By Vladimir D. Liseikin

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**Read Online or Download A Computational Differential Geometry Approach to Grid Generation PDF**

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**Additional resources for A Computational Differential Geometry Approach to Grid Generation**

**Example text**

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).