|Abstract||In this thesis we address the implementation of collocation and Galerkin boundary element methods (BEM) and the numerical evaluation of the arising integrals. The focus is on methods that are based on a NURBS (non-uniform rational b-splines) parametrization of the boundary, which we refer to as NURBS-based methods. The advantage of NURBS-based methods is that a geometric error is avoided, which is induced by the boundary approximation in standard methods and diminishes the convergence of BEM.
The first part of this thesis is devoted to the derivation of new, stable algorithms for the accurate and efficient numerical evaluation of the arising integrals. By exploiting the special structure of the NURBS parametrization and by interpolating parts of the kernel functions by Legendre polynomials, we are able to evaluate the boundary integral operators in a stable way.
The singular integrals arising in the assembly of the Galerkin matrices are regularized with coordinate transformations and evaluated with adapted quadrature rules. For all arising integrals, an exponential convergence of the error is proven and rigorous error bounds are derived. We use these bounds for the estimation of the consistency errors and the a priori computation of the quadrature orders for Galerkin methods.
The algorithms for the numerical integration are used for the implementation of NURBS-based methods in the second part of this dissertation. Our implementation is the first, which is known to us, that can be used for solving boundary integral equations arising from Laplace, Lame, and Helmholtz problems with collocation and Galerkin methods on exact boundary parametrizations. Furthermore, it allows the use of different basis functions. The final numerical experiments show that even for high degrees (p < 128) of the polynomial basis functions accurate results are obtained and practice-relevant problems can be efficiently solved.||dc.description.abstract