A first order conservation law formulation for fast solid...

Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions

A first order conservation law formulation for fastsolid dynamics in OpenFOAM

Jibran Haider a, Antonio J. Gil a, Javier Bonet a, Chun Hean Lee a, Antonio Huerta b

a Zienkiewicz Centre for Computational Engineering (ZC2E)College of Engineering, Swansea University, UK

b Laboratory of Computational Methods and Numerical Analysis (LaCáN)Polytechnic University of Catalonia, Spain

ACME 201523rd conference on Computational Mechanics, UK

Solids & Structures 5

Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015

Outline

1 IntroductionMotivationOpenFOAM

2 Reversible elastodynamicsGoverning equations

3 Numerical techniqueSpace-time discretisationContact fluxInvolution

4 Numerical resultsMesh convergence2D results3D results

5 Conclusions

Outline

5 Conclusions

Introduction

Motivation

• Fast solid dynamics

• Large strain deformation

• Shock propagation

• Impact mechanics

• Solid dynamics formulations

Displacementbased

formulations

Mixedformulation

• Incompressibility × X• Performance in

bending problems × X• Shock capturing

ability × X• Convergence of

stresses/strains × X• Conservation of

angular momentum X -

0 0.5 1

X-Coordinate

t=0.03s

-0.5 0 0.5 1 1.5

X-Coordinate

t=0.0006s

0 0.5 1

X-Coordinate

t=0.03s

-0.5 0 0.5 1 1.5

X-Coordinate

t=0.0006s

Introduction

Motivation

• Fast solid dynamics

• Large strain deformation

• Shock propagation

• Impact mechanics

• Solid dynamics formulations

Displacementbased

formulations

Mixedformulation

• Incompressibility × X• Performance in

bending problems × X• Shock capturing

ability × X• Convergence of

stresses/strains × X• Conservation of

angular momentum X -

0 0.5 1

X-Coordinate

t=0.03s

-0.5 0 0.5 1 1.5

X-Coordinate

t=0.0006s

0 0.5 1

X-Coordinate

t=0.03s

-0.5 0 0.5 1 1.5

X-Coordinate

t=0.0006s

Introduction

OpenFOAM

• An open source CFD software package in C++

• Based on Cell Centered Finite Volume Method

• Limited functionality for solid dynamics

v = 500 m/s

(0.5, 0.5, 0.5)

(−0.5,−0.5,−0.5)

Tensile rubber cube

Existing solid solver in OpenFOAM

× Linear elastic material

× Small strain deformation

Mixed Formulation in OpenFOAM

X Hyper elastic material

X Large strain deformation

[Animation - 3D]

Introduction

OpenFOAM

v = 500 m/s

(0.5, 0.5, 0.5)

(−0.5,−0.5,−0.5)

Tensile rubber cube

[Animation - 3D]

Introduction

OpenFOAM

v = 500 m/s

(0.5, 0.5, 0.5)

(−0.5,−0.5,−0.5)

Tensile rubber cube

[Animation - 3D]

Introduction

OpenFOAM

v = 500 m/s

(0.5, 0.5, 0.5)

(−0.5,−0.5,−0.5)

Tensile rubber cube

[Animation - 3D]

Outline

5 Conclusions

Governing equations

First order conservation formulation

• Conservation of linear momentum:∂ρ0v∂t−∇0 · P(F) = ρ0b

• Conservation of deformation gradient:∂F∂t−∇0 · (v⊗ I) = 0

• Conservation of energy:∂E∂t−∇0 ·

(PT v− Q

• Or in standard form:

∂U∂t

+∇0 ·F(U) = S

A constitutive model is needed to complete the coupled system A first order hyperbolic system similar to the one in CFD Our aim is to develop low order numerical schemes for Total Lagrangian fast solid dynamics

Governing equations

(PT v− Q

∂U∂t

+∇0 ·F(U) = S

Governing equations

(PT v− Q

∂U∂t

+∇0 ·F(U) = S

Governing equations

(PT v− Q

∂U∂t

+∇0 ·F(U) = S

Governing equations

(PT v− Q

∂U∂t

+∇0 ·F(U) = S

Outline

5 Conclusions

Finite Volume Methodology

Space-time discretisation

• The conservation laws are spatially semi-discretised by astandard Cell Centered Finite Volume Method(CCFVM):

dt= −

∑f∈e

[FCN ]f Af

• An explicit 2 step Total Variation DiminishingRunge-Kutta (TVD-RK) time integrator is utilised:

U∗n+1 = Un + ∆t Un

U∗n+2 = U∗n+1 + ∆t U∗n+1

Un+1 =12

(Un + U∗n+2

)with stability constraint:

∆t = αCFLhmin

Up,max

[FCN ]f

Just one Gauss point per face!

Finite Volume Methodology

Space-time discretisation

• The conservation laws are spatially semi-discretised by astandard Cell Centered Finite Volume Method(CCFVM):

dt= −

∑f∈e

[FCN ]f Af

• An explicit 2 step Total Variation DiminishingRunge-Kutta (TVD-RK) time integrator is utilised:

U∗n+1 = Un + ∆t Un

U∗n+2 = U∗n+1 + ∆t U∗n+1

Un+1 =12

(Un + U∗n+2

)with stability constraint:

∆t = αCFLhmin

Up,max

[FCN ]f

Just one Gauss point per face!

Riemann Solver

Generalised Riemann Solver

• Rankine-Hugoniot relation:

U J ρ0v K = −J P KN

U J F K = −J v K⊗ N

U J E K = −J PT v K · N

where JU K = U+c − U

• Linear reconstruction procedure to enhance thespatial discretisation:

U+,−c = Ue + Ge · (Xc − Xe)

where the local gradient operator is defined as:

[∑α∈e

νeα ⊗ νeα

]−1 ∑α∈e

(Uα − Ue

)νeα

() Time

Time = 0

= = −

Riemann Solver

U+,−c = Ue + Ge · (Xc − Xe)

[∑α∈e

νeα ⊗ νeα

]−1 ∑α∈e

(Uα − Ue

)νeα

() Time

Time = 0

= = −

Riemann Solver

U+,−c = Ue + Ge · (Xc − Xe)

[∑α∈e

νeα ⊗ νeα

]−1 ∑α∈e

(Uα − Ue

)νeα

() Time

Time = 0

= = −

Riemann Solver

Contact flux

• Contact flux at an interface:

FCN(U+,U−) =

−tC−vC ⊗ N−tC · vC

• Contact values for a homogeneous body(Up = U−p = U+

p , Us = U−s = U+s ):

vC =12

[v− + v+

]︸︷︷︸

unstable flux

Up(n ⊗ n) +

(I − n ⊗ n)] [

P+ − P−]

N︸︷︷︸stabilising term

tC =12

[P− + P+

]N︸︷︷︸

unstable flux

2[Up(n ⊗ n) + Us(I − n ⊗ n)] [v+ − v−]︸︷︷︸

stabilising term

v+ = 0, U+p = U+

s = ∞

v−(t)

v+n = 0, U+p = ∞, U+

v−(t)

v+t = 0, U+p = 0, U+

s = ∞

v−(t)

U+p = U+

v−(t)

Riemann Solver

Contact flux

• Contact flux at an interface:

FCN(U+,U−) =

−tC−vC ⊗ N−tC · vC

• Contact values for a homogeneous body(Up = U−p = U+

p , Us = U−s = U+s ):

vC =12

[v− + v+

]︸︷︷︸

unstable flux

Up(n ⊗ n) +

(I − n ⊗ n)] [

P+ − P−]

N︸︷︷︸stabilising term

tC =12

[P− + P+

]N︸︷︷︸

unstable flux

2[Up(n ⊗ n) + Us(I − n ⊗ n)] [v+ − v−]︸︷︷︸

stabilising term

v+ = 0, U+p = U+

s = ∞

v−(t)

v+n = 0, U+p = ∞, U+

v−(t)

v+t = 0, U+p = 0, U+

s = ∞

v−(t)

U+p = U+

v−(t)

Involution

Local constraint preserving scheme• Compatibility condition

[ Torrilhon(2005), Lee-Gil-Bonet(2013) ]:

∇0 × F = 0

• An adapted curl-free updated scheme:

dt=∑a∈e

[va ⊗∇0Na]

• Bilinear shape functions:

Na =18

(1 + ξξa)(1 + ηηa)(1 + µµa)

• Nodal velocity:

va = vRe + G(vC

) · (Xa − Xe)

where:

1∑f∈e

∑f∈e

[dfevC

]; dfe = |XC − Xe|

G(vC) =

∑f∈e

νfe ⊗ νfe

−1 ∑f∈e

vCf − vR

vaG (vC )

1 = Linear reconstruction + Riemann solver

2 = Velocity gradient (Least square minimisation)

3 = Interpolation/Extrapolation to nodes

Involution

∇0 × F = 0

dt=∑a∈e

[va ⊗∇0Na]

Na =18

(1 + ξξa)(1 + ηηa)(1 + µµa)

• Nodal velocity:

va = vRe + G(vC

) · (Xa − Xe)

where:

1∑f∈e

∑f∈e

[dfevC

]; dfe = |XC − Xe|

G(vC) =

∑f∈e

νfe ⊗ νfe

−1 ∑f∈e

vCf − vR

vaG (vC )

Involution

∇0 × F = 0

dt=∑a∈e

[va ⊗∇0Na]

Na =18

(1 + ξξa)(1 + ηηa)(1 + µµa)

• Nodal velocity:

va = vRe + G(vC

) · (Xa − Xe)

where:

1∑f∈e

∑f∈e

[dfevC

]; dfe = |XC − Xe|

G(vC) =

∑f∈e

νfe ⊗ νfe

−1 ∑f∈e

vCf − vR

vaG (vC )

Involution

∇0 × F = 0

dt=∑a∈e

[va ⊗∇0Na]

Na =18

(1 + ξξa)(1 + ηηa)(1 + µµa)

• Nodal velocity:

va = vRe + G(vC

) · (Xa − Xe)

where:

1∑f∈e

∑f∈e

[dfevC

]; dfe = |XC − Xe|

G(vC) =

∑f∈e

νfe ⊗ νfe

−1 ∑f∈e

vCf − vR

vaG (vC )

Involution

∇0 × F = 0

dt=∑a∈e

[va ⊗∇0Na]

Na =18

(1 + ξξa)(1 + ηηa)(1 + µµa)

• Nodal velocity:

va = vRe + G(vC

) · (Xa − Xe)

where:

1∑f∈e

∑f∈e

[dfevC

]; dfe = |XC − Xe|

G(vC) =

∑f∈e

νfe ⊗ νfe

−1 ∑f∈e

vCf − vR

vaG (vC )

Outline

5 Conclusions

1D Cable

Smooth Sinusoidal Loading: 1D mesh convergenceProblem description: Linear elastic material, ρ0 = 1 kg/m3, E = 1 GPa, ν = 0.3 αCFL = 0.5,

P(t) = 1× 10−3 [sin ((πt)/20− π/2) + 1] N

L = 10m

10−2 10−1 10010−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Grid Size (m)

Contact Velocity at t = 34.4757 sec

L1 norm error (Forward Euler, 1st Order)

L1 norm error (2−Step RK, 2nd Order)

L2 norm error (2−step RK, 2nd Order)

Slope = 1Slope = 2

10−2 10−1 10010−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Grid Size (m)

Contact Stress at t = 34.4757 sec

Slope = 1Slope = 2

X 1D convergence analysis by means of the L1 & L2 norm has been carried out at t ≈ 34.5 sX Demonstrates the expected accuracy of the available schemes for all variables

1D Cable

Smooth Sinusoidal Loading: 1D mesh convergenceProblem description: Linear elastic material, ρ0 = 1 kg/m3, E = 1 GPa, ν = 0.3 αCFL = 0.5,

P(t) = 1× 10−3 [sin ((πt)/20− π/2) + 1] N

L = 10m

10−2 10−1 10010−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Grid Size (m)

Contact Velocity at t = 34.4757 sec

Slope = 1Slope = 2

10−2 10−1 10010−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Grid Size (m)

Contact Stress at t = 34.4757 sec

Slope = 1Slope = 2

X 1D convergence analysis by means of the L1 & L2 norm has been carried out at t ≈ 34.5 sX Demonstrates the expected accuracy of the available schemes for all variables

2D Spinning Plate

Spinning Plate

Problem description: Unit side plate, nearly incompressible hyperelastic Neo-Hookean material,ρ0 = 1100 kg/m3, E = 17 MPa, ν = 0.45,Discretisation = 20×20 cells, ∆t = 1× 10−4s, αCFL = 0.5, ω0 = 105 rad/s

0 0.05 0.1 0.15 0.2−500

Time (sec)

.s−1

Linear momentum xLinear momentum yLinear momentum zAngular momentum

X Demonstrates the conservation of angular momentum

2D Spinning Plate

Spinning Plate

Problem description: Unit side plate, nearly incompressible hyperelastic Neo-Hookean material,ρ0 = 1100 kg/m3, E = 17 MPa, ν = 0.45,Discretisation = 20×20 cells, ∆t = 1× 10−4s, αCFL = 0.5, ω0 = 105 rad/s

0 0.05 0.1 0.15 0.2−500

Time (sec)

.s−1

Linear momentum xLinear momentum yLinear momentum zAngular momentum

X Demonstrates the conservation of angular momentum

3D Bending Column

Bending dominated scenarioProblem description: Nearly incompressible hyperelastic Neo-Hookean material, ρ0 = 1100 kg/m3,

E = 17 MPa, ν = 0.45, αCFL = 0.3, V = 10 m/s

L = 6m

v0 = [V y/L, 0, 0]T

[Animation - 3D]

X The formulation eliminates the bending difficulty

3D Bending Column

Bending dominated scenarioProblem description: Nearly incompressible hyperelastic Neo-Hookean material, ρ0 = 1100 kg/m3,

E = 17 MPa, ν = 0.45, αCFL = 0.3, V = 10 m/s

L = 6m

v0 = [V y/L, 0, 0]T

[Animation - 3D]

X The formulation eliminates the bending difficulty

3D Twisting Column

Highly non-linear problem

Problem description: Nearly incompressible hyperelastic Neo-Hookean material, ρ0 = 1100 kg/m3,E = 17 MPa, ν = 0.45, αCFL = 0.3, Ω = 105 rad/s

ω0 = [0,Ωsin(πy/2L), 0]T

(−0.5, 0,−0.5)

(0.5, 6, 0.5)

[Animation - 3D]

X Demonstrates the robustness of the numerical scheme

3D Twisting Column

Highly non-linear problem

Problem description: Nearly incompressible hyperelastic Neo-Hookean material, ρ0 = 1100 kg/m3,E = 17 MPa, ν = 0.45, αCFL = 0.3, Ω = 105 rad/s

ω0 = [0,Ωsin(πy/2L), 0]T

(−0.5, 0,−0.5)

(0.5, 6, 0.5)

[Animation - 3D]

X Demonstrates the robustness of the numerical scheme

Outline

5 Conclusions

Conclusions and further research

Conclusions• An upwind first order conservation law formulation based on the finite volume method, has been

presented for fast solid dynamic simulations within the OpenFOAM environment

• Linear elements can be used without usual volumetric and bending difficulties

• Velocities (or displacements) and stresses (or strains) display the same rate of convergence

On-going work• Introduction of an additional conservation law for the Jacobian of the deformation gradient

• Introduction of a plasticity model

References· 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 118 (2013) 13-38.

· M. Aguirre, A. J. Gil, J. Bonet and A. Arranz Carreño. A vertex centred Finite Volume Jameson-Schmidt-Turkel (JST)algorithm for a mixed conservation formulation in solid dynamics, Journal of Computational Physics, 259 (2014)672-699.

· A. J. Gil, C. H. Lee, J. Bonet and M. Aguirre. A stabilised Petrov-Galerkin formulation for linear tetrahedral elementsin compressible, nearly incompressible and truly incompressible fast dynamics, Computer Methods in AppliedMechanics and Engineering, 279 (2014) 659-690

· J. Bonet, A. J. Gil, C. H. Lee, M. Aguirre and R. Ortigosa. A first order hyperbolic framework for large straincomputational solid dynamics: Part 1 Total Lagrangian Isothermal Elasticity, 283 (2015) 689-732.

· J. Haider, A. J. Gil, J. Bonet and C. H. Lee. A first order conservation law formulation for fast solid dynamics inOpenFOAM, Journal of Computational Physics, In preparation

Conclusions and further research

Conclusions• An upwind first order conservation law formulation based on the finite volume method, has been

presented for fast solid dynamic simulations within the OpenFOAM environment

• Linear elements can be used without usual volumetric and bending difficulties

• Velocities (or displacements) and stresses (or strains) display the same rate of convergence

On-going work• Introduction of an additional conservation law for the Jacobian of the deformation gradient

• Introduction of a plasticity model

References· 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 118 (2013) 13-38.

· M. Aguirre, A. J. Gil, J. Bonet and A. Arranz Carreño. A vertex centred Finite Volume Jameson-Schmidt-Turkel (JST)algorithm for a mixed conservation formulation in solid dynamics, Journal of Computational Physics, 259 (2014)672-699.

· A. J. Gil, C. H. Lee, J. Bonet and M. Aguirre. A stabilised Petrov-Galerkin formulation for linear tetrahedral elementsin compressible, nearly incompressible and truly incompressible fast dynamics, Computer Methods in AppliedMechanics and Engineering, 279 (2014) 659-690

· J. Bonet, A. J. Gil, C. H. Lee, M. Aguirre and R. Ortigosa. A first order hyperbolic framework for large straincomputational solid dynamics: Part 1 Total Lagrangian Isothermal Elasticity, 283 (2015) 689-732.

· J. Haider, A. J. Gil, J. Bonet and C. H. Lee. A first order conservation law formulation for fast solid dynamics inOpenFOAM, Journal of Computational Physics, In preparation

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