Updated Lagrangian SPH

1. Outline Introduction First order conservation laws Numerical method Numerical results Summary A Robust Updated Lagrangian Smooth Particle Hydrodynamics Algorithm For Fast Solid Dynamics Paulo R. Refachinho de Campos∗, Antonio J. Gil∗, Chun Hean Lee†, Antonio Huerta‡, Matteo Giacomini‡ ∗ Zienkiewicz Centre for Computational Engineering (ZCCE) † Glasgow Computational Engineering Centre (GCEC) ‡ Laboratori de Càlcul Numèric (LaCàN) UKACM 2021 Conference on Computational Mechanics MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 1/27

2. Outline Introduction First order conservation laws Numerical method Numerical results Summary Outline 1 Introduction Fast solid dynamics Computational approach 2 First order conservation laws Isothermal process 3 Numerical method Spatial discretisation 4 Numerical results Isothermal elasticity Isothermal plasticity 5 Summary MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 2/27

4. Outline Introduction First order conservation laws Numerical method Numerical results Summary Engineering Applications for fast solid dynamics · Short-duration dynamic events where large deformations, high energy impacts, fracture and fragmentation can take place. · Commercial solutions: VPS/PAM-CRASH, LS-DYNA, Abaqus/Explicit, RADIOSS . . . MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 4/27

5. Outline Introduction First order conservation laws Numerical method Numerical results Summary Engineering Applications for fast solid dynamics · Short-duration dynamic events where large deformations, high energy impacts, fracture and fragmentation can take place. O&G blowout preventer Explosive forming Drop test Crashworthiness Bird impact Ballistic Space debris impact Shock-wave lithotripsy · Commercial solutions: VPS/PAM-CRASH, LS-DYNA, Abaqus/Explicit, RADIOSS . . . MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 4/27

6. Outline Introduction First order conservation laws Numerical method Numerical results Summary Engineering Applications for fast solid dynamics · Short-duration dynamic events where large deformations, high energy impacts, fracture and fragmentation can take place. O&G blowout preventer Explosive forming Drop test Crashworthiness Bird impact Ballistic Space debris impact Shock-wave lithotripsy · Commercial solutions: VPS/PAM-CRASH, LS-DYNA, Abaqus/Explicit, RADIOSS . . . MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 4/27

7. Outline Introduction First order conservation laws Numerical method Numerical results Summary Limitations of current technology · Mesh-based methods can potentially suffer from mesh entanglement when simulating very large deformations · A wealth of alternative meshfree methods have emerged over the years · Smooth Particle Hydrodynamics (SPH) · Simple and versatile: · Complex geometries · Large deformation · Failure representation · Classical SPH formulation: · Not robust · Non-physical modes · Lack of consistency · Reduced accuracy for stresses and strains MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 5/27

8. Outline Introduction First order conservation laws Numerical method Numerical results Summary Limitations of current technology · Mesh-based methods can potentially suffer from mesh entanglement when simulating very large deformations · A wealth of alternative meshfree methods have emerged over the years · Smooth Particle Hydrodynamics (SPH) · Simple and versatile: · Complex geometries · Large deformation · Failure representation · Classical SPH formulation: · Not robust · Non-physical modes · Lack of consistency · Reduced accuracy for stresses and strains Mesh entanglement MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 5/27

9. Outline Introduction First order conservation laws Numerical method Numerical results Summary Limitations of current technology · Mesh-based methods can potentially suffer from mesh entanglement when simulating very large deformations · A wealth of alternative meshfree methods have emerged over the years · Smooth Particle Hydrodynamics (SPH) · Simple and versatile: · Complex geometries · Large deformation · Failure representation · Classical SPH formulation: · Not robust · Non-physical modes · Lack of consistency · Reduced accuracy for stresses and strains Mesh entanglement Non-physical modes MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 5/27

11. Outline Introduction First order conservation laws Numerical method Numerical results Summary Kinematic description · Consider the deformation of an isothermal solid body: F := ∇X φ, H := 1 2 F F , J := 1 3 H : F Tensor-cross product property [Bonet et al., 2016]: (A B)ij = εiklεjmnAkmBln MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 7/27

12. Outline Introduction First order conservation laws Numerical method Numerical results Summary Kinematic description · Consider the deformation of an isothermal solid body: F := ∇X φ, H := 1 2 F F , J := 1 3 H : F Tensor-cross product property [Bonet et al., 2016]: (A B)ij = εiklεjmnAkmBln MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 7/27

13. Outline Introduction First order conservation laws Numerical method Numerical results Summary Continuum balance principles · Total Lagrangian conservation laws for solid dynamics: ∂p ∂t − DIVP = b0 ∂F ∂t − DIV(v ⊗ I) = 0 ∂H ∂t − CURL (v F ) = 0 ∂J ∂t − DIV HT v = 0 CURLF = 0 and DIVH = 0 · Written in hyperbolic form: EI := Ith unit vector of the Cartesian basis ∂U ∂t + 3 X I=1 ∂FI ∂XI = S U =    p F H J    , FI = −    P EI v ⊗ EI F (v ⊗ EI ) H : (v ⊗ EI )    , S =    b0 0 0 0    MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 8/27

16. Outline Introduction First order conservation laws Numerical method Numerical results Summary Kinematic description based on multiplicative decomposition F := fFχ, H := hHχ, J := jJχ MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 9/27

17. Outline Introduction First order conservation laws Numerical method Numerical results Summary Continuum balance principles · INCREMENTAL Updated Lagrangian conservation laws for solid dynamics: J−1 χ ∂U ∂t + 3 X i=1 ∂ FH−1 χ i ∂χi = J−1 χ S, in ΩV χ J−1 χ ∂p ∂t − divχ P H−1 χ = J−1 χ b0 J−1 χ ∂F ∂t − divχ v ⊗ H−1 χ = 0 J−1 χ ∂H ∂t − divχ h F v ⊗ H−1 χ i = 0 J−1 χ ∂J ∂t − divχ H−T χ HT v = 0 · Fluxes are highly nonlinear MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 10/27

18. Outline Introduction First order conservation laws Numerical method Numerical results Summary Continuum balance principles · INCREMENTAL Updated Lagrangian conservation laws for solid dynamics: J−1 χ ∂U ∂t + 3 X i=1 ∂ FH−1 χ i ∂χi = J−1 χ S, in ΩV χ J−1 χ ∂p ∂t − divχ P H−1 χ = J−1 χ b0 J−1 χ ∂F ∂t − divχ v ⊗ H−1 χ = 0 J−1 χ ∂H ∂t − divχ h F v ⊗ H−1 χ i = 0 J−1 χ ∂J ∂t − divχ H−T χ HT v = 0 · Fluxes are highly nonlinear MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 10/27

19. Outline Introduction First order conservation laws Numerical method Numerical results Summary Continuum balance principles · INCREMENTAL Updated Lagrangian conservation laws for solid dynamics: ∂pχ ∂t − divχσχ = bχ ∂f ∂t − divχ (v ⊗ I) = 0 ∂h ∂t − curlχ (v f) = 0 ∂j ∂t − divχ hT v = 0 curlχf = 0 and divχh = 0 · Written in hyperbolic form: ei := ith unit vector of the Cartesian basis ∂Uχ ∂t + 3 X i=1 ∂Fi χ ∂χi = Sχ Uχ =    pχ f h j    , F i χ = −    σχei v ⊗ ei f (v ⊗ ei) h: (v ⊗ ei)    , Sχ =    bχ 0 0 0    MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 11/27

20. Outline Introduction First order conservation laws Numerical method Numerical results Summary Continuum balance principles · INCREMENTAL Updated Lagrangian conservation laws for solid dynamics: ∂pχ ∂t − divχσχ = bχ ∂f ∂t − divχ (v ⊗ I) = 0 ∂h ∂t − curlχ (v f) = 0 ∂j ∂t − divχ hT v = 0 curlχf = 0 and divχh = 0 · Written in hyperbolic form: ei := ith unit vector of the Cartesian basis ∂Uχ ∂t + 3 X i=1 ∂Fi χ ∂χi = Sχ Uχ =    pχ f h j    , F i χ = −    σχei v ⊗ ei f (v ⊗ ei) h: (v ⊗ ei)    , Sχ =    bχ 0 0 0    MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 11/27

21. Outline Introduction First order conservation laws Numerical method Numerical results Summary Incremental deformation · How to define the reference configuration? MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 12/27

22. Outline Introduction First order conservation laws Numerical method Numerical results Summary Incremental deformation · How to define the reference configuration? MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 12/27

23. Outline Introduction First order conservation laws Numerical method Numerical results Summary Incremental deformation · How to define the reference configuration? … MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 12/27

26. Outline Introduction First order conservation laws Numerical method Numerical results Summary Incremental deformation · How to define the reference configuration? … … MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 12/27

27. Outline Introduction First order conservation laws Numerical method Numerical results Summary Incremental deformation · How to define the reference configuration? … … MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 12/27

29. Outline Introduction First order conservation laws Numerical method Numerical results Summary Smooth Particle Hydrodynamics Method · Pioneers [Gingold, Monaghan, 1977; Lucy, 1977] · Lagrangian scheme · Meshfree method · Simple implementation · Low computational cost · Suitable for Large deformation · Foundations in interpolation theory: v (X) = Z ΩV v X0 δ X − X0 dV ≈ Z ΩV v X0 W X − X0 , h dV · SPH approximation: v (Xa) = X b∈Λb a VbvbWX ba , ∇X v (Xa) = X b∈Λb a Vb(vb − va) ⊗ ∇X WX ba MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 14/27

34. Outline Introduction First order conservation laws Numerical method Numerical results Summary Classical SPH correction · SPH does not fulfil the Kronecker-Delta property · Corrected SPH kernel approximation [Bonet et al., 1999] v (Xa) = P b∈Λb a VbvbW̃X ba, W̃X ba = KX a WX ba · Consistency conditions X b∈Λb a VbW̃X ba = 1, X b∈Λb a Vb (Xa − Xb) W̃X ba = 0 · Corrected gradient evaluation [Bonet et al., 1999] ∇X v (Xa) = P b∈Λb a Vb(vb − va) ⊗ ˜ ∇X WX ba , ˜ ∇X WX ba = GX a ∇X WX ba · Rigid body invariance X b∈Λb a Vb (Xb − Xa) ⊗ ˜ ∇X WX ba = I MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 15/27

39. Outline Introduction First order conservation laws Numerical method Numerical results Summary Smooth Particle Hydrodynamics Method Material support MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 16/27

40. Outline Introduction First order conservation laws Numerical method Numerical results Summary Smooth Particle Hydrodynamics Method Material support Updated support MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 16/27

41. Outline Introduction First order conservation laws Numerical method Numerical results Summary Smooth Particle Hydrodynamics Method Material support Updated support Anisotropic support MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 16/27

42. Outline Introduction First order conservation laws Numerical method Numerical results Summary Smooth Particle Hydrodynamics Method Anisotropic support MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 17/27

43. Outline Introduction First order conservation laws Numerical method Numerical results Summary Smooth Particle Hydrodynamics Method Anisotropic support · Option 1: Wχ ba = J−1 χ,bKX a WX ba ∇χWχ ba = J−1 χ,bF −T χ,a GX a ∇X WX ba where WX ba = W (Xa − Xb, hb) , hb = fh max j∈Λ j b (kXj − Xbk) MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 17/27

44. Outline Introduction First order conservation laws Numerical method Numerical results Summary Smooth Particle Hydrodynamics Method Anisotropic support · Option 2: W̃χ ba = Kχ a J−1 χ,bWX ba ˜ ∇χWχ ba = Gχ a J−1 χ,bF −T χ,a ∇X WX ba where WX ba = W (Xa − Xb, hb) , hb = fh max j∈Λ j b (kXj − Xbk) MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 17/27

45. Outline Introduction First order conservation laws Numerical method Numerical results Summary Smooth Particle Hydrodynamics Method Anisotropic support · Option 3: W̃χ ba = Kχ a J−1 χ,bŴX ba ˜ ∇χWχ ba = Gχ a J−1 χ,bF −T χ,a ∇X ŴX ba where ŴX ba = Ŵ Fχ,ab, χa − χb, ĥb ĥb = fh max j∈Λ̂ j b F −1 χ,bjkχj − χbk Xa − Xb ≈ F −1 χ,ab (χa − χb) , Fχ,ab = 1 2 Fχ,a + Fχ,b MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 17/27

46. Outline Introduction First order conservation laws Numerical method Numerical results Summary Smooth Particle Hydrodynamics Method Anisotropic support · Option 4: W̃χ ba = Kχ a J−1 χ,bŴX ba ˜ ∇χWχ ba = Gχ a J−1 χ,bF −T χ,ab∇X ŴX ba where ŴX ba = Ŵ Fχ,ab, χa − χb, ĥb ĥb = fh max j∈Λ̂ j b F −1 χ,bjkχj − χbk Xa − Xb ≈ F −1 χ,ab (χa − χb) , Fχ,ab = 1 2 Fχ,a + Fχ,b MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 17/27

47. Outline Introduction First order conservation laws Numerical method Numerical results Summary Smooth Particle Hydrodynamics Method · Semi-discrete equations: V χ a dpa χ dt = P b∈Λb a 1 2 σa χCχ ba − σb χCχ ab + V χ a ba χ + Aχ a ta χ+ P b∈Λb a Dχ ab V χ a dfa dt = P b∈Λb a 1 2 (vb − va) ⊗ Cχ ba V χ a dha dt = fa P b∈Λb a 1 2 (vb − va) ⊗ Cχ ba V χ a dja dt = ha : P b∈Λb a 1 2 (vb − va) ⊗ Cχ ba · Pseudo-area vectors: Cχ ba := 2V χ a V χ b ˜ ∇χWχ ba, Cχ ab := 2V χ b V χ a ˜ ∇χWχ ab · Riemann-based numerical dissipation [Ghavamian et al., 2021] · Three-stages Runge-Kutta time integrator [Shu et al., 1988] · Discrete angular momentum preserving algorithm [Lee et al., 2019] MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 18/27

51. Outline Introduction First order conservation laws Numerical method Numerical results Summary Numerical results Isothermal elasticity MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 20/27

52. Outline Introduction First order conservation laws Numerical method Numerical results Summary Stability: Bending column Problem: Bending column, neo-Hookean material, initial density ρ0 = 1100kg/m3 , Young’s modulus E = 17MPa and Poisson’s ratio ν = 0.3. Initial velocity V = 10m/s. Updates performed at every time step. [video] MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 21/27

53. Outline Introduction First order conservation laws Numerical method Numerical results Summary Stability: Bending column Problem: Bending column, neo-Hookean material, initial density ρ0 = 1100kg/m3 , Young’s modulus E = 17MPa and Poisson’s ratio ν = 0.3. Initial velocity V = 10m/s. Updates performed at every time step. [video] MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 21/27

54. Outline Introduction First order conservation laws Numerical method Numerical results Summary Robustness: Twisting column Problem: Twisting column, neo-Hookean material, initial density ρ0 = 1100kg/m3 , Young’s modulus E = 17MPa and Poisson’s ratio ν = 0.4995. Initial angular velocity Ω = 105rad/s. Updates performed at every time step. [video] MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 22/27

55. Outline Introduction First order conservation laws Numerical method Numerical results Summary Numerical results Isothermal plasticity MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 23/27

56. Outline Introduction First order conservation laws Numerical method Numerical results Summary High speed impact: Taylor bar Problem: Taylor bar, Hencky-Mises plasticity, initial density ρ0 = 8930kg/m3 , Young’s modulus E = 117GPa, Poisson’s ratio ν = 0.35, yield stress, τ0 y = 400MPa and linear hardening modulus H = 100MPa. Initial velocity V = −227m/s. [video] MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 24/27

59. Outline Introduction First order conservation laws Numerical method Numerical results Summary Robustness: Necking bar Problem: Necking bar, Hencky-Mises plasticity, initial density ρ0 = 7850kg/m3 , Young’s modulus E = 206.9GPa, Poisson’s ratio ν = 0.29, yield stress τ0 y = 450MPa, residual yield stress τ∞ y = 715MPa, linear hardening modulus H = 129.24MPa and saturation exponent δ = 16.93. Quasi-static 25% elongation. [video] MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 25/27

63. Outline Introduction First order conservation laws Numerical method Numerical results Summary Summary Highlights · A ROBUST Updated Lagrangian SPH framework, laying the foundations for Dynamic Fracture: · System of first order conservation laws · Incremental ULF (multiplicative decomposition) · Anisotropic kernel function · Entropy stable upwinding stabilisation scheme Further work · Extended capabilities: · Thermal plasticity (ongoing) · Dynamic fracture · Contact Acknowledgements This project has received funding from the EU Framework Programme for Research and Innovation Horizon 2020 under Grant Agreement No 764636. Thank you! MSCA-ITN-EJD ProTechTion, Horizon 2020 (Grant Agreement No 764636) April 2021 27/27

Updated Lagrangian SPH

Recommended

Recommended

More Related Content

Similar to Updated Lagrangian SPH

Similar to Updated Lagrangian SPH (20)

Recently uploaded

Recently uploaded (20)

Updated Lagrangian SPH