Generalization of the k-space formulation to elastodynamic scattering problems

A generalized k-space (GKS) formulation is presented for vectorial elastodynamic scattering problems. It represents a generalization of Bojarski’s scalar k-space formulation. From the basic second-order partial differential equation or its integral representation in the space-frequency (r-f) domain, a local equation is derived for the displacement field in the spectral-frequency (k-f) domain. This equation, together with the constitutive equation in the r-f domain, reduces the original scattering problem into two simultaneous local equations with two unknowns (displacement field and the induced source), which are then solved by the conjugate-gradient (CG) method. The connection between the k-f domain and r-f domain is obtained by the spatial fast Fourier transform (FFT) algorithm. The number of complex multiply-add operations per CG iteration is O(N log2 N), and the storage requirement is only O(N), where N is the number of spatial discretization points. This is much more efficient than the conventional method of moment combined with the CG procedures which requires O(N2) operations per iteration and O(N2) storage. In the spectral-time (k-t) domain, however, it is found that four simultaneous local equations have to be used to formulate the k-space algorithm because of the existence of two wave speeds. By virtue of the causality, a new local time-stepping algorithm is derived with the aid of two temporal propagators, i.e., the compressional and shear propagators. The connection between the r-t domain and k-t domain is again obtained by the spatial FFT algorithm. Therefore, in each time step, the number of complex multiply–add operations is O(N log2 N), and the storage requirement is O(N). Most importantly, for the same accuracy, N can be much smaller than that for the conventional finite-difference method.