Evaluation of hypersingular and nearly singular integrals in the Isogeometric Boundary Element Method for acoustics

Special integration routines for hypersingular and nearly singular integrals that arise in an Isogeometric Boundary Element Method (IGABEM) are presented. The IGABEM is applied to an acoustic problem in the frequency domain that corresponds to the Helmholtz equation in three dimensions. A major topic of such an acoustical BEM is the non-uniqueness of the conventional boundary integral equation, which can be corrected by the Burton-Miller formulation at the cost of a hypersingular integral that requires the presented routines. The combination of the BEM with an Isogeometric Analysis (IGA) is advantageous, since the exact CAD description is directly incorporated to the numerical simulation. As the geometry is represented exactly, the integration routines become even more relevant and cannot be masked by the geometrical approximation. For the conventional approximation with plane Lagrange elements, a variety of different integration routines exist, but are not investigated in the scope of an IGA. Therefore, the hypersingular integration scheme of Guiggiani is adapted to be applicable to this new methodology. A direct evaluation of the hypersingular integral is possible, instead of an integration over the complete surface, as in the often used regularized boundary integral equation. Additionally, the sinh-transformation for nearly singular integrals is introduced to achieve more accurate results with the same number of integration points as the conventional Gaussian integration. The new formulation is analyzed under the special case of the oscillatory kernels in acoustics.