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numerical grid generationGrid generation has been around since the dawn of the computational modeling. It is well known that the quality of the computational grid can significantly affect the convergence of the numerical scheme, as well as the final results. Grid generation "by hand" is very expensive in terms or manhours. For complex domains it can take larger part of the total work. This is why automatic grid genereation has been a longstanding goal in industry and science. The oldest numerical grid generation techniques are based upon the solution of elliptic PDE's. Typically, a Poissiontype equation is solved given the boundary grid distribution to generate interior nodal points. The solution domain is often topologically equivalent to a cube in 3D and a square in 2D. One of drawbacks of these methods is that the mapping (usually a conformal mapping of a square onto the curvilinear domain) can have singularities in the corners of the domain. This creates two types of difficulties:
These problems can be solved by changing the "prototype" domain from a square to a geodesic quadrangle on a surface of constant curvature. These are quadrangles that live on surface of a constant curvature (e.g., a sphere or a negativecurvature surface), and have the sides that are geodesics, the shortest paths between two points in the given metric. For example, on a plane the geodesics are straight lines and on sphere the geodesics are the great circles (such as the equator). The geodesic quadrangle P is generated in such a way that it has same angles as the physical domain D, and a conformal mapping between the two is found. Therefore whatever grid we specify in P, will conformally be mapped into D.
For example, we can use this propery for the purpose of grid adaptation, as shown below using circle and a channel with wavy wall as examples (clickable).
The algorithm that we have developed is extendable to multiblock grids, as shown in the figures above (circles) and on the right (multipart wing). The grid around the twopart Boeing airfoil consists of 6 connected blocks, yet it is given by a single mapping. In this case, the prototype domain consists of six different quadrangles on six different surfaces of six different constant curvatures, all "glued" together. Note that our approach guarantees the existence and uniqueness of such a mapping. This means that you cannot get two solutions to the meshing problem for one set of specified parameters. relevant publications


Copyright © 2009 Sergei Chumakov Last modified Jul 29 2010 18:17 Accessed 5479 times 