lagrange
method, which they refer to as the Voronoi fluid-particle model, to the Euler equations.
They formulate a discrete gradient operator with the remarkable property that semi-
discrete free-Lagrange approximation of the Euler equations is exact when the pressure
field is linear. This property avoids artificial numerical instabilities in the limit of zero
viscosity and suggests statistical features that actually correspond to similar features
observed in experiments on Lagrangian tracers in homogeneous fully developed turbu-
lence. We propose putting the VFL method in the context of Serrano et al. (2005) by
applying the method to the Euler equations and compiling, as they did, the distribution
of accelerations in the dynamical equilibrium state of the Voronoi fluid particles.
159
In the next Chapter we describe how ghost particles can be introduced into a
Hamiltonian framework to implement velocity boundary conditions. This is most
simply demonstrated for the 2D rotating shallow water equations through implement-
ation in a more established numerical method, referred to as the Hamiltonian particle
mesh method. It remains an open question as to whether this approach can be adapted
for the VFL method presented here.
160
Chapter 5
A Hamiltonian Particle Mesh
Method for Shallow Water in a
Bounded Domain
Synopsis This Chapter formulates free-slip boundary conditions for the HPM ap-
proximation of 2D rotating shallow water. We introduce ghost or image particles into
the HPM method and show how to modify the basis of the mesh to implicitly rep-
resent them. We also extend the layer depth smoothing matrix for bounded domains
by imposing a Neumann boundary condition on the smoothed layer depth. We finally
present numerical results showing the conservative properties of the HPM method for
shallow water channels and the motion of a vortex pair in rotating shallow water as it
reaches the channel wall.
Overview This Chapter is structured as follows. We begin with a Eulerian de-
scription of rotating shallow water before revisiting the Lagrangian description of the
continuum shallow water equations, only this time on a bounded domain. Section 5.2
reviews the formulation of the Hamiltonian particle-mesh method for 2D shallow water
in a channel and presents the extension of this method to bounded domains. Section
5.3 describes how the layer depth is smoothed on a bounded domain and considers the
1D case first before extending the approach to shallow water in channels and basins.
We refer the reader to the previous Chapter for a description of the time-stepper used
to solve the HPM equations for non-rotating shallow water. The inclusion of a Coriolis
term, however, calls for a modified symplectic integrator which is presented in Sec-
tion 5.2.1. Section 5.4 presents numerical results from four experiments which validate
the stability and conservative properties of the HPM approximation of a bounded 1D
shallow water model, compare the HPM approximation of shallow water in a channel
161
(without rotation) with a spectral-Chebyshev method and show the motion of a pair of
vortices in a channel of rotating shallow water on a f-plane as they reach the channel
wall.
5.1 Introduction
We consider a bounded domain of shallow water rotating with angular velocity ω re-
lative to an inertial frame. We assume that the shallow water is in a rotating plane,
referred to as a f-plane. This plane is tangent to the surface of the earth, across which
the Coriolis force 2ω×u takes its value at the point of tangency and only the component
of the angular velocity vector corresponding to rotations about the unit gravity vector
k is considered. The associated constant Coriolis parameter in the f-plane is f0 = 2|ω|.
The Eulerian description of bounded rotating shallow water provides two equations
for the Eulerian velocity u and the layer depth h and a Dirichlet boundary condition
on the normal component of the velocity
∂u
∂t+ u ∙ ∇u+ f0k× u = −g∇h,
∂h
∂t+ u ∙ ∇h = −h∇ ∙ u,
n ∙ u = 0, x ∈ ∂Ω.
(5.1)
The boundary condition on the component of the Eulerian velocity, normal to the
boundary n ∙ u = 0, is commonly referred to in inviscid fluid dynamics as a free-slip
boundary condition, since the tangential component of the velocity is not constrained.
For the case when f0 = 0, the free-slip boundary condition is equivalent to the Neumann
boundary condition on the layer depth n ∙ ∇h = 0. For the purposes of this Chapter,
the Eulerian description serves little more than to introduce the bounded shallow water
model in the most intuitive form.
Lagrangian description This Chapter shall pursue the formulation and implement-
ation of bounded shallow water in the Lagrangian description instead. Recall from the
previous Chapter, that in this description, each fluid parcel is positioned labelled by
some ` ∈ L ⊂ R2 meaning that the value of the label is the initial position of the par-
cel. The position of each parcel in the fluid container R2 is given by a diffeomorphism
X = X(`, t) where
X : L × R+ → R2.
162
The independent variables in this description are the fluid parcel labels and time and
the shallow water equations are
DU
Dt= −g∇h− f0k×U,
Dh
Dt= −h∇ ∙U,
n ∙ ∇h = 0, x ∈ ∂Ω,
(5.2)
where DDt =
∂∂t + u ∙ ∇ is the material derivative. We impose the zero condition on
the normal component of the gradient of the layer depth. For the case when f0 = 0
(the non-rotating case), this boundary condition on the layer depth is equivalent to the
kinematic form the free-slip boundary condition on the velocity
n ∙D
Dt(U) = n ∙ ∇h = 0. (5.3)
For the rotating case (f0 6= 0) the Neumann boundary condition n ∙ ∇h takes the
form of a zero mass transport condition normal to the boundary
Dh
Dt=∂h
∂t+ (n ∙U) (n ∙ ∇h)
︸ ︷︷ ︸=0
+(τ ∙U)(τ ∙ ∇h). (5.4)
The boundary is regarded as a line of symmetry in an extended domain across which
fluid parcels are symmetric (see Figure 5.1). Parcels reflect off the boundary by moving
across the boundary and exchanging their positions and velocities with parcels on the
other side of the boundary. Because the boundary is a line of symmetry, the normal
component of the gradient of the layer depth vanishes and no mass is transferred across
the boundary.
5.2 The Hamiltonian Particle Mesh Approximation
This Section introduces the HPM approximation for 2D rotating shallow water in a
channel Ω := [0, Lx]× [0, Ly) which is bounded in x and periodic in y. Further details
of this approach, without consideration of the boundary conditions, can be found in
Frank et al. (2002), Frank & Reich (2004).
We follow the standard methodology for deriving particle-mesh methods by adopt-
ing two stages. Firstly, the label space is discretised into N particles which are labelled
by greek indices. The particle coordinates in momentum phase space are
Z := (XT ,PT ) = ([X1, . . . ,Xα, . . . ,XN ]T , [P1, . . . ,Pα, . . . ,PN ]
T ). (5.5)
163
i=n−1i=0i=−1 i=n
x=-L
Ω ΩRΩL
x=0 x=L x=2L
Figure 5.1: This diagram provides the conceptual view of the boundary in the Lag-rangian description of shallow water in a channel. Each boundary is a line of symmetrypartitioning the domains Ω with ΩL and ΩR. In the HPM implementation, it is suffi-cient to extend the basis of cubic B-splines to the shaded region by positioning splinesat i = −1 and i = n. This ensures that the basis forms a partition of unity over Ω.
We introduce one non-standard feature at this stage, namely ghost or image particles.
We will shortly show that these particles are needed to impose the zero Neumann
boundary condition on the layer depth in HPM.
Remark 5.2.0.1 (Terminology). We shall use both names because ghost particles are
frequently referred to in the literature but in the opinion of the author, image better con-
veys that these particles are simply images of particles about each of the four boundaries,
moving with opposite velocities in the x-direction to their pre-images.
The definition of these particles is now given.
Definition 5.2.0.2 (Image Particles). Particles whose phase space coordinates are
given by the total relation Zα 7→ [Z−α,Z+α] where Z−α := ([−Xα, Yα], [−P xα , Pyα ]) and
Z+α := ([2L−Xα, Yα], [−P xα , Pyα ]) and whose mass m±α = mα are referred to as image
or ghost particles.
Gridded layer depth In the second stage of the HPM construction, the layer depth
H is discretised over a fixed mesh whose nodes are denoted by roman indices Ωh =
{xi,j : xi,j = (iΔx, jΔy) , i = 0, . . . , nx − 1, j = 0, . . . , ny − 1}, Ωh ⊆ Ω. We denote
the element of H at xi,j as hi,j and subsequently refer to the set of hi,j on Ωh as the
gridded layer depth.
The gridded layer depth is determined by interpolating particle masses mα at
position Xα using a basis of (cubic) B-splines functions {φi,j , i = −1, . . . , nx, j =
164
−1, . . . , ny}, where
φi,j(Xα) := φ(|xi −Xα|)⊗ φ(|yj − Yα|), (5.6)
is the tensor product of 1D cubic B-splines centred at xi,j . A horizontal Section of this
basis, in the region of the boundary at x = 0, is shown in Figure 5.2. The basis is
extended by k = sx4Δx number of basis functions (where sx denotes the support of the
basis function in the x direction), centred at grid cells beyond the boundary. When
the basis function are cubic B-splines, with a support width of sx = 4Δx, the basis
is extended by k = 1 splines (shown with the dotted line in Figure 5.2) which are
positioned at (−iΔx, jΔy) and (L + iΔx, jΔy), i = 1, . . . , k so that a partition of
unity is formed over Ω
nx−1+k,ny−1∑
i=−k,j=0
φi,j(x) = 1, x ∈ Ω. (5.7)
The partition of unity ensures global conservation of mass when interpolating particle
masses (and their images) to the gridded layer depth using the equation
hi,j =N∑
α=1
mαγ (φi,j(Xα) + φi,j(−Xα) + φi,j(+Xα)) , (5.8)
where mα := mαΔ`aΔ`b, the weighting γ :=∫Ω φ(x)dxdy. From the definiton of the
ghost particles and the symmetry of the basis functions, we observe that the subset of
basis functions φi,j implicity represent the image particles
φi,j(Xα) + φi,j(X−α) + φi,j(X+α) =(φi,j + φ−i,j + φ2(nx−1)−i,j
)(Xα) =: φi,j(Xα).
(5.9)
Note that if any of the basis functions in this subset are not in the basis, defined
above, then they are neglected.
Definition 5.2.0.3 (Gradient of the layer depth). The gradient of the layer depth at
position Xα is defined as
∇h(Xα) =nx−1+k,ny−1∑
i=−k,j=0
hi,j ∇φi,j(Xα). (5.10)
Lemma 5.2.0.4. HPM satisfies the boundary condition n ∙ ∇h = 0 through the sym-
metry properties of the image particles.
Proof 5.2.0.5. We shall consider the boundary xL = [0, y] along which the normal
165
component of the gradient of the layer depth is given by
n ∙ ∇h(xL) =nx−1+k,ny−1∑
i=−k,j=0
hi,j n ∙ ∇φi,j(xL). (5.11)
Given that the basis functions have a support of sx, this expression simplifies to
n ∙ ∇h(xL) =k,ny−1∑
i=−k,j=0
hi,j n ∙ ∇φi,j(xL). (5.12)
Using the property that n ∙∇φi,j(xL) = −n ∙∇φ−i,j(xL) and hi,j = h−i,j, the expression
for the gradient is
n ∙ ∇h(xL) = n ∙ ∇φ0,j(xL) = 0, (5.13)
which is the required boundary condition.
particle
Xα-Xα x=0
Additional basis function
Ghost particle
Rigid wall
ΩL Ω
Basis functions
Figure 5.2: This diagram shows a horizontal Section of the basis of cubic B-splines inthe boundary region around x = 0. There are two distinguishing features of the HPMimplementation to note. Firstly, when the basis function are cubic B-splines, the basisis extended by one spline (shown with the dotted line). This extended basis forms apartition of unity over Ω. This property is needed for HPM to globally conserve mass.Secondly, ghost or image particles are implicitly introduced to impose the boundarycondition n ∙ ∇h = 0. These are images of the particles reflected about the boundary.The image particles have the same mass, but opposite velocities as their pre-images.HPM does not explicitly use image particles but instead uses an appropriate set of basisfunctions to represent them.
166
Canonical HPM equations The canonical Hamiltonian equations of motion for
each particle in rotating shallow water are are
Pα = −c20mα
∑
i,j
hi,j(t)∇Xαφi,j(Xα)− f0k×Pα,
Xα =Pαmα
,
(5.14)
where hi,j denotes a smoothed gridded layer depth and is defined in the next Section.
These equations of motion preserve the discrete Hamiltonian
H =1
2
∑
α
||Pα(t)||2
mα+c202
∑
i,j
hi,j(t)hi,j(t)ΔxΔy, (5.15)
where the first term is the kinetic energy of the particles and the second term is the
potential energy over the gridded layer depth.
5.2.1 Symplectic time stepping
No rotation In the absence of a Coriolis term, the symplectic Stormer-Verlet method
is used as the time-stepper for the HPM shallow water equations as described in Section
4.7 for the 1D VFL shallow water equations.
Rotation The time-stepper for the HPM approximation of rotating shallow water
is adapted from the above symplectic second order explicit integrator to include an
implicit mid-point approximation of the Coriolis terms
Pn+1/2α = Pnα −Δt
2(∇V (Xnα)− f0k× (P
nα +P
n+1/2α )/2),
Xn+1α = Xnα +ΔtPn+1/2α
mα,
Pn+1α = Pn+1/2α −Δt
2(∇V (Xn+1α )− f0k× (P
n+1α +Pn+1/2α )/2).
(5.16)
Other approaches This is a convenient point to comment on the connection between
semi-Lagrangian schemes (Staniforth 1997) which are widely used in meterological ap-
plications and Lagrange-Galerkin finite element methods which have been developed
for a more general class of problems but also includes shallow water (Giraldo 2000).
Essentially, semi-Lagrangian schemes can be regarded as an extension of the above
167
time-stepper by interpolating the particle velocities to the mesh and then resetting
the particle positions to the mesh at the end of the time step. Interpolating back to
the grid renders the method as convenient for computations as Eulerian methods, but
with the added benefit of enhanced stability through computation of the shallow water
equations in the Lagrangian frame.
Semi-Lagrangian methods, however, integrate the particle shallow water equations
along the backward trajectory rather than, as above, the forward trajectory. This has
led to the development of ”remapped” particle-mesh methods for shallow water by
Cotter et al. (2007) which use a leap-frog scheme, similar to the above, but remap the
particles to the grid at the end of the time step.
Of course, the interpolation of velocities and resetting of particle positions destroys
the Hamiltonian structure of the flow field. This, however, may not be a primary
property for resolving significant geophysical dynamics when the equations of motion
are appropriately spatially discretised. Indeed, Giraldo (2000) combines a spectral
element approximation with a semi-Lagrangian scheme to resolve the motion of a pair
of equatorially trapped Rossby soliton waves. The scheme is able to model these waves
without an observed change in their profile or distance apart. The author points out
that the only disadvantage of using an explicit time-stepper is the more stringent CFL
constraint determined by the fastest modes. For this reason, we relax this constraint by
slowing down the gravity waves by introducing a dispersively regularising operator. The
following Section describes how the gridded layer depth is regularised over a bounded
domain.
5.3 Layer Depth Smoothing on a Bounded Domain
The HPM method is constructed using a regularisation operator. In the continuum
limit, the HPM equations for rotating shallow water correspond, not to the Lagrangian
description of the rotating shallow water given by equation (5.2) but the regularised
rotating shallow water equations
DU
Dt= −g∇h− f0k×U,
Dh
Dt= −h∇ ∙U,
n ∙ ∇h = 0, x ∈ ∂Ω,
(5.17)
in which the boundary condition n ∙ ∇h = 0 is imposed on the regularised layer depth.
Following Frank & Reich (2003), the layer depth is regularised by the inverse Helm-
holtz equation h = Sh, where S is a symmetric inverse Helmholtz operator of the form
168
S = (1− α2∇2)−1 with fixed smoothing length α typically of size 2Δx. In this Section,
we shall begin by using the finite element method to approximate the variational (weak)
form of the 1D Helmholtz equation. Neumann boundary conditions on the smoothed
gridded layer depth are imposed naturally, thus preserving the symmetry of the mat-
rix approximation of the inverse Helmholtz operator Sx, referred to as the smoothing
matrix. We will then consider the smoothing of 2D layer depths in a channel and a
basin.
5.3.1 A smoothing operator for 1D shallow water
Definition 5.3.1.1 (Variational Form). The variational form of the 1D Helmholtz
equation, paired with a smooth test function v(x) ∈ V, V := {v :∫Ω[v(x)
2+v′(x)2]dx <
∞}, x ∈ R is given by
∫
Ωv(x)h(x)dx =
∫
Ωv(x)(1− α2∂2x)h(x)dx, ∀v, (5.18)
with boundary condition
∂xh(x) = 0, x ∈ ∂Ω. (5.19)
Integrating equation (5.18) by parts yields
∫
Ωv(x)h(x)dx =
∫
Ωvh(x) + α2∂xv(x)∂xh(x)dx−
=0︷ ︸︸ ︷∫
∂Ωv(x)∂xh(x)dS, ∀v. (5.20)
where the boundary integral is formed over the set ∂Ω := {0, L} and vanishes to
impose the natural boundary condition. The 1D continuous piecewise linear finite
element solution to the variational form is now formulated on a uniform mesh, defined
as before, Ωh := {iΔx, i = 0, . . . , n− 1}.
Definition 5.3.1.2. The space of linear functions V h := {v : v ∈ C0(Ω), v is a linear
function on (xk−1, xk), k = 1, . . . , n} ⊆ V . Define a basis of piecewise linear functions
Nk(xj) = δkj , ∀k, j = 1, . . . , n.
{Nk} forms a nodal basis for V h (see Brenner & Scott 2002). hI(x) ∈ V h is the
interpolant of h(x), where hI(x) =∑
l hlNl(x) using the notation hl := h(xl) are the
nodal values of h(x). Similarly hI(x) =∑
l hlNl(x).
Choose, by construction of the (Ritz-Galerkin) finite-element method (Brenner &
Scott 2002), a test function Nk(x) ∈ V h is introduced so that equation (5.20) becomes
∫ xk+1
xk−1
Nk(x)hI(x)dx =
∫ xk+1
xk−1
Nk(x)hI(x) + α2∂xNk(x)∂xhI(x)dx, ∀k. (5.21)
169
Substituting these expressions into the previous equations yields
∑
l
Mklhl =∑
l
Aklhl, ∀k, (5.22)
where
Mkl =
∫ xk+1
xk−1
Nk(x)Nl(x)dx, Akl =
∫ xk+1
xk−1
Nk(x)Nl(x)+α2∂xNk(x)∂xNl(x)dx. (5.23)
Alternatively, equation (5.22) can be expressed in the matrix form
Mh = Ah. (5.24)
It follows from the form of the elements of A and M , that A and M are symmetric
tridiagonal matrices. The smoothing matrix Sx = A−1M requires the inversion of a
tridiagonal symmetric matrix, which is relatively efficient to compute.
5.3.2 A smoothing operator for 2D shallow water
We now consider the smoothing of the 2D gridded layer depth H, whose elements
are denoted hi,j and are defined on a regular nx by ny grid. We consider two types of
bounded domains: (i) a channel of the form Ω := [0, Lx]× [0, Ly), where the x-direction
is bounded and the y-direction is periodic and (ii) a (rectangular) basin of the form
Ω := [0, Lx]× [0, Ly], where the x and y directions are bounded.
In each case, we introduce a splitting of the smoothing operator into x and y
components, so that the inverse Helmholtz equation becomes
H = SyHSx, (5.25)
where H denotes the smoothed gridded layer depth, Sx and Sy are matrix approxim-
ations of symmetric inverse Helmholtz smoothing operators which each smooth inde-
pendently in the x and y directions respectively with smoothing parameters αx and
αy.
We smooth in two steps. Firstly, the finite element approximation of the x compon-
ent of the smoothing operator is extended to act on H. Substituting a new definition
of the linear interpolant (hI)m =∑
l hmlNl(x) into equation (5.21) gives
170
∫ xk+1
xk−1
Nk(x)(hI)m(x)dx =
∫ xk+1
xk−1
Nk(x)(hI)m(x) + α2∂xNk(x)∂x(hI)m(x)dx, ∀k,m,
(5.26)
or in matrix form
MxHT = AxH
T , (5.27)
where Mx is the mass matrix, Ax is the stiffness matrix and H := HSx = S−1y H is the
layer depth, smoothed in the x-direction.
The second step, for our purposes, is dependent on whether the domain is a channel
or a basin.
Channel The spectral approximation of the y component of the inverse Helmholtz
operator is the diagonal matrix
(Sy)i,i =(1 + α2(kiy)
2)−1
, kiy = −ny
2+ i, i = 1, . . . , ny. (5.28)
{kiy} is the finite set of all wave numbers of the streamwise Fourier modes permissable
over Ωh. Taking the product of this matrix with the discrete Fourier transform of the
finite element smoothed layer depth H gives an expression for the combined smoothing
in each direction
ˆH = Sy
H, (5.29)
or equivalently, using the finite convolution theorem,
H = Sy ∗ H. (5.30)
Basin In the case when the domain is a basin, the smoothing operator in the y
direction is formulated as a finite element approximation of the form
MyHT = AyH
T , (5.31)
which is solved for HT .
Remark 5.3.2.1 (Smoothing operator splitting). The splitting of the smoothing matrix
works well in practice because the boundary conditions are imposed naturally without the
computational overhead of a two dimensional finite element method for the Helmholtz
equation. Note also that Ax and Ay are constant nx × ny matrices and need only be
inverted once. The splitting, however, breaks the rotational symmetry of the Helmholtz
171
operator and therefore violates angular momentum conservation. In order to preserve
the isotropy of the operator, one can formulate a 2D Galerkin finite element approxim-
ation of the smoothing operator. The resulting stiffness matrix is nx×ny in length and
is more computationally expensive to invert. Cotter (2005) addresses this problem by
inverting the matrix using a preconditioned conjugate gradient approach. Alternatively,
one could use the split operator defined above as an ADI preconditioner, a description
of which is given in Miellou & Spiteri (2002).
Numerical simulations of this model are provided in the next Section.
5.4 Numerical Experiments
In this Section, we measure the conservative properties of 1D shallow water and show
the effect on the shallow water dynamics of preserving the symmetry of the smoothing
matrix under a finite element approximation of the Helmholtz equation.
The purpose of the numerical experiments that will now be described is to firstly
measure the long-time energy error, compare a 1D implementation of the HPM method
with a spectral-Chebyshev method and secondly simulate the motion of a vortex pair
in a channel of rotating shallow water as it reaches the channel wall.
The long-time energy properties of the HPM method are most conveniently assessed
through a 1D experiment.
5.4.1 HPM for 1D (non-rotating) shallow water
Experiment 1 This experiment measures the long-time energy error over 3 × 105
time steps using HPM with image particles and the finite element smoother in 1D. The
simulation parameters for this experiment are given in Table 5.1.
Parameter Value
N 256n 64Δt 5× 10−3
α 2ΔxL 2π
Initial conditions mα = 1 + a0e−50((X−L/2)2)/L2
Uα = 0
Table 5.1: Experiment 1: the set of simulation parameters and initial conditions for ex-periment 1 and 2, which measures the energy error and the effectiveness of the smooth-ing operator in the HPM simulation of 1D shallow water on a bounded domain.
172
Parameter Value
Time elapsed 3× 105 time stepsRelative energy error 5× 10−8
Table 5.2: Experiment 1 results: the relative energy error (E − E0)/E0 of the HPMapproximation of 1D shallow water over 3 × 105 time steps.
5.4.2 The 2D (non-rotating) shallow water equations in a channel
Experiment 2 We perform a simulation of shallow water in a channel and com-
pare the HPM method with the spectral-Chebyshev method. Figure 5.3 shows contour
plots at successive simulation times of the layer depth, starting from a Gaussian per-
turbation and Figure 5.4 shows a comparison of the HPM approximation (fixing the
particle-to-grid ratio at 4) of the layer depth and velocity to a spectral-Chebyshev ap-
proximation and compares energy and mass conservation. The parameters for channel
flow simulations, unless otherwise stated, are given in Table 5.3.
Parameter Value
Number of particles N 256Number of grid points n 64Time step 5× 10−4
Smoothing length α 2ΔxDomain length 2π × 2πInitial conditions mα = 1 + a0exp(−800[(Xα(t0)− L/2)2 + (Yα(t0)− L/2)2]/L2)
Uα = 0, Vα = 1Amplitude a0 0.5Coriolis parameter f0 0
Table 5.3: Experiment 2: this Table lists the simulation parameters for 2D shallowwater channel flow together with their values.
173
Figure 5.3: Experiment 3: contour plots of the smoothed layer depth in a 2D channelwith non-zero mean streamwise velocity shown starting from the top left and increasingin reading order at intervals of 20 iterations up to 160. The reason why the layer depthis not rotationally symmetric is because the split smoothing operator is not isotropic.
174
0 1000 2000 3000 4000 50000
0.5
1
1.5
2
2.5x 10
-5
time steps
||hhp
m-h
cheb
|| L2 (Ω)
16x1632x3264x64
0 1000 2000 3000 4000 50000
1
2
3
4
5
6x 10
-5
||uhp
m-u
cheb
|| L2 (Ω)
time steps
16x1632x3264x64
0 200 400 600 800 100010
-12
10-10
10-8
10-6
10-4
10-2
time
Rel
ativ
e en
ergy
err
or
HPMSpectral-Chebyshev
0 200 400 600 800 100010
-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
Rel
ativ
e m
ass
erro
r
time
HPMSpectral-Chebyshev
Figure 5.4: Experiment 3: L2 error measures of the HPM approximation of the (topleft) layer depth and (top right) velocity field to the spectral-chebyshev solution, wherethe norm ||h− h′||2L2(Ω) is defined as ||h− h
′||2L2(Ω) :=1
nxny
∑i,j |hi,j − h
′i,j |2. The error
profiles do not exhibit a definitive convergence trend, especially in the layer depthprofile. (bottom left) Relative energy error and (bottom right) relative mass erroragainst the number of time steps.
175
5.4.3 Rotating shallow water in a channel
Experiment 4 This experiment shows the motion of a vortex pair in a channel of
rotating shallow water as simulated by HPM. The vortex pair moves to the boundary
and then separates, each vortex moving apart along the wall. The vorticity field is
plotted at successive times in Figure 5.5 starting from a geostrophically balanced flow
with a layer depth described by a double Gaussian function. The parameters for this
experiment and the conservative properties of the HPM method are shown in Tables
5.4 and 5.5 respectively.
Parameter value
Nt 6000 (6 days)c0 48L2
f0 4√2π
dt 5× 10−3
L 2πH 1N 252×252nx 64ny 64
α√120(c0Hdt2)/(4 + f20dt
2)
Table 5.4: Experiment 4: the simulation parameters for the rotating shallow waterequations in a channel initialised from a geostrophically balanced flow with the layerdepth described by a double Gaussian function.
Parameter value
Time elapsed 6 daysMean energy 32277.698
Relative energy error 9.929× 10−5
Total mass 16384Total mass error 0
PV(0) 6534.525PV(Nt) 6543.632
Relative PV error 1.393× 10−3
Table 5.5: Experiment 4 results: the conservative properties of the HPM approximationof 2D rotating shallow water in a channel after 8 days.
176
Figure 5.5: Experiment 4: this graph shows the vorticity field of rotating shallowwater in a channel as computed by HPM. The vorticity field is shown at times t ={0, 2, 3, 4, 5, 6, 7, 7.4, 8.2} days, increasing in reading order. We observe that undergeostrophically balanced initial flow conditions, the vortex pair moves to the wall andthen separates. Each vortex then moves further apart along the wall.
177
5.5 Summary and Further Research
The HPM method is an efficient geometric numerical method for modelling shallow wa-
ter without adding numerical dissipation to ensure long-time stability. In this Chapter
we describe the use of image particles to extend the HPM method to rotating shal-
low water in a bounded domain. The Neumann boundary condition on the layer depth
restricts smoothed layer depth solutions to those which are symmetric about the bound-
aries. The HPMmethod imposes this boundary condition by introducing ghost or image
particles as reflections about the boundaries. This can be implemented very efficiently
by modifying the basis of cubic-B-splines to implicitly represent the image particles.
This Chapter only considers planar boundaries which coincide with a uniform mesh
and it is the subject of future research to consider more general boundary geometries
and meshes.
Another necessary step for long-time stability of the method is to approximate the
dispersively regularising Helmholtz operator with a finite element method. Numerical
results in 1D demonstrate that the implementation of the Neumann boundary condition
on the smoothed gridded layer depth must result in a symmetric Helmholtz matrix. The
finite element method imposes Neumann boundary conditions naturally and therefore
satisfies this property. This approach can be easily extended to a channel, by con-
volving the finite element regularisation of spanwise layer depth Fourier modes with
a spectral regularisation of the streamwise layer depth Fourier modes, and to a basin,
by successively smoothing with the 1D finite element method in the x and y direc-
tions. A study of the splitting error introduced by using this computational convenient
approach, should be assessed as further research.
Numerical experiments show that the HPM approximation of (non-rotating) shallow
water in a channel does not distinctly convergence to a spectral-Chebyshev approxima-
tion as the mesh is refined. These experiments should be repeated to study convergence
of the HPM approximation to a spectral-Chebyshev approximation as the particle to
grid ratio is increased, keeping instead the mesh fixed. The numerical results also in-
clude a simulation of a vortex pair in rotating shallow water as it reaches a channel
wall.
Further experiments should compare HPM with other (dissipative numerical meth-
ods) for shallow water in a beta plane with the intent of studying the effect of numerical
dissipation in models of western boundary current intensification (see Stommel 1948).
178
Chapter 6
Summary
6.1 Contribution of this Thesis
Geometric integrators transfer powerful concepts in geometric mechanics to computa-
tional continuum dynamics by preserving properties of the continuous system such as
the geometric structure, symmetries and phase space volume. Holm et al. (1986) presen-
ted a unified geometric approach for the study of idealised Hamiltonian continuum
models (fluids, plasmas, elasticity, etc.) in the material, inverse material, spatial and
convective representations. This unified approach is based on momentum maps, which
carry the Poisson brackets in one representation into another.
6.1.1 Development of a unified framework
In this thesis, we pursued the development of a unified computational framework for de-
riving geometric integrators for the convective and spatial representation of continuum
dynamics. This computational framework transfers the following powerful concepts
given by the unified framework of Holm et al. (1986) , for the convective and spatial
Hamiltonian continuum dynamics, to computational continuum dynamics:
1 Holm et al. (1986) show that the group action for passing between the represent-
ations generates an infinitesimally equivariant momentum map which carry the
Poisson brackets in one representation to those of the other. Using the discrete
Clebsch approach (Cotter & Holm 2006), we give the corresponding (diagonal)
group actions for passing between the representations and their momentum maps
from the cotangent bundle to the dual of the Lie algebra of the group.
2 Holm et al. (1986) show that the equations of continuum motion with advected
quantities are coadjoint orbits for the action of a semi-direct product Lie-algebra
on the dual of a semi-direct product Lie algebra. These orbits are symplectic
179
foliations of the Poisson manifold P defined by the augmented cotangent bundle.
We show that the discrete Clebsch approach gives discrete equations of motion
with advected quantities which define co-adjoint orbits for the action of the semi-
direct product group on the dual of its Lie-algebra. These orbits and are also
symplectic foliations of P . The co-adjoint actions preserves the ± Lie-Poisson
brackets on the dual of this semi-direct product Lie algebra.
3 Holm et al. (1986) show through various examples, that these momentum maps
encode fundamental conservation laws of Hamiltonian continuum dynamics. These
conserved momentummaps are generated from the Noether symmetries for passing
to the spatial and convective representations, the latter of which is referred to
as a particle relabelling symmetry. We show that the discrete Clebsch approach
gives
I conserved momentum maps for the polar decomposed pseudo-rigid body
which are the conserved spatial angular momentum and the discrete Kelvin
circulation theorem generated by the respective rotational and material re-
labelling symmetries.
II conserved momentum maps for the inextensible and geometrically exact
elastic rods which are the total spatial angular momentum generated by
the rotational symmetries.
For the latter case, there is a fundamental difference between the form of the
momentum maps derived from the continuum framework and our computational
framework, however. Our computational framework represents the continuum as
a finite dimensional system of particles. The conserved momentum map then
takes the form of a discrete sum over all particle labels and is only the form of
the conserved momentum map for the continuum, in the continuum limit of the
particle system.
4 Our computational framework gives a prototype MV integrator for the convective
and spatial representations of compressible fluids.
Metric tensors are central to the theory of continuum mechanics. Holm et al.
(1986) consider a compressible fluid flow, in which the passage to the convective
and spatial representation is by reduction under the group of diffeomorphisms,
and show how the metric tensor and densities respectively transform in the differ-
ent representations. Analogously, we consider ellipsoidal motion and demonstrate
how our framework transforms the metric tensor (the Cauchy-Green matrix) and
180
shape matrices under reduction by the group GL(n)+ to the convective and spa-
tial representations. The discrete Clebsch approach gives MV integrators for
these representations in which the metric and shape matrix are respectively the
advected quantities.
6.1.2 Development of new DMV algorithms
This thesis shows that, under a forward in-time finite difference approximation of the
continuous Clebsch constrained action principle for the body representation of the rigid
body, that the discrete Clebsch approach recovers the Moser-Veselov integrator. MV
integrators are computed using the explicit DMV algorithm developed by McLachlan
& Zanna (2005). In parallel with the development of our computational framework,
as described above, we give new DMV algorithms to solve for the MV integrators and
verify their conservative properties by numerical experiment.
• Rigid body motions in the spatial representation We modify the DMV algorithm,
for the body representation of rigid bodies, to solve the spatial MV integrators
derived in Chapter 2. We then provided several numerical experiments to study
the comparative conservative properties of the spatial DMV algorithm. The res-
ults are conclusive and show that there is a negligible difference between the
conservative properties of each algorithm.
• Rigidly Coupled motions We develop a DMV algorithm for the coupled matrix
Ricatti equation which arises from coupled rigid body motion between the free
rigid body motions of the coupled rigid body and the circulatory and rotational
rigid body motions of the polar decomposed pseudo-rigid body. We implement a
model of a Mooney-Rivlin type pseudo-rigid body to describe the stretching and
rotational components of the motion and show that the DMV algorithm conserves
angular momentum and vorticity (relative to the Lagrangian frame) and exhibits
no secular drift in the energy.
• Elastically coupled motions We solve a system of elastically coupled MV integrat-
ors for the elastically coupled director motions of the geometrically exact elastic
rod. The DMV algorithm for an elastically coupled rigid body differs from that
of the rigidly coupled rigid body. In the former case, the coupling is only through
the source term and not, as in the latter case, through the Coriolis term and re-
quires minor modification. The discrete Clebsch approach also gives a variational
integrator for the material representation of the rod centroid positions which is
equivalent to a Stormer-Verlet symplectic integrator. The DMV algorithm for
a rod of 50 sections conserves total spatial angular momentum, total linear mo-
mentum and exhibits no secular drift in the energy.
181
6.1.3 Geometric integrators for shallow water
This thesis pursues geometric integrators for shallow water in a variational framework.
These integrators preserve the canonical symplectic structure of a finite dimensional
system of fluid particles and are partially expressed in terms of the Eulerian quantities
(such as the layer depth).
• Variational free-Lagrange method A variational free-Lagrange method for rotating
shallow water with bottom topography is presented in Chapter 4. We establish
the conservative properties of the semi-discrete shallow water equations and de-
rive a semi-discrete shallow water divergence conservation law and an evolution
equation for the potential vorticity equation with a non-zero right hand side.
The semi-discrete divergence equation is given by the extrema of the discrete
action principle. The potential vorticity equations, however, are formed through
a choice of a discrete curl operator which is not resolved from the extrema of
the action principle. This suggests the need for an additional constraint in the
action principle to constrain the form of the discrete curl operator so that the
semi-discrete potential vorticity equation is satisified too. Numerical results are
also presented which show that the VFL method for rotating shallow water in 1D
exhibits long-time energy and potential vorticity conservative properties in 1D
and exhibits the geostrophic adjustment mechanism of rotating shallow water.
• Boundary conditions in the HPM method We extend the HPMmethod to bounded
rotating shallow water flows by implicity introducing ghost or image particles.
We demonstrate that the HPM approximation of rotating shallow water in a
bounded domain conserves mass, exhibits no secular drift in the energy and re-
mains stable over long-time simulations. We compare the HPM approximation of
(non-rotating) shallow water in a channel to a spectral-Chebyshev approximation
and also use HPM to simulate the motion of a vortex pair as it approaches a
shallow water channel wall.
6.2 Conclusions
In this thesis, we have pursued the development of a unified computational framework
for deriving geometric integrators for the convective and spatial representation of con-
tinuum dynamics. We consider the application of this framework to ellipsoidal motions
and rod models and derive MV integrators for the reduced motions. We solve these in-
tegrators using DMV algorithms and assess their conservative properties by numerical
experiment. The extension of MV integrators to fluids is a challenging open problem
182
and requires the use of a finite dimensional representation of the group of diffeomorph-
isms which has been pursued by Zhong & Scovel (1994), Zeitlin (2004). We turn to
the formulation of a variational free-Lagrange method to represent the Hamiltonian
structure of shallow water in terms of the (material) particle velocities and the spatial
scalar quantity, referred to as the layer depth. We study the conservative properties of
the semi-discrete VFL method and present the semi-discrete divergence and potential
vorticity equations, the latter of which takes the form of a conservation law. We then
assess the conservative properites of the fully discrete VFL method by simple numerical
experiments. We finally consider the formulation of velocity boundary conditions in a
Hamiltonian framework.
Throughout the thesis, we identified limitations in our methodology and results
and proposed future research possibilities to address these limitations, which we now
conclude.
6.2.1 Future directions
Compressible fluid dynamics The unified computational framework seeks applic-
ation to the convective and spatial representations of fluid dynamics. The convective
and spatial representations of (compressible) ellipsoidal motion provide a basic proto-
type for ideal compressible fluid dynamics for reasons which we now state.
• Spatial : Holm et al. (1986) consider a compressible fluid and show that the pas-
sage to the spatial representation is by reduction under the group of diffeomorph-
isms. This group acts on the density by pull-back, but acts trivially on the
metric-tensor on Eulerian space, forming the kinetic energy. Analogously, the
(compressible) ellipsoid reduces to the spatial representation by the right action
of GL(n)+. This group acts on the shape matrix by conjugation and trivially on
the right Cauchy-Green Matrix (the metric-tensor on Eulerian space).
• Convective: Conversely, Holm et al. (1986) show that the passage to the con-
vective representation of compressible fluids is by reduction under the group of
spatial diffeomorphisms. This group acts trivially on the density and on the
metric-tensor on Eulerian space by pull-back, forming the kinetic energy. Ana-
logously, the (compressible) ellipsoid reduces to the convective representation by
the left action of GL(n)+. This group acts trivially on the shape matrix and on
the right Cauchy-Green Matrix by conjugation.
In order to extend the MV integrators for ellipsoidal motion to compressible fluids,
two problems must be addressed, formulation of the finite dimensional representation
of the group action of diffeomorphisms on (i) G, where G is a finite dimensional repres-
entation of the group of diffeomorphisms, and (ii) on V ∗ by pull-back. Use of a finite
183
dimensional representation of the group of diffeomorphisms in computational fluid mod-
els is a challenging open area of research which has made little progress over the last
decade. We refer the reader to Zhong & Scovel (1994), Zeitlin (2004) for further details.
MV integrators for the pseudo-rigid body The stretching component of the dis-
crete pseudo-rigid body motion is currently updated using a discrete Euler-Lagrange
equation. This update does not constrain the stretching matrices to the space of di-
agonal matrices with positive determinant. Further research should consider the for-
mulation of a multiplicative update procedure for the stretching motion. The MV
integrator should also be extended for an anisotropic polar decomposed pseudo-rigid
body in which the shape matrix is advected by both rotations and stretching. In order
to solve this integrator, the DMV algorithm must be extended to solve for stretching
motion.
MV integrators for the geometrically exact rod Future research should address
the extent to which the MV integrator for the geometrically exact rod model is Poisson
with respect to the Lie-Poisson bracket for the continuous SO(3) reduced rod motion
given by Krishnaprasad et al. (1988). The MV integrator for this rod model should
also be adapted to model the director orientations using quaternions, rather than Euler
angles, using the approach described in Chapter 2. The latter is motivated by the
need to model supercoiling and twisting motions of elastic materials such as DNA and
other polymer chains, for which Euler angle parameterisation is not suitable. Boundary
conditions, such as clamped end conditions, should also be investigated.
VFL methods for shallow water In Chapter 4, we find that the form of the
discrete curl operator does not satisfy the property that the discrete curl of the discrete
gradient is zero and results in a violation of the conservation of potential vorticity.
Further research should address how to constrain the semi-discrete curl operator in the
discrete action principle to give a semi-discrete PV conservation law. The question of
whether the VFL method can be formulated for the spatial or convective representation
of shallow water remains an open problem. The existence of a unified computational
approach for modelling the Hamiltonian structure of shallow water using spatial and
convective variables appears to rest on addressing this question.
Boundary conditions In Chapter 5, we show how a Hamiltonian particle mesh
method for rotating shallow water can be extended using ghost particles to impose
velocity boundary conditions. It remains an open problem as to whether this approach
can be integrated into our unified computational framework giving, for example, MV
integrators for rod models with clamped ends. By treating planar boundaries as a
184
particle symmetry, we show how image or ghost particles may be implicitly introduced
into the HPM method. We also derive a finite element approximation of the Helm-
holtz operator which is used to dispersively regularise the layer depth and thus relaxes
the CFL stability constraint. A Neumann boundary condition on the regularised layer
depth is imposed naturally and the resulting Helmholtz matrix is symmetric. Numer-
ical experiments demonstrate that symmetry of this matrix is essential for numerical
stability and further demonstrate the effects of preserving symplectic structure on a
vortex pair in a rotating shallow water basin.
185
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192
Appendix A
Properties of MV Integrators
A.1 Body and Spatial Representations in Continuous
and Discrete Time
Dixon Moser and Veselov Description
Λk Xk Attitude matrix at time tkI0 J Inertia matrixΩk = Λ
Tk−1Λk ωk = X
Tk Xk−1 body angular ’velocity’ at time tk
Mk = I0ΩTk − ΩkI0 Mk = ω
Tk J − Jωk body angular momentum at time tk
mk = Λk−1MkΛTk−1 mk = Xk−1MkX
Tk−1 spatial angular momentum at time tk
Table A.1: This Table provides a comparison of our notation with that of Moser &Veselov (1991).
Property Continuous Discrete
Body attitude Λ(t) ∈ SO(N) Λk ∈ SO(N)Angular velocity Ω = ΛT Λ = −ΩT Ωk+1 = Λ
TkΛk+1
Inertia Matrix I0Angular momentum M = I0Ω− ΩT I0 Mk = I0Ω
Tk − ΩkI0
Equations of motion M = ad∗ΩM Mk+1 = Ad∗ΩkMk
Right momentum map JR = skew(ΛP T ) Jk+1R = skew(ΛkPTk )
Table A.2: A comparison of the terms required to describe the body representation ofthe rigid body in continuous and discrete time as derived using the Clebsch approach.Blank items in the right-hand column indicate that they are identical to their discretetime descriptions.
193
Property Continuous Discrete
Body attitude Λ(t) ∈ SO(N) Λk ∈ SO(N)Angular velocity ω = ΛΛT = −ωT ωk+1 = Λk+1Λ
Tk
Inertia Matrix I = ΛI0ΛT Ik = ΛkI0Λ
Tk
Angular momentum m = Iω − ωT I mk = IkωTk − ωkIk
Equations of motion m = ad∗ωm−∇IL � I = 0, mk+1 = Ad∗ωTk+1
mk + 2∇Ik lIk � Ik+1,
I = [ω, I] Ik+1 = ωk+1IkωTk+1
Left momentum map JL = P � Λ + J � I Jk+1L = Pk � Λk +Gk � Ik
Table A.3: Comparison of the terms required to describe the spatial representation ofthe rigid body in continuous and discrete time.
Property Continuous Discrete
Body attitude Λ(t) ∈ SO(N) Λk ∈ SO(N)Angular velocity Ω = ΛT Λ = −ΩT Ωk+1 = Λ
TkΛk+1
Inertia Matrix I0Angular momentum M = I0Ω− ΩT I0 Mk = I0Ω
Tk − ΩkI0
Orientation of the z-axis Γ = ΛT z Γk = ΛTk z
Equations of motion M = ad∗ΩM +mgΓ � χ, Mk+1 = Ad∗ΩTkMk +mgΓk � χ,
Γ = −ΩΓ Γk+1 = ΩTk+1Γk
Right momentum map JR = P � Λ + Γ � JΓ JRk+1 = Pk � Λk + Γk �Gk
Table A.4: A summary of the terms required to describe the body representation of theheavy top in continuous and discrete time.
194
Property Continuous Discrete
Body i attitude Λi(t) ∈ SO(N) Λki ∈ SO(N)Body i angular velocity Ω = ΛTi Λi = −Ω
Ti Ωk+1i = Λk
T
i Λk+1i
Position of c.o.m. of body i diOrientatn matrix (rel. to b. 1) Λ = ΛT1 Λ2
Mod. inertia matrix of body i Ii = Ii −m2im Dii
C1 C1 = I1 − εAΩ2 Ck+11 = I1 + εAk(Ωk+12 − Id)
C2 C2 = I2 − εATΩ1 Ck+12 = I2 + εAkT (Ωk+11 − Id)
Body 1 angular momentum M1 = C1Ω1 − ΩT1 CT1 Mk
1 = Ck1Ω
kT
1 − Ωk1C
kT
1
Body 2 angular momentum M2 = C2Ω2 − ΩT2 CT2 Mk
2 = Ck2Ω
kT
2 − Ωk2C
kT
2
Equations of motion Mi = ad∗ΩiMi +
δlδΛ � Λ, Mk+1
i = Ad∗ΩkiMki +∇Λk lΛk � Λ
k,
Λ = ΛΩ2 − Ω1Λ Λk+1 = Ωk+1T
1 ΛkΩk+12Right momentum map J iR = Pi � Λi + J � Λ J i
k
R = Pki � Λ
ki +G
k � Λk
Table A.5: A summary of the terms required to describe the body representation of thecoupled rigid body in continuous and discrete time, where i ∈ {1, 2}.
A.2 MV Integrators for the Cayley-Klein Parameterisa-
tion of Rigid Body Motion
Property SU(2) = h(Q) Q
Body attitude Λk ∈ SU(2) qk ∈ QAngular ’velocity’ Ωk+1 = Λ
†kΛk+1 Ωk+1 = qk ∗ qk+1
Moments of inertia Ij , j := 1→ 3
Inertia matrix Ak = Id − IiTr(Ω†kEi
)Ei Ak = [1,Av(tk)]
Av = ıjIjΩ(tk)jAngular momentum Mk+1 = Ak+1Ω
†k+1 − Ωk+1A
†k+1 Mk+1 =
12Ak+1 ∗ Ωk −
12Ωk ∗ Ak+1
Equations of motion Mk+1 = Ad∗ΩTkMk Mk+1 = Ωk ∗Mk ∗ Ωk
Right mom. map Jk+1R = Pk � Λk Jk+1R = pk � qk
Table A.6: A comparison of the terms required to describe the body representation ofdiscrete time rigid body motion as elements of SU(2) and quaternions.
195
A.3 The Spatial DMV Algorithm for the Rigid Body
For completeness, we include the specification of the (explicit) DMV algorithm for
the spatial representation of the rigid body. Following the approach of McLachlan &
Zanna (2005), the algorithm uses the Schur decomposition of the Hamiltonian for the
matrix Ricatti equation to construct a symmetric matrix Sk. Numerical experiments in
Section 2.10 show that there is little difference between the conservative properties of
the spatial and body versions of this algorithm. We observe that the numerical round-
off error differs between the two different versions of the DMV algorithm depending
upon the principal moments of inertia. Numerical experiments, not presented in this
thesis, find negligible difference in the stability and computational performance between
the two versions.
1. For k = 1 to NT
2. mk = ωk−1mk−1ωTk−1 + [Ik−1, ωk−1 + ω
Tk−1]
3. Hk =
(mk2 , Id
(mk2 )2, I2k−1 −
mTk2
)
4. [Rk, Uk] = Schur(Hk)
5. Sk = (Rk)21(Rk)−111
6. ωk = (Sk +mk2 )I
−1k
7. Ik = ωkIk−1ωTk
8. k = k + 1
A.4 A DMV Algorithm for Coupled Rigid Body Motion
Consider a coupled matrix Ricatti equation for discrete coupled rigid body motions of
the form
196
M1 =M′1 + J(Ω2)Ω
T1 − Ω1J(Ω2)
T ,
M2 =M′2 + J(Ω1)Ω
T2 − Ω2J(Ω1)
T ,(A.1)
whereMi and Ω denote the momenta and discrete velocity of each body, each multiplied
by a factor of Δt - the DMV algorithm of McLachlan & Zanna (2005) requires that
these variables are initially rescaled. M ′i denotes an uncoupled matrix Ricatti equation
of the form M ′i = IiΩTi − ΩiIi, for some symmetric positive definite matrix Ii.
[Step 1] Introduce a splitting of the coupled matrix Ricatti equations into two
separate matrix Ricatti equations in Ωi and formulate an iterative procedure for their
solution. The split matrix Ricatti equations are
M1 =M′1 + J(Ω2)Ω
T1 − Ω1J(Ω2)
T ,
M2 =M′1 + J(Ω1)Ω
T2 − Ω2J(Ω1)
T ,(A.2)
where J(Ωi) is a function of the previous value of Ωi and denoted as Ωi. Note that the
modified Coriolis terms are both O(Δt2), an O(Δt) smaller than the other terms.
[Step 2] The DMV algorithm given by McLachlan & Zanna (2005) is then applied
to compute the iterative solutions of equation A.2 which take the form
Ωi = I−1i (−M
′i + S
i), (A.3)
where Si is a symmetric matrix with expressions determined from each of the split
matrix Ricatti equations given by equation A.2.
[Step 3] The iterative solutions give updated values of J(Ωi), and using equation
A.2, updated values of M ′i . [Step 4] Repeat steps 2 and 3 until Ωi both converge to a
specified tolerance in their matrix two-norm and then rescale Mi by 1/Δt.
197
Appendix B
The Variational Description of
Elastic Body and Rod Models
B.1 The Anisotropic Pseudo-Rigid Body
We generalise the Euler-Poincare description of the polar decomposed isotropic pseudo-
rigid body by Sousa-Dias (2002) to the anisotropic free rigid body. This is the case
when the reference configuration is not spherical. In this case, the shape matrix is
advected by rotational and stretching motions.
Following Holm et al. (1986), we consider the Lagrangian defined on TGL(3)+×V ∗
for the anisotropic pseudo-rigid body
LI0 =Tr
2(AI0A
T ), (B.1)
where I0 denotes that I becomes an active parameter under the right actions φ :
TGL(3)+ × V ∗ × GD : (A, I0) ∙ φ := (AD−1, DI0DT ) and φ′ : TGL(3)+ × V ∗ ×
SO(3) | (A, I0)∙φ := (AST , SI0ST ) where GD is the group of positive diagonal matrices.
Polar decomposition of RASTD−1 gives
L =Tr
2(AI0A
T )
=Tr
2
((RTDS +RT DS +RTDS)I0(R
TDS +RT DS +RTDS)T)
=Tr
2
(RT (ΩT + Γ +DωD−1)It(Ω + Γ
T +D−1ωTD)R)
=Tr
2
(ΣItΣ
T),
(B.2)
where It = DSI0STD, ω = SS−1 ∈ so(3), Γ = DD−1 belongs to the Lie algebra of all
positive diagonal matrices gD(3) ' R3+, Ω = RR−1 ∈ so(3) and Σ = ΩT + Γ +m∗ω,
198
for Σ ∈ gl(3) where
g = so(3)⊕ R3+ ⊕m∗so(3). (B.3)
Lemma B.1.0.1. m∗ : g→ h , m∗(ω) = DωD−1 is a Lie algebra isomorphism.
Proof. m∗ is a Lie algebra homomorphism since
m∗[Λ,Γ] = D[Λ,Γ]D−1
= DΛD−1DΓD−1 −DΓD−1DΛD−1
= [m∗Λ,m∗Γ],
(B.4)
for Λ,Γ ∈ gl(3).
The variations of each of the Lie algebras and the inertia matrix
δΓ = ΞD + [ΞD,Γ],
δω∗ = [ΞD, ω∗] +D(Ξs + [ΞS , ω])D
−1,
δΩ = ΞR + [ΞR,Ω],
δIt = ΞDIt + ItΞTD +D[Ξs, Is]D,
(B.5)
are substituted into the Hamilton’s action principle
∫dt〈
δL
δΓ, δΓ〉+ 〈
δL
δω∗, δω∗〉+ 〈
δL
δIt, δIt〉 = 0, (B.6)
to give, after rearrangement of terms,
〈πD(ΣIt), ΞD + [ΞD,Γ]〉+ 〈ΣIt, [ΞD, ω∗] +D(ΞS + [ΞS , ω])D
−1〉
+ 〈skew(ItΣT ), ΞR + [ΞR,Ω]〉+ 〈
DΣTΣD
2,ΞDIt + ItΞ
TD +D[Σs, Is]D〉 = 0.
(B.7)
where mD :=δLδΓ = πD(ΣIt), mR :=
δLδΩ = skew(ItΣ
T ) and m′S := D−1 δLδω∗D. πD de-
notes the diagonal component of the matrix. This gives three Euler-Poincare equations
for evolution of the projected momentum
m′s = ad∗ωm′s +D
δL
δItD � Is,
mR = ad∗ΩmR,
mD = 2ItδL
δIt+ ad∗ω∗
δL
δω∗+ ad∗ΓmD,
(B.8)
together with the auxillary equation for the Lie advected inertia tensor
199
It = ΞDIt + ItΞTD +D[ω, Is]D. (B.9)
These equations describe how the components of the momentum of the pseudo-rigid
body evolve.
This decomposition together with the defined left and right symmetry reductions
gives complex expressions for mD and m′s associated with pure stretching and internal
rotational-stretching deformations.
B.2 The Geometrically Exact Elastic Rod Model
This Appendix provides a variational description of the geometrically exact rod model,
using the Clebsch approach (Holm & Kupershmidt 1983). The derive equations of
motion correspond to the Lie-Poisson equations given by Krishnaprasad et al. (1988).
The application of the Clebsch approach, described here, to the geometrically exact
rod also provides a useful reference point for the application of the discrete Clebsch
approach to the geometrically exact rod model.
The Clebsch constrained Lagrangian density is defined as
ˆ= `+ 〈P , Λ− ΛΩ〉+ 〈P, φ− ΛV〉+ 〈J ,Λ′ − ΛΩ〉+ 〈J, φ′ − Λ(Γ + E3)〉. (B.10)
The pairing 〈., .〉 denotes the product for identifying elements in the tangent space
of a group with elements in the dual space for all labels S. For elements of Diff(C)
describing the rod centroid position and B → SO(3) describing the orientation of a
director, these pairing take the from
〈p, q〉 =∫
Bp(S)T q(S) dS, q ∈ TqDiff(C), p ∈ T
∗qDiff(C), and
〈P, Q〉 =∫
BTr(P (S)T Q(S)
)dS, Q ∈ TQSO(3), P ∈ T ∗QSO(3).
(B.11)
The stationary action principle gives the following Clebsch relations
P = PΩ− J ′ + J ΩT − P ⊗ V − J ⊗ (Γ + E3),
δ`
δV= ΛTP,
δ`
δΓ= ΛTJ,
δ`
δΩ= skew(ΛPT ),
δ`
δΩ= skew(ΛJ T ).
(B.12)
200
Lemma B.2.0.2. The left infinitesimally equivariant momentum map for the cotangent
lifted left action ΨL on T ∗C is the total angular momentum
JL =
∫
Bskew(ΛTP)dS. (B.13)
Proof. The infinitesimal generator ζC = ΨLζ of Ψ
L on C acts on T ∗C by the canonical
action ζT ∗C = ζ ′C . This canonical action is the cotangent lift of ζC to T∗C. By the
theorem on momentum maps for lifted actions (Marsden & Ratiu 1999), it follows that
the infinitesimal generator is Hamiltonian with an associated infinitesimally equivariant
momentum map JL : T ∗C → so(3)∗. We recall that the infinitesimal equivariance
property of JL is
〈JL(αQ(S)), ζ〉 = 〈P(S), ζM (Λ(S))〉 = 〈Λ � P, ζ〉, (B.14)
where Λ(S) belongs to M = SO(3) ⊂ C, P(S) belongs to T ∗ΛM and αQ(S) belongs
to T ∗Q(S)C. This property implies that the momentum map can be expressed in the
standard form
JL = Λ � P (B.15)
which is the statement that JL is a left infinitesimally equivariant momentum map
for the cotangent lifted left action ΨL on T ∗C. We now verify that the computed
expression for the momentum map is equivalent to the standard form, by pairing the
Clebsch relation for the body angular momentum δ`δΩ with its dual ζ ∈ so(3)
∗ to give
〈skew(Λ(S)PT (S)
), ζ〉 =
∫
B−Tr
(skew(Λ(S)PT (S))ζ
)dS
=
∫
B−Tr(PT (S)ζΛ(S)) dS
=
∫
BTr(PT (S)LζΛ(S)
)dS
= 〈Λ � P, ζ〉,
(B.16)
which shows that the computed and standard form for JL are equivalent.
201
The convective representation of the Lagrange-Poincare equations
The final step, given the form of the momentum map, is to derive the SO(3) reduced
equations of motion which are expressed relative to the rotating frames. Taking deriv-
atives with respect to time of δ`δV
∂
∂t(δ`
δV) = −Ω
δ`
δV− ΛTJ ′, (B.17)
and substituting J ′ = Λ′ δ`δΓ + Λ(δ`δΓ
)′into the above equation gives
∂
∂t(δ`
δV) = −Ω
δ`
δV− Ω
δ`
δΓ−
(δ`
δΓ
)′. (B.18)
Repeating this for δ`δΩ gives
∂
∂t
δ`
δΩ= skew(−ΩΛTP)︸ ︷︷ ︸
−ad∗Ωδ`δΩ
−skew(ΛT (J ′ + J Ω + P ⊗ V + J ⊗ (Γ + E3)
). (B.19)
Substituting(δ`
δΩ
)′= skew(ΩΛTJ ) + skew(ΛTJ ′) into the above equation then gives
∂
∂t
δ`
δΩ= −ad∗Ω
δ`
δΩ−
(δ`
δΩ
)′+ skew(ΩΛTJ − ΛTJ Ω)︸ ︷︷ ︸
ad∗Ω
δ`
δΩ
− skew
(δ`
δV⊗ V
)
︸ ︷︷ ︸=0
−skew
(δ`
δΓ⊗ (Γ + E3)
)
.
(B.20)
To summarise, the Euler-Lagrange equation for the material linear momentum in
the rotating frame is
∂
∂t
(δ`
δV
)
= −(Ω)ˆ ×δ`
δV− (Ω)ˆ ×N −N. (B.21)
Note that this equation is just a convenient form of the canonical Euler-Lagrange
equation
δ`
δφ=
d
dt
δ`
δφ+
d
dS
δ`
δφ′. (B.22)
The Lagrange-Poincare equation for the convective angular momentum is
∂
∂t
(δ`
δΩ
)
= −ad∗Ωδ`
δΩ−M ′ + ad∗
ΩM − skew (N ⊗ (Γ + E3)) , (B.23)
202
where M := δ`
δΩand N := δ`
δΓ . These equations are consistent with those presented
in (Krishnaprasad et al. 1988) and (Simo & Vu-Quoc 1988). The last term in the
convective angular momentum equation couples the director motion with the position
of the rod centroid and represents the shearing component of the motion. Indeed, when
the tangent vector of the rod centroid is aligned with the unit vector d3, Γ is zero and
the last term vanishes.
203
Appendix C
Additional aspects of the
Variational Free-Lagrange
Method
We outline two aspects of the VFL method which, though not essential to the material
presented in Chapter 4, respectively enhance the geometric description of the method
and applicability to computational shallow water models.
C.1 Canonical Formulation of the Variational Free-Lagrange
Method
This Section provides an alternative proof that the VFL method conserves energy by
formulating the corresponding semi-discrete canonical Hamiltonian equations on mo-
mentum phase space. Consider the smooth and invertible Legendre transformation from
the Lagrangian defined on the tangent bundle(velocity phase space) to the Hamiltonian
H defined on the cotangent bundle (momentum phase space)
H = 〈Pα, Xα〉 − L(Xα, Xα), (C.1)
where the pairing between elements of the tangent space and its dual 〈, 〉 is given by
〈Pα, Xα〉 =∑
α
Pα ∙ Xα, (C.2)
and Pα =∂L∂Xα. The Hamiltonian takes the form
H =∑
α
1
mα|Pα(t)|
2 + 2Rα ∙Pα − g∑
α
mα
(D(Xα, t)− 2bα
), (C.3)
204
and the canonical Hamiltonian particle equations are
Pα = −∂H∂Xα
,
Xα =∂H∂Pα
.
(C.4)
Evaluating the weak form of the derivatives of the Hamiltonian, analogously to the
derivation of the semi-discrete Euler-Lagrange equations given in Section 4.4, gives the
canonical Hamiltonian free-Lagrange equations
Pα = −gmαgrad(h(Xα, t)
)+ f0k ×Pα,
Xα =Pαmα= Uα.
(C.5)
By construction, these canonical Hamiltonian particle equations are dual to the Euler-
Lagrange equations defined in equation 4.25. Further, conservation of mass in each
cell surrounding a particle implies that the canonical particle equations, on momentum
phase space, are simply a constant scalar multiple of the semi-discrete Euler-Lagrange
equations on velocity phase space as given by equation 4.25.
Expressing the change in the Hamiltonian in terms of the change in momentum
space coordinatesDHDt=∑
α
∂H∂Xα
∙DXαDt
+∂H∂Pα
∙DPαDt
, (C.6)
it follows that particle trajectories on momentum phase space , described by the dual
of the Euler-Lagrange particle equations conserve the Hamiltonian
DHDt=∑
α
∂H∂Xα
∙∂H∂Pα
−∂H∂Pα
∙∂H∂Xα
= 0. (C.7)
This approach assumes that the Legendre transformation is smooth and invertible.
C.2 Representation on a Fixed Grid
It is often convenient to compute the velocity and layer depth fields on a fixed grid.
This Section briefly describes how this is performed in a locally mass conservative way
via a process referred to as ’rezoning’. The Eulerian (spatial) velocity field and layer
depth field are respectively computed over a fixed grid by interpolating the particle
velocities to a velocity grid and taking the average of the layer-depth in a fixed grid
cell from the weighted contribution of each Lagrangian cell intersecting the fixed cell.
The fixed grid cell-average layer depth is
205
Di,j(t) =1
Ai,j
∑
α
∫
Vα(t)⋂Vi,j
D(x, t)dA, (C.8)
where Vi,j denotes the Eulerian grid cell with index i, j and Ai,j denotes its area.
The particle velocities at time tn are interpolated to the velocity grid according to
uni,j =∑
α
Unαφi,j(Xnα). (C.9)
206