Isogeometric boundary element analysis and shape optimization by PSO for 3D axi-symmetric high frequency Helmholtz acoustic problems

Abstract

The boundary element method (BEM) has been widely used for approximating the solution of the Helmholtz acoustic equation due to its superiority for handling unbounded domains by modeling only the inner boundaries and avoiding the truncation error, present in the finite element method (FEM). Non-uniform rational B-splines basis functions (NURBS) are used in an isogeometric setting to describe both the CAD geometries and the physical fields. Isogeometric analysis (IGA) is coupled with BEM forming (IGABEM) which speeds up meshing and gives higher accuracy in comparison with standard BEM. In this work, we consider axi-symmetric domains, where the three-dimensional problem is simplified in BEM from a surface integral to a combination of two 1D integrals. The first is the line integral similar to a two-dimensional BEM problem. The second integral is performed over the angle of revolution. The discretization is applied only to the former integration. This leads to significant computational savings and, consequently, better treatment for higher frequencies over the full three-dimensional models. We first validate the capability of IGABEM for axi-symmetric problems to handle high frequencies and verify it by means of manufactured analytical solutions. Secondly, we show the performance of the proposed approach in shape optimization problems by using a particle swarm optimization (PSO) method. Coupling PSO with IGABEM in the optimization models benefits from the IGA feature of representing the three different models: shape design, analysis and optimization models using a single set of control points. The proposed method for the shape optimization of a horn is compared with reference solutions for BEM and FEM.