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 g_{ij} defined by Eq. (III5), 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 quasilinear, secondorder 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. TwoDimensional 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. (XI71). 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. (XI70), we have,
The generation system based only on orthogonality then is

(12) 
Finally, for the smoothness integral, Eq, (XI69), 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 threedimensional 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. (III37), 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) 