Mesh Generation

By Marshall Bern, Paul E. Plassmann. Handbook of Computational Geometry, 2020

483 citations recorded by the source corpus.

An essential step in scientific computing is to find a proper discretization of a continuous domain. This is the problem of mesh generation. Once we have a discretization or sometimes we just say a " mesh " , differential equations for flow, waves, and heat distribution are then approximated by finite difference or finite element formulations. However, not all meshes are equally good numerically. Discretization errors depend on the geometric shape and size of the elements while the computa tional complexity for finding the numerical solution depends on the number of elements in the mesh and often the overall geometric quality of the mesh as well. The most general and versatile mesh is an unstructured triangular mesh. Such a mesh is simply a triangulation of the input domain (e.g., a polygon), along with some extra vertices, called Steiner points. A triangulation is a decomposition of a space into a collection of interior disjoint simplices so that two simplices can only intersect at a lower dimensional simplex. We all know that in two dimensions, a simplex is a triangle and in three dimensions a simplex is a tetrahedron. A triangulation can be obtained by triangulating a point set, that form the vertices of the triangulation. Even among triangulations some are better than others. Numerical errors depend on the quality of the triangulation, meaning the shapes and sizes of triangles. In order for a mesh to be useful in approximating partial differential equations, it is necessary that discrete functions generated from the mesh (such as the piecewise linear functions) be capable of approximating the solutions sought. Classical finite element theory [18] shows that a sufficient condition for optimal approximation results to hold is that the minimum angle or the aspect ratio of each simplex in the triangulation be bounded independently of the mesh used; however, Babuska [4] shows that while this is sufficient, it is not a necessary condition. See Figure ?? for a triangulation whose minimum degree is at least 20 degree. In summary, properties of a good mesh are: • Fills the space Non-overlapping (� areas = total area) • • Conforming Mesh (every edge shared by exactly 2 triangles) • High Quality Elements (approximately equilateral triangles) Automatic mesh generation is a relatively new field. No mathematically sound procedure for obtaining the 'best' mesh distribution is available. The criteria are usually heuristic. The input description of physical domain has two …

Subfields: Computational Mechanics, Computer Graphics and Computer-Aided Design

View Paper: https://www.semanticscholar.org/paper/31245c869c615d9ecaf1ad6ca848d94775ac7d75

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Computer Graphics and Computer-Aided Design

Citations in the corpus

Isotropic Remeshing of Surfaces: A Local Parameterization Approach (2003, unknown): “The shape, angles or size of these elements must match certain criteria (see [4, 5]).”
Gmsh: A 3‐D finite element mesh generator with built‐in pre‐ and post‐processing facilities (2009, methodology): “To quickly evaluate the adequation between the mesh and the prescribed mesh size field, we defined an efficiency index τ [13] as”
Continuous Mesh Framework Part I: Well-Posed Continuous Interpolation Error (2011, background): “These considerations are based on the concept of a unit element and a unit mesh [17], recalled hereafter.”
Studied X-FEM enrichment to handle material interfaces with higher order finite element (2010, methodology): “Several methods of adaptive h-refinements were developed (for example [9,25,30]), by subdividing elements or remeshing.”
Geometric Shortest Paths and Network Optimization (2000, background): “(See the chapter of Bern and Plassman 66] in this handbook, or the survey by Bern 63] for more information on triangulations.)”
Large-scale 3D random polycrystals for the finite element method: Generation, meshing and remeshing (2011, unknown): “Free meshing methods typically involve two steps: a preliminary meshing and mesh optimization, during which topological operations are applied to improve element quality [31,32,34,35].”