Some spherical boundary elements and a discretization error indicator for acoustic problems with spherical surfaces

In this paper, three types of quadratic interpolated boundary elements are presented to investigate acoustic problems. These elements, called the spherical elements, are specially designed for the discretization of spherical surfaces, which are directly constructed on the real geometry of spherical surface without any error due to geometrical approximation. The first one is a triangular spherical element, while the other two are quadrilateral spherical elements. These spherical elements are applied to analyze the acoustic problems, wherein the Burton–Miller formulation is used to build the boundary element formulation. Furthermore, in order to guide the mesh refinement and evaluate the accuracy of numerical results, an error indicator based on the gradient distribution of the sound pressure is presented. Several examples are presented to illustrate the applicability of these elements, including the pulsating sphere, scattering of a rigid sphere and the multi-sphere scattering. Numerical results indicate that not only the spherical elements are effective and accurate, but also the error indicator proposed is quite helpful to evaluate the accuracy of numerical results.