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Large strain solid dynamics in OpenFOAM

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Jibran Haider

The document describes a numerical methodology for simulating large strain solid dynamics using OpenFOAM. It proposes using a total Lagrangian formulation and first-order conservation laws similar to computational fluid dynamics to model solid mechanics problems involving large deformations. A cell-centered finite volume method is used for spatial discretization along with Riemann solvers and linear reconstruction to capture fluxes. A two-stage Runge-Kutta scheme is employed for time integration. Results are presented demonstrating the method's ability to handle problems involving mesh convergence, enhanced reconstruction, highly nonlinear behavior, plasticity, contact, unstructured meshes, and complex geometries.

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Introduction Governing equations Numerical methodology Results Conclusions Large strain solid dynamics in OpenFOAM 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 The 4th Annual OpenFOAM User Conference (11th - 13th October 2016) 12 th October 2016 http://www.jibranhaider.weebly.com Funded by the Erasmus Mundus SEED PhD Programme and ESI Group Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 1
Introduction Governing equations Numerical methodology Results Conclusions Research group at Swansea University Dr. Antonio J. Gil Associate Professor Dr. Chun Hean Lee Research Fellow Prof. Javier Bonet University of Greenwich Prof. Antonio Huerta UPC BarcelonaTech Dr. Rogelio Ortigosa Postdoc Jibran Haider Research Assistant Osama I. Hassan Research Assistant Roman Poya Research Assistant Emilio G. Blanco Research Assistant Ataollah Ghavamian Research Assistant Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 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) 4th OpenFOAM User Conference (Cologne, Germany) 3
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) 4th OpenFOAM User Conference (Cologne, Germany) 4
Introduction Governing equations Numerical methodology Results Conclusions Fast transient dynamics Objectives • Simulate fast-transient solid dynamic problems. • Develop an industry-driven library of low order numerical schemes. Solid dynamics in OpenFOAM [Jasak & Weller, 2000] × Standard displacement based implicit dynamics × Linear elastic material with small strain deformation × Locking in nearly incompressible scenarios × First order convergence for stresses and strains × Poor performance in shock dominated scenarios OpenFOAM solid mechanics community [Ivankovic et al.] • [Cardiff et al., 2012; 2014; 2016] −→ displacement based + pressure instabilities + moderate strains + .... Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 5
Introduction Governing equations Numerical methodology Results Conclusions Proposed solid formulation • 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. TOUCH scheme [Haider et al., 2016; Lee et al., 2013] Mixed explicit dynamics Complex constitutive models Large strain deformation No bending and volumtric locking Second order convergence for stresses and strains v = 100 m/s (0.5, 0.5, 0.5) (−0.5, −0.5, −0.5) [Punch cube] Aim is to bridge the gap between CFD and computational solid dynamics. Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 6
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) 4th OpenFOAM User Conference (Cologne, Germany) 7
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) 4th OpenFOAM User Conference (Cologne, Germany) 8
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) 4th OpenFOAM User Conference (Cologne, Germany) 9
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) 4th OpenFOAM User Conference (Cologne, Germany) 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) 4th OpenFOAM User Conference (Cologne, Germany) 11
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) 4th OpenFOAM User Conference (Cologne, Germany) 12
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) 4th OpenFOAM User Conference (Cologne, Germany) 13
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) 4th OpenFOAM User Conference (Cologne, Germany) 14
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 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 How do we obtain U−,+ ? Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 15
Introduction Governing equations Numerical methodology Results Conclusions Godunov’s method • Piecewise constant representation in every cell. • Methodology is first order accurate in space. x y U Ue Uα1 Uα4 Uα2 Uα3 (a) Piecewise constant values × x y U Uα4 Uα3 Uα2 Uα1 Ue Uα3 Uα4 (b) Linear reconstruction × First order simulations suffer from excessive numerical dissipation. A linear reconstruction procedure is essential to increase spatial accuracy. Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 16
Introduction Governing equations Numerical methodology Results Conclusions Linear reconstruction procedure Gradient operator: • Classical least squares minimisation procedure. Ge =   α∈Λα e ˆdeα ⊗ ˆdeα   −1 α∈Λα e Uα − Ue deα ˆdeα Linear extrapolation to flux integration point: U{f,a} = Ue + Ge · X{f,a} − Xe de1α2 e1 α1 α2 α3 α4 αf1 αf2 αf3αf4 αf5 e2 de2α4 Gradient correction procedure: • Necessary for the satisfaction of monotonicity through Barth and Jespersen limiter (φe). U{f,a} = Ue + φe Ge(Ue, Uα) · X{f,a} − Xe Ensures that the spatial discretisation is second order accurate. Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 17
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) 4th OpenFOAM User Conference (Cologne, Germany) 18
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) 4th OpenFOAM User Conference (Cologne, Germany) 19
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) 4th OpenFOAM User Conference (Cologne, Germany) 20
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) 4th OpenFOAM User Conference (Cologne, Germany) 21
Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Enhanced reconstruction Highly non-linear problem Von-Mises plasticity Contact problems Unstructured meshes Complex geometries 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 22
Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Enhanced reconstruction Highly non-linear problem Von-Mises plasticity Contact problems Unstructured meshes Complex geometries 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 23
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) 4th OpenFOAM User Conference (Cologne, Germany) 24 Problem description: Unit side cube, linear elastic material, ρ0 = 1100 kg/m3 , E = 17 MPa, ν = 0.3 [Haider et al., 2016] and αCFL = 0.3. [Aguirre et al., 2014]
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) 4th OpenFOAM User Conference (Cologne, Germany) 25 Problem description: Unit side cube, linear elastic material, ρ0 = 1100 kg/m3 , E = 17 MPa, ν = 0.3 [Haider et al., 2016] and αCFL = 0.3. [Aguirre et al., 2014]
Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Enhanced reconstruction Highly non-linear problem Von-Mises plasticity Contact problems Unstructured meshes Complex geometries 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 26
Introduction Governing equations Numerical methodology Results Conclusions Bending dominated scenario X, x Y, y (−0.5, 0, 0.5) (0.5, 6, −0.5) Z, z L = 6m v0 = [V Y/L, 0, 0]T [Bending column] Mesh convergence at t = 1.5 s Pressure (Pa) Eliminates bending difficulty. Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 27 Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1100 kg/m3 , [Haider et al., 2016] E = 17 MPa, ν = 0.45, αCFL = 0.3 and V = 10 m/s.
Introduction Governing equations Numerical methodology Results Conclusions Bending dominated scenario Time = 0.5 s (a) OpenFOAM least square gradient (b) Enhanced least square gradient Pressure (Pa) Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 28 Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1100 kg/m3 , [Haider et al., 2016] E = 17 MPa, ν = 0.45, αCFL = 0.3 and V = 10 m/s.
Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Enhanced reconstruction Highly non-linear problem Von-Mises plasticity Contact problems Unstructured meshes Complex geometries 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 29
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 - Refinement] [Twisting column - Comparison] 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) 4th OpenFOAM User Conference (Cologne, Germany) 30 Problem description: Nearly incompressible neo-Hookean material, ρ0 = 1100 kg/m3 , E = 17 MPa, [Haider et al., 2016] ν = 0.45, αCFL = 0.3 and Ω = 105 rad/s. [Gil et al., 2014]
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) 4th OpenFOAM User Conference (Cologne, Germany) 31 Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1100 kg/m3 , [Haider et al., 2016] E = 17 MPa, ν = 0.495, αCFL = 0.3 and Ω = 105 rad/s. [Lee et al., 2016]
Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Enhanced reconstruction Highly non-linear problem Von-Mises plasticity Contact problems Unstructured meshes Complex geometries 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 32
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] [Taylor impact - Radius] 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) 4th OpenFOAM User Conference (Cologne, Germany) 33 Problem description: Hyperelastic-plastic material, ρ0 = 8930 kg/m3 , E = 117 GPa, ν = 0.35, [Aguirre et al., 2014] αCFL = 0.3, ¯τ0 y = 0.4 GPa, H = 0.1 GPa and v0 = −227 m/s. [Lee et al., 2014]
Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Enhanced reconstruction Highly non-linear problem Von-Mises plasticity Contact problems Unstructured meshes Complex geometries 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 34
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) 4th OpenFOAM User Conference (Cologne, Germany) 35 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.
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) 4th OpenFOAM User Conference (Cologne, Germany) 36 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.
Introduction Governing equations Numerical methodology Results Conclusions Torus impact [Torus impact] t = 2 ms t = 4 ms t = 8 ms t = 17 ms t = 28 ms t = 28 ms Pressure (Pa) Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 37 Problem description: Neo-Hookean material, ρ0 = 1000 kg/m3 , E = 1 MPa, ν = 0.4, αCFL = 0.3 and v0 = −3 m/s.
Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Enhanced reconstruction Highly non-linear problem Von-Mises plasticity Contact problems Unstructured meshes Complex geometries 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 38
Introduction Governing equations Numerical methodology Results Conclusions Spinning plate: Structured vs unstructured elements X, x Y, y (0.5, 0.5, 0) ω0 = [0, 0, Ω]T (−0.5, −0.5, 0) Time = 0.15 s (a) Structured 20 × 20 cells (b) Unstructured 484 cells Pressure (Pa) Demonstrates the ability of the framework to handle unstructured grids. Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 39 Problem description: Unit side square, nearly incompressible hyperelastic neo-Hookean material, [Haider et al., 2016] ρ0 = 1000 kg/m3 , E = 17 MPa, ν = 0.45 and αCFL = 0.3 and Ω = 105 rad/s.
Introduction Governing equations Numerical methodology Results Conclusions Spinning plate: Structured vs unstructured elements X, x Y, y (0.5, 0.5, 0) ω0 = [0, 0, Ω]T (−0.5, −0.5, 0) Displacement of point X = [0.5, 0.5, 0]T 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 −1.5 −1.25 −1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 1 Time (sec) Displacement(m) ux structured u y structured u x unstructured u y unstructured Demonstrates the ability of the framework to handle unstructured grids. Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 40 Problem description: Unit side square, nearly incompressible hyperelastic neo-Hookean material, [Haider et al., 2016] ρ0 = 1000 kg/m3 , E = 17 MPa, ν = 0.45, αCFL = 0.3 and Ω = 105 rad/s.
Introduction Governing equations Numerical methodology Results Conclusions Flapping device 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Enhanced reconstruction Highly non-linear problem Von-Mises plasticity Contact problems Unstructured meshes Complex geometries 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 41
Introduction Governing equations Numerical methodology Results Conclusions Flapping device t = 0 ms t = 25 ms t = 50 ms t = 75 ms t = 100 ms t = 125 ms t = 175 ms t = 200 ms Pressure (Pa) [Flapping device] Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 42 Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1000 kg/m3 , E = 17 MPa, ν = 0.45, αCFL = 0.3.
Introduction Governing equations Numerical methodology Results Conclusions Complex twisting [Complex twisting] t = 5 ms t = 10 ms t = 15 ms t = 20 ms t = 25 ms t = 30 ms Pressure (Pa) Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 43 Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1000 kg/m3 , E = 17 MPa, ν = 0.45, αCFL = 0.3.
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) 4th OpenFOAM User Conference (Cologne, Germany) 44
Introduction Governing equations Numerical methodology Results Conclusions Conclusions and on-going work Conclusions • Upwind cell centred 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. • Extension to fluid-structure interaction problems. Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 45
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. • 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 (2016); 311: 71-111. • 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, O. I. Hassan, J. Bonet and S. Kulasegaram. "An efficient Streamline Upwind Petrov-Galerkin Smooth Particle Hydrodynamics algorithm for large strain explicit fast dynamics, CMAME (2016). In preparation • J. Haider, C. H. Lee, A. J. Gil, A. Huerta and J. Bonet. "Contact dynamics in OpenFOAM, JCP. • A. J. Gil, J. Bonet, C. H. Lee, J. Haider and A. Huerta. "Adapted Roe’s Riemann solver in explicit fast solid dynamics, JCP. More information at: http://www.jibranhaider.weebly.com/research Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 46

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Large strain solid dynamics in OpenFOAM

  • 1. Introduction Governing equations Numerical methodology Results Conclusions Large strain solid dynamics in OpenFOAM 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 The 4th Annual OpenFOAM User Conference (11th - 13th October 2016) 12 th October 2016 http://www.jibranhaider.weebly.com Funded by the Erasmus Mundus SEED PhD Programme and ESI Group Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 1
  • 2. Introduction Governing equations Numerical methodology Results Conclusions Research group at Swansea University Dr. Antonio J. Gil Associate Professor Dr. Chun Hean Lee Research Fellow Prof. Javier Bonet University of Greenwich Prof. Antonio Huerta UPC BarcelonaTech Dr. Rogelio Ortigosa Postdoc Jibran Haider Research Assistant Osama I. Hassan Research Assistant Roman Poya Research Assistant Emilio G. Blanco Research Assistant Ataollah Ghavamian Research Assistant Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 2
  • 3. 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) 4th OpenFOAM User Conference (Cologne, Germany) 3
  • 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) 4th OpenFOAM User Conference (Cologne, Germany) 4
  • 5. Introduction Governing equations Numerical methodology Results Conclusions Fast transient dynamics Objectives • Simulate fast-transient solid dynamic problems. • Develop an industry-driven library of low order numerical schemes. Solid dynamics in OpenFOAM [Jasak & Weller, 2000] × Standard displacement based implicit dynamics × Linear elastic material with small strain deformation × Locking in nearly incompressible scenarios × First order convergence for stresses and strains × Poor performance in shock dominated scenarios OpenFOAM solid mechanics community [Ivankovic et al.] • [Cardiff et al., 2012; 2014; 2016] −→ displacement based + pressure instabilities + moderate strains + .... Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 5
  • 6. Introduction Governing equations Numerical methodology Results Conclusions Proposed solid formulation • 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. TOUCH scheme [Haider et al., 2016; Lee et al., 2013] Mixed explicit dynamics Complex constitutive models Large strain deformation No bending and volumtric locking Second order convergence for stresses and strains v = 100 m/s (0.5, 0.5, 0.5) (−0.5, −0.5, −0.5) [Punch cube] Aim is to bridge the gap between CFD and computational solid dynamics. Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 6
  • 7. 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) 4th OpenFOAM User Conference (Cologne, Germany) 7
  • 8. 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) 4th OpenFOAM User Conference (Cologne, Germany) 8
  • 9. 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) 4th OpenFOAM User Conference (Cologne, Germany) 9
  • 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) 4th OpenFOAM User Conference (Cologne, Germany) 10
  • 11. 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) 4th OpenFOAM User Conference (Cologne, Germany) 11
  • 12. 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) 4th OpenFOAM User Conference (Cologne, Germany) 12
  • 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) 4th OpenFOAM User Conference (Cologne, Germany) 13
  • 14. 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) 4th OpenFOAM User Conference (Cologne, Germany) 14
  • 15. 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 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 How do we obtain U−,+ ? Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 15
  • 16. Introduction Governing equations Numerical methodology Results Conclusions Godunov’s method • Piecewise constant representation in every cell. • Methodology is first order accurate in space. x y U Ue Uα1 Uα4 Uα2 Uα3 (a) Piecewise constant values × x y U Uα4 Uα3 Uα2 Uα1 Ue Uα3 Uα4 (b) Linear reconstruction × First order simulations suffer from excessive numerical dissipation. A linear reconstruction procedure is essential to increase spatial accuracy. Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 16
  • 17. Introduction Governing equations Numerical methodology Results Conclusions Linear reconstruction procedure Gradient operator: • Classical least squares minimisation procedure. Ge =   α∈Λα e ˆdeα ⊗ ˆdeα   −1 α∈Λα e Uα − Ue deα ˆdeα Linear extrapolation to flux integration point: U{f,a} = Ue + Ge · X{f,a} − Xe de1α2 e1 α1 α2 α3 α4 αf1 αf2 αf3αf4 αf5 e2 de2α4 Gradient correction procedure: • Necessary for the satisfaction of monotonicity through Barth and Jespersen limiter (φe). U{f,a} = Ue + φe Ge(Ue, Uα) · X{f,a} − Xe Ensures that the spatial discretisation is second order accurate. Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 17
  • 18. 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) 4th OpenFOAM User Conference (Cologne, Germany) 18
  • 19. 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) 4th OpenFOAM User Conference (Cologne, Germany) 19
  • 20. 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) 4th OpenFOAM User Conference (Cologne, Germany) 20
  • 21. 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) 4th OpenFOAM User Conference (Cologne, Germany) 21
  • 22. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Enhanced reconstruction Highly non-linear problem Von-Mises plasticity Contact problems Unstructured meshes Complex geometries 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 22
  • 23. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Enhanced reconstruction Highly non-linear problem Von-Mises plasticity Contact problems Unstructured meshes Complex geometries 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 23
  • 24. 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) 4th OpenFOAM User Conference (Cologne, Germany) 24 Problem description: Unit side cube, linear elastic material, ρ0 = 1100 kg/m3 , E = 17 MPa, ν = 0.3 [Haider et al., 2016] and αCFL = 0.3. [Aguirre et al., 2014]
  • 25. 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) 4th OpenFOAM User Conference (Cologne, Germany) 25 Problem description: Unit side cube, linear elastic material, ρ0 = 1100 kg/m3 , E = 17 MPa, ν = 0.3 [Haider et al., 2016] and αCFL = 0.3. [Aguirre et al., 2014]
  • 26. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Enhanced reconstruction Highly non-linear problem Von-Mises plasticity Contact problems Unstructured meshes Complex geometries 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 26
  • 27. Introduction Governing equations Numerical methodology Results Conclusions Bending dominated scenario X, x Y, y (−0.5, 0, 0.5) (0.5, 6, −0.5) Z, z L = 6m v0 = [V Y/L, 0, 0]T [Bending column] Mesh convergence at t = 1.5 s Pressure (Pa) Eliminates bending difficulty. Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 27 Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1100 kg/m3 , [Haider et al., 2016] E = 17 MPa, ν = 0.45, αCFL = 0.3 and V = 10 m/s.
  • 28. Introduction Governing equations Numerical methodology Results Conclusions Bending dominated scenario Time = 0.5 s (a) OpenFOAM least square gradient (b) Enhanced least square gradient Pressure (Pa) Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 28 Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1100 kg/m3 , [Haider et al., 2016] E = 17 MPa, ν = 0.45, αCFL = 0.3 and V = 10 m/s.
  • 29. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Enhanced reconstruction Highly non-linear problem Von-Mises plasticity Contact problems Unstructured meshes Complex geometries 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 29
  • 30. 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 - Refinement] [Twisting column - Comparison] 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) 4th OpenFOAM User Conference (Cologne, Germany) 30 Problem description: Nearly incompressible neo-Hookean material, ρ0 = 1100 kg/m3 , E = 17 MPa, [Haider et al., 2016] ν = 0.45, αCFL = 0.3 and Ω = 105 rad/s. [Gil et al., 2014]
  • 31. 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) 4th OpenFOAM User Conference (Cologne, Germany) 31 Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1100 kg/m3 , [Haider et al., 2016] E = 17 MPa, ν = 0.495, αCFL = 0.3 and Ω = 105 rad/s. [Lee et al., 2016]
  • 32. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Enhanced reconstruction Highly non-linear problem Von-Mises plasticity Contact problems Unstructured meshes Complex geometries 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 32
  • 33. 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] [Taylor impact - Radius] 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) 4th OpenFOAM User Conference (Cologne, Germany) 33 Problem description: Hyperelastic-plastic material, ρ0 = 8930 kg/m3 , E = 117 GPa, ν = 0.35, [Aguirre et al., 2014] αCFL = 0.3, ¯τ0 y = 0.4 GPa, H = 0.1 GPa and v0 = −227 m/s. [Lee et al., 2014]
  • 34. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Enhanced reconstruction Highly non-linear problem Von-Mises plasticity Contact problems Unstructured meshes Complex geometries 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 34
  • 35. 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) 4th OpenFOAM User Conference (Cologne, Germany) 35 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.
  • 36. 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) 4th OpenFOAM User Conference (Cologne, Germany) 36 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.
  • 37. Introduction Governing equations Numerical methodology Results Conclusions Torus impact [Torus impact] t = 2 ms t = 4 ms t = 8 ms t = 17 ms t = 28 ms t = 28 ms Pressure (Pa) Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 37 Problem description: Neo-Hookean material, ρ0 = 1000 kg/m3 , E = 1 MPa, ν = 0.4, αCFL = 0.3 and v0 = −3 m/s.
  • 38. Introduction Governing equations Numerical methodology Results Conclusions Scheme of presentation 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Enhanced reconstruction Highly non-linear problem Von-Mises plasticity Contact problems Unstructured meshes Complex geometries 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 38
  • 39. Introduction Governing equations Numerical methodology Results Conclusions Spinning plate: Structured vs unstructured elements X, x Y, y (0.5, 0.5, 0) ω0 = [0, 0, Ω]T (−0.5, −0.5, 0) Time = 0.15 s (a) Structured 20 × 20 cells (b) Unstructured 484 cells Pressure (Pa) Demonstrates the ability of the framework to handle unstructured grids. Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 39 Problem description: Unit side square, nearly incompressible hyperelastic neo-Hookean material, [Haider et al., 2016] ρ0 = 1000 kg/m3 , E = 17 MPa, ν = 0.45 and αCFL = 0.3 and Ω = 105 rad/s.
  • 40. Introduction Governing equations Numerical methodology Results Conclusions Spinning plate: Structured vs unstructured elements X, x Y, y (0.5, 0.5, 0) ω0 = [0, 0, Ω]T (−0.5, −0.5, 0) Displacement of point X = [0.5, 0.5, 0]T 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 −1.5 −1.25 −1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 1 Time (sec) Displacement(m) ux structured u y structured u x unstructured u y unstructured Demonstrates the ability of the framework to handle unstructured grids. Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 40 Problem description: Unit side square, nearly incompressible hyperelastic neo-Hookean material, [Haider et al., 2016] ρ0 = 1000 kg/m3 , E = 17 MPa, ν = 0.45, αCFL = 0.3 and Ω = 105 rad/s.
  • 41. Introduction Governing equations Numerical methodology Results Conclusions Flapping device 1. Introduction 2. Governing equations 3. Numerical methodology 4. Results Mesh convergence Enhanced reconstruction Highly non-linear problem Von-Mises plasticity Contact problems Unstructured meshes Complex geometries 5. Conclusions Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 41
  • 42. Introduction Governing equations Numerical methodology Results Conclusions Flapping device t = 0 ms t = 25 ms t = 50 ms t = 75 ms t = 100 ms t = 125 ms t = 175 ms t = 200 ms Pressure (Pa) [Flapping device] Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 42 Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1000 kg/m3 , E = 17 MPa, ν = 0.45, αCFL = 0.3.
  • 43. Introduction Governing equations Numerical methodology Results Conclusions Complex twisting [Complex twisting] t = 5 ms t = 10 ms t = 15 ms t = 20 ms t = 25 ms t = 30 ms Pressure (Pa) Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 43 Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1000 kg/m3 , E = 17 MPa, ν = 0.45, αCFL = 0.3.
  • 44. 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) 4th OpenFOAM User Conference (Cologne, Germany) 44
  • 45. Introduction Governing equations Numerical methodology Results Conclusions Conclusions and on-going work Conclusions • Upwind cell centred 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. • Extension to fluid-structure interaction problems. Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 45
  • 46. 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. • 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 (2016); 311: 71-111. • 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, O. I. Hassan, J. Bonet and S. Kulasegaram. "An efficient Streamline Upwind Petrov-Galerkin Smooth Particle Hydrodynamics algorithm for large strain explicit fast dynamics, CMAME (2016). In preparation • J. Haider, C. H. Lee, A. J. Gil, A. Huerta and J. Bonet. "Contact dynamics in OpenFOAM, JCP. • A. J. Gil, J. Bonet, C. H. Lee, J. Haider and A. Huerta. "Adapted Roe’s Riemann solver in explicit fast solid dynamics, JCP. More information at: http://www.jibranhaider.weebly.com/research Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 46