Some Mathematical Problems In Mesh Generation

The problem of mesh generation plays a very important role in the field of numerical computation,In the reach of NASA's CFD vision2030,“Mesh generation and its adaptive techniques” listed as one of the six important research fields in the future.In fact,the first step in numerical computation is to generate a suitable computational mesh.The computational problem on the continuous manifold is discretized so that it can be calculated on the grid cell,for example,two-dimensional manifolds are usually discretized into triangular mesh,quadrilateral mesh or polygonal mesh,while three-dimensional problems usually use tetrahedral or hexahedral meshes.In practical problems,the quality of mesh directly affects the accuracy of numerical calculation results,and with the introduction of high-precision and high-resolution format,the computing format requires more and more meshs.Mesh generation technology involves many mathematical disciplines,such as topology,computer graphics,computational geometry,computational mechanics and so on.In the area of mesh generation,there are a number of mathematical problems that are still unresolved.For examplem,it is well-known there exist 3d indecomposable polyhedra whose interiors cannot be triangulated without new vertices added.The famous example of Sch ¨onhardt(known as the Sch ¨onhardt polyhedron)shows that a twisted non-convex triangular prism cannot be triangulated without adding new vertices,so-called Steiner points.Since 1911,various examples of such polyhedra have been constrcuted.Rambau first showed that any non-convex twisted prisms over an n-gon(n ? 3)cannot be triangulated without Steiner points.Furthermore,he showed that the non-triangulability of such polyhedra does not depend on how much it is twisted.This generalised Sch ¨onhardt polyhedron into a family of polyhedra with such property.We call polyhedra of this family Rambau polyhedra,The original Sch ¨onhardt polyhedron is the simplest case of a Rambau polyhedron.So,are all the non-convex triangular prism cannot be triangulated? The answer is no,the first part of this paper is considering a more general class of prismatoids and studying the question whether they can be triangulated without Steiner points or not.We call a prismatoid decomposable if it can be cut into two smaller prismatoids(which have smaller volumes)without using additional points.Otherwise it is indecomposable.The indecomposable property implies the non-triangulable property of a prismatoid but not vice versa.we show that the set of boundary edges of a prismatoid is related to a knot(a closed curve)or a link of two knots which appear on the surface of a torus.We then show that the question “whether a prismatoid is decomposable or not” is equivalent to the question “whether there exists a compatible separation of its corresponding knot(or link)into a set of unknots or not”.The second part of this paper is to varify two embeddings of a piecewise linear topological space in the Euclidean space to be isotropic,the main tool in that research is the Haefliger-Wu invariants which is put forward in 1978 by Mr Wu,he provided some invariants in[1],some of them is the necessary conditions of two embeddings to be isotropic,they called Haefliger-Wu invariants.Haefliger-Wu invariants is computed based on the homology groups of the deleted product of a manifold,it can be used to verify whether two embeddings are isotopic or not,which is an important topic in algebraic topology.Our work proposes an algorithm of computing Haefliger-Wu invariants based on some algebraic topological methods.Given a simplicial complex embedded in Euclidean space,the deleted product of it is the direct product with diagonal removed.The Gauss map transforms deleted product to the unit sphere.Pull-back of the generator of cohomology group of the sphere defines characteristic class of the isotopy of the embedding.By using Mayer Vietoris sequence and K ¨unneth theorem,the authors prove the ranks of homology groups of the deleted product of a closed surface and give a construction of the generators of the homology groups of the deleted product.Numerical experimental results show the efficiency and efficacy of the proposed method.The reason to the importance of mesh generation in CAD is its important role in CFD.The modern problem in CFD includes some complex physical problems.Examples include in particular convection-dominated problems whose solutions have,e.g.,layers,shocks,or corner and edge singularities.Anisotropic meshes have great importance in numerical methods to solve partial differential equations.They improve the accuracy of solution and decrease the computational cost.So the generation of anisotropic mesh is an important process to research.The third part of this paper we propose a method which is able to generate anisotropic meshes according to the given metric tensors using quasi-conformal mappings.The inputs of our algorithm are a metric tensor and a surface,our goal is to construct a metric-adapted mesh of the surface.We convert the problem of the construction of a quasi conformal parameterization from the given surface to a canonical domain,whose Beltrami coefficients are related to the metric tensor.Then the generation of the metric-adapted mesh of the surface is converted to the generation of an isotropic mesh on a canonical domain,which has mature solutions.We then generate the isotropic mesh,then map it to the target surface.Our method has its corresponding theory to support,so the image mesh of the mapping can be guaranteed to be of high quality.Our method is different from these classical metricbased or high dimensional embedding mesh adaptation methods,while similar with them,it also can deal with complicated metric tensor cases.