Numerical Grid Generation
Foundations and Applications

By: Joe E. Thompson, Z.U.A. Warsi and C. Wayne Mastin

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APPENDIX B

EULER EQUATIONS

1. Variational Principle in Transformed Space

          Consider the integral

where is the covariant metric tensor, with elements g_ij defined by Eq. (III-5), and w() is a weight function dependent on .

A. Grid Generation System

          The Euler equations then are given by

(2)

as has been noted. Since

and F depends on (x_i)^_j only through the elements of the metric tensor, , we have

(3)

where _i is the unit vector in the x_i-direction. Here the operation indicated by the notation, _i, is the simple replacement of _j by _i in F. Also, since F depends on _j only through , we have

or

Therefore,

          Since F depends on only through the weight function we have

Then the Euler Equations can be written as

or as the vector equation

(6)

(Note that the symmetric elements of the metric tensor, g_jk = g _kj, are to be left as distinct elements in F until after the differentiation has been performed.)

          Expanding the ^j-derivative, we then have

But also

so that

          Thus we have the grid generation system, with written as F',

(7)

where

(8)

This is a quasi-linear, second-order partial differential equation for the cartesian coordinates .

          If the weight function depends directly on , instead of on in Eq. (1), then in Eq. (2). Also in his case, the that appears on p. 439 and in the development that leads to Eq. (7) is replaced by simply w^_j . Then Eq. (7) is replaced by

(9)

for a weight function w() in Eq. (1).

B. Two-Dimensional Examples

          In two dimensions, the generation system (7) becomes (with ¹ = and ² = )

If the weight function depends on , rather than on x, the terms and in Eq. (10) become w and w, respectively, and the last term, -- 1/2 F'w, vanishes.

          As an example, consider F_w from Eq. (XI-71). Then we have

Then the generation system based on concentration by Eq. (7) is

(11)

          With F taken to be a measure of orthogonality, i.e., F_o from Eq. (XI-70), we have,

The generation system based only on orthogonality then is

(12)

          Finally, for the smoothness integral, Eq, (XI-69), the derivatives needed are

          The complete generation system is then obtained as the linear combination of the concentration system, Eq. (11), the orthogonality system, Eq. (12), and the smoothness system which is formed by substituting the above relations into the general equations (7). The three-dimensional case follows in an analogous fashion.

2. Variational Principle in Physical Space

          With the variational problem formulated in the physical space, consider the integral

(13)

where is the contravariant metric tensor, i.e., with elements g^ij from Eq. (III-37), and the weight function is a function of .

A. Grid Generation System

          Then for the Euler equations, we have

(14)

Now,

and F depends on (ⁱ)_xj only through . Then

          Also, since F depends on ⁱ only through g^ik (k = 1,2,3) we have

Therefore,

          Also, since F depends on only through the weight function, we have

Then the Euler equations can be written

or

Now

and

Then

or,

Now

and

          Then the generation system is, with written as F',

(15)

where

(16)

This can also be written as

	(17)
	(18)

Then

(19)

where C_ik is the signed cofactor of A_ki.

          If the weight function in the integral (13) is a function of , rather than , then in the Euler equation (14), and Eq. (15) is replaced by

(20)

In this case S_i of Eq. (18) are redefined as

(21)