Isogeometric indirect boundary element method for solving the 3D acoustic problems

Abstract This paper provides an Isogeometric Indirect Boundary Element Method (IGIBEM) based on NURBS (Non-Uniform Rational B-Splines) and PHT-splines (polynomial splines over hierarchical T-meshes) for analyzing the three-dimensional (3D) acoustic problems. In the classical procedure, the geometries are discretized by Lagrange polynomials elements, which leads to both substantial geometrical error and time-consuming meshing steps. However, these deficiencies can be eliminated by the isogeometric analysis (IGA) directly incorporating the geometry description generated from the CAD (Computer Aided Design) software into CAE (Computer Aided Engineering) analysis. Unlike the DBEM (direct boundary element method), the IBEM (indirect boundary element method) allows different types of boundary conditions on the two sides of a surface. Moreover, the hypersingular integrals in IBEM can be transformed to a weakly singular form. In addition, the PHT-based IBEM is used to investigate the influence of local refinement on the accuracy of solutions. Finally, the non-uniqueness problem is solved, which is a fatal defect in the acoustic BEM for the exterior problem. Four different methods to handle the non-uniqueness problem are discussed and compared. The results obtained by the proposed method were compared with analytical solutions and the results computed by Lagrange-based IBEM. Several benchmark examples demonstrate: (1) the present method, i.e. IGIBEM, has super accuracy over conventional IBEM for the acoustic problems; (2) local refinement has a significant influence on the convergence rate of the solutions, and the numerical accuracy is relevant to the distance to the boundary where the local refinement acted; (3) as for the non-uniqueness problem, the imposition of specific interior boundary conditions not only obtains the best calculation result over the entire range of frequencies, but also has a simple integral formulation.

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