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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

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

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 ].

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

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

X, x

Y, y

(−0.5,0,0.5)m

(0.5,6,−0.5)m

Z, z

ω0= Ω [0,sin(πY /2H),0]Trad/s

H= 6 m

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

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