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

Abstract

The isogeometric approach to computational engineering analysis makes use of Non-Uniform Rational B-splines (NURBS) 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 NURBS geometric definition, the boundary element method is an ideal choice for isogeometric analysis of solids in 3-D. 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 M2L operators has been hitherto based on Singular Value Decomposition (SVD), which can be very time consuming for large 3-D problems, and this motivates the present work. We introduce the Proper Generalized Decomposition (PGD) method as an alternative approach, and this is demonstrated to enhance efficiency in comparison with schemes that rely on the SVD. 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 CAD models.