PhD Defence: Approximate Geometric Continuity for Numerical Simulation and Surface Reconstruction on Multi-patch Domains

Approximate Geometric Continuity for Numerical Simulation and Surface Reconstruction on Multi-patch Domains

In the context of geometric modeling and isogeometric analysis, more complex geometries are represented as multi-patch domains that consist of several tensor-product spline patches. Naturally, smoothness across the patch interfaces is an important issue, for the design both of the multi-patch surface itself as well as for functions that are defined on the entire domain. This gives rise to different coupling methods, some of which we study in this thesis.

Exact parametric smoothness is hard to achieve and in most cases requires restrictive assumptions on the parameterization of the geometry. Moreover, in real-world applications approximate smoothness often is sufficient. We consider coupling approaches that work on general domains but only provide approximately C1-smooth isogeometric functions or approximate G1-smoothness of a multi-patch surface across patch interfaces.

There are two main possibilities how to encourage smoothness of an approximate solution to a partial differential equation on a multi-patch domain: Firstly, one can make use of the fact that the given problem is fulfilled on all individual patches. The patch-wise terms are summed up and different smoothness penalty terms can be added. One instance of such a method is the discontinuous Galerkin method, to which we devote the first part of this thesis. The discretized problem contains integrals of test functions along the patch interfaces. Their evaluation is crucial. Without matching interface parameterizations, two main difficulties arise in this framework, which we tackle with reparameterizations and suitable quadrature techniques. Secondly, one can use globally smooth functions on the whole domain as test functions in a continuous Galerkin scheme. Here, the essential part is the construction of such functions. In the second part of this thesis we present an approach to the construction of approximately C1-smooth isogeometric functions. Starting from globally C0-smooth functions, the central idea is to bound their gradient jumps across the patch interface. Numerical examples suggest that the resulting functions are sufficiently smooth to solve higher-order problems such as the biharmonic equation and maintain full approximation power. Finally, the third part of this thesis considers smooth transitions between surface patches. In order to improve the overall smoothness of a multi-patch spline surface, we consider the simultaneous approximation of point and normal data. If the normal data to be approximated by one patch is taken from the boundary of its neighbors, this controls the behavior of the resulting spline patch along the boundary and ensures approximate G1-smoothness of the composite surface.