A Discontinuous Galerkin Scheme for Elastic Waves in Nearly Incompressible Materials

Two discontinuous Galerkin schemes for linear elastic waves in two and three dimensions with Dirichlet and optional Neumann boundary conditions are formulated: a pure displacement formulation and a mixed formulation with the displacement and a pressure parameter as variables. The schemes are implemented for two dimensions using Concepts with piecewise linear orthogonal basis functions for the displacements and piecewise constant functions for the pressure parameter, both on unstructured triangular meshes. Symmetric and nonsymmetric variants are tested in static and dynamic test cases, using implicit and explicit timestepping schemes and different linear solvers. All symmetric schemes are found to be locking free, while the nonsymmetric ones behave badly for reasons made plausible. The non-mixed scheme has optimal convergence, but computation costs depend on compressibility. The mixed scheme can overcome this problem, but only in the stationary case.