Phase of the finite element method

1. Meshing Methods

A significant phase of the finite element method for numerical computation is mesh generation. Meshing is defined as substituting of solid geometry model with a set of distinct points, lines, panels, elements. The approach to cut whole flow domain in to small elements is called meshing technique. Particular portions of the domain require small elements in order that the computation is more precise. The meshing methods include structured, unstructured, hybrid, adaptive etc.

1.1.1 Structured Meshing

It is defined as in which the elements are laid in a regular grid acknowledged as block. Structured meshing requires more elements and saves constant factor in runtime. It makes use of hexahedral elements in 3D and quadrilateral elements in 2D in a computationally rectangular selection. In addition to it, it develops elliptic equations in order to optimize the outline of the mesh intended for orthogonality and uniformity. The mesh can be formed to body fitted by means of stretching and twisting of the block.

Multi block unstructured mesh generation used for solution domains with complex geometries which involves a complex solution domain partitioned into simpler sub-domains. Hereafter, mesh is produced in apiece sub-domain and matching routine which bear a resemblance to the sub-domains and correspond with the individual mesh at the boundaries of the sub-domains.

Structured meshing has several advantages as compared to the other meshing methods which follow as:-

  1. High degree of freedom since control points and edges interacts which results in total freedom in positioning the mesh
  2. Hexahedral and quadrilateral elements sustain high amount of skewness and stretching which yields in condensation of points in the area of high gradient in the flowfield and enlarges to less dense.
  3. The structured mesh is often flow-aligned

1.1.2 Unstructured Meshing

It is defined as arrangement of elements with no discernible pattern acknowledged as unstructured meshing. It uses random assembly of elements in order to pile up the domain and utilize triangles in 2D and tetrahedral in 3D. Unstructured meshing offer more flexibility as compared to the structured mesh and hence is very useful in finite element and finite volume methods. It permits automatic adaptive refinement based on the pressure gradient or regions interested. However, it has several disadvantages which include limitation to largely isotropic due to the triangle and tetrahedral elements ability of twisting and stretching.

Unstructured mesh techniques depend upon the features of the Delaunay triangulation and voronoi diagram. Delaunay triangulation is defined as set of triangles of the points in plane such that no point is within the circumcircle of a triangle. The triangulation is distinctive on stipulation that no four points are on the same circle and no three points are on the same line. In addition to it, a related definition holds for higher dimensions, with tetrahedral replacing triangles in 3D.

1.1.3 Hybrid Meshing

Utilization of structured mesh in the local area whereas unstructured mesh in the bulk of the domain known as hybrid meshing (quasi structured). It consists of triangles and quadrilateral elements in 2D and hexahedral, tetrahedral, prismatic and pyramidal elements in 3D. Hybrid meshing has the aptitude to manipulate the shape and the division of the mesh which yields immense mesh.

  1. Hexahedral elements are immense where the field flow gradients are high and a greater extent of control but consumes time to get produced
  2. Tetrahedral elements are utilized to fill up the remaining volume.
  3. Pyramid elements are utilized to alteration from hexahedral to tetrahedral elements.
  4. Prismatic meshes are produced by defining the surface mesh and marching off the surface to generate the 3D elements. Prismatic elements defined as triangles extruded into section are utilized for determining nearby wall gradients, however unable to gather in the lateral directions because of underlying triangular structure.

1.1.4 Adaptive Meshing

In adaptive mesh, the algorithm begins with a structured base coarse grid. The individual grid cells are filtered by means of enhanced mesh is overlayed on the coarse. Subsequent to refinement, particularized mesh pieces which are on a specific stage of refinement are conceded to an integrator which develops cells within time.

Enhanced meshes and sub-mesh are recursively advanced in anticipation of maximum stage of refinement is achieved. However, the concentration of refinement at certain points in a cell is higher than needed; the high determination mesh will be replaced with a coarser grid.

Adaptive meshing is categorized into three types which follow as:-

  • r-refinement: - Characterized as alteration of mesh determination without changing the number of nodes exhibit in a mesh. Moving the mesh points into the areas of movement increases the mesh determination which yields in greater scattering of points in areas. However, the nodes movement can be controlled by deforming the mesh.
  • h-refinement: - Defined as alteration of mesh determination by varying the mesh connectivity. Although, it would not result in change in number of overall mesh cells. The simplest strategy for this type of refinement subdivides cells, while more complex procedures may insert or remove nodes (or cells) to change the overall mesh topology.
  • p-refinement: - It attains increased mesh determination by means of increasing the order of accuracy of the polynomial in each element (or cell).

Element types that can be used in adaptive meshing pursue as: -

  1. 2-D Structural Solids: - 2-D 6-Node Triangular Solid, Axisymmetric Harmonic Solid, 2-D 4-Node Isoperimetric Solid
  2. 3-D Structural Solids:- 3-D 8-Node Isoperimetric Solid, 3-D Anisotropic Solid, 3-D 8-Node Solid with Rotational DOF
  3. 3-D Structural Shells:- Plastic Quadrilateral Shell, Elastic Quadrilateral Shell, 8-Node Isoparametric Shell
  4. 2-D Thermal Solids:- 2-D 6-Node Triangular Solid, Axisymmetric Harmonic Solid 2-D 4-Node Isoparametric Solid
  5. 3-D Thermal Shells :- Plastic Quadrilateral Shell

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