Responsible: V.A. Garanzha, Doctor of Science

Computational mesh generation is an integral component and an important stage in numerical simulation of fluid and gas flows around bodies of complex shape. Geometric models of bodies and computational domains in the area of computational aerodynamics and structural analysis, as a rule, are created using "heavy" CAD packages such as Catia, Solidworks, etc. To work with such models one have to use special libraries, which are usually referred to as "geometric kernels". Currently, majority of advanced CAD/CAE software tools use proprietary Parasolid geometric kernel. Freeware geometric kernels such as OpenCascade have also achieved reasonable degree of usability and versatility.

In problems of engineering analysis and production quality models are reconstructed from 3d scan data using special reconstruction techniques. The methods of reconstruction are also widely used in computer graphics, biology, medicine, architecture, and in many other applied and fundamental areas. Rather typical situation in these fields is that the geometric model may contain a variety of defects, inaccuracies, contradictions both from a geometrical point of view, as well as topologically.

Thus, the mesh generation stage is usually preceded by a stage of geometry clean-up which outputs correct solid or surface geometric model. Mathematical software used to clean up geometrical models is quite sophisticated and expensive. One should not that modern mesh generation algorithms can handle moderate defects in the description of the geometry, thus in many cases avoiding the use of specialized software for geometry correction.

## Methods and algorithms for mesh generation

For the problems of numerical simulation of fluid and gas flows, one can use different types of computational grids, including a block-structured curvilinear grids, tetrahedral meshes, and hybrid grids consisting of tetrahedra with layers of prismatic cells in the boundary and shear layers, the general unstructured mesh consisting of tetrahedra, prisms , pyramids, and hexahedra, polyhedral mesh, consisting of non-convex polyhedral cells with an arbitrary number of faces, as well as adaptive Cartesian grid with a hierarchical structure based on octal trees and the use of truncated cells.

Each type of meshes you has its own advantages and drawbacks. For tetrahedral meshes, there are fast and reliable algorithms in the case of bodies of complex shape, but they are not the most effective tool in the presence of boundary layers and mixing layers. A relatively simple and effective way to correct the basic flaws of tetrahedral meshes is based on the construction of highly anisotropic layers of prisms in the boundary layers and other regions of the anisotropy of solutions. Curvilinear block-structured grids allow to obtain the numerical solutions with a high degree of accuracy and reliability, but their construction is still not amenable to automation and remains time-consuming and labor intensive. Methods of constructing a fully unstructured hexahedral meshes are currently being actively developed, but the problem of automatic construction has not yet been resolved.

Our group focuses on the variational methods of construction, untangling and optimization of computational grids. The main applications are the construction of block structured grids with a relatively small number of blocks for bodies of complex shape, the optimization of meshes of various types, such as tetrahedral, hexahedral and polyhedral, mapping of surfaces, and mapping of meshes onto irregular surfaces. Another important area is the construction of tetrahedral meshes based on the methods of self-organization based on inaccurate and inconsistent geometric models. In this method, one can specify the following basic components:

- definition of computational domain via implicit function;
- construction of tetrahedral meshes in implicit domains;
- self-organization of the mesh vertices via repulsive forces;
- projection of the approximate boundary vertices on the exact boundary;
- sharpening stage for "manifestation" of sharp edges on the boundary;
- mesh optimization for the removal of flat tetrahedra (slivers).

Methods and algorithms for constructing a structured and block-structured meshes

- variational principle for the construction of quasi-isometric mappings;
- iterative minimization algorithm;
- implementation of slip boundary conditions on the boundaries of implicit domains;
- untangling and optimization of hexahedral meshes:
- comprehensive tests for untangling and optimization.

Current status of development of mesh generator and visualization tool and work plans

In 2011, it was implemented a complete technological mesh generation cycle albeit with reduced functionality:

- a model in the STEP format -> tessellated model -> model cut into blocks -> surface mesh generator -> volume mesh generator;
- CAD models manipulation using OpenCascade geometric kernel ;
- in the current version of the elongated bodies of revolution with control elements (rudders, wings, etc.) are considered, cut into blocks produced by planes;
- in the current version is internal representation of block structured meshes is supported, computational kernel of mesh optimizer also support block data structure;
- mesh adaptation to the curvature of the surface is implemented albeit in the research phase;
- visualization tool displays surface and volume meshes, calculated fields, contours, isosurfaces and supports stereo regime using glasses.

Priorities for the development of mesh generator and visualization tool.

- implementation of a stable version surface mesh generation with adaptation to surface curvature in the presence of geometrical noise;
- implementation of semi-automatic block structured surface mesh generation which requires integration with a graphical user interface;
- implementation of the 3d block structured generator with quasi-2d block partitioning, in particularly layered mesh generator around block structured surface meshes;
- interactive tool for 3d block partitioning;
- implementation of tet mesher based on prescribed boundary triangulation;
- support of manipulation with block structured meshes in visualization tool.