A sparse integral equation method for acoustic scattering

Numerical calculations of acoustic scattering may be based on either differential equations or on corresponding integral equations. The differential equations generate a sparse matrix problem involving (possibly implicitly) an M by M matrix. Integral equations generate a matrix of size N by N, where N is much smaller than M. In the past the matrix resulting from an integral equation was necessarily full. A method for greatly reducing the storage required is presented here. The full N by N matrix resulting from the discretization of the Helmholtz integral is transformed into a sparse N by N matrix using the interaction matrix localization (IML) method. IML is a general method for integral equations describing wave phenomena. It produces a sparse matrix by partitioning the scatterer into several regions and using groups of basis functions each of which is nonzero over an entire region. The basis functions for a region are chosen so that each radiates a narrow beam in a different direction. The number of important physical interactions is greatly reduced. A large number of matrix elements have a relative magnitude of less than 10−4, and these are approximated by zero. This method is implemented as a modification to existing boundary element method codes. Details of the formulation are given for scattering from rigid bodies of revolution. Memory and storage requirements are reduced from N2 to approximately 100N, while the solution time for the matrix problem is reduced from O(N3) to O(N2). The solution method uses a sparse approximate inverse which allows an iterative method to converge to machine accuracy in less than five iterations. Results are given for scattering from a hemispherically endcapped cylinder of L/a=22, for ka=1 to ka=17.