3D isogeometric boundary element analysis and structural shape optimization for Helmholtz acoustic scattering problems

Abstract

The capability of boundary element methods (BEM) for solving three-dimensional time harmonic Helmholtz acoustic scattering problems is presented in the framework of the isogeometric analysis (IGA). Both the CAD geometry and the physical boundary variables are approximated using Non-uniform Rational B-splines basis functions (NURBS) in an isogeometric setting. A detailed comparison between two BEM methods: the conventional boundary integral equation (CBIE) and Burton–Miller (BM) is provided including the computational cost. The proposed models are enhanced with a modified collocation scheme with offsets to Greville abscissae to avoid placing collocation points at the corners. Placing collocation points on a smooth surface enables accurate evaluation of normals for BM formulation in addition to straightforward prediction of jump-terms and avoids singularities in O(1∕r) integrals eliminating the need for polar integration. Furthermore, no additional special treatment is required for the hyper-singular integral while collocating on highly distorted elements, such as those containing sphere poles.Acoustic shape optimization in different mediums (air and water) is performed with Particle Swarm Optimization (PSO) and the results are compared with the benchmark solutions from the literature. The reference solutions are obtained with BM which deals with higher order singularities and gradient-based optimization, and requires more complicated sensitivity analysis. The obtained results indicate that, CBIE with PSO is a feasible alternative (except for a small number of fictitious frequencies) which is easier to implement.