A first order conservation law framework

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

SWANSEAUNIVERSITYCollegeofEngineeringProf.J.Bonet
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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SWANSEAUNIVERSITYSchoolofEngineeringProf.J.Bonet
 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 1
2 2 2
1
2
1
1
; ;
n n nn n n
a 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 a
e e e
N dV N dv N dVT P E f
0
( , )
( , ,...)
:
p
tx X
v x
a v x
F
P F F
F
v

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
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0
DIV
t
v
P f
0 0 0
0 ;
d
dV dV dA
dt
v f t t PN
( ,...)
( ,...)
F
P P F
F

 Large strain polyconvex strain energy functions satisfy (Ball):
 Neo-Hookean (or 2-D):
 Mooney-Rivlin:
 Nearly incompressible forms can be derived using isochoric
components of F and H (Schroder et al.)
Polyconvex elasticity
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convex( ,...) ( , , ,...);W J W
d d
d d
dv JdV
F F H
x F X
a H A
1
2
( , ) : ( )NHW J f JF F F
( , , ) : : ( )MRW J f JF H F F H H

Geometric Conservation Laws
 Conservation of deformation gradient:
 Conservation of Jacobian:
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0 0
0 0
0 dV d
d
dV d
dt
F F A
F v A
0 0
: ( )
d d
dv d J dV d
dt dt
v a H v A
( )DIV
t
F
v I 0
( ) 0TJ
DIV
t
H v

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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T
JH F
1
2
1
2iI ijk IJK jJ kK ijk IJK j kK
J
H F F x F
X
0 0
0iI ijk IJK j kK
J
iI ijk IJK jJ k K
H v F
X
d
H dV F v dA
dt

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
[ ( )] iK
iI IJK
J
A
CURL
X
A
0 0
( )
( )
d
dV d
dt
CURL
t
H F v A
H
v F 0
x
x

Conservation laws for solid dynamics
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 The complete set of conservation laws is:
 With involutions:
 And constitutive model
0
( )
( )
( )
0T
DIV
t
DIV
t
J
DIV
CURL
t
t
H
v
P f
F
v I
0
0
H v
v Fx
;CURL DIVF 0 H 0
( , , ,...)JP P HF

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
I
I
I
IJ
v PE
F v E
H F v E
E
H v E
x
;
0
I
It X
f
0
0
0

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
1
02
( , , ); : ( )I IS E W Jv v F H P v E
J
S F
H
v

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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; ; J
W W W
JF H
F 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

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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1
2
; TI
I I I
I
S
t X
0 0
0
0
0
1
2
1
: 0
1
( )
2
J
J
CUDIV
t
t
f J
RL
t
t
F
H
H
F F
F v 0
v
H f
v 0
H v
x

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
H
xX
H
yX
H H
xX yX
H
xY
H
yY
H H
xY yY
1
yY yX
xY xX
F F
F F
F
H
F

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-GALERKIN
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st st
0
0;
T
T T I
I
dV
X
0 0 0
0
T T T
I I
T
I
I
I
dV dV N dA
t
dV
X
Variational
Multiscale
stabilisation

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 a
b
ab b a B a
b
ab
ab b a B
B a
a
b
b a
b
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 a
a a a a
a a
a 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

Stabilised Conservation Variables
 The stabilised conservation variables are:
 Typically
 In practice:
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st
st
st
0
0
0
0
( )
( )
( )
( )
( ) )
(det )[ ( ) ]T
J
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
2
e
p
h
U
0; ; 0.05 0.1
2
e
J J
t
v F H F H

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
1
0 0
T
I I
I I
dV
X t X
1/2
2 2
1 1 1
0 0 0
2 X Y
h
1
; ;
( ) ( )
B B
B J J
B
F Fv v N N
I HN HN t PN

Edge Based Vertex Centred Upwind FV
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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 1
1 22 2
n n n
n n n
n n n
t
t
min
max
n
h
t CFL
U
1 1( )
2n n n n
t
x x v v

2D SWINGING PLATE: MESH CONVERGENCE
25
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Velocity @ t =
0.012s
Stress @ t = 0.012s
Analytical solution of the form
𝒖 = 𝑈0 cos
𝑐 𝑑 𝜋𝑡
2
sin
𝜋𝑋1
2
cos
𝜋𝑋2
2
− cos
𝜋𝑋1
2
sin
𝜋𝑋2
2
; 𝑐 𝑑 =
𝜇
𝜌0
Problem description: Unit square plate, 𝜌0 = 1.1 × 103 𝑘𝑔/𝑚3, 𝐸 = 1.7 × 107 𝑃𝑎, 𝜈 = 0.45, 𝛼 𝐶𝐹𝐿 = 0.4, 𝑈0 = 5 ×
10−4, 𝜏 𝐹 = 0.5 Δ𝑡, 𝜏 𝑝 = 𝜁 𝐹 = 0, lumped mass matrix

2D TENSILE PLATE II
26
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Avoid spurious pressure modes in near incompressibility limit
Effectiveness of PG formulation using linear triangular
mesh

27
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2D TENSILE PLATE I
27
Conservation and entropy formulations yield practically identical
results
Problem description: Unit square plate @ 𝑡 = 0.001𝑠, 𝜌0 = 7𝑀𝑔/𝑚3, 𝐸 = 21 𝐺𝑃𝑎, 𝜈 = 0.3, 𝛼 𝐶𝐹𝐿 ≈ 0.3, 𝑉𝑝𝑢𝑙𝑙 =
500𝑚/𝑠, PG stabilisation: 𝜏 𝐹 = Δ𝑡, 𝜁 𝐹 = 0.1, 𝜏 𝑝 = 0, lumped mass matrix

2D COLUMN I
28
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Examine the bending capability of the PG formulation
 Given initial constant velocity:
𝑉0 = 10 𝑚/𝑠
 Experiences bending locking
phenomena

2D COLUMN II
29
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Performance of the PG in bending dominated scenario
Problem description: Column 1 × 6 @ 𝑡 = 0.45𝑠, 𝜌0 = 1.1 × 103 𝑘𝑔/𝑚3, 𝐸 = 1.7 × 107 𝑃𝑎, 𝜈 = 0.45, 𝛼 𝐶𝐹𝐿 ≈ 0.3,
𝑉0= 10 𝑚/𝑠, 𝜏 𝐹= 0.5 Δ𝑡, 𝜁 𝐹 = 0.05, 𝜏 𝑝 = 0, lumped mass matrix

2D COLUM III
30
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Effectiveness with only 1 element across the thickness
Pressure contour plot

3D BENDING COLUMN I
31
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Hu-Washizu type
variational
formulation
JST p-F vertex
centred FVM
Stabilised 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, 0 𝑇 where 𝑉0 = 10 𝑚/𝑠, compressible Mooney-Rivlin model (𝛼 = 𝛽 =
𝜇
4
)

3D SWINGING PLATE I
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Analytical solution of the form
𝒖 = 𝑈0 cos
3
2
𝑐 𝑑 𝜋𝑡
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
; 𝑐 𝑑 =
𝜇
𝜌0
Problem description: Unit solid cube, 𝜌0 = 1.1 × 103 𝑘𝑔/𝑚3, 𝐸 = 1.7 × 107 𝑃𝑎, 𝜈 = 0.45, 𝛼 𝐶𝐹𝐿 = 0.3, 𝑈0 = 5 ×
10−4, 𝜏 𝐹 = 0.5 Δ𝑡, 𝜏 𝑝 = 𝛼 = 0, 𝐴 = 𝐵 = 1, 𝐶 = −2 , lumped mass matrix
Stresses @ t = 0.002s Velocity @ t = 0.002s

33
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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, 0 𝑇 where 𝑉0 = 10 𝑚/𝑠, compressible Mooney-Rivlin model (𝛂 = 𝛃 =
𝛍
𝟒
)

34
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3D TWISTING COLUMN IV
34
Square Hollow Section
Assess the robustness of the proposed PG formulation

3D L-SHAPED BLOCK I
35
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Initial Condition
Components of the angular momentum evolution

3D L-SHAPED BLOCK III
36
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Ability of the algorithm to preserve angular momentum
Pressure distribution of a L-shaped block

3D L-SHAPED BLOCK II
37
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Problem description: L-shaped block, 𝜌0 = 1 𝑀𝑔/𝑚3, 𝐸 = 50046 𝑃𝑎, 𝜈 = 0.3, compressible Neo-Hookean model
(𝛼 =
𝜇
2
, 𝛽 = 0), 𝛼 𝐶𝐹𝐿 = 0.3, lumped mass contribution
Norm of the velocity distribution 𝐯
Stabilised p-F-H-J PG formulation
Stabilised p-F-J PG formulation

3D TENSILE CUBE I
38
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Problem description: Unit cube, 𝜌0 = 7 𝑀𝑔/𝑚3, 𝐸 = 21 𝑃𝑎, 𝜈 = 0.3, compressible Neo-Hookean model (𝛼 =
𝜇
2
Pressure distribution @ t = 0.0016s

3D TENSILE CUBE III
39
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Avoid the appearance of spurious pressure modes
Pressure distribution of a tensile cube

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
Time integrated stabilisation 𝜻 𝑱 = 𝟎.5
𝝁
𝜿

3D TAYLOR IMPACT BAR I
41
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Initial radius 𝑟0 = 0.0032 𝑚 and length 𝐿0 =
0.0324 𝑚
Compressible and nearly incompressible NH model
Young’s modulus 𝐸 = 117 𝐺𝑃𝑎
Density 𝜌0 = 8930 𝑘𝑔/𝑚3
Velocity 𝑉0 = 1000 𝑚/𝑠

3D TAYLOR IMPACT BAR II
42
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3D TAYLOR IMPACT BAR II
42Compressible and nearly Incompressible NH models
Pressure distribution of an Impact bar

1D CABLE: SHOCK CAPTURING SCHEME
43
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Problem description: 𝐿 = 10m , 𝜌0 = 8 𝑀𝑔/𝑚3 , 𝐸 = 200 𝐺𝑃𝑎 , 𝜈 = 0 , 𝛼 𝐶𝐹𝐿 = 0.3 , 𝑃 𝐿, t = −5 × 107 𝑁 ,
𝜏 𝐹= 0.5 Δ𝑡, 𝜏 𝑝 = 𝜁 𝐹 = 0, lumped mass contribution
Velocity @ mid-bar
Stress @ mid-bar

SOD’s Shock Test
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44

SOD’s Test solution (Acustic Riemann Solver)
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45

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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

ACKOWLDEGEMENTS
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47

Publications
WCCM-BARCELONA-2014
48

A first order conservation law framework

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