Numerical Grid Generation
Foundations and Applications
##### By: Joe E. Thompson, Z.U.A. Warsi and C. Wayne Mastin

APPENDIX B

EULER EQUATIONS

1. Variational Principle in Transformed Space

Consider the integral

where is the covariant metric tensor, with elements gij 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 (xi)j only through the elements of the metric tensor, , we have

 (3)

where i is the unit vector in the xi-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, gjk = 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 wj . 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 Fw 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., Fo 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 gij 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 gik (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 Cik is the signed cofactor of Aki.

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 Si of Eq. (18) are redefined as

 (21)