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An upwind cell centred Finite Volume Method for nearly incompressible explicit solid dynamics

Authors:
Jibran Haider at Swansea University
Jibran Haider
Chun Hean Lee at University of Glasgow
Chun Hean Lee
Antonio Javier Gil at Swansea University
Antonio Javier Gil
Antonio Huerta
Antonio Huerta
Antonio Huerta
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Abstract

My presentation at the European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS) 2018 in Glasgow, Scotland.
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Motivation Governing equations Methodology Implementation Results Conclusions
An upwind cell centred Finite Volume Method
for nearly incompressible explicit solid dynamics
Jibran Haider a, Chun Hean Lee a, Antonio J. Gil a, Antonio Huerta b& Javier Bonetc
aZienkiewicz Centre for Computational Engineering (ZCCE),
College of Engineering, Swansea University, UK
bLaboratory of Computational Methods and Numerical Analysis (LaCàN),
Universitat Politèchnica de Catalunya (UPC BarcelonaTech), Spain
cUniversity of Greenwich, London, UK
ECCOMAS ECCM-ECFD Conference 2018 (11th -15th June 2018)
http://www.jibranhaider.weebly.com
June 20, 2018
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 1
Motivation Governing equations Methodology Implementation Results Conclusions
Outline
1. Motivation
2. Governing equations
3. Numerical methodology
4. OpenFOAM implementation
5. Results
6. Conclusions
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 2
Motivation Governing equations Methodology Implementation Results Conclusions
Outline
1. Motivation
2. Governing equations
3. Numerical methodology
4. OpenFOAM implementation
5. Results
6. Conclusions
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 3
Motivation Governing equations Methodology Implementation Results Conclusions
Large strain solid dynamics
Objectives:
•Simulate fast-transient solid dynamic problems.
•Develop an efficient low order numerical scheme.
Displacement based FEM/FVM formulations:
•Linear tetrahedral elements suffer from
×Volumetric locking in nearly incompressible materials.
×Reduced order of convergence for stresses and strains.
×Poor performance in bending and shock scenarios.
Proposed mixed formulation:
•First order conservation laws in a Total Lagrangian formalism.
•An upwind cell-centred Finite Volume scheme.
•Entitled TOtal Lagrangian Upwind Cell-centred FVM for
Hyperbolic conservation laws (TOUCH).
•Implemented using the open-source OpenFOAM code.
[Mixed formulation in OpenFOAM]
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 4
Motivation Governing equations Methodology Implementation Results Conclusions
Large strain solid dynamics
Objectives:
•Simulate fast-transient solid dynamic problems.
•Develop an efficient low order numerical scheme.
Displacement based FEM/FVM formulations:
•Linear tetrahedral elements suffer from
×Volumetric locking in nearly incompressible materials.
×Reduced order of convergence for stresses and strains.
×Poor performance in bending and shock scenarios.
Proposed mixed formulation:
•First order conservation laws in a Total Lagrangian formalism.
•An upwind cell-centred Finite Volume scheme.
•Entitled TOtal Lagrangian Upwind Cell-centred FVM for
Hyperbolic conservation laws (TOUCH).
•Implemented using the open-source OpenFOAM code.
[Mixed formulation in OpenFOAM]
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 4
Motivation Governing equations Methodology Implementation Results Conclusions
Large strain solid dynamics
Objectives:
•Simulate fast-transient solid dynamic problems.
•Develop an efficient low order numerical scheme.
Displacement based FEM/FVM formulations:
•Linear tetrahedral elements suffer from
×Volumetric locking in nearly incompressible materials.
×Reduced order of convergence for stresses and strains.
×Poor performance in bending and shock scenarios.
Proposed mixed formulation:
•First order conservation laws in a Total Lagrangian formalism.
•An upwind cell-centred Finite Volume scheme.
•Entitled TOtal Lagrangian Upwind Cell-centred FVM for
Hyperbolic conservation laws (TOUCH).
•Implemented using the open-source OpenFOAM code.
[Mixed formulation in OpenFOAM]
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 4
Motivation Governing equations Methodology Implementation Results Conclusions
Outline
1. Motivation
2. Governing equations
3. Numerical methodology
4. OpenFOAM implementation
5. Results
6. Conclusions
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 5
Motivation Governing equations Methodology Implementation Results Conclusions
Total Lagrangian formulation
Conservation laws
•Linear momentum
∂p
∂t=DIV P(F) + ρ0b;p=ρ0v
•Deformation gradient
∂F
∂t=DIV p
ρ0
⊗I
Additional equations
•Cofactor of deformation
∂H
∂t=CURL p
ρ0
F
•Jacobian of deformation
∂J
∂t=DIV HTp
ρ0
•Total energy
∂E
∂t=DIV 1
ρ0
PTp−Q+s
1
x,
1
X
3
x,
3
X
2
x,
2
X
)t,X(φ=x
dV
JdV=dv
Xd
XdF=xd
AdH=ad
Ad
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 6
Motivation Governing equations Methodology Implementation Results Conclusions
Total Lagrangian formulation
Conservation laws
•Linear momentum
∂p
∂t=DIV P(F) + ρ0b;p=ρ0v
•Deformation gradient
∂F
∂t=DIV p
ρ0
⊗I
Additional equations
•Cofactor of deformation
∂H
∂t=CURL p
ρ0
F
•Jacobian of deformation
∂J
∂t=DIV HTp
ρ0
•Total energy
∂E
∂t=DIV 1
ρ0
PTp−Q+s
1
x,
1
X
3
x,
3
X
2
x,
2
X
)t,X(φ=x
dV
JdV=dv
Xd
XdF=xd
AdH=ad
Ad
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 6
Motivation Governing equations Methodology Implementation Results Conclusions
Total Lagrangian formulation
Conservation laws
•Linear momentum
∂p
∂t=DIV P(F) + ρ0b;p=ρ0v
•Deformation gradient
∂F
∂t=DIV p
ρ0
⊗I
Additional equations
•Cofactor of deformation
∂H
∂t=CURL p
ρ0
F
•Jacobian of deformation
∂J
∂t=DIV HTp
ρ0
•Total energy
∂E
∂t=DIV 1
ρ0
PTp−Q+s
1
x,
1
X
3
x,
3
X
2
x,
2
X
)t,X(φ=x
dV
JdV=dv
Xd
XdF=xd
AdH=ad
Ad
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 6
Motivation Governing equations Methodology Implementation Results Conclusions
Total Lagrangian formulation
Conservation laws
•Linear momentum
∂p
∂t=DIV P(F) + ρ0b;p=ρ0v
•Deformation gradient
∂F
∂t=DIV p
ρ0
⊗I
Additional equations
•Cofactor of deformation
∂H
∂t=CURL p
ρ0
F
•Jacobian of deformation
∂J
∂t=DIV HTp
ρ0
•Total energy
∂E
∂t=DIV 1
ρ0
PTp−Q+s
1
x,
1
X
3
x,
3
X
2
x,
2
X
)t,X(φ=x
dV
JdV=dv
Xd
XdF=xd
AdH=ad
Ad
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 6
Motivation Governing equations Methodology Implementation Results Conclusions
Total Lagrangian formulation
Conservation laws
•Linear momentum
∂p
∂t=DIV P(F) + ρ0b;p=ρ0v
•Deformation gradient
∂F
∂t=DIV p
ρ0
⊗I
Additional equations
•Cofactor of deformation
∂H
∂t=CURL p
ρ0
F
•Jacobian of deformation
∂J
∂t=DIV HTp
ρ0
•Total energy
∂E
∂t=DIV 1
ρ0
PTp−Q+s
1
x,
1
X
3
x,
3
X
2
x,
2
X
)t,X(φ=x
dV
JdV=dv
Xd
XdF=xd
AdH=ad
Ad
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 6
Motivation Governing equations Methodology Implementation Results Conclusions
Hyperbolic system
First order conservation laws
∂U
∂t=∂FI
∂XI
+S;∀I=1,2,3
U=
p
F
H
J
E
;FN=FINI=
P N
1
ρ0p⊗N
F1
ρ0p⊗N
H:1
ρ0p⊗N
1
ρ0PTp·N−Q·N
;S=
ρ0b
0
0
0
s
Aims to bridge the gap between CFD and computational solid dynamics.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 7
Motivation Governing equations Methodology Implementation Results Conclusions
Hyperbolic system
First order conservation laws
∂U
∂t=∂FI
∂XI
+S;∀I=1,2,3
U=
p
F
H
J
E
;FN=FINI=
P N
1
ρ0p⊗N
F1
ρ0p⊗N
H:1
ρ0p⊗N
1
ρ0PTp·N−Q·N
;S=
ρ0b
0
0
0
s
Aims to bridge the gap between CFD and computational solid dynamics.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 7
Motivation Governing equations Methodology Implementation Results Conclusions
Hyperbolic system
First order conservation laws
∂U
∂t=∂FI
∂XI
+S;∀I=1,2,3
U=
p
F
H
J
E
;FN=FINI=
P N
1
ρ0p⊗N
F1
ρ0p⊗N
H:1
ρ0p⊗N
1
ρ0PTp·N−Q·N
;S=
ρ0b
0
0
0
s
Aims to bridge the gap between CFD and computational solid dynamics.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 7
Motivation Governing equations Methodology Implementation Results Conclusions
Hyperbolic system
First order conservation laws
∂U
∂t=∂FI
∂XI
+S;∀I=1,2,3
U=
p
F
H
J
E
;FN=FINI=
P N
1
ρ0p⊗N
F1
ρ0p⊗N
H:1
ρ0p⊗N
1
ρ0PTp·N−Q·N
;S=
ρ0b
0
0
0
s
Aims to bridge the gap between CFD and computational solid dynamics.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 7
Motivation Governing equations Methodology Implementation Results Conclusions
Hyperbolic system
First order conservation laws
∂U
∂t=∂FI
∂XI
+S;∀I=1,2,3
U=
p
F
H
J
E
;FN=FINI=
P N
1
ρ0p⊗N
F1
ρ0p⊗N
H:1
ρ0p⊗N
1
ρ0PTp·N−Q·N
;S=
ρ0b
0
0
0
s
Aims to bridge the gap between CFD and computational solid dynamics.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 7
Motivation Governing equations Methodology Implementation Results Conclusions
Outline
1. Motivation
2. Governing equations
3. Numerical methodology
Spatial discretisation
Flux computation
Involutions
4. OpenFOAM implementation
5. Results
6. Conclusions
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 8
Motivation Governing equations Methodology Implementation Results Conclusions
Outline
1. Motivation
2. Governing equations
3. Numerical methodology
Spatial discretisation
Flux computation
Involutions
4. OpenFOAM implementation
5. Results
6. Conclusions
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 9
Motivation Governing equations Methodology Implementation Results Conclusions
Finite volume methodology
Conservation equations for an arbitrary element:
dUe
dt =1
Ωe
0ZΩe
0
3
X
I=1
∂FI
∂XI
dΩ0+Se=1
Ωe
0Z∂Ωe
0
FNdA +Se
Standard Godunov-type CC-FVM
dUe
dt =1
Ωe
0
X
f∈Λf
e
FC
Nef kCef k
+Se
eFC
Nef
kCef kΩe
0
Alternative nodal CC-FVM
dUe
dt =1
Ωe
0
X
a∈Λa
e
FC
Nea kCeak
+Se
FC
Nea
kCeak
Ωe
0
e
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 10
Motivation Governing equations Methodology Implementation Results Conclusions
Finite volume methodology
Conservation equations for an arbitrary element:
dUe
dt =1
Ωe
0ZΩe
0
3
X
I=1
∂FI
∂XI
dΩ0+Se=1
Ωe
0Z∂Ωe
0
FNdA +Se
Standard Godunov-type CC-FVM
dUe
dt =1
Ωe
0
X
f∈Λf
e
FC
Nef kCef k
+Se
eFC
Nef
kCef kΩe
0
Alternative nodal CC-FVM
dUe
dt =1
Ωe
0
X
a∈Λa
e
FC
Nea kCeak
+Se
FC
Nea
kCeak
Ωe
0
e
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 10
Motivation Governing equations Methodology Implementation Results Conclusions
Finite volume methodology
Conservation equations for an arbitrary element:
dUe
dt =1
Ωe
0ZΩe
0
3
X
I=1
∂FI
∂XI
dΩ0+Se=1
Ωe
0Z∂Ωe
0
FNdA +Se
Standard Godunov-type CC-FVM
dUe
dt =1
Ωe
0
X
f∈Λf
e
FC
Nef kCef k
+Se
eFC
Nef
kCef kΩe
0
Alternative nodal CC-FVM
dUe
dt =1
Ωe
0
X
a∈Λa
e
FC
Nea kCeak
+Se
FC
Nea
kCeak
Ωe
0
e
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 10
Motivation Governing equations Methodology Implementation Results Conclusions
Finite volume methodology
Conservation equations for an arbitrary element:
dUe
dt =1
Ωe
0ZΩe
0
3
X
I=1
∂FI
∂XI
dΩ0+Se=1
Ωe
0Z∂Ωe
0
FNdA +Se
Standard Godunov-type CC-FVM
dUe
dt =1
Ωe
0
X
f∈Λf
e
FC
Nef kCef k
+Se
eFC
Nef
kCef kΩe
0
Alternative nodal CC-FVM
dUe
dt =1
Ωe
0
X
a∈Λa
e
FC
Nea kCeak
+Se
FC
Nea
kCeak
Ωe
0
e
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 10
Motivation Governing equations Methodology Implementation Results Conclusions
Finite volume methodology
Standard Godunov-type CC-FVM
dUe
dt =1
Ωe
0
X
f∈Λf
e
FC
Nef kCef k
+Se
w
dpe
dt =1
Ωe
0X
f∈Λf
e
tC
fkCef k+ρ0be
dFe
dt =1
Ωe
0X
f∈Λf
e
pC
f
ρ0
⊗Cef
dHe
dt =Fe
1
Ωe
0X
f∈Λf
e
pC
f
ρ0
⊗Cef
dJe
dt =He:1
Ωe
0X
f∈Λf
e
pC
f
ρ0
⊗Cef
dEe
dt =1
Ωe
0X
f∈Λf
e
pC
f
ρ0
·tC
f!kCef k
Alternative nodal CC-FVM
dUe
dt =1
Ωe
0
X
a∈Λa
e
FC
Nea kCeak
+Se
w
dpe
dt =1
Ωe
0X
a∈Λa
e
tC
ea kCeak+ρ0be
dFe
dt =1
Ωe
0X
a∈Λa
e
pC
a
ρ0
⊗Cea
dHe
dt =Fe
1
Ωe
0X
a∈Λa
e
pC
a
ρ0
⊗Cea
dJe
dt =He:1
Ωe
0X
a∈Λa
e
pC
a
ρ0
⊗Cea
dEe
dt =1
Ωe
0X
a∈Λa
e pC
a
ρ0
·tC
ea!kCea k
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 11
Motivation Governing equations Methodology Implementation Results Conclusions
Finite volume methodology
Standard Godunov-type CC-FVM
dUe
dt =1
Ωe
0
X
f∈Λf
e
FC
Nef kCef k
+Se
w
dpe
dt =1
Ωe
0X
f∈Λf
e
tC
fkCef k+ρ0be
dFe
dt =1
Ωe
0X
f∈Λf
e
pC
f
ρ0
⊗Cef
dHe
dt =Fe
1
Ωe
0X
f∈Λf
e
pC
f
ρ0
⊗Cef
dJe
dt =He:1
Ωe
0X
f∈Λf
e
pC
f
ρ0
⊗Cef
dEe
dt =1
Ωe
0X
f∈Λf
e
pC
f
ρ0
·tC
f!kCef k
Alternative nodal CC-FVM
dUe
dt =1
Ωe
0
X
a∈Λa
e
FC
Nea kCeak
+Se
w
dpe
dt =1
Ωe
0X
a∈Λa
e
tC
ea kCeak+ρ0be
dFe
dt =1
Ωe
0X
a∈Λa
e
pC
a
ρ0
⊗Cea
dHe
dt =Fe
1
Ωe
0X
a∈Λa
e
pC
a
ρ0
⊗Cea
dJe
dt =He:1
Ωe
0X
a∈Λa
e
pC
a
ρ0
⊗Cea
dEe
dt =1
Ωe
0X
a∈Λa
e pC
a
ρ0
·tC
ea!kCea k
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 11
Motivation Governing equations Methodology Implementation Results Conclusions
Outline
1. Motivation
2. Governing equations
3. Numerical methodology
Spatial discretisation
Flux computation
Involutions
4. OpenFOAM implementation
5. Results
6. Conclusions
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 12
Motivation Governing equations Methodology Implementation Results Conclusions
Godunov-type flux
where flux Jacobian matrix ANef := ∂FNef
∂U
pC
f=1
2(p−
ef +p+
ef ) + 1
2St
ef (t+
ef −t−
ef )
tC
f=1
2(t−
ef +t+
ef ) + 1
2Sp
ef (p+
ef −p−
ef )
St
ef =1
cp
(nef ⊗nef ) + 1
cs
(I−nef ⊗nef )
Sp
ef =cp(nef ⊗nef ) + cs(I−nef ⊗nef )
Interface states U−,+
f:
Least-square gradient operator and Barth & Jespersen slope limiter.
XEnsures second order spatial accuracy.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 13
Riemann solver:
FC
Nef U−
f,U+
f=1
2hFNef (U−
f) + FNef (U+
f)i
| {z }
average
Motivation Governing equations Methodology Implementation Results Conclusions
Godunov-type flux
where flux Jacobian matrix ANef := ∂FNef
∂U
pC
f=1
2(p−
ef +p+
ef ) + 1
2St
ef (t+
ef −t−
ef )
tC
f=1
2(t−
ef +t+
ef ) + 1
2Sp
ef (p+
ef −p−
ef )
St
ef =1
cp
(nef ⊗nef ) + 1
cs
(I−nef ⊗nef )
Sp
ef =cp(nef ⊗nef ) + cs(I−nef ⊗nef )
Interface states U−,+
f:
Least-square gradient operator and Barth & Jespersen slope limiter.
XEnsures second order spatial accuracy.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 13
Riemann solver:
FC
Nef U−
f,U+
f=1
2hFNef (U−
f) + FNef (U+
f)i
| {z }
Unstable flux
X
Motivation Governing equations Methodology Implementation Results Conclusions
Godunov-type flux
where flux Jacobian matrix ANef := ∂FNef
∂U
pC
f=1
2(p−
ef +p+
ef ) + 1
2St
ef (t+
ef −t−
ef )
tC
f=1
2(t−
ef +t+
ef ) + 1
2Sp
ef (p+
ef −p−
ef )
St
ef =1
cp
(nef ⊗nef ) + 1
cs
(I−nef ⊗nef )
Sp
ef =cp(nef ⊗nef ) + cs(I−nef ⊗nef )
Interface states U−,+
f:
Least-square gradient operator and Barth & Jespersen slope limiter.
XEnsures second order spatial accuracy.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 13
Riemann solver:
FC
Nef U−
f,U+
f=1
2hFNef (U−
f) + FNef (U+
f)i
| {z }
Unstable flux
−1
2ZU+
f
U−
fANef dU
| {z }
Upwinding stabilisation
X
Motivation Governing equations Methodology Implementation Results Conclusions
Godunov-type flux
where flux Jacobian matrix ANef := ∂FNef
∂U
pC
f=1
2(p−
ef +p+
ef ) + 1
2St
ef (t+
ef −t−
ef )
tC
f=1
2(t−
ef +t+
ef ) + 1
2Sp
ef (p+
ef −p−
ef )
St
ef =1
cp
(nef ⊗nef ) + 1
cs
(I−nef ⊗nef )
Sp
ef =cp(nef ⊗nef ) + cs(I−nef ⊗nef )
Interface states U−,+
f:
Least-square gradient operator and Barth & Jespersen slope limiter.
XEnsures second order spatial accuracy.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 13
Acoustic Riemann solver:
FC
Nef U−
f,U+
f=1
2hFNef (U−
f) + FNef (U+
f)i
| {z }
Unstable flux
−1
2ANef U+
f−U−
f
| {z }
Upwinding stabilisation
X
Motivation Governing equations Methodology Implementation Results Conclusions
Godunov-type flux
where flux Jacobian matrix ANef := ∂FNef
∂U
pC
f=1
2(p−
ef +p+
ef ) + 1
2St
ef (t+
ef −t−
ef )
tC
f=1
2(t−
ef +t+
ef ) + 1
2Sp
ef (p+
ef −p−
ef )
St
ef =1
cp
(nef ⊗nef ) + 1
cs
(I−nef ⊗nef )
Sp
ef =cp(nef ⊗nef ) + cs(I−nef ⊗nef )
Interface states U−,+
f:
Least-square gradient operator and Barth & Jespersen slope limiter.
XEnsures second order spatial accuracy.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 13
Acoustic Riemann solver:
FC
Nef U−
f,U+
f=1
2hFNef (U−
f) + FNef (U+
f)i
| {z }
Unstable flux
−1
2ANef U+
f−U−
f
| {z }
Upwinding stabilisation
X
Motivation Governing equations Methodology Implementation Results Conclusions
Godunov-type flux
where flux Jacobian matrix ANef := ∂FNef
∂U
pC
f=1
2(p−
ef +p+
ef ) + 1
2St
ef (t+
ef −t−
ef )
tC
f=1
2(t−
ef +t+
ef ) + 1
2Sp
ef (p+
ef −p−
ef )
St
ef =1
cp
(nef ⊗nef ) + 1
cs
(I−nef ⊗nef )
Sp
ef =cp(nef ⊗nef ) + cs(I−nef ⊗nef )
Interface states U−,+
f:
Least-square gradient operator and Barth & Jespersen slope limiter.
XEnsures second order spatial accuracy.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 13
Acoustic Riemann solver:
FC
Nef U−
f,U+
f=1
2hFNef (U−
f) + FNef (U+
f)i
| {z }
Unstable flux
−1
2ANef U+
f−U−
f
| {z }
Upwinding stabilisation
X
Motivation Governing equations Methodology Implementation Results Conclusions
Outline
1. Motivation
2. Governing equations
3. Numerical methodology
Spatial discretisation
Flux computation
Involutions
4. OpenFOAM implementation
5. Results
6. Conclusions
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 14
Motivation Governing equations Methodology Implementation Results Conclusions
Involutive constraints
Conservation of deformation gradient:∂F
∂t=DIV p
ρ0
⊗ICURL ˙
F=0
Conservation of cofactor of deformation:∂H
∂t=CURL p
ρ0
FDIV ˙
H=0
Three involution-free methodologies
XGodunov-type constrained transport algorithm C-TOUCH 1.
XGodunov-type penalisation based scheme P-TOUCH 1.
XAlternative nodal scheme X-GLACE [Kluth & Desprès 2010; Georges et al. 2017 ].
1J. Haider, C. H. Lee, A. J. Gil and J. Bonet. “A first order hyperbolic framework for large strain computational solid
dynamics: An upwind cell centred Total Lagrangian scheme”, IJNME (2017), 109(3): 407–456.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 15
Motivation Governing equations Methodology Implementation Results Conclusions
Involutive constraints
Conservation of deformation gradient:∂F
∂t=DIV p
ρ0
⊗ICURL ˙
F=0
Conservation of cofactor of deformation:∂H
∂t=CURL p
ρ0
FDIV ˙
H=0
Three involution-free methodologies
XGodunov-type constrained transport algorithm C-TOUCH 1.
XGodunov-type penalisation based scheme P-TOUCH 1.
XAlternative nodal scheme X-GLACE [Kluth & Desprès 2010; Georges et al. 2017 ].
1J. Haider, C. H. Lee, A. J. Gil and J. Bonet. “A first order hyperbolic framework for large strain computational solid
dynamics: An upwind cell centred Total Lagrangian scheme”, IJNME (2017), 109(3): 407–456.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 15
Motivation Governing equations Methodology Implementation Results Conclusions
Involutive constraints
Conservation of deformation gradient:∂F
∂t=DIV p
ρ0
⊗ICURL ˙
F=0
Conservation of cofactor of deformation:∂H
∂t=CURL p
ρ0
FDIV ˙
H=0
Three involution-free methodologies
XGodunov-type constrained transport algorithm C-TOUCH 1.
XGodunov-type penalisation based scheme P-TOUCH 1.
XAlternative nodal scheme X-GLACE [Kluth & Desprès 2010; Georges et al. 2017 ].
1J. Haider, C. H. Lee, A. J. Gil and J. Bonet. “A first order hyperbolic framework for large strain computational solid
dynamics: An upwind cell centred Total Lagrangian scheme”, IJNME (2017), 109(3): 407–456.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 15
Motivation Governing equations Methodology Implementation Results Conclusions
Involutive constraints
Conservation of deformation gradient:∂F
∂t=DIV p
ρ0
⊗ICURL ˙
F=0
Conservation of cofactor of deformation:∂H
∂t=CURL p
ρ0
FDIV ˙
H=0
Three involution-free methodologies
XGodunov-type constrained transport algorithm C-TOUCH 1.
XGodunov-type penalisation based scheme P-TOUCH 1.
XAlternative nodal scheme X-GLACE [Kluth & Desprès 2010; Georges et al. 2017 ].
1J. Haider, C. H. Lee, A. J. Gil and J. Bonet. “A first order hyperbolic framework for large strain computational solid
dynamics: An upwind cell centred Total Lagrangian scheme”, IJNME (2017), 109(3): 407–456.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 15
Motivation Governing equations Methodology Implementation Results Conclusions
Outline
1. Motivation
2. Governing equations
3. Numerical methodology
4. OpenFOAM implementation
5. Results
6. Conclusions
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 16
Motivation Governing equations Methodology Implementation Results Conclusions
OpenFOAM for solids
A high-end object oriented software package in C++ widely used in the CFD community.
Existing solid mechanics in OpenFOAM
×Displacement based implicit dynamics.
×Linear elastic material with small/moderate
strain deformation.
×Poor performance in bending and shock
dominated scenarios.
Contribution to OpenFOAM
XA novel solid dynamics tool-kit.
XAdvanced constitutive models.
XParallel computing (excellent scalability).
XSoon to be open-sourced.
Explicit solid dynamics
applications
solvers
solidFoam
utilities
initialConditions
src
boundaryConditions
models
schemes
tutorials
Allwmake
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 17
Motivation Governing equations Methodology Implementation Results Conclusions
OpenFOAM for solids
A high-end object oriented software package in C++ widely used in the CFD community.
Existing solid mechanics in OpenFOAM
×Displacement based implicit dynamics.
×Linear elastic material with small/moderate
strain deformation.
×Poor performance in bending and shock
dominated scenarios.
Contribution to OpenFOAM
XA novel solid dynamics tool-kit.
XAdvanced constitutive models.
XParallel computing (excellent scalability).
XSoon to be open-sourced.
Explicit solid dynamics
applications
solvers
solidFoam
utilities
initialConditions
src
boundaryConditions
models
schemes
tutorials
Allwmake
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 17
Motivation Governing equations Methodology Implementation Results Conclusions
OpenFOAM for solids
A high-end object oriented software package in C++ widely used in the CFD community.
Existing solid mechanics in OpenFOAM
×Displacement based implicit dynamics.
×Linear elastic material with small/moderate
strain deformation.
×Poor performance in bending and shock
dominated scenarios.
Contribution to OpenFOAM
XA novel solid dynamics tool-kit.
XAdvanced constitutive models.
XParallel computing (excellent scalability).
XSoon to be open-sourced.
Explicit solid dynamics
applications
solvers
solidFoam
utilities
initialConditions
src
boundaryConditions
models
schemes
tutorials
Allwmake
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 17
Motivation Governing equations Methodology Implementation Results Conclusions
Outline
1. Motivation
2. Governing equations
3. Numerical methodology
4. OpenFOAM implementation
5. Results
Convergence
Momentum preservation
Benchmarking
Parallel performance
Algorithm robustness
6. Conclusions
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 18
Motivation Governing equations Methodology Implementation Results Conclusions
Outline
1. Motivation
2. Governing equations
3. Numerical methodology
4. OpenFOAM implementation
5. Results
Convergence
Momentum preservation
Benchmarking
Parallel performance
Algorithm robustness
6. Conclusions
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 19
Motivation Governing equations Methodology Implementation Results Conclusions
Low dispersion cube
Velocity
10−2 10−1 100
10−7
10−6
10−5
10−4
Grid Size (m)
L2 Norm Error
vx(C-T OUCH)
vy(C-T OUCH)
vz(C-T OUCH)
vx(P-TO UCH)
vy(P-TO UCH)
vz(P-TO UCH)
vx(X-GL ACE)
vy(X-GL ACE)
vz(X-GL ACE)
slope = 2
Stress
10−2 10−1 100
10−7
10−6
10−5
10−4
Grid Size (m)
L2 Norm Error
Pxx (C-TO UCH)
Pyy (C-T OUCH)
Pzz (C-T OUCH)
Pxx (P-TO UCH)
Pyy (P-TO UCH)
Pzz (P-TO UCH)
Pxx (X-GLA CE)
Pyy (X-GL ACE)
Pzz (X-G LACE)
slope = 2
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 20
Problem: Unit side cube, linear elastic material, ρ0=1100 kg/m3,E=17 MPa and ν=0.3.
Motivation Governing equations Methodology Implementation Results Conclusions
Outline
1. Motivation
2. Governing equations
3. Numerical methodology
4. OpenFOAM implementation
5. Results
Convergence
Momentum preservation
Benchmarking
Parallel performance
Algorithm robustness
6. Conclusions
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 21
Motivation Governing equations Methodology Implementation Results Conclusions
Spinning top
Global angular momentum
0 0.25 0.5 0.75 1
-1
0
1
2
3
410-4
Global linear momentum
0 0.25 0.5 0.75 1
-4
-2
0
2
4
610-17
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 22
Problem: Neo-Hookean material, ρ0=1000 kg/m3,E=50.05 kPa, ν=0.3and Ω = 40 rad/s.
Motivation Governing equations Methodology Implementation Results Conclusions
Outline
1. Motivation
2. Governing equations
3. Numerical methodology
4. OpenFOAM implementation
5. Results
Convergence
Momentum preservation
Benchmarking
Parallel performance
Algorithm robustness
6. Conclusions
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 23
Motivation Governing equations Methodology Implementation Results Conclusions
Twisting column
t=0.09 s
Pressure (Pa)
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 24
Problem: Neo-Hookean material, ρ0=1100 kg/m3,E=17 MPa, ν=0.499 and Ω = 105 rad/s.
Motivation Governing equations Methodology Implementation Results Conclusions
Twisting column
Height of column
0 0.2 0.4 0.6 0.8 1
0.6
0.8
1
1.2
1.4
Time (s)
Height (h/H)
4×24×4
8×48×8
16×9 6×16
Numerical dissipation
0 0.2 0.4 0.6 0.8 1
40
50
60
70
80
90
100
110
Time (s)
(ENumerical /E)×1 00 (%)
4×24×4
8×48×8
16 ×96×16
conserve d
XConvergence proved as the mesh is refined.
XLesser numerical dissipation with increase in mesh density.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 25
Problem: Neo-Hookean material, ρ0=1100 kg/m3,E=17 MPa, ν=0.499 and Ω = 200 rad/s.
Motivation Governing equations Methodology Implementation Results Conclusions
Twisting column
Height of column
0 0.2 0.4 0.6 0.8 1
0.6
0.8
1
1.2
1.4
Time (s)
Height (h/H)
4×24×4
8×48×8
16×9 6×16
Numerical dissipation
0 0.2 0.4 0.6 0.8 1
40
50
60
70
80
90
100
110
Time (s)
(ENumerical /E)×1 00 (%)
4×24×4
8×48×8
16 ×96×16
conserve d
XConvergence proved as the mesh is refined.
XLesser numerical dissipation with increase in mesh density.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 25
Problem: Neo-Hookean material, ρ0=1100 kg/m3,E=17 MPa, ν=0.499 and Ω = 200 rad/s.
Motivation Governing equations Methodology Implementation Results Conclusions
Twisting column: Comparison against alternative schemes
t=0.1s
C-TOUCH P-TOUCH X-GLACE B-bar Taylor Hood JST-SPH SUPG-SPH
Pressure (Pa)
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 26
Problem: Neo-Hookean material, ρ0=1100 kg/m3,E=17 MPa, ν=0.495 and Ω = 105 rad/s.
Motivation Governing equations Methodology Implementation Results Conclusions
Outline
1. Motivation
2. Governing equations
3. Numerical methodology
4. OpenFOAM implementation
5. Results
Convergence
Momentum preservation
Benchmarking
Parallel performance
Algorithm robustness
6. Conclusions
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 27
Motivation Governing equations Methodology Implementation Results Conclusions
Scalability
Speedup ratio =Tserial
Tparallel
100101102103
100
101
102
103
Parallel efficiency =Speedup
Ncores
0 64 128 192 256 320 384 448 512
0
20
40
60
80
100
120
140
XSpeedup of over 200 achieved on 512 cores on the cluster at Swansea University.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 28
Motivation Governing equations Methodology Implementation Results Conclusions
Scalability
Speedup ratio =Tserial
Tparallel
100101102103
100
101
102
103
Parallel efficiency =Speedup
Ncores
0 64 128 192 256 320 384 448 512
0
20
40
60
80
100
120
140
XSpeedup of over 200 achieved on 512 cores on the cluster at Swansea University.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 28
Motivation Governing equations Methodology Implementation Results Conclusions
Outline
1. Motivation
2. Governing equations
3. Numerical methodology
4. OpenFOAM implementation
5. Results
Convergence
Momentum preservation
Benchmarking
Parallel performance
Algorithm robustness
6. Conclusions
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 29
Motivation Governing equations Methodology Implementation Results Conclusions
Stent-like structure
X, x
Y, y
Z, z
T= 0.1mm
Do= 10 mm
tb= [0,0, T ]TkPa
L= 20 mm
[Stent-like structure]
t=500µs
Pressure (Pa)
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 30
Problem: Neo-Hookean material, ρ0=1100 kg/m3,E=17 MPa, ν=0.45 and T=−100 kPa.
Motivation Governing equations Methodology Implementation Results Conclusions
Stent-like structure
X, x
Y, y
Z, z
T= 0.1mm
Do= 10 mm
tb= [0,0, T ]TkPa
L= 20 mm
[Stent-like structure]
t=500µs
Pressure (Pa)
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 30
Problem: Neo-Hookean material, ρ0=1100 kg/m3,E=17 MPa, ν=0.499 &T=−100 kPa.
Motivation Governing equations Methodology Implementation Results Conclusions
Imploding bottle
X, x
Y, y
Z, z
Do=102 mm
H=192 mm
X, x
Y, y
Z, z
15mm35mm
10mm
40mm
140mm
T=1 mm
p
[Imploding bottle]
t=15 ms t=19 ms
Pressure (Pa)
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 31
Problem: Neo-Hookean material, ρ0=1100 kg/m3,E=17 MPa, ν=0.3&p=2000 Pa.
Motivation Governing equations Methodology Implementation Results Conclusions
Imploding bottle: Mesh refinement
t=19 ms
251896 cells 435960 cells 251896 cells 435960 cells
Deformation
Pressure (Pa)
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 32
Problem: Neo-Hookean material, ρ0=1100 kg/m3,E=17 MPa, ν=0.3&p=2000 Pa.
Motivation Governing equations Methodology Implementation Results Conclusions
Imploding bottle: Mesh refinement
t=19 ms
251896 cells 435960 cells 251896 cells 435960 cells
Deformation
Pressure (Pa)
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 32
Problem: Neo-Hookean material, ρ0=1100 kg/m3,E=17 MPa, ν=0.3&p=2000 Pa.
Motivation Governing equations Methodology Implementation Results Conclusions
Crushing of a thin cylinder
X, x
Y, y
Z, z
Do=200 mm
H=113.9mm
T=0.247 mm
db= [0, dmax,0]T
0 1 2 3 4 5
10-3
0
0.5
1
1.5
2
[Crushing cylinder]
t=1.8ms t=2.2ms
t=2.6ms t=5.0ms
Pressure (Pa)
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 33
Problem: Neo-Hookean material, ρ0=1000 kg/m3,E=5.56 GPa, ν=0.3,D0/t≈800 &dmax =−4.5mm.
Motivation Governing equations Methodology Implementation Results Conclusions
Crushing of a thin cylinder
X, x
Y, y
Z, z
Do=200 mm
H=113.9mm
T=0.247 mm
db= [0, dmax,0]T
0 1 2 3 4 5
10-3
0
0.5
1
1.5
2
[Crushing cylinder]
t=1.8ms t=2.2ms
t=2.6ms t=5.0ms
Pressure (Pa)
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 33
Problem: Neo-Hookean material, ρ0=1000 kg/m3,E=5.56 GPa, ν=0.3,D0/t≈800 &dmax =−4.5mm.
Motivation Governing equations Methodology Implementation Results Conclusions
Crushing of a thin cylinder: Mesh refinement
t=5ms
90000 cells 160000 cells 250000 cells
Pressure (Pa)
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 34
Problem: Neo-Hookean material, ρ0=1000 kg/m3,E=5.56 GPa, ν=0.3,D0/t≈800 &dmax =4.5mm.
Motivation Governing equations Methodology Implementation Results Conclusions
Outline
1. Motivation
2. Governing equations
3. Numerical methodology
4. OpenFOAM implementation
5. Results
6. Conclusions
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 35
Motivation Governing equations Methodology Implementation Results Conclusions
Conclusions
Summary:
•A Total Lagrangian cell centred finite volume scheme for explicit fast solid dynamic applications.
•An acoustic upwind Riemann solver is employed for the evaluation of contact fluxes.
•Velocities and stresses display the same rate of convergence.
•Robust scheme without locking and pressure checker-boarding.
Complex problems Plasticity Contact mechanics Thin-walled structures
On-going work:
•Open-source release of the solid dynamics tool-kit on GitHub.
•Thermo-mechanical constitutive model.
•Advanced Roe’s Riemann solver with robust shock capturing.
•Ability to handle tetrahedral/polyhedral elements.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 36
Motivation Governing equations Methodology Implementation Results Conclusions
Conclusions
Summary:
•A Total Lagrangian cell centred finite volume scheme for explicit fast solid dynamic applications.
•An acoustic upwind Riemann solver is employed for the evaluation of contact fluxes.
•Velocities and stresses display the same rate of convergence.
•Robust scheme without locking and pressure checker-boarding.
Complex problems Plasticity Contact mechanics Thin-walled structures
On-going work:
•Open-source release of the solid dynamics tool-kit on GitHub.
•Thermo-mechanical constitutive model.
•Advanced Roe’s Riemann solver with robust shock capturing.
•Ability to handle tetrahedral/polyhedral elements.
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 36
Motivation Governing equations Methodology Implementation Results Conclusions
Journal publications
•J. Haider, C. H. Lee, A. J. Gil, A . Huerta and J. Bonet. “An upwind cell centred Total Lagrangian finite volume
algorithm for nearly incompressible explicit solid dynamic applications”, CMAME (Accepted).
•J. Haider, C. H. Lee, A. J. Gil and J. Bonet. “A first order hyperbolic framework for large strain computational solid
dynamics: An upwind cell centred Total Lagrangian scheme”, IJNME (2017), 109(3): 407–456.
•C. H. Lee, A. J. Gil, J. Bonet and S. Kulasegaram. "An variationally consistent Streamline Upwind Petrov-Galerkin
Smooth Particle Hydrodynamics algorithm for large strain explicit fast dynamics,CMAME (2017), 318: 514–536.
•A. J. Gil, C. H. Lee, J. Bonet and R. Ortigosa. "A first order hyperbolic framework for large strain computational solid
dynamics. Part II: Total Lagrangian compressible, nearly incompressible and truly incompressible elasticity",
CMAME (2016); 300: 146-181.
•J. Bonet, A. J. Gil, C. H. Lee, M. Aguirre and R. Ortigosa. "A first order hyperbolic framework for large strain
computational solid dynamics. Part I: Total Lagrangian isothermal elasticity", CMAME (2015); 283: 689-732.
•M. Aguirre, A. J. Gil, J. Bonet and C. H. Lee. "An upwind vertex centred Finite Volume solver for Lagrangian solid
dynamics", JCP (2015); 300: 387-422.
•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 (2013); 118: 13-38.
•J. Haider, C. H. Lee, A. J. Gil, A . Huerta and J. Bonet. “An open source OpenFOAM solver for large strain explicit
solid dynamics”, Computer Physics Communications (In preparation).
•J. Bonet, A. J. Gil, C. H. Lee, A. Huerta and J. Haider. "Adapted Roe’s Riemann solver in explicit fast solid
dynamics,JCP (In preparation).
Jibran Haider ( Swansea University, UK ) Explicit solid dynamics in OpenFOAM 37
ResearchGate has not been able to resolve any citations for this publication.
A first order hyperbolic framework for large strain computational solid dynamics: An upwind cell centred Total Lagrangian scheme
Article
Full-text available
  • May 2016
This paper builds on recent work developed by the authors for the numerical analysis of large strain solid dynamics, by introducing an upwind cell centred hexahedral Finite Volume framework implemented within the open source code OpenFOAM [http://www.openfoam.com/http://www.openfoam.com/]. In Lee, Gil and Bonet [1], a first order hyperbolic system of conservation laws was introduced in terms of the linear momentum and the deformation gradient tensor of the system, leading to excellent behaviour in two dimensional bending dominated nearly incompressible scenarios. The main aim of this paper is the extension of this algorithm into three dimensions, its tailor-made implementation into OpenFOAM and the enhancement of the formulation with three key novelties. First, the introduction of two different strategies in order to ensure the satisfaction of the underlying involutions of the system, that is, that the deformation gradient tensor must be curl-free throughout the deformation process. Second, the use of a discrete angular momentum projection algorithm and a monolithic Total Variation Diminishing Runge-Kutta time integrator combined in order to guarantee the conservation of angular momentum. Third, and for comparison purposes, an adapted Total Lagrangian version of the Hyperelastic-GLACE nodal scheme of Kluth and Despres [2] is presented. A series of challenging numerical examples are examined in order to assess the robustness and accuracy of the proposed algorithm, benchmarking it against an ample spectrum of alternative numerical strategies developed by the authors in recent publications.
View
A first order hyperbolic framework for large strain computational solid dynamics. Part II: Total Lagrangian compressible, nearly incompressible and truly incompressible elasticity
Article
Full-text available
  • Nov 2015
  • COMPUT METHOD APPL M
In Part I of this series, Bonet et al. (2015) introduced a new computational framework for the analysis of large strain isothermal fast solid dynamics, where a mixed set of Total Lagrangian conservation laws was presented in terms of the linear momentum and an extended set of strain measures, namely the deformation gradient, its co-factor and its Jacobian. The main aim of this paper is to expand this formulation to the case of nearly incompressible and truly incompressible materials. The paper is further enhanced with three key novelties. First, the use of polyconvex nearly incompressible strain energy functionals enables the definition of generalised convex entropy functions and associated entropy fluxes. Two variants of the same formulation can then be obtained, namely, conservation-based and entropy-based, depending on the unknowns of the system. Crucially, the study of the eigenvalue structure of the system is carried out in order to demonstrate its hyperbolicity and, thus, obtain the correct time step bounds for explicit time integrators. Second, the development of a stabilised Petrov-Galerkin framework is presented for both systems of hyperbolic equations, that is, when expressed in terms of either conservation or entropy variables. Third, an adapted fractional step method, built upon the work presented in Gil et al. (2014), is presented to extend the range of applications towards the incompressibility limit. Finally, a series of numerical examples are presented in order to assess the applicability and robustness of the proposed formulation. The overall scheme shows excellent behaviour in compressible, nearly incompressible and truly incompressible scenarios, yielding equal order of convergence for velocities and stresses.
View
A first order hyperbolic framework for large strain computational solid dynamics. Part I: Total Lagrangian isothermal elasticity
Article
Full-text available
  • Jan 2014
This paper introduces a new computational framework for the analysis of large strain fast solid dynamics. The paper builds upon previous work published by the authors (Gil et al., 2014) [1], where a first order system of hyperbolic equations is introduced for the simulation of isothermal elastic materials in terms of the linear momentum, the deformation gradient and its Jacobian as unknown variables. In this work, the formulation is further enhanced with four key novelties. First, the use of a new geometric conservation law for the co-factor of the deformation leads to an enhanced mixed formulation, advantageous in those scenarios where the co-factor plays a dominant role. Second, the use of polyconvex strain energy functionals enables the definition of generalised convex entropy functions and associated entropy fluxes for solid dynamics problems. Moreover, the introduction of suitable conjugate entropy variables enables the derivation of a symmetric system of hyperbolic equations, dual of that expressed in terms of conservation variables. Third, the new use of a tensor cross product (de Boer, 1982) greatly facilitates the algebraic manipulations of expressions involving the co-factor of the deformation. Fourth, the development of a stabilised Petrov–Galerkin framework is presented for both systems of hyperbolic equations, that is, when expressed in terms of either conservation or entropy variables. As an example, a polyconvex Mooney-Rivlin material is used and, for completeness, the eigen-structure of the resulting system of equations is studied to guarantee the existence of real wave speeds. Finally, a series of numerical examples is presented in order to assess the robustness and accuracy of the new mixed methodology, benchmarking it against an ample spectrum of alternative numerical strategies, including implicit multi-field Fraeijs de Veubeke-Hu-Washizu variational type approaches and explicit cell and vertex centred Finite Volume schemes.
View
A variationally consistent Streamline Upwind Petrov Galerkin Smooth Particle Hydrodynamics algorithm for large strain solid dynamics
Article
  • Feb 2017
  • COMPUT METHOD APPL M
This paper presents a new Smooth Particle Hydrodynamics (SPH) computational framework for explicit fast solid dynamics. The proposed methodology explores the use of the Streamline Upwind Petrov Galerkin (SUPG) stabilisation methodology as an alternative to the Jameson-Schmidt-Turkel (JST) stabilisation recently presented by the authors in [1] in the context of a conservation law formulation of fast solid dynamics. The work introduced in this paper puts forward three advantageous features over the recent JST-SPH framework. First, the variationally consistent nature of the SUPG stabilisation allows for the introduction of a locally preserving angular momentum procedure which can be solved in a monolithic manner in conjunction with the rest of the system equations. This differs from the JST-SPH framework, where an a posteriori projection procedure was required to ensure global angular momentum preservation. Second, evaluation of expensive harmonic and bi-harmonic operators , necessary for the JST stabilisation, is circumvented in the new SUPG-SPH framework. Third, the SUPG-SPH framework is more accurate (for the same number of degrees of freedom) than its JST-SPH counterpart and its accuracy is comparable to that of the robust (but computationally more demanding) Petrov Galerkin Finite Element Method (PG-FEM) technique explored by the authors in [2–5], as shown in the numerical examples included. A series of numerical examples are analysed in order to benchmark and assess the robustness and effectiveness of the proposed algorithm. The resulting SUPG-SPH framework is therefore accurate, robust and computationally efficient, three key desired features that will allow the authors in forthcoming publications to explore its applicability in large scale simulations.
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An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics
Article
  • Nov 2015
A vertex centred Jameson–Schmidt–Turkel (JST) finite volume algorithm was recently introduced by the authors (Aguirre et al., 2014 [1]) in the context of fast solid isothermal dynamics. The spatial discretisation scheme was constructed upon a Lagrangian two-field mixed (linear momentum and the deformation gradient) formulation presented as a system of conservation laws [2–4]. In this paper, the formulation is further enhanced by introducing a novel upwind vertex centred finite volume algorithm with three key novelties. First, a conservation law for the volume map is incorporated into the existing two-field system to extend the range of applications towards the incompressibility limit (Gil et al., 2014 [5]). Second, the use of a linearised Riemann solver and reconstruction limiters is derived for the stabilisation of the scheme together with an efficient edge-based implementation. Third, the treatment of thermo-mechanical processes through a Mie– Grüneisen equation of state is incorporated in the proposed formulation. For completeness, the study of the eigenvalue structure of the resulting system of conservation laws is carried out to demonstrate hyperbolicity and obtain the correct time step bounds for non-isothermal processes. A series of numerical examples are presented in order to assess the robustness of the proposed methodology. The overall scheme shows excellent behaviour in shock and bending dominated nearly incompressible scenarios without spurious pressure oscillations, yielding second order of convergence for both velocities and stresses.
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