A Computational Differential Geometry Approach to Grid by Vladimir D. Liseikin

By Vladimir D. Liseikin

The strategy of breaking apart a actual area into smaller sub-domains, referred to as meshing, enables the numerical answer of partial differential equations used to simulate actual platforms. In an up-to-date and multiplied moment variation, this monograph supplies a close therapy according to the numerical resolution of inverted Beltramian and diffusion equations with admire to observe metrics for producing either based and unstructured grids in domain names and on surfaces.

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N . , n . 10) For example, the normal base vector V~i is expanded through the base tangential vectors Xt;j, j = 1, ... , n, by the following formula: i i a~i ae k i,j,k,=l,···,n. 11) = 1, ... 12) i, j = 1, ... , n , and consequently b = biV~i = (b· Xt;i)V~i, i = 1,···,n. 13) These components bi , i = 1"", n, of the vector b are called covariant. In particular, the base tangential vector Xt;i is expressed through the base normal vectors v~j, j = 1, ... , n, as follows: Xt;i = . ax k ax k . (Xt;i 'Xt;j)Ve = a~i a~j Ve, i,j,k = 1, ..

B . Thus the volume of the parallelepiped determined by the vectors a, b, and c equals the Jacobian of the matrix formed by the components of these vectors. 3 Relation to Base Vectors Applying the operation of the cross product to two base tangential vectors xI;' and x~=, we find that the vector X~l x Xt;= is a normal to the coordinate surface ~i = ~b with (i, I, m) cyclic. e. VC = C(Xf,1 X Xt;m) . 32), Xt;i, using the operation of the dot 1 = c J, and therefore . 33) = j(Xf,1 X Xt;m). 18)) can also be found through the tangential vectors X~i by the formula ..

J=l The intermediate transformation s(e) = [sl(e), ... 13) by changing mutually dependent and independent variables. dB s - ~ p a~i' 2,J=1 k = 1, ... 16) 2=1 where g~: is the (ij)th element of the contravariant metric tensor of sxn in the grid coordinates ~n . A two-dimensional Laplace system which implied the parametric coordinates to be solutions in the logical domain 52 was introduced by Godunov and Prokopov (1967), Barfield (1970), and Amsden and Rirt (1973). A general two-dimensional elliptic system for generating structured grids was considered by Chu (1971).

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