Research And Application Of Adaptive Finite Element Mesh Generation Algorithm

The Finite Element Methods (FEM) is one of the general engineering numerical methods.However, for most complex applications, the engineer should have a lot of practicalexperience and time to succefully use the FEM to solve engineering problems. For this reasonmany researchers are investigating ways to automate the finite element method, thus allowingan improved productivity, more accurate solution, and used by less trained personnel. Thefinite element method can be divided into several sub-steps, often the most time consumingand experience requiring task faced by an analyst is the discretization of a general geometricdefinition of the problem into a valid and well conditioned finite element mesh. FiniteElement Analysis of the accuracy and the cost depend directly on the element size, shape andnumber. Automatic quality finite element mesh generation can make FEM easier to be apowerful tool for the general engineer. Therefore, automation of the mesh generation is animportant prerequisite for the complete integration of the FEM with design processes incomputer aided engineering (CAE) and manufacturing (CAM) systems.In this paper, an improved adaptive triangle and tetrahedral adaptive mesh generator has beendeveloped, which includes two-dimensional Riemannian metric based adaptive meshgeneration, complex 3D surface adaptive mesh generation and adaptive 3-D tetrahedral meshgeneration. It also introduces the interfaces of B-Rep which was used to exchange databetween mesh generator and various CAD software. Topology-Based mesh data structuresand the procedures of subdivision point location algorithm are also given here.The Riemannian metric based adaptive mesh generation algorithm can generate not onlytwo-dimensional isotropic elements but also the anisotropic elements, thereby satisfying theneeds of computational fluid analysis. For the combination of three-dimensional complexsurfaces, an extended Advancing Front Technique with shift operations and Riemann metricnamed as shifting-AFT is presented for generating finite element meshes on 3D surfaces,especially 3D closed surfaces. Riemann metric is used to govern the size and shape of thetriangles in parametric space. The shift operators are employed to insert a floating spacebetween real space and parametric space during 2D parametric space mesh generation.Combining the shift operators, the advancing front technique kernel is extended to overcomethe mesh quality-worsening problem in closed surface mesh generation due to introducingvirtual boundaries into 2D open parametric domains generally mapped from closed surfaces.The shifting-AFF can generate high-quality meshes and guarantee convergence in both opensurfaces and closed surfaces. For the shifting-AFT, it is not necessary to introduce virtualboundaries manually or automatically while meshing a closed surface, so that the boundarydiscretization procedure is simplified very much, and moreover, better-shaped triangles will be generated because there are no additional interior constrains yielded by virtual boundaries.Comparing with direct methods, the shifting-AFT avoids carrying out costly and unstable 3Dgeometrical computations in real space. The examples demonstrate the advantages of theshift-AFT in 3D surface mesh generation, especially for closed surfaces.In Chapter 8, a reliable and effective tetrahedral meshing algorithm is also proposed based onadvancing front method. The operators such as insert query and delete like a database areimplemented by using topology based mesh data structures which accelerates the wholealgorithm. Instead of preparing a background mesh for mesh spacing control, this informationis estimated at the beginning of each layer at each node from the area of connecting triangleson the front and a user-specified stretching factor. A cell searcher is prepared to correct themesh spacing information and to perform geometric search efficiently. During rolling backthe advancing path is changed by changing preferential factor of front, as a result the times ofrolling back is decreased significantly. Node inserting based on linear programming techniqueguarantees the convergence of the algorithm. At the end of the mesh generation process,unwanted node removing and angle-based smoothing are employed to enhance the resultingmesh quality. The examples demonstrate that high quality tetrahedral meshes can begenerated within a reasonable time limitIn Chapter 9, Based on the basic transform template from tetrahedron to hexahedron, a seriesof flexible extended transform template is presented. The number and the density of thehexahedral mesh transformed from the tetrahedral mesh can be controlled using the variedtemplates and their assemblies. Thus, it is needless that the initial tetrahedron mesh isgenerated very finely. According to different type of boundary mesh nodes, using differentmap method to modify new generated nodes' coordinates. This finds that it could make thenew generated nodes, which are nearest to the boundary; locate the entity's boundaryaccurately. The essential steps of this scheme are described by means of flow, which based onCAD platformPoint location is one of the most basic searching problems in the computational geometry. Ithas been largely studied on the aspect of the query time in the worst-case and many methodshave been proposed such as Counter Clockwise Wise Search and Barycentric CoordinatesSearch. However most of them aim at the convex field. For a non convex problem such as aconcave field or a convex field with holes, these searching schemes may fail. The numericalexperience shows that even for convex problem, the searching path may lead to an infiniteloop for some special case and can not find the element containing the query point. In Chapter10 a robust backward search method based on Walk-through algorithm is proposed to dealwith the searching problems in non-convex fields and to avoid the problems of infinite loop.Another important improvement is to locate the query point on a 3D surface mesh. Several examples demonstrate that the present method is efficient and robust for the workpieces ofcomplex geometry.The author would like to appreciate the joint supports to this project by the National NaturalScience Foundation of China (10572032, 10421002), Outstanding Young ScientistsFoundation (10225212) and French Foreign Ministry Eiffel PhD scholarship (Bourse d'Excellence EIFFEL).