least square approach to simulate wave propagation in irregular profiles using the indirect boundary element method

in earthquake engineering and seismology it is of interest to know the surface motion at a given site due to the incoming and scattered seismic waves by surface geology. this can be formulated in terms of diffraction of elastic waves and then the indirect boundary element method (ibem) for dynamic elasticity is used. it is based on the explicit construction of diffracted waves at the boundaries from which they radiate. this provides the analyst with insight on the physics of diffraction. the ibem has been applied to study the amplification of elastic waves in irregular soil profiles. from the strong or weak satisfaction of boundary conditions and a simple analytical discretization scheme a linear system of equations for the boundary sources is obtained. here, we explore the use of a weak discretization strategy with more collocation points than force densities. the least squares enforcement of boundary conditions leads to a system with reduced number of unknowns. this approach naturally allows one to use both coarser and finer boundary discretizations for smooth and rapidly varying profiles, respectively. a well studied semicircular canyon under incident p or sv in-plane waves is used to calibrate this method. several benefits are obtained using mixed meshing that leads to the least squares condensation of the ibem.