Mesh generation

Builds on Triangulation Simplicial complex

Definition

Let $d \in \set{2, 3}$ and let $\Omega \subset \R^d$ be a bounded open set whose closure $\overline{\Omega}$ is a polygon ( $d = 2$ ) or a polyhedron ( $d = 3$ ). A conforming mesh of $\overline{\Omega}$ is a finite set $\mathcal{T}$ of closed cells, each a simplex or each a convex quadrilateral or hexahedron, such that:

the union of the cells of $\mathcal{T}$ is $\overline{\Omega}$ ;
the interiors of two distinct cells are disjoint;
the intersection of two distinct cells is empty or a common face of both, that is, a common vertex, edge or facet.

Mesh generation is the task of computing a conforming mesh of $\overline{\Omega}$ from a description of its boundary. Usually it must meet conditions on the size $h_K = \operatorname{diam} K$ and the shape of each cell $K$ . A typical shape condition is an upper bound on $h_K / \rho_K$ , where $\rho_K$ is the radius of the largest ball contained in $K$ .

Remarks

When all cells are simplices, a conforming mesh is a triangulation of $\overline{\Omega}$ , the underlying space of a simplicial complex. Example: the unit square $[0,1]^2$ with the two triangles of vertices $(0,0), (1,0), (1,1)$ and $(0,0), (1,1), (0,1)$ is a conforming mesh. The triangles cover the square, their interiors are disjoint, and they meet along the common edge from $(0,0)$ to $(1,1)$ . Non-example: cover $[0,2] \times [0,1]$ by the squares $[0,1] \times [0,1]$ , $[1,2] \times [0, \tfrac12]$ and $[1,2] \times [\tfrac12, 1]$ . Clauses 1 and 2 hold, but clause 3 fails: the first and second squares meet in the segment $\set{1} \times [0, \tfrac12]$ . This segment is an edge of the second square but not of the first. The point $(1, \tfrac12)$ is called a hanging node. A family of meshes with $\max_K h_K \to 0$ and $h_K / \rho_K$ bounded uniformly is called shape regular. Finite element error estimates are usually proved for such families.

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Simplicial complex concept
Triangulation concept
Mesh generation definition

Tag	`QQ3PBZS9J8`
URI	`m2://QQ3PBZS9J8`
Kind	definition
Writing	W0 stub
MSC	65N50, 65M50, 65D18
Defines	Mesh generation
Uses	Triangulation, Simplicial complex

Mesh generation

Definition

Remarks

Reading path

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