A spacetime discontinuous Galerkin finite-element method for elastodynamic analysis

We present a spacetime discontinuous Galerkin finite element method to solve elastodynamic problems with high accuracy and efficiency, especially for problems involving shocks. The new numerical model is based on a new spacetime continuum formulation of elastodynamics, which is a integral statement of momentum balance in spacetime and is equivalent to a system consisting of a local balance equation in rate form that applies within smooth regions as well as jump conditions that describe discontinuous response across shocks and material interfaces. The continuum formulation is invariant; it does not rely on an arbitrary metric between space and time. The spacetime discontinuous Galerkin formulation derived from the continuum formulation is in a simple Bubnov-Galerkin format that does not include any stabilization term or extra numerical damping. The new finite element method supports arbitrary spacetime grids and can solve elastodynamic problems using highly efficient element-by-element or patch-by-patch approaches when spacetime meshes conform to certain cone constraints.; We use the new spacetime discontinuous Galerkin finite element model to investigate a series of elastodynamic problems in both one and two spatial dimensions. The first study focuses on the problems that exhibit discontinuities or shocks. The new method delivers high accurate results. The second study investigates the convergence properties of the numerical model. Numerical results report that the method delivers optimal convergence rate. The third study involves a crack-tip wave scattering problem, in which a step loading is applied to a solid medium of finite dimension containing a crack. The loading sends a stress wave into the medium and the wave is scattered by the crack. The dynamic stress intensity factor is obtained by virtual crack method.