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Automatic mesh generation is a challenging problem that is motivated by the need for efficient and accurate simulations of physical phenomena. Many demanding problems require mesh elements to have appropriate anisotropy (size and stretching), orientation, and quality to ensure that a solver can converge reasonably and accurately capture the physics involved. One example of this is large deformation structural analysis, where the progressive distortion of mesh elements in many cases causes issues with solution accuracy and convergence — including premature termination. Another example domain is high speed fluid flow. Here, potentially high aspect ratios and suitable orientation are necessary to efficiently capture strong flow features such as shocks, boundary layers, wakes, and free shear layers. These and other mesh characteristics may also help mitigate the formation of numerical artifacts in the solution.
Unfortunately, exact mesh requirements are generally not known a priori, but user input, knowledge of the geometry, and a posteriori solution-derived information can all be considered in order to control orientation and anisotropy. It turns out that other applications, such as pattern placement for product design and structure generation for additive manufacturing, can similarly benefit from this.
This thesis presents a framework using Riemannian metric tensor fields for controlling orientation and anisotropy information over surfaces and volumes, as well as recipes for employing this framework to solve problems in various application domains: design, analysis, and manufacturing. Within the framework, metric tensors are interpolated (to form metric fields), generated from constraints, analyzed, and conditioned as necessary. Furthermore, we present novel mesh generation schemes that exploit metric fields to produce boundary-aligned or anisotropic meshes in 2D and 3D. This is all embedded in an anisotropic mesh adaptation process to iteratively improve fluid dynamics simulations.
Finally, the proposed methods are demonstrated through applications in graphic & product design, structural & fluid dynamics, and additive manufacturing.