Sphere elements in the BEM for the analysis of elastic bodies with spherical voids or inclusions

In this paper, a series of novel sphere elements are proposed in the boundary element method (BEM). These elements are designed as isoparametric closure elements to simulate spherical geometries with greater accuracy and fewer nodes than conventional boundary elements. Constructed similarly to multi-dimensional Lagrange elements, these sphere elements utilize trigonometric bases for each dimension. To avoid zero Jacobians at polar nodes, poleless sphere elements combined with triangular elements are employed to approximate spheres. The evaluation methods of boundary integrals over these elements, including singular and nearly singular integrals, are derived using degenerated element techniques and adaptive subdivision techniques, respectively. Three numerical examples are employed to underscore the advantages of the proposed elements, showing that with only 50 nodes per sphere, results align closely with those obtained using 290 nodes per sphere with conventional boundary elements, effectively reducing degrees of freedom without sacrificing accuracy.