The mesh generation

The mesh generation

Describe general methods structured unstructured hybrid adaptive etc and discuss their key features and applications

A key step of the finite element method for numerical computation is mesh generation One is given a domain such as a polygon or polyhedron more realistic versions of the problem allow curved domain boundaries and must partition it into simple elements meeting in welldefined ways There should be few elements but some portions of the domain may need small elements so that the computation is more accurate there All elements should be well shaped which means different things in different situations but generally involves bounds on the angles or aspect ratio of the elements One distinguishes structured and unstructured meshes by the way the elements meet a structured mesh is one in which the elements have the topology of a regular grid Structured meshes are typically easier to compute with saving a constant factor in runtime but may require more elements or worseshaped elements Unstructured meshes are often computed using quadtrees or by Delaunay triangulation of point sets however there are quite varied approaches for selecting the points to be triangulated

The simplest algorithms directly compute nodal placement from some given function These algorithms are referred to as algebraic algorithms Many of the algorithms for the generation of structured meshes are descendents of numerical grid generation algorithms in which a differential equation is solved to determine the nodal placement of the grid In many cases the system solved is an elliptic system so these methods are often referred to as elliptic methods

It is difficult make general statements about unstructured mesh generation algorithms because the most prominent methods are very different in nature The most popular family of algorithms is those based upon Delaunay triangulation but other methods such as quadtreeoctree approaches are also used

Delaunay Methods

Many of the commonly used unstructured mesh generation techniques are based upon the properties of the Delaunay triangulation and its dual the Voronoi diagram Given a set of points in a plane a Delaunay triangulation of these points is the set of triangles such that no point is inside the circumcircle of a triangle The triangulation is unique if no three points are on the same line and no four points are on the same circle A similar definition holds for higher dimensions with tetrahedral replacing triangles in 3D

QuadtreeOctree Methods

Mesh adaptation often referred to as Adaptive Mesh Refinement AMR refers to the modification of an existing mesh so as to accurately capture flow features Generally the goal of these modifications is to improve resolution of flow features without excessive increase in computational effort We shall discuss in brief on some of the concepts important in mesh adaptation

Mesh adaptation strategies can usually be classified as one of three general types rrefinement hrefinement or prefinement Combinations of these are also possible for example hprefinement and hrrefinement We summarise these types of refinement below

rrefinement is the modification of mesh resolution without changing the number of nodes or cells present in a mesh or the connectivity of a mesh The increase in resolution is made by moving the grid points into regions of activity which results in a greater clustering of points in those regions The movement of the nodes can be controlled in various ways On common technique is to treat the mesh as if it is an elastic solid and solve a system equations suject to some forcing that deforms the original mesh Care must be taken however that no problems due to excessive grid skewness arise

hrefinement is the modification of mesh resolution by changing the mesh connectivity Depending upon the technique used this may not result in a change in the overall number of grid cells or grid points The simplest strategy for this type of refinement subdivid

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