A spectral alternating method for elastostatic problems with multiple spherical cavities

The problem of an infinite elastic solid containing an arbitrary number of non-overlapping spherical cavities of arbitrary sizes and locations and with arbitrary boundary tractions is considered. The numerical procedure to solve this boundary value problem is based on a spectral method in which the boundary displacements and tractions are represented by truncated series of surface spherical harmonics. By using an alternating method, the problem containing multiple spherical cavities is replaced by a sequence of problems for a single spherical cavity with the boundary conditions adjusted iteratively to account for the cavity interactions. A least squares approximation is used to determine the unknown coefficients of surface spherical harmonics representing these cavity interactions. Several examples are given to illustrate the approach.