I present a tetrahedral mesh improvement method that creates meshes whose worst tetrahedra have a level of quality substantially better than those produced by any previous method for tetrahedral mesh generation or "mesh clean-up." Mesh optimization methods often get stuck in bad local optima (poor-quality meshes) because their repertoire of mesh transformations is weak. In contrast, my mesh improvement software employs a broader palette of operations than any other. Alongside the best traditional topological and smoothing operations, I introduce a topological transformation that inserts a new vertex, as well as methods for smoothing vertices on the boundary of the mesh. My implementation routinely improves meshes so that all the dihedral angles lie between 34 and 131 degrees. It also allows a user to locally control the sizes and grading of the tetrahedra, and to generate anisotropic meshes with local control of the orientations and eccentricities of the tetrahedra. With the same operations, I develop a dynamic mesh improvement method for simulations of deforming materials that updates a mesh at each timestep to maintain the quality of its tetrahedra. The dynamic mesher strikes a balance between maintaining high element quality and minimizing the error introduced through artificial diffusion.





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