Universal meshes for two-dimensional domains with piecewise C2-regular boundaries
- We introduce a method to deform a mesh to conform to a domain with piecewise C2-regular boundary, by only moving a fraction of vertices of the given mesh. The method preserves all vertices and connectivities of the mesh, and only small regions near the boundary are deformed, which makes it computationally efficient. The most appealing application of the method is when it is needed to mesh a group of domains, such as the case of moving boundary problems. For such problems, a single mesh that does not require any conformity to the boundary can be generated, which is then used to obtain conforming meshes for a number of domains by simply deforming it. The challenge for such an application is the robustness of the method, which allows the method to be automated without any user intervention. We provide the method with conditions that guarantee its success, which can be either satisfied or checked a priori. The outline of the method is as follows. The boundary is first divided into a set of C2-regular curves, called segments. Some vertices near the endpoints of each segment are perturbed so that each endpoint has a vertex assigned to it. Then, a set of edges, called domain edges, are selected from the mesh, and the domain edges or their vertices are projected onto the boundary. Some vertices near the curve are perturbed to relax any distorted triangles. A crucial necessary condition for the method to render a conforming mesh with elements of good quality is that the projection of domain edges onto the boundary be a homeomorphism, which involves both choosing an appropriate set of domain edges and projection map. In our method, we choose the set of domain edges individually for each segment. This set is "roughly" the set of all edges joining the two vertices on a given side of the curve, of each triangle that has a third vertex on the other side. The domain edges are then projected to their corresponding segment via the closest point projection. The highlight of the strategy is the algorithm to assign an orientation to each segment, which determines the set of domain edges for the segment. The algorithm prevents domain edges of distinct segments from intersecting at point other than the endpoints of a segment. Certain conditions are also required on the mesh and the curve for the projection to be a homeomorphism. Some angles in the mesh are required to be acute, and triangles near the boundary should be sufficiently small with respect to its local radius of curvature of the curve. We analyze the method and present theorems that show the conditions that guarantee the homeomorphism of the projection. The conditions solely depend on the features of the given boundary, and can be computed in advance to generate the background mesh, if needed.
|Type of resource
|electronic resource; remote; computer; online resource
|1 online resource.
|Degree committee member
|Degree committee member
|Stanford University, Department of Mechanical Engineering.
|Statement of responsibility
|Submitted to the Department of Mechanical Engineering.
|Thesis Ph.D. Stanford University 2018.
- © 2018 by Kyuwon Kim
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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