Simulations of waves in elastic solids of cubic symmetry by the Conservation Element and Solution Element method

This paper reports a new numerical approach to simulate waves in elastic solids with cubic symmetry. The governing equation includes the equation of motion and the constitutive relation of the elastic medium. With velocity and stress components as the unknowns, the equations are a set of nine, first-order, hyperbolic partial differential equations. To aid numerical simulation, the characteristic form of the equations is derived. By using the Schur complement in linear algebra, the one-dimensional equations are shown to be equivalent to the Christoffel equations without using the harmonic plane-wave solution. To solve the governing equations by the Conservation Element and Solution Element (CESE) method, we first use Gauss' theorem to recast the equations into a space–time integral form. By integrating the integral equation, space–time flux conservation is imposed over Conservation Elements (CEs). Numerical integration is aided by using prescribed linear discretization of the unknowns in Solution Elements (SEs). A convergence test shows that the CESE method employed is second-order accurate. To demonstrate the capabilities of the present approach, reported numerical results include one-dimensional resonant waves, collinear impact of two blocks, and two-dimensional wave expansion from a point source. Additional results of waves interacting with an interface separating two media with different lattice orientations are also reported. Results compared well with the available analytical solutions. All results show salient features of waves in solids of cubic symmetry.