Large strain solid dynamics in OpenFOAM Jibran Haider a, b , Dr. Chun Hean Lee a , Dr. Antonio J. Gil a , Prof. Javier Bonet c & Prof. Antonio Huerta b a Zienkiewicz Centre for Computational Engineering, Swansea University, UK b Laboratori de C` alcul Num` eric (LaC` aN), UPC BarcelonaTech, Spain c University of Greenwich, London, UK Research outline Objectives: • Simulate fast-transient solid dynamic problems. • Develop a fast and efficient low order numerical scheme. Key features: X An upwind cell-centred FVM Total Lagrangian scheme (TOUCH). X Utilises an explicit Runge-Kutta time integrator. X Programmed in the open-source CFD software OpenFOAM. X Overcomes the shortcomings of linear tetrahedral elements in standard displacement based FEM/FVM formulations: • Equal order of convergence for velocities and stresses. • No volumetric locking for nearly incompressible materials. • Excellent performance in bending and shock dominated scenarios. Q1-P 0 FEM Proposed FVM First order conservation laws 1. Linear momentum: 2. Deformation gradient: 3. Total energy: d dt Z Ω 0 p dΩ 0 = Z ∂ Ω 0 t dA + Z Ω 0 ρ 0 b dΩ 0 d dt Z Ω 0 F dΩ 0 = Z ∂ Ω 0 p ρ 0 ⊗ N dA d dt Z Ω 0 E dΩ 0 = Z ∂ Ω 0 p ρ 0 · t dA - Z ∂ Ω 0 Q · N dA + Z Ω 0 sdΩ 0 • Hyperbolic laws in differential form: ∂ U ∂t = ∂ F I ∂X I + S , ∀ I =1, 2, 3 Cell centred FVM discretisation Standard face-based CC-FVM e F C N ef C ef Ω e 0 dU e dt = 1 Ω e 0 X f ∈Λ f e F C N ef (U - f , U + f ) kC ef k Node-based CC-FVM F C N ea C ea Ω e 0 e dU e dt = 1 Ω e 0 X a∈Λ a e F C N ea (U - a , U + a ) kC ea k • Gradient calculation through least squares minimisation -→ G e • Satisfaction of monotonicity through Barth and Jespersen limiter -→ φ e • Linear reconstruction procedure for second order spatial accuracy -→ U +,- (φ e ,G e ) Lagrangian contact dynamics Contact flux: F C N = F I N I = t C 1 ρ 0 p C ⊗ N 1 ρ 0 p C · t C - Q · N Acoustic Riemann solver: F C N = F C N Ave + F C N Stab = 1 2 h F N (U - f )+ F N (U + f ) i - 1 2 Z U + f U - f |A N | dU | {z } Upwinding stabilisation 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 + p c - p Time t = 0 Time t Explicit time integration Total Variation Diminishing Runge-Kutta scheme: 1 st RK stage -→ U ? e = U n e +Δt ˙ U n e (U n e ,t n ) 2 nd RK stage -→ U ?? e = U ? e +Δt ˙ U ? e (U ? e ,t n+1 ) U n+1 e = 1 2 (U n e + U ?? e ) with stability criterion: Δt = α CFL h min c max p Numerical results Shock scenario 6 7 8 9 10 x 10 -3 -7.5 -5 -2.5 0 2.5 5 x 10 7 Time (sec) Stress (Pa) Analytical TOUCH (1st order) TOUCH (2nd order w/o limiter) TOUCH (2nd order with limiter) JST VCFVM Mesh convergence 10 -2 10 -1 10 0 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 Grid Size (m) Stress Error slope = 1 L 1 norm (1st order) L 2 norm (1st order) slope = 2 L 1 norm (2nd order) L 2 norm (2nd order) Structured vs Unstructured Pressure (Pa) Complex twisting Pressure (Pa) Flapping structure Pressure (Pa) Von Mises plasticity Constrained-TOUCH Penalised-TOUCH Hyperelastic-GLACE Plastic strain Bar rebound Pressure (Pa) Torus impact Pressure (Pa) On-going work 1. An advanced Roe’s Riemann solver. 2. Robust shock capturing algorithm. 3. Ability to handle tetrahedral elements. Future work 1. Extension to Fluid-Structure Interaction (FSI) problems. 2. Implementation of Arbitrary Lagrangian-Eulerian (ALE) formulation. References [1] 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, International Journal for Numerical Methods in Engineering, 109(3) : 407–456, 2017. [2] 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 : 13–38, 2013. Website: http://www.jibranhaider.weebly.com Email:{m.j.haider,c.h.lee,a.j.gil}@swansea.ac.uk