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Introduction Governing equations Numerical methodology Results Conclusions
Large strain computational solid dynamics:
An u...
Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Gover...
Introduction Governing equations Numerical methodology Results Conclusions
Fast transient solid dynamics
Displacement base...
Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Gover...
Introduction Governing equations Numerical methodology Results Conclusions
Total Lagrangian formulation
Conservation laws
...
Introduction Governing equations Numerical methodology Results Conclusions
Hyperbolic system
First order conservation laws...
Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Gover...
Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Gover...
Introduction Governing equations Numerical methodology Results Conclusions
Spatial discretisation
Conservation equations f...
Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Gover...
Introduction Governing equations Numerical methodology Results Conclusions
Lagrangian contact dynamics
Rankine-Hugoniot ju...
Introduction Governing equations Numerical methodology Results Conclusions
Acoustic Riemann solver
Jump condition for line...
Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Gover...
Introduction Governing equations Numerical methodology Results Conclusions
Godunov-type FVM
Standard FV update (CURL F = 0...
Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Gover...
Introduction Governing equations Numerical methodology Results Conclusions
Time integration
Two stage Runge-Kutta time int...
Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Gover...
Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Gover...
Introduction Governing equations Numerical methodology Results Conclusions
Low dispersion cube
X, x
Y, y
Z, z
(0, 0, 0)
(1...
Introduction Governing equations Numerical methodology Results Conclusions
Low dispersion cube: Mesh convergence
Velocity ...
Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Gover...
Introduction Governing equations Numerical methodology Results Conclusions
Twisting column
X, x
Y, y
(−0.5, 0, 0.5)
(0.5, ...
Introduction Governing equations Numerical methodology Results Conclusions
Comparison of various alternative numerical sch...
Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Gover...
Introduction Governing equations Numerical methodology Results Conclusions
Taylor impact
X, x
Y, y
v0
(−0.0032, 0, 0)
(0.0...
Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Gover...
Introduction Governing equations Numerical methodology Results Conclusions
Bar rebound
X, x
Y, y
v0
(−0.0032, 0, 0)
(0.003...
Introduction Governing equations Numerical methodology Results Conclusions
Bar rebound
X, x
Y, y
v0
(−0.0032, 0, 0)
(0.003...
Introduction Governing equations Numerical methodology Results Conclusions
Torus impact
[Torus impact]
Jibran Haider (Swan...
Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Gover...
Introduction Governing equations Numerical methodology Results Conclusions
Conclusions and further research
Conclusions
• ...
Introduction Governing equations Numerical methodology Results Conclusions
References
Published / accepted
• J. Haider, C....
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Large strain computational solid dynamics: An upwind cell centred Finite Volume Method

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Presented our research at the 12th World Congress on Computational Mechanics (WCCM) and 6th Asia Pacific Congress on Computational Mechanics (APCOM) at the COEX Convention Center in Seoul, Korea.

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Large strain computational solid dynamics: An upwind cell centred Finite Volume Method

  1. 1. Introduction Governing equations Numerical methodology Results Conclusions Large strain computational solid dynamics: An upwind cell centred Finite Volume Method Jibran Haider a, b , Chun Hean Lee a , Antonio J. Gil a , Javier Bonet c & Antonio Huerta b a Zienkiewicz Centre for Computational Engineering (ZCCE), College of Engineering, Swansea University, UK b Laboratory of Computational Methods and Numerical Analysis (LaCàN), Universitat Politèchnica de Catalunya (UPC BarcelonaTech), Spain c University of Greenwich, London, UK World Congress in Computational Mechanics (24th - 29th July 2016) MS 703: Advances in Finite Element Methods for Tetrahedral Mesh Computations http://www.jibranhaider.weebly.com Funded by the Erasmus Mundus Programme and International Association for Computational Mechanics Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 1
  2. 2. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 2
  3. 3. Introduction Governing equations Numerical methodology Results Conclusions Fast transient solid dynamics Displacement based FEM/FVM formulations • Linear tetrahedral elements suffer from: × Locking in nearly incompressible materials. × First order for stresses and strains. × Poor performance in shock scenarios. Proposed mixed formulation [Haider et al., 2016] • First order conservation laws similar to the one used in CFD community. • Entitled TOtal Lagrangian Upwind Cell-centred FVM for Hyperbolic conservation laws (TOUCH). Programmed in the open-source CFD software OpenFOAM. 0 0.5 1 0 0.5 1 1.5 X-Coordinate Y-Coordinate t=0.03s -1 -0.5 0 0.5 1x 10 7 -0.5 0 0.5 1 1.5 0 0.5 1 1.5 X-Coordinate Y-Coordinate t=0.0006s Q1-P0 FEM 0 0.5 1 0 0.5 1 1.5 X-Coordinate Y-Coordinate t=0.03s -1 -0.5 0 0.5 1x 10 7 -0.5 0 0.5 1 1.5 0 0.5 1 1.5 X-Coordinate Y-Coordinate t=0.0006s Upwind FVM Aim is to bridge the gap between CFD and computational solid dynamics. Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 3
  4. 4. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 4
  5. 5. Introduction Governing equations Numerical methodology Results Conclusions Total Lagrangian formulation Conservation laws • Linear momentum ∂p ∂t = 0 · P(F) + ρ0b; p = ρ0v • Deformation gradient ∂F ∂t = 0 · 1 ρ0 p ⊗ I ; CURL F = 0 Additional equations • Total energy ∂E ∂t = 0 · 1 ρ0 PT p − Q + s An appropriate constitutive model is required to close the system. Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 5
  6. 6. Introduction Governing equations Numerical methodology Results Conclusions Hyperbolic system First order conservation laws ∂U ∂t = 0 · F(U) + S U =     p F E     ; F =     P(F) 1 ρ0 p ⊗ I 1 ρ0 PT p − Q     ; S =     ρ0b 0 s     • Geometry update ∂x ∂t = 1 ρ0 p; x = X + u Adapt CFD technology to the proposed formulation. Develop an efficient low order numerical scheme for transient solid dynamics. Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 6
  7. 7. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology Spatial discretisation Flux computation Involutions Evolution 4. Results 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 7
  8. 8. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology Spatial discretisation Flux computation Involutions Evolution 4. Results 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 8
  9. 9. Introduction Governing equations Numerical methodology Results Conclusions Spatial discretisation Conservation equations for an arbitrary element dUe dt = 1 Ωe 0 Ωe 0 ∂FI ∂XI dΩ0 −→ ∀ I = 1, 2, 3; = 1 Ωe 0 ∂Ωe 0 FINI FN dA (Gauss Divergence theorem) ≈ 1 Ωe 0 f∈Λf e FC Nef Cef e FC Ne f Ce f Ωe 0 Traditional cell centred Finite Volume Method dUe dt = 1 Ωe 0    f∈Λf e FC Nef Cef    ; FC Nef =      tC 1 ρ0 pC ⊗ N 1 ρ0 tC · pC      Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 9
  10. 10. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology Spatial discretisation Flux computation Involutions Evolution 4. Results 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 10
  11. 11. Introduction Governing equations Numerical methodology Results Conclusions Lagrangian contact dynamics Rankine-Hugoniot jump conditions c U = F N where = + − − c p = t c F = 1 ρ0 p ⊗ N c E = 1 ρ0 PT p · N X, x Y, y Z, z Ω+ 0 Ω− 0 N+ N− n− n+ Ω+(t) Ω−(t) φ+ φ− n− n+ c− s c+ s c+ pc− p Time t = 0 Time t Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 11
  12. 12. Introduction Governing equations Numerical methodology Results Conclusions Acoustic Riemann solver Jump condition for linear momentum c p = t Normal jump → cp pn = tn Tangential jump → cs pt = tt p+ n , t+ np− n , t− n c+ pc− p pC n , tC n x t Normal jump p+ t , t+ tp− t , t− t c+ sc− s pC t , tC t x t Tangential jump Upwinding numerical stabilisation p C = c− p p− n + c+ p p+ n c− p + c+ p + c− s p− t + c+ s p+ t c− s + c+ s pC Ave + t+ n − t− n c− p + c+ p + t+ t − t− t c− s + c+ s pC Stab t C = c+ p t− n + c− p t+ n c− p + c+ p + c+ s t− t + c− s t+ t c− s + c+ s tC Ave + c− p c+ p (p+ n − p− n ) c− p + c+ p + c− s c+ s (p+ t − p− t ) c− s + c+ s tC Stab Linear reconstruction procedure + limiter (monotonicity) for U−,+ . Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 12
  13. 13. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology Spatial discretisation Flux computation Involutions Evolution 4. Results 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 13
  14. 14. Introduction Governing equations Numerical methodology Results Conclusions Godunov-type FVM Standard FV update (CURL F = 0) dFe dt = 1 Ωe 0 f∈Λ f e pC f ρ0 ⊗ Cef X Constrained FV update (CURL F = 0) [Dedner et al., 2002; Lee et al., 2013] dFe dt = 1 Ωe 0 f∈Λ f e ˜pC f ρ0 ⊗ Cef • Algorithm is entitled ’C-TOUCH’. pe pC f −→ ˜pe Ge  ˜pC f ←− pa Constrained transport schemes are widely used in Magnetohydrodynamics (MHD). Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 14
  15. 15. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology Spatial discretisation Flux computation Involutions Evolution 4. Results 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 15
  16. 16. Introduction Governing equations Numerical methodology Results Conclusions Time integration Two stage Runge-Kutta time integration 1st RK stage −→ U∗ e = Un e + ∆t ˙U n e(Un e, tn ) 2nd RK stage −→ U∗∗ e = U∗ e + ∆t ˙U ∗ e (U∗ e , tn+1 ) Un+1 e = 1 2 (Un e + U∗∗ e ) with stability constraint: ∆t = αCFL hmin cp,max ; cp,max = max a (ca p) An explicit Total Variation Diminishing Runge-Kutta time integration scheme. Monolithic time update for geometry. Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 16
  17. 17. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Highly non-linear problem Von-Mises plasticity Contact problems 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 17
  18. 18. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Highly non-linear problem Von-Mises plasticity Contact problems 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 18
  19. 19. Introduction Governing equations Numerical methodology Results Conclusions Low dispersion cube X, x Y, y Z, z (0, 0, 0) (1, 1, 1) Displacements scaled 300 times t = 0 s t = 2 ms t = 4 ms t = 6 ms Pressure (Pa) Boundary conditions 1. Symmetric at: X = 0, Y = 0, Z = 0 2. Skew-symmetric at: X = 1, Y = 1, Z = 1 Analytical solution u(X, t) = 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      Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 19 Problem description: Unit side cube, linear elastic material, ρ0 = 1100 kg/m3 , E = 17 MPa, ν = 0.3 and αCFL = 0.3.
  20. 20. Introduction Governing equations Numerical methodology Results Conclusions Low dispersion cube: Mesh convergence Velocity at t = 0.004 s 10 −2 10 −1 10 0 10 −7 10 −6 10 −5 10 −4 Grid Size (m) L2NormError vx vy vZ Slope = 2 Stress at t = 0.004 s 10 −2 10 −1 10 0 10 −7 10 −6 10 −5 10 −4 Grid Size (m) L2NormError Pxx Pyy Pzz Slope = 2 Demonstrates second order convergence for velocities and stresses. Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 20 Problem description: Unit side cube, linear elastic material, ρ0 = 1100 kg/m3 , E = 17 MPa, ν = 0.3 and αCFL = 0.3.
  21. 21. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Highly non-linear problem Von-Mises plasticity Contact problems 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 21
  22. 22. Introduction Governing equations Numerical methodology Results Conclusions Twisting column X, x Y, y (−0.5, 0, 0.5) (0.5, 6, −0.5) Z, z ω0 = [0, Ω sin(πY/2L), 0]T L [Twisting column] Mesh refinement at t = 0.1 s (a) 4 × 24 × 4 (b) 8 × 48 × 8 (c) 40 × 240 × 40 (a) 4 × 24 × 4 (b) 8 × 48 × 8 (c) 40 × 240 × 40 Pressure (Pa) Demonstrates the robustness of the numerical scheme Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 22 Problem description: Nearly incompressible neo-Hookean material, ρ0 = 1100 kg/m3 , E = 17 MPa, ν = 0.45, αCFL = 0.3 and Ω = 105 rad/s.
  23. 23. Introduction Governing equations Numerical methodology Results Conclusions Comparison of various alternative numerical schemes t = 0.1 s C-TOUCH P-TOUCH B-bar Taylor Hood PG-FEM Hu-Washizu JST-SPH SUPG-SPH Pressure (Pa) Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 23 Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1100 kg/m3 , E = 17 MPa, ν = 0.495, αCFL = 0.3 and Ω = 105 rad/s.
  24. 24. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Highly non-linear problem Von-Mises plasticity Contact problems 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 24
  25. 25. Introduction Governing equations Numerical methodology Results Conclusions Taylor impact X, x Y, y v0 (−0.0032, 0, 0) (0.0032, 0.0324, 0) Z, z r0 [Taylor impact] Evolution of pressure wave t = 0.1 µs t = 0.2 µs t = 0.3 µs t = 0.4 µs t = 0.5 µs t = 0.6 µs Pressure (Pa) Demonstrates the ability of the algorithm to simulate plastic behaviour. Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 25 Problem description: Hyperelastic-plastic material, ρ0 = 8930 kg/m3 , E = 117 GPa, ν = 0.35, αCFL = 0.3, ¯τ0 y = 0.4 GPa, H = 0.1 GPa and v0 = −227 m/s.
  26. 26. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Highly non-linear problem Von-Mises plasticity Contact problems 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 26
  27. 27. Introduction Governing equations Numerical methodology Results Conclusions Bar rebound X, x Y, y v0 (−0.0032, 0, 0) (0.0032, 0.0324, 0) Z, z r0 0.004 [Bar rebound] t = 3 ms t = 6 ms t = 12 ms t = 18 ms t = 27 ms Pressure (Pa) Demonstrates the ability of the algorithm to simulate contact problems. Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 27 Problem description: Nearly incompressible neo-Hookean material, ρ0 = 8930 kg/m3 , E = 585 MPa, [Lahiri et al., 2010] ν = 0.45, αCFL = 0.3 and v0 = −150 m/s.
  28. 28. Introduction Governing equations Numerical methodology Results Conclusions Bar rebound X, x Y, y v0 (−0.0032, 0, 0) (0.0032, 0.0324, 0) Z, z r0 0.004 y Displacement of the points X = [0, 0.0324, 0]T and X = [0, 0, 0]T 0 0.5 1 1.5 2 2.5 3 x 10 −4 −20 −16 −12 −8 −4 0 4 8 x 10 −3 Time (sec) yDispacement(m) Top (2880 cells) Top (23040 cells) Bottom (2880 cells) Bottom (23040 cells) Demonstrates the ability of the algorithm to simulate contact problems. Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 28 Problem description: Nearly incompressible neo-Hookean material, ρ0 = 8930 kg/m3 , E = 585 MPa, [Lahiri et al., 2010] ν = 0.45, αCFL = 0.3 and v0 = −150 m/s.
  29. 29. Introduction Governing equations Numerical methodology Results Conclusions Torus impact [Torus impact] Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 29 Problem description: Neo-Hookean material, ρ0 = 1000 kg/m3 , E = 1 MPa, ν = 0.4, αCFL = 0.3 and v0 = −3 m/s.
  30. 30. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 30
  31. 31. Introduction Governing equations Numerical methodology Results Conclusions Conclusions and further research Conclusions • Upwind CC-FVM is presented for fast solid dynamic simulations within the OpenFOAM environment. • Linear elements can be used without usual locking. • Velocities and stresses display the same rate of convergence. On-going work • Investigation into an advanced Roe’s Riemann solver with robust shock capturing algorithm. • Extension to multiple body and self contact. • Ability to handle tetrahedral elements. Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 31
  32. 32. Introduction Governing equations Numerical methodology Results Conclusions References Published / 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 (2016), DOI: 10.1002/nme.5293. • 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. Under review • C. H. Lee, A. J. Gil, G. Greto, S. Kulasegaram and J. Bonet. "A new Jameson-Schmidt-Turkel Smooth Particle Hydrodynamics algorithm for large strain explicit fast dynamics, CMAME. • C. H. Lee, A. J. Gil, J. Bonet and S. Kulasegaram. "An efficient Streamline Upwind Petrov-Galerkin Smooth Particle Hydrodynamics algorithm for large strain explicit fast dynamics, CMAME. In preparation • J. Haider, C. H. Lee, A. J. Gil, A. Huerta and J. Bonet. "Contact dynamics in OpenFOAM, JCP. • J. Bonet, A. J. Gil, C. H. Lee, A. Huerta and J. Haider. "Adapted Roe’s Riemann solver in explicit fast solid dynamics, JCP. http://www.jibranhaider.weebly.com/research Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 32

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