Efficient Space-Time Finite Elements for Thermo-Mechanical Problems

This talk discusses efficient and reliable Finite Element Methods to simulate the thermo-mechanical response of high explosives. A key motivation is the modelling of the initiation of shear bands in materials such as HMX. The localised plastic deformation associated with a shear band leads to the formation of hot spots and can subsequently lead to thermal runaway. Standard finite element methods struggle to accurately resolve the sharp variations associated with these thermal and mechanical features which may lead to unphysical predictions of the numerical models. The numerical methods presented in this talk aim to provide efficient and reliable tools towards modelling the initiation of shear banding and thermal runaway. We consider two approaches: adaptively generated meshes based on mathematically rigorous estimates of the numerical errors, and enriched finite elements. They are illustrated for thermal and elastic problems, as they arise in reduced models. First, we present results based on adaptive finite elements for non-linear thermal problems. Steep temperature gradients are resolved by appropriate mesh refinement procedures. Steered by indicators for the accuracy of the solution, the algorithm automatically resolves hot spots on a refined mesh, significantly reducing computational costs, see for example [2]. Secondly, we con-sider enriched space-time finite elements (also known as generalised finite elements) which include a priori physical information into the numerical method. This a priori information could represent localised of wave-like features, which are added to a coarse approximation space. The modelling can effectively capture features occurring at different spatial and temporal scales [4, 5]. Here we consider a first order formulation of the wave equation [1] and choose plane-wave enrichments [6]. Future work aims to address the full, coupled thermo-mechanical system, as well as to combine the adaptive and enriched approaches of Iqbal et al. [3].[1] H. Barucq, H. Calandra, J. Diaz, and E. Shishenina. Space-Time Trefftz - Discontinuous Galerkin Approximation for Elasto-Acoustics. (RR-9104), 2017.[2] Heiko Gimperlein and Jakub Stocek. Space–time adaptive finite elements for nonlocal parabolic variational inequalities. Computer Methods in Applied Mechanics and Engineering, 352:137 – 171, 2019.[3] M. Iqbal, D. Stark, H. Gimperlein, M.S. Mohamed, and O. Laghrouche. Local adaptive q-enrichments and generalized finite elements for transient heat diffusion problems. Computer Methods in Applied Mechanics and Engineering, 372:113359, 2020.[4] O. Laghrouche, P. Bettess, E. Perrey-Debain, and J. Trevelyan. Wave interpolation finite elements for Helmholtz problems with jumps in the wave speed. Computer Methods in Applied Mechanics and Engineering, 194(2):367 – 381, 2005.[5] E. Perrey-Debain, J. Trevelyan, and P. Bettess. On wave boundary elements for radiation and scattering problems with piecewise constant impedance. IEEE Transactions on Antennas and Propagation, 53(2):876–879, 2005[6] Steffen Petersen, Charbel Farhat, and Radek Tezaur. A space–time discontinuous galerkin method for the solution of the wave equation in the time domain. International Journal for Numerical Methods in Engineering, 78(3):275–295, 2009