| In this thesis, we study the generation and optimization methodology of anisotropic Delaunay triangulation meshes and applications. Anisotropic is a physical property that has di?erent values while measured in di?erent directions, such as the tree’s growth rate di?erences along the vertical direction and horizontal direction. Anisotropic problems appear in various ?elds of scienti?c and engineering applications, e.g. anisotropic materials, porous media, petroleum engineering,viscous ?ows simulations, image processing. Given an anisotropic problem, the key step is domain partition in the ?nite element method. It is natural to choose an anisotropic triangulation mesh for the anisotropic problem. Many anisotropic properties can be described by metric tensors which are symmetric positive de?-nite matrices. The main contents of this thesis are de?ning a suitable metric for the given anisotropic problem, and designing a new anisotropic triangulation mesh generation method.Firstly, we introduce a new mesh generation algorithm when the metric is a constant matrix. We use advancing front method to insert new points, meanwhile smoothing the front and mesh based on Anisotropic Centroidal Voronoi Tessellation(ACVT). The result mesh is not only best conforming the boundary of geometry but also has high quality with most of elements being equilateral triangles on the metric. Based on anisotropic triangulation meshes, we use ?nite element method(FEM) to solve the anisotropic elliptic partial di?erential equations(PDEs), the anisotropic di?usion matrix of which is a constant matrix. We show the inverse of the di?usion matrix is a natural metric tensor, the corresponding anisotropic mesh is a matched mesh with three important observations as follows.The condition numbers of the resulting discrete algebraic system are smaller than others. The numerical solution is more accurate on the matched mesh. More importantly, there is superconvergence on the nodes of matched mesh with the convergence rate O(h2+α), α ≈ 0.5. We conclude that the inverse of the constant coe?cient matrix is a suitable metric tensor. Many numerical examples validate our conclusion.Secondly, while the metrics are variable, we develop a new algorithm to generate anisotropic Delaunay triangulation meshes. Based on an anisotropic background mesh, combining modi?ed anisotropic Delaunay criterion, we insert new points or delete existing non-?xed nodes, and move the nodes’ positions using force-based equilibrium smoothing function during each iteration. The average quality of the resulting anisotropic Delaunay mesh is high. We also apply the anisotropic meshes to anisotropic elliptic PDEs with variable coe?cient matrices.Using the matched anisotropic meshes with variable metrics to solve equations, we observe there similar results: better discrete algebraic systems with smaller condition numbers, more accurate ?nite element solutions with much smaller errors,and superconvergence in l2 norm on the nodes. We generalize the conclusion in the case of constant coe?cient matrix. Choosing the inverse of coe?cient matrix as a matched metric tensor is also suitable for the variable anisotropic elliptic PDEs.Thirdly, based on the explicit polynomial recovery(EPR) technique to reconstruct the numerical approximations, we ?nd the reconstruction solution on the matched anisotropic meshes is more accurate than the numerical solution before. It indicates the EPR technique can improve the accuracy. We introduce an anisotropic mesh adaption with EPR-based error estimator. Numerical tests are presented to demonstrate its e?cient.Finally, we consider a Sandia benchmark problem that is an anisotropic heat conduction in a thermal battery. The material coe?cients are large orthotropic jumps. Compared the numerical solutions of matched anisotropic meshes and uniform meshes, the analysis further con?rms that using the inverse of coe?cient matrix as a suitable metric is reasonable. The practical problem illustrates the anisotropic mesh is widely applied. |
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