USNCCM13

Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions

A Total Lagrangian Hydrocode For Linear TetrahedralElements In Compressible, Nearly Incompressible and Truly

Incompressible Fast Solid Dynamics

Chun Hean Lee1, Antonio J. Gil, Javier Bonet

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

13th U.S. National Congress on Computational Mechanics

Advanced Finite Elements for Complex-Geometry Computations: Tetrahedral Algorithms and Related Methods

1 http://www.swansea.ac.uk/staff/academic/engineering/leeheanchun/

CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

Outline

1 Motivation

2 Reversible elastodynamicsBalance principlesHydrodynamics formulation

3 Petrov-Galerkin formulationPetrov-Galerkin spatial discretisationTemporal discretisation

4 Numerical resultsSwinging cubeL-shaped blockTwisting columnTaylor impact bar

5 Conclusions and further research

Outline

1 Motivation

Motivation[Bending-mixed formulation]

Fast transient solid dynamics• Wide variety of industrial applications

• Explicit displacement based softwares (ANSYS, AltairHyperWorks, LS-DYNA, ABAQUS, . . .)

• Linear tetrahedral elements attractive: lowcomputational cost + meshing, but...· Poor bending behaviour· Hourglassing and pressure instabilities· First order for strains and stresses· Difficulties for shock capturing

• In contrast in the CFD community:· Robust techniques for linear tetrahedra· Equal orders in velocity and pressure· Robust shock capturing

• Aims:· First order conservation laws for solid dynamics· Adapt CFD technology to the proposed formulation· Introduce hydrodynamics frameworkCHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

Outline

1 Motivation

Large strain kinematics: F,H, J

)t,X(φ=x

JdV=dv

XdF=xd

AdH=ad

F = ∇0x; H = JF−T; J = detF

Balance principles

First order conservation formulation

• Consider the standard dynamic equilibrium equation:

∂p∂t− DIVP = ρ0b

• Supplemented with a set of geometric conservation laws [Bonet et al., 2015]:

∂F∂t− DIV

p⊗ I)

∂H∂t− CURL

∂J∂t− DIV

P = P (F, . . .) =∂Ψ(F, . . .)

∂FHowever, energy function is not convex

Balance principles

∂F∂t− DIV

p⊗ I)

∂H∂t− CURL

∂J∂t− DIV

P = P (F, . . .) =∂Ψ(F, . . .)

Balance principles

∂F∂t− DIV

p⊗ I)

∂H∂t− CURL

∂J∂t− DIV

• With Involutions:CURLF = 0; DIVH = 0

• Alternatively:∂U∂t

+ DIVF = S

Balance principles

∂F∂t− DIV

p⊗ I)

∂H∂t− CURL

∂J∂t− DIV

P = P (F, . . .) =∂Ψ(F, . . .)

Balance principles

Polyconvex elasticity

• Large strain polyconvex strain energy function [Ball, 1976] satisfy:

Ψ(∇0x) = W(F,H, J)

dx = F dXda = H dAdv = J dV

W is convex with F, H and J1x,1X

)t,X(φ=x

JdV=dv

XdF=xd

AdH=ad

• Nearly incompressible models can be derived using isochoric components of Fand H [Schroder et al., 2011]:

· Mooney Rivlin:

W = αJ−2/3F : F + βJ−2(H : H)3/2 + U(J)

· Neo Hookean:

2J−2/3F : F + U(J); U(J) =

2(J − 1)2

• Compressible Neo Hookean and Mooney Rivlin models [Bonet et al., 2015]

Balance principles

Polyconvex elasticity

• Large strain polyconvex strain energy function [Ball, 1976] satisfy:

Ψ(∇0x) = W(F,H, J)

dx = F dXda = H dAdv = J dV

W is convex with F, H and J1x,1X

)t,X(φ=x

JdV=dv

XdF=xd

AdH=ad

• Nearly incompressible models can be derived using isochoric components of Fand H [Schroder et al., 2011]:

· Mooney Rivlin:

W = αJ−2/3F : F + βJ−2(H : H)3/2 + U(J)

· Neo Hookean:

2J−2/3F : F + U(J); U(J) =

2(J − 1)2

• Compressible Neo Hookean and Mooney Rivlin models [Bonet et al., 2015]

Hydrodynamics formulation

Convex entropy extension

• Consider the following convex entropy function:

S(p,F,H, J) =1

2ρ0p · p + W(F,H, J)

• Define the set of entropy variables:

V =∂S∂U

ΣFΣHΣJ

[HS] =∂V∂U

=∂2S

∂U∂U=

0 [HW ]

S(p,F,H, J) =1

2ρ0p · p + W(F,H, J)

V =∂S∂U

ΣFΣHΣJ

[HS] =∂V∂U

=∂2S

∂U∂U=

0 [HW ]

S(p,F,H, J) =1

2ρ0p · p + W(F,H, J)

V =∂S∂U

ΣFΣHΣJ

• And a symmetric positive definite Hessian operator:

[HS] =∂V∂U

=∂2S

∂U∂U=

0 [HW ]

WFF WFH WFJ

WHF WHH WHJ

WJF WJH WJJ

S(p,F,H, J) =1

2ρ0p · p + W(F,H, J)

V =∂S∂U

ΣFΣHΣJ

• Compressible Mooney Rivlin:

[HS] =∂V∂U

=∂2S

∂U∂U=

0 [HW ]

WFF 0 0

0 WHH 0

0 0 WJJ

S(p,F,H, J) =1

2ρ0p · p + W(F,H, J)

V =∂S∂U

ΣFΣHΣJ

• Nearly incompressible Mooney Rivlin:

[HS] =∂V∂U

=∂2S

∂U∂U=

0 [HW ]

WFF 0 WFJ

0 WHH WHJ

WJF WJH WJJ

Entropy system

• The system of conservation laws can be written in terms of entropy variables:

∂V∂t

+ [HS] DIVF = [HS]S; [DIVF ]α =∂FαI

• For Mooney Rivlin material and the use of ΣJ = ΣJ + p:

∂v∂t

DIVP +1ρ0

∂ΣF

∂t= (WFF + WFJ ⊗ HΣ) : ∇0v

∂ΣH

∂t= (WHH FΣ + WHJ ⊗ HΣ) : ∇0v

∂ΣJ

∂t=(WJF + WJH FΣ + WJJHΣ

): ∇0v

∂p∂t

= κ (HΣ : ∇0v)

• The first Piola-Kirchhoff stress:

P = P(ΣF,ΣH, ΣJ , p)

Entropy system

∂V∂t

∂v∂t

DIVP +1ρ0

∂ΣF

∂t= (WFF + WFJ ⊗ HΣ) : ∇0v

∂ΣH

∂ΣJ

): ∇0v

∂p∂t

= κ (HΣ : ∇0v)

Entropy system

∂V∂t

∂v∂t

DIVP +1ρ0

∂ΣF

∂t= (WFF + WFJ ⊗ HΣ) : ∇0v

∂ΣH

∂ΣJ

): ∇0v

∂p∂t

= κ (HΣ : ∇0v)

Entropy system

∂V∂t

∂v∂t

DIVP +1ρ0

∂ΣF

∂t= (WFF + WFJ ⊗ HΣ) : ∇0v

∂ΣH

∂ΣJ

): ∇0v

∂p∂t

= κ (HΣ : ∇0v)

Outline

1 Motivation

Petrov-Galerkin spatial discretisation

Stabilised Petrov-Galerkin formulation

• Standard Bubnov-Galerkin weak formulation (unstable) [Morgan and Peraire, 1998]:∫V0

δU ·R dV = 0; R = [HS]S − [HS] (DIVF)−∂V∂t

• Stabilised Petrov Galerkin [Hughes et al., 1986]:

δU st ·R dV = 0; δU st = δU + τ∂F I

∂U∂δU∂XI

; δU st =

• Assuming τ a diagonal matrix gives:

δpst = δp− τpDIVδP(δF, δH, δJ)

δFst = δF−τF

ρ0(∇0δp)

δHst = δH −τH

ρ0(FΣ ∇0δp)

δJst = δJ −τJ

ρ0(HΣ : ∇0δp)

• Standard Bubnov-Galerkin is recovered by setting τ = 0CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

∂U∂δU∂XI

; δU st =

δFst = δF−τF

ρ0(∇0δp)

δHst = δH −τH

ρ0(FΣ ∇0δp)

δJst = δJ −τJ

ρ0(HΣ : ∇0δp)

∂U∂δU∂XI

; δU st =

δFst = δF−τF

ρ0(∇0δp)

δHst = δH −τH

ρ0(FΣ ∇0δp)

δJst = δJ −τJ

ρ0(HΣ : ∇0δp)

∂U∂δU∂XI

; δU st =

δFst = δF−τF

ρ0(∇0δp)

δHst = δH −τH

ρ0(FΣ ∇0δp)

δJst = δJ −τJ

ρ0(HΣ : ∇0δp)

Petrov Galerkin spatial discretisation

• Using linear tetrahedra for entropy variables and its virtual conjugates:

v =4∑

vaNa; δp =4∑

δpaNa; ΣF =4∑

ΣaFNa; δF =

4∑a=1

δFaNa; . . .

• Gives:

Mabvb =

ρ0f 0 dV +

∫∂V

ρ0tB dA−

P(Σst

F,ΣstH,Σ

stJ , p

st)∇0Na dV

MabΣbF =

Na (WFF + WFJ ⊗ HΣ) : ∇0v dV

MabΣbH =

Na (WHH FΣ + WHJ ⊗ HΣ) : ∇0v dV

Mab˙Σb

Na(WJF + WJH FΣ + WJJHΣ

): ∇0v dV

Mabpb =

∫∂V

Na κ vB · (HΣN) dA−∫

Vκ vst · (HΣ∇0Na) dV

v =4∑

vaNa; δp =4∑

δpaNa; ΣF =4∑

ΣaFNa; δF =

4∑a=1

δFaNa; . . .

• Gives:

Mabvb =

ρ0f 0 dV +

∫∂V

ρ0tB dA−

P(Σst

F,ΣstH,Σ

stJ , p

st)∇0Na dV

MabΣbF =

MabΣbH =

Mab˙Σb

): ∇0v dV

Mabpb =

∫∂V

v =4∑

vaNa; δp =4∑

δpaNa; ΣF =4∑

ΣaFNa; δF =

4∑a=1

δFaNa; . . .

• Gives:

Mabvb =

ρ0f 0 dV +

∫∂V

ρ0tB dA−

P(Σst

F,ΣstH,Σ

stJ , p

st)∇0Na dV

MabΣbF =

MabΣbH =

Mab˙Σb

): ∇0v dV

Mabpb =

∫∂V

Petrov Galerkin stabilisation

• The stabilised entropy variables are:

ΣstF = ΣF + τF

[(WFF + WFJ ⊗ HΣ) : ∇0v−

∂ΣF

]+ ζF [WFF : (Fx − FΣ)]

ΣstH = ΣH + τH

[(WHH FΣ + WHJ ⊗ HΣ) : ∇0v−

∂ΣH

]+ ζH [WHH : (Hx − HΣ)]

ΣstJ = ΣJ + τJ

[(WJF + WJH FΣ + WJJHΣ

): ∇0v−

∂ΣJ

]+ ζJ

[WJJ : (Jx − JΣ)

]pst = p + τp

[κ (HΣ : ∇0v)−

∂p∂t

]+ ζp

[Jx − 1−

]vst = v + τv

DIVP +1ρ0

f 0 −∂v∂t

• To reduce implicitness of the formulation:

τp = 0; τF = {τH, τJ}; ζF = ζH; ζJ = ζp

• In practice only four stabilising parameters involved: {τv, τF, ζF, ζJ}

ΣstF = ΣF + τF

[(WFF + WFJ ⊗ HΣ) : ∇0v−

∂ΣF

]+ ζF [WFF : (Fx − FΣ)]

ΣstH = ΣH + τH

∂ΣH

]+ ζH [WHH : (Hx − HΣ)]

ΣstJ = ΣJ + τJ

): ∇0v−

∂ΣJ

]+ ζJ

[WJJ : (Jx − JΣ)

]pst = p + τp

[κ (HΣ : ∇0v)−

∂p∂t

]+ ζp

[Jx − 1−

]vst = v + τv

DIVP +1ρ0

f 0 −∂v∂t

ΣstF = ΣF + τF

[(WFF + WFJ ⊗ HΣ) : ∇0v−

∂ΣF

]+ ζF [WFF : (Fx − FΣ)]

ΣstH = ΣH + τH

∂ΣH

]+ ζH [WHH : (Hx − HΣ)]

ΣstJ = ΣJ + τJ

): ∇0v−

∂ΣJ

]+ ζJ

[WJJ : (Jx − JΣ)

]pst = p + τp

[κ (HΣ : ∇0v)−

∂p∂t

]+ ζp

[Jx − 1−

]vst = v + τv

DIVP +1ρ0

f 0 −∂v∂t

Time Integration

• Integration in time is achieved by means of an explicit two-stage Total VariationDiminishing Runge-Kutta time integrator:

V(1)n+1 = Vn + ∆tVn

V(2)n+2 = V(1)

n+1 + ∆tV(1)n+1

Vn+1 =12

(Vn + V(2)

)with a stability constraint

∆t = αCFLhmin

; Unmax = max

• Geometry increment:

xn+1 = xn +∆t2

(vn + vn+1)

• Angular momentum conserving algorithm is introduced [Lee et al., 2014]

• Fractional time stepping used for truly incompressible materials [Gil et al., 2014]:

Predict V int −→ Compute pn+1 via a Poisson-like equation −→ Update Vn+1

Time Integration

• Integration in time is achieved by means of an explicit two-stage Total VariationDiminishing Runge-Kutta time integrator:

V(1)n+1 = Vn + ∆tVn

V(2)n+2 = V(1)

n+1 + ∆tV(1)n+1

Vn+1 =12

(Vn + V(2)

)with a stability constraint

∆t = αCFLhmin

; Unmax = max

• Geometry increment:

xn+1 = xn +∆t2

(vn + vn+1)

• Angular momentum conserving algorithm is introduced [Lee et al., 2014]

• Fractional time stepping used for truly incompressible materials [Gil et al., 2014]:

Predict V int −→ Compute pn+1 via a Poisson-like equation −→ Update Vn+1

Outline

1 Motivation

Swinging cube

Mesh convergence analysis

Convergence behaviour

• Analytical displacement field

u = U0 cos

2cdπt

)A sin

)cos(πX2

)cos(πX3

)B cos

)sin(πX2

)cos(πX3

)C cos

)cos(πX2

)sin(πX3

• Symmetric BC at X1 = 0, X2 = 0 and X3 = 0

• Skew symmetric BC at X1 = 1, X2 = 1 and X3 = 1

• Parameters: A = 2, B = −1, C = −1, U0 = 5 × 10−4

• E = 0.017 GPa, ρ0 = 1100 kg/m3, ν = 0.3 and cd =√

L-shaped block

Angular momentum analysis

Problem description: Materials ρ0 = 1000 kg/m3, E = 5.005× 104 Pa, ν = 0.5,αCFL = 0.3, η0 = [150, 300, 450]T .

T(3,3,3)

T(0,10,3)

T(6,0,0)

[Hydrocode-L Shaped Block]

F1(t) = −F2(t) =

tη0, 0 ≤ t < 2.5(5 − t)η0, 2.5 ≤ t < 50, t ≥ 5

Truly incompressible NH model

Preservation of momentum within a system

Linear momentum Angular momentum

Twisting column

Robustness of the methodology

Problem description: Column 1× 1× 6, ρ0 = 1100 kg/m3, E = 0.017 GPa, ν = 0.499,αCFL = 0.3, lumped mass.

T(1,1,6)

T(1,1,0)

v0= ω × X; ω =

(0, 0, 100 sin

Nearly incompressible MR modelHigh nonlinear problemLocking-free behaviour

[Hydrocode-Twisting Column]

HuWashizu P1/P1 Hex. Hydrocode

Pressure instabilities in P1/P1 Hexahedra

Taylor impact bar

Pressure instability

r0 = 0.0032 m and L0 = 0.0324 m

Young’s modulus E = 117 GPa

Density ρ0 = 8930 kg/m3

Velocity V0 = 1000 m/s

Truly incompressible MR model

Avoidance of volumetric locking

Eliminate non-physical hydrostatic pressure fluctuations

[Hydrocode-Taylor Impact]

τv = 0

τv = 0.2∆t

With and without velocity correction in pressure evolution

Outline

1 Motivation

Conclusions and further research

Conclusions

• A stabilised hydrocode is presented for solid dynamics in large deformations

• Linear tetrahedra can be used without volumetric and bending difficulties

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

On-going works

• Shock capturing technique [Scovazzi et al., 2007]

• Thermoelasticity

• Updated Lagrangian Hydrocode [Scovazzi et al., 2012]

• Fracture and explosion modelling

Publications

Journal publications[J-1] 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.

[J-2] I. A. Karim, C. H. Lee, A. J. Gil and J. Bonet. A Two-Step Taylor Galerkin formulation for fast dynamics,Engineering Computations 31 (2014) 366-387.

[J-3] C. H. Lee, A. J. Gil and J. Bonet. Development of a stabilised Petrov-Galerkin formulation for a mixedconservation law formulation in fast solid dynamics, CMAME 268 (2013) 40-64.

[J-4] 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, JCP 259 (2014) 672-699.

[J-5] A. J. Gil, C. H. Lee, J. Bonet and M. Aguirre. A stabilised Petrov-Galerkin formulation for linear tetrahedralelements in compressible, nearly incompressible and truly incompressible fast dynamics, CMAME 276 (2014)659-690.

[J-6] J. Bonet, A. J. Gil, C. H. Lee, M. Aguirre and R. Ortigosa. A first order hyperbolic framework for large straincomputational solid dynamics. Part I: Total Lagrangian Isothermal Elasticity, CMAME 283 (2015) 689-732.

[J-7] M. Aguirre, A. J. Gil, J. Bonet and C. H. Lee. An edge based vertex centred upwind finite volume method forLagrangian solid dynamics. JCP. In Press. DOI:10.1016/j.jcp.2015.07.029.

Under review

[U-1] A. J. Gil, C. H. Lee, J. Bonet and R. Ortigosa. A first order hyperbolic framework for large strain computationalsolid dynamics. Part II: Total Lagrangian compressible, nearly Incompressible and truly incompressibleelasticity. CMAME. Under review.

[U-2] A. J. Gil and R. Ortigosa. A new framework for polyconvex large strain electromechanics: Variationalformulation and material characterisation. CMAME. Under review.

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