Asociaciones y organizaciones del sector

The untangling has recently received greater attention in research activities in the area of unstructured mesh quality improvement. The term untangling refers to the process of elimination of tangled (concave) elements from a mesh. It is applied to those mesh nodes, which represent concave corners of tangled cells. Current position of such a node is clearly unsatisfactory as it creates at least one concave (i.e. invalid) mesh element. Therefore, a new position which corrects all tangled cells adjacent to the node while preserving validity of the rest should be found. The process of untangling derives this position directly from the concavity/convexity criteria applied to each affected cell. In other words, the feasible set or the kernel (a complete set of points satisfying convexity condition for all surrounding cells) is found and the node is placed somewhere within this region (see Figure 2.14).

For instance, Freitag and Plassman [46] solve the problem of maximization of minimum area/volume of elements attached to the node being moved. Knupp [73] uses untangling based on optimization of an objective function constructed using determinants of Jacobian matrices of elements surrounding the node.

(a) Cells around a node. (b) The kernel of a node.

Figure 2.14: Untangling.

A new combination of Laplacian and optimization-based smoothing methods with an un-tangling approach constitutes one of the topics of the present thesis.

2.3 Viscous mesh generation

Up-to-date techniques for viscous flow analysis require high quality of computational meshes in regions where viscous effects are essential. These requirements include smoothness of cell sizes of viscous grids, control of position of the first mesh point close to wall (depending on type of turbulence model [87, 112]), limitation of stretching ratio hi+1/hi that must range within certain values (in the work by Spalart [112] it ranges between 1.2 and 1.3).

Ever since viscous computations became possible, various viscous mesh generation ap-proaches have been developed. In case of boundary-conforming structured grids, the only

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difference between viscous and inviscid regions is the grid spacing in normal-to-wall di-rection. Unfortunately, structured grids tend to contain too many cells in regions where it is not necessary, which is typically due to the limited ability of generators to produce grids with strongly non-uniform cell size distribution. In order to generate more cells in a certain region, new grid lines should be introduced along the entire computational domain in corresponding direction, which usually results in overpopulation of cells in regions where it is not desired. Block-structured methods can resolve this drawback to some degree but they still provide limited flexibility within one block. Therefore, up-to-date unstructured methods require new approaches for viscous meshing.

Essentially, families of approaches that address the issue of high quality anisotropic mesh generation can be classified as follows [48]:

• Direct generation of anisotropic meshes by combining isotropic meshing methods with domain transformation.

• Transformation of initial isotropic mesh by node repositioning and local mesh modi-fications.

• Building anisotropic elements by special methods (advancing layer methods) and isotropic meshing of the rest of the domain.

The first group of methods related to direct generation [18,19,28,29,50] had been attempted together with the Delaunay and advancing front approaches to triangular and tetrahedral (simplicial) unstructured mesh generation. Due to their nature, these approaches produce isotropically graded meshes by default. Certain modifications of algorithms are required in order to allow for generation of highly stretched elements.

Mavriplis [86] generated a metric field consisting of stretching vectors specified at each point. This metric is used to map an existing computational domain to a control domain, where an isotropic mesh is generated via the Delaunay method. Backward conversion of this mesh into original space results in an anisotropic mesh in the domain.

Borouchaki et al [18,19,50] generalized the idea of anisotropic mesh generation via Delaunay method based on the metric specification. Castro-Diaz et al [28,29] extended this approach by recognizing that the mesh orthogonality near the wall as well as the fixed first cell size are of prime importance for successful viscous computations. The metric was modified in the near-wall region to take these criteria into account.

This family of methods, however, generates purely simplicial meshes. This restriction poses a problem with element quality, as presence of low angles is unavoidable. Besides, these meshes contain too many elements with respect to hexahedral or hybrid meshes with prismatic layers.

The approach of modification of initial isotropic mesh is well represented in work by Hassan et al [55]. This method is based on the advancing front technique. Layers of cells are

2.3. Viscous mesh generation 21

generated along an advancing front using isotropic criteria. After generation of another layer, its elements are compressed towards boundary via node repositioning to result in anisotropic elements along the wall. The procedure is repeated until the required number of anisotropic layers is generated along the boundary. After that, the rest of the domain is meshed using a standard advancing front technique. Although this method shows good results in 2-D, its 3-D version is prone to problems. One of the problems is the fact that the method tends to generate more points on the surface of the first layer than on the boundary, thus resulting in degraded mesh quality after node repositioning.

The third group of methods for viscous mesh generation consists in generation of layers of anisotropic cells along boundaries prior to volume isotropic meshing. The layers are generated based on certain criteria. As the next step, the rest of the domain is meshed isotropically using one of the available techniques. Methods of this family are often referred to as the advancing layer methods [100].

Kallinderis et al [65–67] have developed a method that generates layers of prisms around a body. The height of prisms increases with distance from the wall according to user-specification. After specified layers are generated, the remaining domain is meshed using a combination of Octree and advancing front techniques. Sharov and Nakahashi [109]

suggest a similar approach but with some modifications to produce higher quality cells.

Their method generates purely tetrahedral meshes. Generally, use of prisms produces less elements. Problems with element quality near sharp corners appear to be common for this family of methods.

Lohner [80] also uses a similar technique. In his method, layers of tetrahedronized (sub-divided into tetrahedra according to a predefined pattern) prisms are generated along a surface, after which an advancing front method is used to mesh the rest of the domain.

This method is equipped with powerful quality control and improvement tools. However, in his later work [81], Lohner advocates use of anisotropic refinement of isotropic mesh us-ing the Delaunay criterion. Inability of the advancus-ing layer methods to deal with complex geometrical models is the reason for focusing on a new technique.

Pirzadeh [100] proposed a similar approach called the Advancing Layer Method (ALM), in which layers of anisotropic prisms are generated along boundaries. A certain procedure is applied to provide consistent diagonalization of prisms. It is necessary that prism diagonals are consistently related to each other in order to allow for subdivision of a prism into tetrahedra. An example of indivisible prism is shown in Figure 2.8.

Finally, Garimella provided a generalization of the advancing layer methods applied to the problem of anisotropic tetrahedral mesh generation in his PhD thesis [48].

The issue of generation of unstructured hexahedral meshes for viscous flows has not yet been addressed widely in literature. Quite simply, hexahedral methods for unstructured automated meshing are relatively young and still pose many problems related to isotropic meshing.

Research presented herein addresses the problem of anisotropic unstructured hexahedral mesh generation for viscous flow applications.

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