Multi-field spacetime discontinuous Galerkin methods for linearized elastodynamics

We extend the single-field spacetime discontinuous Galerkin (SDG) method for linearized elastodynamics of Abedi et al. [1] to multi-field versions. A three-field method, in displacement, velocity and strain, is derived by invoking a Bubnov–Galerkin weighted residuals procedure on the system of spacetime field equations and the corresponding jump conditions. A two-field formulation, in displacement and velocity, and the one-field displacement formulation of [1] are obtained from the three-field model through strong enforcement of kinematic compatibility relations. All of these formulations balance linear and angular momentum at the element level, and we prove that they are energy-dissipative and unconditionally stable. As in [1], we implement the SDG models using a causal, advancing-front meshing procedure that enables a patch-by-patch solution procedure with linear complexity in the number of spacetime elements. Numerical results show that the three-field formulation is most efficient, wherein all interpolated fields converge at the optimal, O(hp+1)O(hp+1), rate. For a given mesh size, the three-field model delivers error values that are more than an order of magnitude smaller than those of the one- and two-field models. The three-field formulation’s efficiency is also superior, independent of whether the comparison is based on matching polynomial orders or matching convergence rates.