Universal meshes [electronic resource] : computing conforming triangulations
Keywords: Mechanical Engineering.

Three-dimensional realistic simulations are often unsteady and involve moving geometries. Problems of this type include design-shape optimization, crack propagation, hydraulic fracturing, solid dynamics for large deformations, phase transformations, and fluid-structure interaction, among others. Finite element methods for such problems fall in two different classes: 1) deforming mesh methods, and 2) immersed boundary methods. Deforming mesh methods require remeshing or updating the mesh for finite element calculations every few time steps/instances. Immersed boundary methods approximate the changing domain through a fixed background mesh. In this thesis we present a mesh generation algorithm that retains the feature of generating conforming meshes for changing domains, and is computationally efficient like the immersed boundary methods. The method we propose can be used with any off-the-shelf solver for moving domain three dimensional simulations. We introduce a method to mesh the boundary of a smooth, open domain in three dimensions immersed in a mesh of tetrahedra. The mesh follows by mapping a specific collection of triangular faces in the mesh to . Two types of surface meshes follow: (a) a mesh that exactly meshes the surface, and (b) meshes that approximate the surface to any order, by interpolating the map over the selected faces; i.e., an isoparametric approximation to the surface. The map we use to deform the faces is the closest point projection onto the surface. We formulate conditions for the closest point projection to define a homeomorphism between each face and its image. These are conditions on some of the tetrahedra intersected by the boundary, and they essentially state that each such tetrahedra should: (a) have a small enough diameter, and (b) have two of its dihedral angles be acute. We provide explicit upper bounds on the mesh size, and these can be computed on the fly. We extend the method to mesh the boundary, to generate a conforming mesh of tetrahedra for the open domain and bounded domain in three dimensions. Similarly to the case of surface meshes, we generate tetrahedral meshes that approximate the domain to any order, by interpolating the map over the selected tetrahedra. Through the perturbation of selected vertices in the background mesh, we construct a mapping that ensures tetrahedra with positive measure in the approximating tetrahedralization. An approximating tetrahedral mesh for a given domain is constructed by : a) selecting a set of vertices in a neighborhood of the surface and b) perturbing them. We need the tetrahedra in the neighborhood of the surface to satisfy the conditions necessary for quality-ensured surface meshes. The perturbation is described in two different ways: 1) explicit mapping, and 2) quality optimization based perturbations. We present an efficient implementation of the algorithm to generate approximate tetrahedralization that takes advantage of the conjecture that the set of faces used for the surface mesh is a connected set. We present several numerical experiments that are consistent with this conjecture. These experiments suggest that the complexity of the presented algorithm is linearly proportional to the number of triangles in the surface mesh of the surface. We showcase the usage of the algorithm with various examples of moving domain problems. We further extend the notion of universal meshes to generate a conforming triangulation for a given smooth curve from a generalized anisotropic background mesh. In particular method allows for triangles with obtuse angles in the background mesh. The novelty of the approach lies in the way the set of faces are selected to discretize the curve, and in the projection map constructed for the selected faces that discretize the given curve. We discuss two particular cases where the proposed method fails to generate conforming meshes. We generate these conforming meshes without altering the connectivity of the background mesh. Furthermore, we present preliminary results of using the same method for continuous curves, and anisotropic tetrahedral meshes for three-dimensional geometries. The universal meshes algorithm presented here is advantageous for a few important reasons. The method does not require that any global problem be solved; the background mesh is only generated once; and the connectivity of the background mesh is left unaltered. Hence, the computational complexity of mesh generation is observed to be proportional to the number of triangles on the surface. The examples and discussion presented here indicate that universal meshes can be a useful tool in simulating realistic engineering problems.