By Vladimir D. Liseikin
The strategy of breaking apart a actual area into smaller sub-domains, often called meshing, enables the numerical answer of partial differential equations used to simulate actual structures. In an up-to-date and extended moment version, this monograph supplies a close therapy in response to the numerical resolution of inverted Beltramian and diffusion equations with recognize to watch metrics for producing either based and unstructured grids in domain names and on surfaces.
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Additional info for A Computational Differential Geometry Approach to Grid Generation
3) depends upon a single parameter and therefore describes a curve. This curve is referred to as the ~i curvilinear coordinate line. 2 Coordinate Lines, Tangential Vectors, and Grid Cells 35 The vector-valued function x(e) generates the nodes, edges, faces, etc. of the cells of the coordinate grid in the domain xn. ::1i x = x(e + hiei) - x(e) , where hi is the step size of the uniform grid in the ~i direction in the logical domain En. 4) e, is the vector tangential to the coordinate curve and t is a residual vector whose length does not exceed the following quantity: 1 2 "2maxlxE,iE,il(hi), i=l, ...
C) and respectively. The infinitesimal distance PQ denoted by ds is called the element of length or the line element. In the Cartesian coordinates the line element is the length of the diagonal of the elementary parallelepiped whose edges are dXl, ... , dxn, where .. a~. dx' = x'(e + de) - x'(e) = ~de u~J + o(ldW, i,j = 1, ... , n, (see Fig. 4). Therefore ds = V(dX 1)2 + ... id~i. jd~j + o(ldel) = Jgijd~id~j + o(ldel), Thus the length s of the curve in xn, i,j = 1,···,n. prescribed by the parametrization x[e(t)] : [a, b] -+ Xn , is computed by the formula i,j=l, ...
N, are the elements of the contravariant tensor in the logical coordinates ~ 1 , ... , ~n of a monitor metric g0'e imposed on En, gme = det(g0'e), while gtJ is the covariant Eucledian metric tensor of 5 n in the coordinates This functional was proposed for n = 2 by Godunov and Prokopov (1967) for generating conformal and quasi-conformal grids in 52. In their consideration the elements i, j = 1,2, of the monitor metric should be dependent on e and some, in general vector-valued parameter T. Belinskii et al.
A Computational Differential Geometry Approach to Grid Generation by Vladimir D. Liseikin