Isogeometric Analysis for boundary integral equations

Taus, Matthias Franz
Journal Title
Journal ISSN
Volume Title

Since its emergence, Isogeometric Analysis (IgA) has initiated a revolution within the field of Finite Element Methods (FEMs) for two reasons: (i) geometry descriptions originating from Computer Aided Design (CAD) can be used directly for analysis purposes, and (ii) the availability of smooth exact geometry descriptions and smooth basis functions can be used to develop new, highly accurate and highly efficient numerical methods. Whereas in FEMs the first issue is still open, it has already been shown that Isogeometric BEMs (IBEMs) provide a complete design-through-analysis framework. However, in contrast to FEMs, the effect of smoothness provided by IgA has not yet been explored in IBEMs. In this dissertation, we address this aspect of IgA. We show that the smoothness and exactness properties provided by the IgA framework can be used to design highly accurate and highly efficient BEMs which are not accessible with conventional BEMs. We develop Collocation IBEMs on piecewise smooth geometries. This allows us to show that IBEMs converge in the expected rates and result in system matrices with mesh-independent condition numbers. The latter property is particularly beneficial for large-scale problems that require iterative linear solvers. However, using conventional Collocation BEMs, this approach is not accessible because hyper-singular integrals have to be evaluated. In contrast, using Collocation IBEMs, the smoothness properties of the IgA framework can be used to regularize the hyper-singular integrals and reduce them to weakly singular integrals which can be evaluated using well-known techniques. We perform several numerical examples on canonical shapes to show these results. In addition, we use well-known mathematical results to develop a sound theoretical foundation to some of our methods, a result that is very rare for Collocation discretizations. Finally, using the exactness of IgA geometry descriptions, we design Patch Tests that allow one to rigorously test IBEM implementations. We subject our implementation to these Patch Tests which not only shows the reliability of our method but also shows that IBEMs can be as accurate as machine precision. We apply our IBEMs to Laplace's equation and the equations of linear elasticity. In addition, input files for our implementation can be automatically obtained from commercial CAD packages. These practical aspects allow us to apply IBEMs to analyze a propeller under a wind load.