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Beschreibung:
Even today, the treatment of industrial-grade geometries is a huge challenge in the field of numerical simulations. The geometries that are created by computer aided design (CAD) are often very complex and contain many flaws. Hence the discretization by mesh-based methods like the finite element method (FEM) is very time consuming and can take several months when human interaction is required. Therefore, a growing interest in so-called meshfree methods arose in the scientific community over the last few decades. One such meshfree method is the partition of unity method (PUM), which is very promising because of its flexibility due to its very abstract formulations. But even though the PUM is meshfree in its core, the treatment of complex geometries is still lacking. In this thesis we develop methods to close that gap. First we propose a post-processing step to the original cover construction algorithm employed in the PUM, that guarantees that stable approximation spaces can be constructed for arbitrary geometries in two and three space-dimensions. Then, we tackle the problem of efficient and robust integration in 2D, by proposing a monotone decomposition of the input geometry. By exploiting properties of the resulting decomposition, we can prove that all required intersection operations can be implemented reliably. By adding all inflection points of the domain's boundary when constructing local decompositions of the integration domains, we can prove that the resulting curved triangles always form a valid decomposition. In 3D, we propose to create a linear approximation of the input geometry. The linear representation allows all subsequent operations to be performed reliably and fast. Then, we develop a method to estimate the domain approximation error and relate that error to the approximation error of the PUM discretization. Refinement controlled by those error estimates then yields a method that can overall converge with optimal rates. All methods proposed throughout that thesis are validated by numerical ...