A Boundary Algebraic Formulation for Plane Strain Elastodynamic Scattering

Abstract

Solving of elastodynamic problems arises in many scientific fields such as wave propagation in the ground, nondestructive testing, vibration design of buildings, or vibroacoustics in general. An integral formulation based on boundary algebraic equations is presented here. This formulation leads to a numerical method with a discretized boundary. An important advantage of the method over the standard boundary element method is that no contour (in two-dimensional problems) or surface (in three-dimensional problems) integral needs to be computed. This feature is helpful in order to obtain a discrete version of the combined field integral equations (designed to damp numerically the fictitious eigenfrequencies) without difficulties caused by the evaluation of hypersingular integrals. The key aspects are (i) the approach deals with discrete equations from the very beginning; (ii) discrete (instead of continuous) tensor Green's functions are considered (the methodology to evaluate them is demonstrated); (iii) the boundary must be described by means of a regular square grid. In order to overcome the drawback of this third condition the boundary integral is coupled, if needed, with a thin layer of finite elements. This improves the description of curved geometries and reduces numerical errors. The properties of the method are demonstrated by means of numerical examples: the scattering of waves by objects and holes in an unbounded elastic medium, and an interior elastic problem.