Accelerating isogeometric boundary element analysis for 3‐dimensional elastostatics problems through black‐box fast multipole method with proper generalized decomposition

Abstract

SummaryThe isogeometric approach to computational engineering analysis makes use of nonuniform rational B‐splines to discretise both the geometry and the analysis field variables, giving a higher‐fidelity geometric description and leading to improved convergence properties of the solution over conventional piecewise polynomial descriptions. Because of its boundary‐only modelling, with no requirement for a volumetric nonuniform rational B‐spline geometric definition, the boundary element method is an ideal choice for isogeometric analysis of solids in 3D. An isogeometric boundary element analysis (IGABEM) algorithm is presented for the solution of such problems in elasticity and is accelerated using the black‐box fast multipole method (bbFMM). The bbFMM scheme is of O(n) complexity, giving a general kernel‐independent separation that can be easily integrated into existing conventional IGABEM codes with little modification. In the bbFMM scheme, an important process of obtaining a low‐rank approximation of moment‐to‐local operators has been hitherto based on singular value decomposition, which can be very time consuming for large 3D problems, and this motivates the present work. We introduce the proper generalized decomposition method as an alternative approach, and this is demonstrated to enhance efficiency in comparison with schemes that rely on the singular value decomposition. In the worst case, a factor of approximately 2 performance gain is achieved. Numerical examples show the performance gains that are achievable in comparison to standard IGABEM solutions and demonstrate that solution accuracy is not affected. The results illustrate the potential of this numerical technique for solving arbitrary large scale elastostatics problems directly from computer‐aided design models.