Triangular Spectral Element simulation of two-dimensional elastic wave propagation using unstructured triangular grids

A Triangular Spectral Element Method (TSEM) is presented to simulate elastic wave propagation using an unstructured triangulation of the physical domain. TSEM makes use of a variational formulation of elastodynamics based on unstructured straight-sided triangles that allow enhanced flexibility when dealing with complex geometries and velocity structures. The displacement field is expanded into a high-order polynomial spectral approximation over each triangular subdomain. Continuity between the subdomains of the triangulation is enforced using a multidimensional Lagrangian interpolation built on a set of Fekete points of the triangle. High-order accuracy is achieved by resorting to an analytical computation of the associated internal product and bilinear forms leading to a non-diagonal mass matrix formulation. Therefore, the time stepping involves the solution of a sparse linear algebraic system even in the explicit case. In this paper the accuracy and the geometrical flexibility of the TSEM is explored. Comparison with classical spectral elements on quadrangular grids shows similar results in terms of accuracy and stability even for long simulations. Surface and interface waves are shown to be accurately modelled even in the case of complex topography with the TSEM. Numerical results are presented for 2-D canonical examples as well as more specific problems, such as 2-D elastic wave scattering by a cylinder embedded in an homogeneous half-space. They all illustrate the enhanced geometrical flexibility of the TSEM. This clearly suggests the need of further investigations in computational seismology specifically targeted towards efficient implementations of the TSEM both in the time and the frequency domain.