A spectral multipole method for efficient solution of large‐scale boundary element models in elastostatics

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Abstract
In this paper we introduce a method to reduce the solution cost for Boundary Element (BE) models from O(N3)operations to O(N2logN) operations (where N is the number of elements in the model). Previous attempts to achieve such an improvement in efficiency have been restricted in their applicability to problems with regular geometries defined on a uniform mesh. We have developed the Spectral Multipole Method (SMM) which can be used not only for problems with arbitrary geometries but also with a variety of element types. The memory necessary to store the required influence coefficients for the spectral multipole method is O(N) whereas the memory required for the traditional Boundary Element method is O(N2). We demonstrate the savings in computational speed and fast memory requirements in some numerical examples. We have established that the break‐even point for the method can be as low as 500 elements, which implies that the method is not only suitable for extremely large‐scale problems, but that it also provides a useful bridge between the small‐scale and large‐scale problems. We also demonstrate the performance of the multipole algorithm on the solution of large‐scale granular assembly models. The large‐scale BE capacity provided by this algorithm will not only prove to be useful in large macroscopic models but it will also make it possible to model microscopic damage processes that form the fundamental mechanisms in plastic flow and brittle fracture.

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