Development of a low order stabilised Petrov-Galerkin formulation for a mixed conservation law formulation in fast dynamics

Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions

Development of a low-order stabilised Petrov-Galerkinformulation for a mixed conservation law formulation in fast

dynamics

Chun Hean Lee, Antonio J. Gil, Javier Bonet, Miquel Aguirre

Civil and Computational Engineering Centre (C2EC)College of Engineering, Swansea University, UK

12th U.S. National Congress on Computational Mechanics (USNCCM12)

1 http://swansea.academia.edu/ChunHeanLee

CHL-AJG-JB-MA (Raleigh, North Carolina) 22nd - 25th July 2013

http://swansea.academia.edu/ChunHeanLee


Outline

1 Motivation

2 Reversible elastodynamicsBalance principlesConservation laws

3 Petrov-Galerkin formulationPetrov-Galerkin spatial discretisationPerturbed test function spaceTemporal discretisationAngular momentum preserving scheme

4 Numerical results

5 Conclusions and further research



Outline

1 Motivation



4 Numerical results




Motivation

• Standard solid dynamic formulations:

× Linear tetrahedral elements behave poorly in nearly incompressibleand bending dominated scenarios

× Uniform and selective reduced integrated linear hexahedral elementssuffer from respected hourglassing and pressure instabilities

× Convergence of stresses and strains is only first order

× Shock capturing technologies are poorly developed

X Time integrators are robust and preserve angular momentum

X Extensive availability of commercial packages (ANSYS, AltairHyperWorks, LS-DYNA, . . .)

• Mixed conservation law formulation:

X Express as first order conservation laws enabling the use ofstandard CFD discretisation process

X Permits the use of linear tetrahedra, as well as enhanced linearhexahedra, for solid dynamics without locking difficulties

X Achieves optimal convergence with equal orders in velocities andstresses

X Take advantage of the conservative formulation to introducestate-of-the-art discontinuity-capturing operator

× Enhance existing time integrators to preserve angular momentum



Outline

1 Motivation



4 Numerical results




Balance principles

First order conservation formulation

• Consider the dynamic equilibrium equation:

∂ρ0v∂t−∇0 · P(F ) = ρ0b

• Which represents the conservation of linear momentum

ddt

∫V0

ρ0v dV0 =

∫V0

ρ0b dV0 +

∫∂V0

t dA; t = PN

• A mixed formulation can be developed by writing a conservation equation for thedeformation gradient:

∂F∂t

= ∇0v =⇒ddt

∫V0

F dV0 =

∫∂V0

v ⊗ N dA

∂F∂t−∇0 · (v ⊗ I) = 0

Constitutive model is needed to complete the coupled system



Governing equation

Conservation laws

• The mixed equations can be written as a system of first order conservation laws:

∂

∂t

[ρ0vF

]+ ∇0 ·

[−P(F )−v ⊗ I

]=

[ρ0b

0

]• More generally, if the energy equation is added:

∂

∂t

ρ0vFE

+ ∇0 ·

−P(F )−v ⊗ I

Q − PT v

=

ρ0b0s

• Or in standard form:

∂U∂t

+∇0·F(U) = S; U =

ρ0vFE

; F =

−P(F )−v ⊗ I

Q − PT v

; S =

ρ0b0s

Our aim is to develop a library of low order numerical schemes for a mixedconservation law formulation of fast solid dynamics using existing CFD technologies



Governing equation

CFD formulations for fast solid dynamics

• Following are the stabilised numerical methodologies recently developed for fastsolid dynamics using mixed formulation:

Swansea University Research Group (Lead by Prof. Javier Bonet and Dr.Antonio J. Gil)

· Two-Step Taylor-Galerkin (2TG) Formulation [Karim, Lee, Gil and Bonet,2011]

· Total Variation Diminishing (TVD) Upwind Cell Centred Finite VolumeMethod (CCFVM) [Lee, Gil and Bonet, 2012]

· Jameson-Schmidt-Turkel (JST) Vertex Centred Finite Volume Method(VCFVM) [Aguirre, Gil, Bonet and Carreño, 2013]

· Stabilised Petrov-Galerkin (PG) Finite Element Method [Lee, Gil andBonet, 2013]

M.I.T Research Group (Lead by Prof. Jaime Peraire)

· Hybridizable Discontinuous Galerkin (HDG) Method [Nguyen and Peraire,2012]



Outline

1 Motivation



4 Numerical results




Petrov-Galerkin spatial discretisation

Stabilised Petrov-Galerkin formulation

• Variational statement of Bubnov-Galerkin formulation (unstable):∫V0

δV ·R dV = 0; R =∂U∂t

+ ∇0 ·F − S; δV =

[δvδP

]• Integration by parts gives:

∫V0

δV ·∂U∂t

dV =

∫V0

F : ∇0δV dV −∫∂V0

δV ·FN dA +

∫V0

δV · S dV

• Define stabilised Petrov-Galerkin (PG) formulation that satisfy Second Law ofThermodynamics:

∫V0

δVst ·R dV = 0; δVst =

[δvst

δPst

]



Perturbed test function space

Perturbation

• Stabilised test function space is generally defined by

δVst = δV + τT(∂F I

∂U

)T ∂δV∂XI︸︷︷︸

Perturbation

• Define flux Jacobian matrix:

∂F I

∂U=

(− ∂P I∂(ρ0v)

− ∂P I∂F

− 1ρ0

∂(v⊗I I )∂v − ∂(v⊗I I )

∂F

)=

(03×3 −CI− 1ρ0

I I 09×9

)• Assuming τ (intrinsic time scale) a diagonal matrix for simplicity:

δVst :=

[δvst

δPst

]=

[δv − τp

ρ0∇0 · δP

δP − τFC : ∇0δv

]; δP = C : δF

• Bubnov-Galerkin is recovered by setting stabilisation τ = 0




Petrov-Galerkin stabilisation

• Weak statement of stabilised Petrov-Galerkin (PG) formulation:

0 =

∫V0

[δV +

(∂F∂U

τ

)T∇0δV

]·R dV

=

∫V0

δV ·R dV︸︷︷︸Bubnov-Galerkin

+

∫V0

[∂F∂U

τR]

: ∇0δV dV︸︷︷︸Petrov-Galerkin stabilisation

• Integration by parts gives:

∫V0

δV·∂U∂t

dV =

∫V0

[F −

∂F∂U

τR]

︸︷︷︸Fst

: ∇0δV dV−∫∂V0

δV·FN dA+

∫V0

δV·S dV

• The stabilised flux Fst can be more generally defined as (equivalent toVariational Multi-Scale (VMS) stabilisation):

Fst = F(Ust ); Ust = U + U ′; U ′ = −τR




Finite element discretisation

• Using standard linear finite element interpolation for velocity and deformationgradient renders:

v =∑

avaNa; F =

∑a

F aNa

∑b

Mabpb =

∫∂V

NatB dA +

∫V0

Naρ0b dV −∫

V0

P(F st )∇0Na dV

∑b

MabF b =

∫∂V

Na(vB ⊗ N) dA−∫

V0

vst ⊗∇0Na dV

• By construction the stabilised deformation gradient and velocity are:

F st = F + τF

(∇0v − F

)vst = v +

τp

ρ0(∇0 · P + ρ0b − p)

• To prevent zero-energy modes an additional residual-based artificial diffusionassociated with non-physical compatibility mechanisms can be addedCHL-AJG-JB-MA (Raleigh, North Carolina) 22nd - 25th July 2013


Explicit time marching scheme

Time Integration

• Integration in time is achieved by means of an explicit two-stage Total VariationDiminishing Runge-Kutta (TVD-RK) time integrator:

U (1)n+1 = Un + ∆tUn

U (2)n+2 = U (1)

n+1 + ∆tU (1)n+1

Un+1 =12

(Un + U (2)

n+2

)

together with a stability constraint

∆t = αCFLhmin

Unmax

; Unmax = max

a

(Un

p,a)



Conservation of angular momentum

Lagrange multiplier correction procedure

• The scheme presented thus far does not necessarily preserve total angularmomentum of a system

• Introduce a discrete functional Π assuming homogeneous Neumann boundaryconditions defined by [Lee, Gil, Bonet, 2013]

Π(pC ,λ) =

12

∑a,b

Mab

(pC

a − pa

)·(

pCb − pb

)︸︷︷︸

Objective function

−λ ·∑a,b

Mab

(xa × pC

b

)︸︷︷︸

Constraint

• The rate of linear momentum in general can be corrected:

pCa = pa +λ×xa; λ =

∑a,b

Mab

(xa · xb) I −xb ⊗ xa︸︷︷︸vanish in 2D

−1∑

a,b

Mab (pb × xa)

• Such additional operation can be computed efficiently using diagonal (or lumped)mass contribution



Outline

1 Motivation



4 Numerical results




1D Cable

1D mesh convergence

· Problem description: L = 10m, ρ0 = 1Kg/m3, E = 1Pa, ν = 0, αCFL = 0.5, P = 1 × 10−3EXP(−0.1(t − 13)2)N,τF = 0.5∆t , τp = α = 0

2

1

2

1

� 1D convergence analysis by means of the L2 norm has been carried out at t = 40s

� Demonstrates the expected accuracy of the available schemes for all variables

� The use of both slope limiter and lumped mass matrix maintains the expected order ofconvergence



2D Swinging plate

2D mesh convergence· Problem description: Unit square plate, ρ0 = 1.1Mg/m3, E = 0.017GPa, ν = 0.45, αCFL = 0.4, U0 = 5 × 10−4,τF = 0.5∆t, τp = α = 0

1X

2X

Analytical solution of the form:

u(t) = U0cos

(cdπt√

2

) sin(πX1

2

)cos(πX2

2

)−cos

(πX1

2

)sin(πX2

2

) ; cd =

√µ

ρ0

10-1

100

10-5

10-4

10-3

10-2

Grid Size

L2 Norm Error

Velocity errors at t=0.012s

Horizontal Velocity

Vertical Velocity

2

1

10-1

100

101

102

103

104

Grid Size

L2 Norm Error

Stress errors at t=0.012s

P11

P22

2

1



2D Short column

Bending dominated problem· Problem description:Column 1 × 6 at t = 0.45s, ρ0 = 1.1Mg/m3, E = 0.017GPa, ν = 0.45, αCFL = 0.4, V0 = 10m/s,τF = 0.5∆t , α = 0.05, τp = 0, consistent mass contribution

Performance of the Petrov-Galerkin formulation in bending dominated scenario

[movie]

Standard 3-Node Tri. 4-Node Quad. (P1-P0) Petrov-Galerkin formulation

C. H. Lee, A. J. Gil, J. Bonet. Development of a stabilised low-order finite element methodology for a mixed conservationlaw formulation in Lagrangian fast solid dynamics, CMAME. Under Review.



3D Low dispersion cube

Problem statement

Problem description: Unit side cube, linear elasticity,ρ0 = 1.1Mg/m3, E = 1.7× 107Pa, ν = 0.45, αCFL = 0.3,τF = ∆t , α = 0.1, τp = 0.2∆t , lumped mass contribution

• Given analytical solution of the form

u = U0 cos

(√3

2cdπt

)A sin

(πX1

2

)cos(πX2

2

)cos(πX3

2

)B cos

(πX1

2

)sin(πX2

2

)cos(πX3

2

)C cos

(πX1

2

)cos(πX2

2

)sin(πX3

2

)

• Impose IC at t=0, F 0(X) = F (X , t = 0)

• Symmetric BC at X1 = 0, X2 = 0 and X3 = 0

• Skew symmetric BC at X1 = 1, X2 = 1 and X3 = 1

• Problem parameters: A = 2, B = −1, C = −1,U0 = 5× 10−4, cd =

√µρ0

[MOVIE] Solution plotted with displacements scaled 200 times

Symmetric BC

1X 2X

3X

T(1,1,1)= 03X

= 02X= 01X

Skew symmetric BC

1X 2X

3X

T(1,1,1)= 13X

= 12X= 11X



3D Low dispersion cube

3D Mesh convergence

· Problem description: Unit side cube, linear elasticity, ρ0 = 1.1Mg/m3, E = 1.7 × 107Pa, ν = 0.45, αCFL = 0.3,

τF = ∆t , α = 0.1, τp = 0.2∆t , A = B = 1, C = −2, U0 = 5 × 10−4, lumped mass contribution

2

1

2

1

� 3D convergence analysis by means of the L2 norm has been carried out at t = 0.002s

� Demonstrates the expected accuracy of the stabilised Petrov-Galerkin formulation usinglumped mass matrix



3D L-shaped block

Angular momentum preserving example

· Problem description: L-shaped block, ρ0 = 1000Kg/m3, E = 5.005 × 104Pa, ν = 0.3, αCFL = 0.3, τF = ∆t , α = 0.1,τp = 0, lumped mass contribution

1X

2X

3X

T(3,3,3)

T(0,10,3)

T(6,0,0)

)t(1F

)t(2F

J. C. Simo, N. Tarnow, K. K. Wong. Exact energy-momentumconserving algorithms and symplectic schemes for nonlinear

dynamics, CMAME 63-116 (1992)

• Imposed external forces at faces {X1 = 6,X2 = 10} described as

· F1(t) = − F2(t) = η(t) (150, 300, 450)T

η(t) =

t, 0 ≤ t < 2.55 − t, 2.5 ≤ t < 50, t ≥ 5

• Free BC at all sides

• Suitable for long term dynamic response

· Angular Momentum· Total energy (summation of kinetic and

potential energies)

[MOVIE]

� Study the conservation properties of the proposed formulation



3D Taylor impact bar

Classical benchmark impact problem

· Problem description: Copper bar, L0 = 3.24 cm, r0 = 0.32 cm, v0 = (0, 0,−227) m/s. von Mises hyperelastic-plastic

material with ρ0 = 8930Kg/m3, E = 117GPa, ν = 0.35, τ0y = 0.4GPa, H = 0.1GPa αCFL = 0.4, 1361 nodes, lumped

mass contribution

0V

= 03X

0L

0r

Radius and length (in cm) at t = 80µs

Methods Radius LengthStandard 4-Node Tet. 0.555 -8-Node Hex. (P1/P0) 0.695 2.1484-Node ANP Tet. (P1/P1-projection) 0.699 -4-Node Mixed Tet. (P1/P1-stabilised) 0.700 2.156

J. Bonet, A. Burton. A simple average nodal pressure tetrahedral element forincompressible and nearly incompressible dynamic explicit applications, COMMUN

NUMER METH EN 14, 437-449 (1998)

O. C. Zienkiewicz, J. Rojek, R. L. Taylor, M Pastor. Triangles and tetrahedra in explicitdynamic codes for solids, INT J NUMER METH ENG 43, 565-583 (1998)

[MOVIE]

� Assess the performance within the context of contact/impact mechanics



Outline

1 Motivation



4 Numerical results




Conclusions and further research

Conclusions

• A stabilised Petrov-Galerkin formulation is presented for the numerical simulations of fastdynamics in large deformations

• Linear tetrahedral elements can be used without usual volumetric and bending difficulties

• Velocities (or displacements) and stresses display the same rate of convergence

On-going works

• Investigate alternative angular momentum preserving time integrator

• Standard CFD techniques for discontinuity capturing operator can be incorporated

• Combine the Geometric Conservation Law (GCL) into a mixed formulation to effectively solvenearly (and fully) incompressible media.

• Sophisticated constitutive models (Mie-Gruneisen) can be employed by relating internalenergy and temperature to the first Piola-Kirchhoff stress

Journal publications· C. H. Lee, A. J. Gil and J. Bonet. Development of a cell centred upwind finite volume algorithm for a new conservation law

formulation in structural dynamics, Computers and Structures 118 (2013) 13-38.

· I. A. Karim, C. H. Lee, A. J. Gil and J. Bonet. A Two-Step Taylor Galerkin formulation for fast dynamics, Engineering Computations,2013. In Press.

· C. H. Lee, A. J. Gil and J. Bonet. Development of a stabilised Petrov-Galerkin formulation for a mixed conservation law formulationin fast solid dynamics, Computer Methods in Applied Mechanics and Engineering, 2013. Under Review.

· M. Aguirre, A. J. Gil, J. Bonet and A. Arranz Carreño. A vertex centred Finite Volume Jameson-Schmidt-Turkel (JST) algorithm for amixed conservation formulation in solid dynamics, Journal of Computational Physics, 2013. Submitted.


Documents

Development of a low order stabilised Petrov-Galerkin formulation for a mixed conservation law formulation in fast dynamics