Anisotropic geometry-conforming d-simplicial meshing via isometric embeddings

We develop a dimension-independent, Delaunay-based anisotropic mesh generation algorithm suitable for integration with adaptive numerical solvers. As such, the mesh produced by our algorithm conforms to an anisotropic metric prescribed by the solver as well as the domain geometry, given as a piecewise smooth complex. Motivated by the work of Lévy and Dassi [10-12,20], we use a discrete manifold embedding algorithm to transform the anisotropic problem to a uniform one. This work differs from previous approaches in several ways. First, the embedding algorithm is driven by a Riemannian metric field instead of the Gauss map, lending itself to general anisotropic mesh generation problems. Second we describe our method for computing restricted Voronoi diagrams in a dimension-independent manner which is used to compute constrained centroidal Voronoi tessellations. In particular, we compute restricted Voronoi simplices using exact arithmetic and use data structures based on convex polytope theory. Finally, since adaptive solvers require geometry-conforming meshes, we offer a Steiner vertex insertion algorithm for ensuring the extracted dual Delaunay triangulation is homeomorphic to the input geometries. The two major contributions of this paper are: a method for isometrically embedding arbitrary mesh-metric pairs in higher dimensional Euclidean spaces and a dimension-independent vertex insertion algorithm for producing geometry-conforming Delaunay meshes. The former is demonstrated on a two-dimensional anisotropic problem whereas the latter is demonstrated on both 3d and 4d problems. Keywords: Anisotropic mesh generation; metric; Nash embedding theorem; isometric; geometry-conforming; restricted Voronoi diagram; constrained centroidal Voronoi tessellation; Steiner vertices; dimension-independen

An adaptive framework for high-order, mixed-element numerical simulations

Thesis: S.M., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2014.Cataloged from PDF version of thesis.Includes bibliographical references (pages 99-103).This work builds upon an adaptive simulation framework to allow for mixed-element meshes in two dimensions. Contributions are focused in the area of mesh generation which employs the Lk norm to produce various mesh types. Mixed-element meshes are obtained by first using the L[alpha] norm to create a right-triangulation which is then combined to form quadrilaterals through a graph-matching approach. The resulting straight-sided mesh is then curved using a nonlinear elasticity analogy. Since the element sizes and orientations are prescribed to the mesh generator through a field of Riemannian metric tensors, the adaptation algorithm used to compute this field is also discussed. The algorithm is first tested through the L² error control of isotropic and anisotropic problems and shows that optimal mesh gradings can be obtained. Problems drawn from aerodynamics are then used to demonstrate the ability of the algorithm in practical applications. With mixed element meshes, the adaptation algorithm works well in practice, however, improvements can be made in the cost and error models. In fact, using the L[alpha]-generated meshes inherits the same properties of traditional triangulations while adding structure to the mesh. The use of the L[alpha]-norm in generating tetrahedral meshes is worth pursuing in the future.by Philip Claude Delhaye Caplan.S.M

Four-dimensional anisotropic mesh adaptation for spacetime numerical simulations

This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Thesis: Ph. D., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2019Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (pages [135]-142).Engineers and scientists are increasingly relying on high-fidelity numerical simulations. Within these simulations, mesh adaptation is useful for obtaining accurate predictions of an output of interest subject to a computational cost constraint. In the quest for accurately predicting outputs in problems with time-dependent solution features, a fully unstructured coupled spacetime approach has been shown to be useful in reducing the cost of the overall simulation. However, for the simulation of unsteady three-dimensional partial differential equations (PDEs), a four-dimensional mesh adaptation tool is needed. This work develops the first anisotropic metric-conforming four-dimensional mesh adaptation tool for performing adaptive numerical simulations of unsteady PDEs in three dimensions. The theory and implementation details behind our algorithm are first developed alongside an algorithm for constructing four-dimensional geometry representations.We then demonstrate our algorithm on three-dimensional benchmark cases and it appears to outperform existing implementations, both in metric-conformity and expected tetrahedra counts. We study the utility of the mesh adaptation components to justify the design of our algorithm. We then develop four-dimensional benchmark cases and demonstrate that metric-conformity and expected pentatope counts are also achieved. This is the first time anisotropic four-dimensional meshes have been presented in the literature. Next, the entire mesh adaptation framework, Mesh Optimization via Error Sampling and Synthesis (MOESS), is extended to the context of finding the optimal mesh to represent a function of four variables. The mesh size and aspect ratio distributions of the optimized meshes match the analytic ones, thus verifying our framework.Finally, we apply MOESS in conjunction with the mesh adaptation tool to perform the first four-dimensional anisotropic mesh adaptation for the solution of the advection-diffusion equation. The optimized meshes effectively refine the solution features corresponding to both a boundary layer solution as well as an expanding spherical wave.by Philip Claude Caplan.Ph. D.Ph.D. Massachusetts Institute of Technology, Department of Aeronautics and Astronautic