New hybrid quadrature schemes for weakly singular kernels applied to isogeometric boundary elements for 3D Stokes flow

This work proposes four novel hybrid quadrature schemes for the efficient and accurate evaluation of weakly singular boundary integrals (1/r kernel) on arbitrary smooth surfaces. Such integrals appear in boundary element analysis for several partial differential equations including the Stokes equation for viscous flow and the Helmholtz equation for acoustics. The proposed quadrature schemes apply a Duffy transform-based quadrature rule to surface elements containing the singularity and classical Gaussian quadrature to the remaining elements. Two of the four schemes additionally consider a special treatment for elements near to the singularity, where refined Gaussian quadrature and a new moment-fitting quadrature rule are used. The hybrid quadrature schemes are systematically studied on flat B-spline patches and on NURBS spheres considering two different sphere discretizations: An exact single-patch sphere with degenerate control points at the poles and an approximate discretization that consist of six patches with regular elements. The efficiency of the quadrature schemes is further demonstrated in boundary element analysis for Stokes flow, where steady problems with rotating and translating curved objects are investigated in convergence studies for both, mesh and quadrature refinement. Much higher convergence rates are observed for the proposed new schemes in comparison to classical schemes

A transformation approach for efficient evaluation of oscillatory surface integrals arising in three-dimensional boundary element methods

We propose a method for efficient evaluation of surface integrals arising in boundary element methods for three-dimensional Helmholtz problems (with real positive wavenumber k), modelling wave scattering and/or radiation in homogeneous media. To reduce the number of degrees of freedom required when k is large, a common approach is to include in the approximation space oscillatory basis functions, with support extending across many wavelengths. A difficulty with this approach is that it leads to highly oscillatory surface integrals whose evaluation by standard quadrature would require at least O(k2) quadrature points. Here, we use equivalent contour integrals developed for aperture scattering in optics to reduce this requirement to O(k), and possible extensions to reduce it further to O(1)are identified. The contour integral is derived for arbitrary shaped elements, but its application is limited to planar elements in many cases. In addition, the transform regularises the singularity in the surface integrand caused by the Green’s function, including for the hyper-singular case under appropriate conditions. An open-source Matlab™ code library is available to demonstrate our routines

Sound Radiation from Railway Wheels including Ground Reflections: A half-space formulation for the Fourier Boundary Element Method

Current models for the acoustic radiation from railway wheels assume free field radiation. However, slab tracks are increasingly used for new railway lines. The acoustically hard surface of those tracks makes a re-evaluation of the free field assumption relevant, as such a surface can affect the radiation efficiency of an acoustic radiator. The wheel as the acoustic radiator is most conveniently described in a cylindrical coordinate system, thus making use of its axisymmetry. While this is a viable solution for the structural vibrations, for instance by using the curved Waveguide Finite Element formulation, the axisymmetry breaks when including a reflective plane in the calculation of the acoustic radiation. A convenient method to include an infinitely large, reflective plane is by using half-space Green’s functions in combination with the Boundary Element method. This method can be formulated in cylindrical coordinates using the Fourier series BEM (FBEM). However, the FBEM has not yet been combined with half-space Green’s functions. This paper provides a half-space formulation for the FBEM, which enables e.g. the evaluation of sound radiation of railway wheels over reflective surfaces. Finally, it is shown that the assumption of free field radiation for railway wheels is valid, as there is no major contribution of the reflective plane to the radiation efficiency of the wheel. The developed method is validated against laboratory measurements as well as analytical models

The Boundary Element Method in Acoustics

The boundary element method (BEM) is a powerful tool in computational acoustic analysis. The Boundary Element Method in Acoustics serves as an introduction to the BEM and its application to acoustic problems and goes on to complete the development of computational models. Software implementing the methods is available