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A first order conservation law framework
1. 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
2. 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
WCCM-BARCELONA-2014
2
3. SWANSEAUNIVERSITYCollegeofEngineeringProf.J.Bonet
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
WCCM-BARCELONA-2014
3
4. 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
WCCM-BARCELONA-2014
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
5. SWANSEAUNIVERSITYCollegeofEngineeringProf.J.Bonet
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
WCCM-BARCELONA-2014
5
6. SWANSEAUNIVERSITYCollegeofEngineeringProf.J.Bonet
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
WCCM-BARCELONA-2014
6
7. SWANSEAUNIVERSITYCollegeofEngineeringProf.J.Bonet
Consider the conservation of linear momentum:
In differential form:
Constitutive model:
However, energy function is not convex
Conservation of momentum
WCCM-BARCELONA-2014
7
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
8. SWANSEAUNIVERSITYCollegeofEngineeringProf.J.Bonet
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
WCCM-BARCELONA-2014
8
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
10. SWANSEAUNIVERSITYCollegeofEngineeringProf.J.Bonet
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!
WCCM-BARCELONA-2014
10
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
11. SWANSEAUNIVERSITYCollegeofEngineeringProf.J.Bonet
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:
WCCM-BARCELONA-2014
11
[ ] ; [ ]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
12. SWANSEAUNIVERSITYCollegeofEngineeringProf.J.Bonet
Conservation laws for solid dynamics
WCCM-BARCELONA-2014
12
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
14. SWANSEAUNIVERSITYCollegeofEngineeringProf.J.Bonet
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:
WCCM-BARCELONA-2014
14
0I
I
S
t X
1
02
( , , ); : ( )I IS E W Jv v F H P v E
J
S F
H
v
15. SWANSEAUNIVERSITYCollegeofEngineeringProf.J.Bonet
Conjugate Stresses
The conjugate stresses to geometric conservation variables are:
The relationship between Piola-Kirchhoff and these stresses is:
For Mooney-Rivlin:
WCCM-BARCELONA-2014
15
; ; 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
16. SWANSEAUNIVERSITYCollegeofEngineeringProf.J.Bonet
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:
WCCM-BARCELONA-2014
16
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
17. SWANSEAUNIVERSITYCollegeofEngineeringProf.J.Bonet
Symmetric Flux Matrix – 2D
In 2-D:
WCCM-BARCELONA-2014
17
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
18. SWANSEAUNIVERSITYCollegeofEngineeringProf.J.Bonet
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
WCCM-BARCELONA-2014
18
20. SWANSEAUNIVERSITYCollegeofEngineeringProf.J.Bonet
Petrov-Galerkin Stabilised Discretisation
Using standard linear FE discretisation for conservation
variables and virtual entropy variables:
Gives:
WCCM-BARCELONA-2014
20
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
21. SWANSEAUNIVERSITYCollegeofEngineeringProf.J.Bonet
Stabilised Conservation Variables
The stabilised conservation variables are:
Typically
In practice:
WCCM-BARCELONA-2014
21
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
22. SWANSEAUNIVERSITYCollegeofEngineeringProf.J.Bonet
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:
WCCM-BARCELONA-2014
22
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
24. SWANSEAUNIVERSITYCollegeofEngineeringProf.J.Bonet
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:
WCCM-BARCELONA-2014
24
(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
25. SWANSEAUNIVERSITYSchoolofEngineeringProf.J.Bonet
2D SWINGING PLATE: MESH CONVERGENCE
25
WCCMXI2014
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
46. SWANSEAUNIVERSITYCollegeofEngineeringProf.J.Bonet
WCCM-BARCELONA-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