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WCCM - BARCELONA - 2014
A first order conservation law framework for
solid dynamics
J. BONET
College of Engineering
Collaborators: A.J. Gil, C.H. Lee, M. Aguirre, R. Ortigosa
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OUTLINE
Motivation
Standard FE Formulations
Aims
Conservation laws
Momentum and energy
Geometric conservation laws
Polyconvex constitutive models
Entropy variables
Conjugate stresses
Conservation laws in symmetric form
Discretisation
Possible CFD techniques
SUPG
Examples and validation
Concussions
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MOTIVATION – Fast Transient Dynamics
Computational Solid Dynamics is a well established and mature
subject and there is extensive software available.
Standard FE formulation based on
Explicit codes
Hexahedral elements
Updated Lagrangian formulations
Co-rotational stress updates
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Consider the motion of a discretised solid:
Equilibrium can be defined by :
Time integration:
0 a a a aDIV ma P f a E T
MOTIVATION – Standard Solid Formulation
a a
01 1,X x
3 3,X x
2 2,X x
4
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1 1 12 2 2
12
1
1
; ;n n nn n na a a a a a
n n n
t t
t
v v a x x v
0 0
0 ;e e e
a a a a ae e e
N dV N dv N dVT P E f
0
( , )
( , ,...)
:
p
tx X
v x
a v x
F
P F FF
v
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Motivation
Standard solid dynamic formulations are have a number of
difficulties:
Linear tetrahedral elements behave poorly in incompressible and
bending dominated problems – ad hoc solutions using nodal
elements are available (Bonet, Dohrman, Gee, Scovazzi,…)
Under integrated hexahedral elements suffer from hourglass
modes
Convergence of stresses and strains is only first order
Shock capturing technologies are poorly developed
In contrast in the CFD community:
Many robust techniques are available for linear triangles and
tetrahedra
Convergence of pressure and velocity at same rate
Robust shock capturing
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Aims
To derive a mixed formulation for Lagragian solid dynamics
as a set of first order conservation laws so as to permit the
use of CFD technology
To obtain the convex entropy extension, the set of
conjugate entropy variables to conservation variables and
the symmetric form of the conservation laws
To explore several CFD discretisation techniques applied to
Lagrangian Solid Dynamics in conservation form
To assess the advantages and disadvantages of the
proposed conservation formulation
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Consider the conservation of linear momentum:
In differential form:
Constitutive model:
However, energy function is not convex
Conservation of momentum W
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0 DIVt
vP f
0 0 0
0 ;d
dV dV dAdt
v f t t PN
( ,...)( ,...)
FP P F
F
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Neo-Hookean (or 2-D):
Mooney-Rivlin:
Nearly incompressible forms can be derived using isochoric
components of F and H (Schroder et al.)
Polyconvex elasticityW
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convex( ,...) ( , , ,...);W J W
d d
d d
dv JdV
F F H
x F X
a H A
12
( , ) : ( )NHW J f JF F F
( , , ) : : ( )MRW J f JF H F F H H
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Geometric Conservation Laws
Conservation of deformation gradient:
Conservation of Jacobian:
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0 0
0 0
0 dV d
ddV d
dt
F F A
F v A
0 0
: ( )d ddv d J dV d
dt dtv a H v A
( )DIVt
Fv I 0
( ) 0TJDIV
tH v
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Area Map
The area map tensor is usually evaluated via Nanson’s rule:
The time derivative of this equation does not lead to a useful
conservation law.
Alternative forms using alternating tensor:
Giving conservation laws (Qin 98, Wagner 2008):
Notation highly cumbersome!
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TJH F
12
1
2iI ijk IJK jJ kK ijk IJK j kKJ
H F F x FX
0 0
0iI ijk IJK j kKJ
iI ijk IJK jJ k K
H v FX
dH dV F v dA
dt
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Tensor Cross Product Notation
Define cross product of a vector by a tensor:
Cross product of two tensors:
Curl of a tensor:
With this notation:
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[ ] ; [ ]iI ijk j kK iI IJK iJ Kv A A Vv A A Vx x
[ ]iI ijk IJK jJ kKA BA Bx
[ ( )] iKiI IJK
J
ACURL
XA
0 0
( )
( )
ddV d
dt
CURLt
H F v A
Hv F 0
x
x
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Conservation laws for solid dynamicsW
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The complete set of conservation laws is:
With involutions:
And constitutive model
0
( )
( )
( )
0T
DIVt
DIVt
JDIV
CURLt
t
H
vP f
Fv I
0
0
H v
v Fx
;CURL DIVF 0 H 0
( , , ,...)JP P HF
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System of conservation laws
Using the combined notation:
The system can be expressed in standard form:
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0
1,2,3
1 0 0
; ; 0 , 1 , 0
0 0 1: )
( )
(
I
II
I
IJ
v PE
F v E
H F v EE
H v E
x
;
0
I
It X
f
00
0
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Convex entropy extension
The system has a convex entropy extension function and
associated fluxes such that (Wagner):
Where for non-thermal problems:
Define the set of entropy variables:
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0I
I
S
t X
102
( , , ); : ( )I IS E W Jv v F H P v E
J
S F
H
v
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Conjugate Stresses
The conjugate stresses to geometric conservation variables are:
The relationship between Piola-Kirchhoff and these stresses is:
For Mooney-Rivlin:
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; ; JW W W
JF HF H
: ( )
( , , )
: :
: : ( ) :
:
J
J
J
W J
JF H
F H
F H
P F F
F H
F H
F F F H F
F H F
x
x
JF HP F Hx
2 , 2 , ( )J f JF HF H
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Symmetric System
The system of conservation laws can be written in symmetric
form in terms of entropy variables:
For Mooney-Rivlin material this gives the symmetric system:
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; TII I I
I
S
t X
0 0
0
0
0
1
2
1: 0
1
( )
2
J
J
CUDIVt
t
f J
RL
t
t
F
H
H
F F
F v 0
vH f
v 0
H v
x
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Symmetric Flux Matrix – 2D
In 2-D:
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0 0 1 0 0 0
0 0 0 0 1 0
1 0 0 0 0 0 0
0 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0
0 0 0 1 0 0
0 0 0 0 0 1
0 0 0 0 0 0 0
1 0 0 0 0 0 0
0 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 0 0 0
X
Y
HxXHyX
H HxX yX
HxYHyY
H HxY yY
1
yY yX
xY xX
F F
F F
F
H
F
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CFD Formulations for Solid Dynamics
Given a first order conservation formulation of solid dynamics,
the following discretisation techniques are available:
2 Step Taylor-Galerkin:
I. Karim, C.H. Lee, A.J. Gil & J. Bonet, 2011
Upwind Cell Centred Finite Volume:
C.H. Lee, A.J. Gil & J. Bonet, 2012
Hibridazable Discontinuous Galerkin: Nguyen & Peraire, 2012
Jameson-Schmidt-Turkel Vertex Centred FV:
M. Aguirre. A.J. Gil & J. Bonet, 2013
Petrov-Galerkin, CH Lee, AJ Gil, J Bonet, 2013
Fractional Step Petrov-Galerkin,
AJ Gil, CH Lee, J Bonet & M Aguirre, 2014
Upwind Vertex Centred FV, M. Aguirre. A.J. Gil & J. Bonet, 2014
SUPG Stabilised FE
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Stabilised Petrov-Galerkin
Integrating by parts
STABILISED PETROV-GALERKINW
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st st
0
0;T
T T I
I
dVX
0 0 0
0
T T TI I
TI
II
dV dV N dAt
dVX
Variational
Multiscale
stabilisation
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Petrov-Galerkin Stabilised Discretisation
Using standard linear FE discretisation for conservation
variables and virtual entropy variables:
Gives:
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st st st
st
t
st
s
0 00
0 0
0
0 0
0
0
0
0
0
( , , )
: ( ) )
( ) ( )
: (
t
ab b a B a ab
ab b a B ab
ab
ab b a B
B a
ab
b ab
M N dV N dA J N d
M N
V
M N d N dV
M J N d N d
d N dV
V
v f t P F H
F v A v
H F v A F v
H v A H v
x x
= ...
0 0 ; ; ;
; ; ;
a a a a a a a aa a a a
a aa a a a
a a a
N N N J J N
N N NF F H H
v v F F H H
v v
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Stabilised Conservation Variables
The stabilised conservation variables are:
Typically
In practice:
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st
st
st
0
0
0
0
( )
( )
( )
( )
( ) )
(det )[ ( ) ]TJ
t
J
s DIV
CURL
DIV JJ JJ
F
H x
v
F
H
f P v
v F
v F H
x F
H
v v
F F
H H
H
H
xv
x
2e
p
h
U
0; ; 0.05 0.12e
J J
tv F H F H
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SUPG in Entropy Variables
Symmetric system can be discretised using SUPG in entropy
variables:
Where
And both entropy variables and virtual entropy variables are
interpolated in the same FE space
Boundary conditions can only be enforced in strong form:
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0
10 0
T
I II I
dVX t X
1/22 21 1 10 0 02 X Y
h
1
; ;
( ) ( )
B BB J J
B
F Fv v N N
I HN HN t PN
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Edge Based Vertex Centred Upwind FVW
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TIME INTEGRATION
Integration in time is achieved by means of an explicit Total
Variational Diminishing (TVD) Runge-Kutta scheme:
with a stability constraint:
Fractional time stepping (implicit in pressure component) used
for Incompressible and Nearly incompressible materials
Geometry increment:
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(1)1
(2) (1) (1)2 1 1
(2)1 11 22 2
n n n
n n n
n n n
t
t
min
maxn
ht CFL
U
1 1( )2n n n nt
x x v v
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2D SWINGING PLATE: MESH CONVERGENCE
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XI 2
014Velocity @ t =
0.012s
Stress @ t = 0.012s
Analytical solution of the form
𝒖 = 𝑈0 cos𝑐𝑑𝜋𝑡
2
sin𝜋𝑋12
cos𝜋𝑋22
−cos𝜋𝑋12
sin𝜋𝑋22
; 𝑐𝑑 =𝜇
𝜌0
Problem description: Unit square plate, 𝜌0 = 1.1 × 103𝑘𝑔/𝑚3, 𝐸 = 1.7 × 107𝑃𝑎, 𝜈 = 0.45, 𝛼𝐶𝐹𝐿 = 0.4, 𝑈0 = 5 ×10−4, 𝜏𝐹 = 0.5 Δ𝑡, 𝜏𝑝 = 𝜁𝐹 = 0, lumped mass matrix
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2D TENSILE PLATE II
26
WC
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XI 2
014
Avoid spurious pressure modes in near incompressibility limit
Effectiveness of PG formulation using linear triangular
mesh
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WC
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XI 2
014
2D TENSILE PLATE I
27Conservation and entropy formulations yield practically identical
results
Problem description: Unit square plate @ 𝑡 = 0.001𝑠, 𝜌0 = 7𝑀𝑔/𝑚3, 𝐸 = 21 𝐺𝑃𝑎, 𝜈 = 0.3, 𝛼𝐶𝐹𝐿 ≈ 0.3, 𝑉𝑝𝑢𝑙𝑙 =
500𝑚/𝑠, PG stabilisation: 𝜏𝐹 = Δ𝑡, 𝜁𝐹 = 0.1, 𝜏𝑝 = 0, lumped mass matrix
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2D COLUMN I
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WC
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XI 2
014
Examine the bending capability of the PG formulation
Given initial constant velocity:
𝑉0 = 10 𝑚/𝑠 Experiences bending locking
phenomena
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2D COLUMN II
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WC
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XI 2
014
Performance of the PG in bending dominated scenario
Problem description: Column 1 × 6 @ 𝑡 = 0.45𝑠, 𝜌0 = 1.1 × 103𝑘𝑔/𝑚3, 𝐸 = 1.7 × 107𝑃𝑎, 𝜈 = 0.45, 𝛼𝐶𝐹𝐿 ≈ 0.3,
𝑉0= 10 𝑚/𝑠, 𝜏𝐹= 0.5 Δ𝑡, 𝜁𝐹 = 0.05, 𝜏𝑝 = 0, lumped mass matrix
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2D COLUM III
30
WC
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XI 2
014
Effectiveness with only 1 element across the thickness
Pressure contour plot
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3D BENDING COLUMN I
31
WC
CM
XI 2
014
Hu-Washizu type
variational
formulation
JST p-F vertex
centred FVMStabilised p-F-H-J PG
formulation
Problem description: Bending column 1 ×1×6, 𝜌0 = 1.1 𝑀𝑔/𝑚3, 𝐸 = 0.017𝐺𝑃𝑎, 𝜈 = 0.3, linear variation in velocity
field v0 = 𝑉0 𝑋3/𝐿, 0, 0𝑇 where 𝑉0 = 10 𝑚/𝑠, compressible Mooney-Rivlin model (𝛼 = 𝛽 =
𝜇
4)
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3D SWINGING PLATE I
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WC
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XI 2
014
Analytical solution of the form
𝒖 = 𝑈0 cos3
2𝑐𝑑𝜋𝑡
𝐴 sin𝜋𝑋12
cos𝜋𝑋22
cos𝜋𝑋32
B cos𝜋𝑋12
sin𝜋𝑋22
cos𝜋𝑋32
C cos𝜋𝑋12
cos𝜋𝑋22
sin𝜋𝑋32
; 𝑐𝑑 =𝜇
𝜌0
Problem description: Unit solid cube, 𝜌0 = 1.1 × 103𝑘𝑔/𝑚3, 𝐸 = 1.7 × 107𝑃𝑎, 𝜈 = 0.45, 𝛼𝐶𝐹𝐿 = 0.3, 𝑈0 = 5 ×10−4, 𝜏𝐹 = 0.5 Δ𝑡, 𝜏𝑝 = 𝛼 = 0, 𝐴 = 𝐵 = 1, 𝐶 = −2 , lumped mass matrix
Stresses @ t = 0.002s Velocity @ t = 0.002s
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WC
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XI 2
014
3D TWISTING COLUMN I
33
Problem description: Twisting column 1 ×1×6, 𝜌0 = 1.1 𝑀𝑔/𝑚3, 𝐸 = 0.017𝐺𝑃𝑎, 𝜈 = 0.3, linear variation in velocity
field v0 = 𝑉0 𝑋3/𝐿, 0, 0𝑇 where 𝑉0 = 10 𝑚/𝑠, compressible Mooney-Rivlin model (𝛂 = 𝛃 =
𝛍
𝟒)
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WC
CM
XI 2
014
3D TWISTING COLUMN IV
34
Square Hollow Section
Assess the robustness of the proposed PG formulation
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3D L-SHAPED BLOCK I
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014
Initial Condition
Components of the angular momentum evolution
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3D L-SHAPED BLOCK III
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014
Ability of the algorithm to preserve angular momentum
Pressure distribution of a L-shaped block
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3D L-SHAPED BLOCK II
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014
Problem description: L-shaped block, 𝜌0 = 1 𝑀𝑔/𝑚3, 𝐸 = 50046 𝑃𝑎, 𝜈 = 0.3, compressible Neo-Hookean model
(𝛼 =𝜇
2, 𝛽 = 0), 𝛼𝐶𝐹𝐿 = 0.3, lumped mass contribution
Norm of the velocity distribution 𝐯
Stabilised p-F-H-J PG formulation
Stabilised p-F-J PG formulation
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3D TENSILE CUBE I
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XI 2
014
Problem description: Unit cube, 𝜌0 = 7 𝑀𝑔/𝑚3, 𝐸 = 21 𝑃𝑎, 𝜈 = 0.3, compressible Neo-Hookean model (𝛼 =𝜇
2, 𝛽 = 0), 𝛼𝐶𝐹𝐿 = 0.3, lumped mass contribution
Pressure distribution @ t = 0.0016s
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3D TENSILE CUBE III
39
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014
Avoid the appearance of spurious pressure modes
Pressure distribution of a tensile cube
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40
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3D TENSILE CUBE II
40
Problem description: Unit cube, 𝜌0 = 7 𝑀𝑔/𝑚3, 𝐸 = 21 𝑃𝑎, 𝜈 = 0.3, compressible Neo-Hookean model (𝛼 =𝜇
2, 𝛽 = 0), 𝛼𝐶𝐹𝐿 = 0.3, lumped mass contribution
Time integrated stabilisation 𝜻𝑱 = 𝟎.5 𝝁
𝜿
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3D TAYLOR IMPACT BAR I
41
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014
Initial radius 𝑟0 = 0.0032 𝑚 and length 𝐿0 =0.0324 𝑚Compressible and nearly incompressible NH model
Young’s modulus 𝐸 = 117 𝐺𝑃𝑎
Density 𝜌0 = 8930 𝑘𝑔/𝑚3
Velocity 𝑉0 = 1000 𝑚/𝑠
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3D TAYLOR IMPACT BAR II
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XI 2
014
3D TAYLOR IMPACT BAR II
42Compressible and nearly Incompressible NH models
Pressure distribution of an Impact bar
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1D CABLE: SHOCK CAPTURING SCHEME
43
WC
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XI 2
014
Problem description: 𝐿 = 10m , 𝜌0 = 8 𝑀𝑔/𝑚3 , 𝐸 = 200 𝐺𝑃𝑎 , 𝜈 = 0 , 𝛼𝐶𝐹𝐿 = 0.3 , 𝑃 𝐿, t = −5 × 107𝑁 ,
𝜏𝐹= 0.5 Δ𝑡, 𝜏𝑝 = 𝜁𝐹 = 0, lumped mass contribution
Velocity @ mid-bar
Stress @ mid-bar
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SOD’s Shock TestW
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SOD’s Test solution (Acustic Riemann Solver)W
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WC
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-2014
46
SUMMARY & CONCLUSIONS
A first order conservation formulation can be used to derive
mixed type of solutions in Lagrangian solid dynamics
Equations can be written in conservation or symmetric form
Entropy variables are the velocity and a new set of
conjugate stresses
Linear triangles and tetrahedra can be used without the
usual volumetric and bending difficulties
Standard CFD discretisation techniques can be used
Cell centred Finite volume
SUPG in conservation and entropy variables
Fractional step integration for Incompressible materials
Vertex centred Finite Volume
Only 2-step explicit TVD R-K time integration has been used
Convergence of velocities/displacements and stresses at
equal rates – avoidance of locking
, , JF H
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ACKOWLDEGEMENTSW
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PublicationsW
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