USNCCM13

1. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions A Total Lagrangian Hydrocode For Linear Tetrahedral Elements 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

2. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions Outline 1 Motivation 2 Reversible elastodynamics Balance principles Hydrodynamics formulation 3 Petrov-Galerkin formulation Petrov-Galerkin spatial discretisation Temporal discretisation 4 Numerical results Swinging cube L-shaped block Twisting column Taylor impact bar 5 Conclusions and further research CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

4. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions Motivation [Bending-mixed formulation] Fast transient solid dynamics • Wide variety of industrial applications • Explicit displacement based softwares (ANSYS, Altair HyperWorks, LS-DYNA, ABAQUS, . . .) • Linear tetrahedral elements attractive: low computational 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 framework CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

8. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions Large strain kinematics: F, H, J 1x,1X 3x,3X 2x,2X )t,X(φ=x dV JdV=dv Xd XdF=xd AdH=ad Ad F = 0x; H = JF −T ; J = detF CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

9. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions 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 1 ρ0 p ⊗ I = 0 ∂H ∂t − CURL 1 ρ0 p F = 0 ∂J ∂t − DIV 1 ρ0 HT p = 0 P = P (F, . . .) = ∂Ψ(F, . . .) ∂F However, energy function is not convex CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

11. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions 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 1 ρ0 p ⊗ I = 0 ∂H ∂t − CURL 1 ρ0 p F = 0 ∂J ∂t − DIV 1 ρ0 HT p = 0 • With Involutions: CURLF = 0; DIVH = 0 • Alternatively: ∂U ∂t + DIVF = S CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

13. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions Balance principles Polyconvex elasticity • Large strain polyconvex strain energy function [Ball, 1976] satisfy: Ψ( 0x) = W(F, H, J) dx = F dX da = H dA dv = J dV W is convex with F, H and J 1x,1X 3x,3X 2x,2X )t,X(φ=x dV JdV=dv Xd XdF=xd AdH=ad Ad • Nearly incompressible models can be derived using isochoric components of F and H [Schroder et al., 2011]: · Mooney Rivlin: W = αJ−2/3 F : F + βJ−2 (H : H)3/2 + U(J) · Neo Hookean: W = µ 2 J−2/3 F : F + U(J); U(J) = κ 2 (J − 1)2 • Compressible Neo Hookean and Mooney Rivlin models [Bonet et al., 2015] CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

14. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions Balance principles Polyconvex elasticity • Large strain polyconvex strain energy function [Ball, 1976] satisfy: Ψ( 0x) = W(F, H, J) dx = F dX da = H dA dv = J dV W is convex with F, H and J 1x,1X 3x,3X 2x,2X )t,X(φ=x dV JdV=dv Xd XdF=xd AdH=ad Ad • Nearly incompressible models can be derived using isochoric components of F and H [Schroder et al., 2011]: · Mooney Rivlin: W = αJ−2/3 F : F + βJ−2 (H : H)3/2 + U(J) · Neo Hookean: W = µ 2 J−2/3 F : F + U(J); U(J) = κ 2 (J − 1)2 • Compressible Neo Hookean and Mooney Rivlin models [Bonet et al., 2015] CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

15. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions Hydrodynamics formulation Convex entropy extension • Consider the following convex entropy function: S(p, F, H, J) = 1 2ρ0 p · p + W(F, H, J) • Deﬁne the set of entropy variables: V = ∂S ∂U =     v ΣF ΣH ΣJ     [HS] = ∂V ∂U = ∂2S ∂U∂U =   1 ρ0 I 0 0 [HW ]   =   1 ρ0 I 0 0   CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

16. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions Hydrodynamics formulation Convex entropy extension • Consider the following convex entropy function: S(p, F, H, J) = 1 2ρ0 p · p + W(F, H, J) • Deﬁne the set of entropy variables: V = ∂S ∂U =     v ΣF ΣH ΣJ     [HS] = ∂V ∂U = ∂2S ∂U∂U =   1 ρ0 I 0 0 [HW ]   =   1 ρ0 I 0 0   CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

17. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions Hydrodynamics formulation Convex entropy extension • Consider the following convex entropy function: S(p, F, H, J) = 1 2ρ0 p · p + W(F, H, J) • Deﬁne the set of entropy variables: V = ∂S ∂U =     v ΣF ΣH ΣJ     • And a symmetric positive deﬁnite Hessian operator: [HS] = ∂V ∂U = ∂2S ∂U∂U =   1 ρ0 I 0 0 [HW ]   =           1 ρ0 I 0 0 WFF WFH WFJ WHF WHH WHJ WJF WJH WJJ           CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

18. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions Hydrodynamics formulation Convex entropy extension • Consider the following convex entropy function: S(p, F, H, J) = 1 2ρ0 p · p + W(F, H, J) • Deﬁne the set of entropy variables: V = ∂S ∂U =     v ΣF ΣH ΣJ     • Compressible Mooney Rivlin: [HS] = ∂V ∂U = ∂2S ∂U∂U =   1 ρ0 I 0 0 [HW ]   =           1 ρ0 I 0 0 WFF 0 0 0 WHH 0 0 0 WJJ           CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

19. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions Hydrodynamics formulation Convex entropy extension • Consider the following convex entropy function: S(p, F, H, J) = 1 2ρ0 p · p + W(F, H, J) • Deﬁne the set of entropy variables: V = ∂S ∂U =     v ΣF ΣH ΣJ     • Nearly incompressible Mooney Rivlin: [HS] = ∂V ∂U = ∂2S ∂U∂U =   1 ρ0 I 0 0 [HW ]   =           1 ρ0 I 0 0 WFF 0 WFJ 0 WHH WHJ WJF WJH WJJ           CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

20. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions Hydrodynamics formulation Entropy system • The system of conservation laws can be written in terms of entropy variables: ∂V ∂t + [HS] DIVF = [HS] S; [DIVF]α = ∂FαI ∂XI • For Mooney Rivlin material and the use of ΣJ = ˆΣJ + p: ∂v ∂t = 1 ρ0 DIVP + 1 ρ0 f0 ∂ΣF ∂t = (WFF + WFJ ⊗ HΣ) : 0v ∂ΣH ∂t = (WHH FΣ + WHJ ⊗ HΣ) : 0v ∂ ˆΣJ ∂t = WJF + WJH FΣ + ˆWJJHΣ : 0v ∂p ∂t = κ (HΣ : 0v) • The ﬁrst Piola-Kirchhoff stress: P = P(ΣF, ΣH, ˆΣJ, p) CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

25. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions 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]: V0 δUst · R dV = 0; δUst = δU + τ ∂FI ∂U ∂δU ∂XI ; δUst =     δpst δFst δHst δJst     • 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 = 0 CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

29. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions Petrov-Galerkin spatial discretisation Petrov Galerkin spatial discretisation • Using linear tetrahedra for entropy variables and its virtual conjugates: v = 4 a=1 vaNa; δp = 4 a=1 δpaNa; ΣF = 4 a=1 Σa FNa; δF = 4 a=1 δFaNa; . . . • Gives: b Mab ˙vb = V Na ρ0 f0 dV + ∂V Na ρ0 tB dA − V 1 ρ0 P Σst F, Σst H, Σst J , pst 0Na dV b Mab ˙Σb F = V Na (WFF + WFJ ⊗ HΣ) : 0v dV b Mab ˙Σb H = V Na (WHH FΣ + WHJ ⊗ HΣ) : 0v dV b Mab ˙ˆΣb J = V Na WJF + WJH FΣ + ˆWJJHΣ : 0v dV b Mab ˙pb = ∂V Na κ vB · (HΣN) dA − V κ vst · (HΣ 0Na) dV CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

32. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions Petrov-Galerkin spatial discretisation Petrov Galerkin stabilisation • The stabilised entropy variables are: Σst F = ΣF + τF (WFF + WFJ ⊗ HΣ) : 0v − ∂ΣF ∂t + ζF [WFF : (Fx − FΣ)] Σst H = ΣH + τH (WHH FΣ + WHJ ⊗ HΣ) : 0v − ∂ΣH ∂t + ζH [WHH : (Hx − HΣ)] Σst J = ΣJ + τJ WJF + WJH FΣ + ˆWJJHΣ : 0v − ∂ ˆΣJ ∂t + ζJ ˆWJJ : (Jx − JΣ) pst = p + τp κ (HΣ : 0v) − ∂p ∂t + ζp Jx − 1 − p κ vst = v + τv 1 ρ0 DIVP + 1 ρ0 f0 − ∂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} CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

35. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions Time Integration Time Integration • Integration in time is achieved by means of an explicit two-stage Total Variation Diminishing Runge-Kutta time integrator: V (1) n+1 = Vn + ∆t ˙Vn V (2) n+2 = V (1) n+1 + ∆t ˙V (1) n+1 Vn+1 = 1 2 Vn + V (2) n+2 with a stability constraint ∆t = αCFL hmin Un max ; Un max = max a Un p,a • Geometry increment: xn+1 = xn + ∆t 2 (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 Vint −→ Compute pn+1 via a Poisson-like equation −→ Update Vn+1 CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

36. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions Time Integration Time Integration • Integration in time is achieved by means of an explicit two-stage Total Variation Diminishing Runge-Kutta time integrator: V (1) n+1 = Vn + ∆t ˙Vn V (2) n+2 = V (1) n+1 + ∆t ˙V (1) n+1 Vn+1 = 1 2 Vn + V (2) n+2 with a stability constraint ∆t = αCFL hmin Un max ; Un max = max a Un p,a • Geometry increment: xn+1 = xn + ∆t 2 (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 Vint −→ Compute pn+1 via a Poisson-like equation −→ Update Vn+1 CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

38. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions Swinging cube Mesh convergence analysis Convergence behaviour • Analytical displacement ﬁeld u = U0 cos √ 3 2 cdπt        A sin πX1 2 cos πX2 2 cos πX3 2 B cos πX1 2 sin πX2 2 cos πX3 2 C cos πX1 2 cos πX2 2 sin πX3 2        • 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 = µ ρ0 CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

39. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions 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 . 1X 2X 3X T(3,3,3) T(0,10,3) T(6,0,0) )t(1F )t(2F [Hydrocode-L Shaped Block] F1(t) = −F2(t) =    tη0, 0 ≤ t < 2.5 (5 − t)η0, 2.5 ≤ t < 5 0, t ≥ 5 Truly incompressible NH model Preservation of momentum within a system Linear momentum Angular momentum CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

40. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions 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) 3X 2X1X 0ω v 0 = ω × X; ω = 0, 0, 100 sin πX3 2L T Nearly incompressible MR model High nonlinear problem Locking-free behaviour [Hydrocode-Twisting Column] HuWashizu P1/P1 Hex. Hydrocode Pressure instabilities in P1/P1 Hexahedra CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

41. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions Taylor impact bar Pressure instability 0V = 03X 0L 0r 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 CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

43. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions 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 CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

44. Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions 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 mixed conservation 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 tetrahedral elements 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 strain computational 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 for Lagrangian 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 computational solid dynamics. Part II: Total Lagrangian compressible, nearly Incompressible and truly incompressible elasticity. CMAME. Under review. [U-2] A. J. Gil and R. Ortigosa. A new framework for polyconvex large strain electromechanics: Variational formulation and material characterisation. CMAME. Under review. CHL-AJG-JB (Advanced Finite Elements for Complex-Geometry Computations) 26th - 30th July 2015

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