Page 81 - Basic Structured Grid Generation

70 Basic Structured Grid Generation

using properties of determinants and suffixes to denote partial differentiation. We
deduce that
LN − M 2
κ G =
EG − F 2
 
 r uu · r vv [11, 1][11, 2] r uv · r uv [12, 1][12, 2] 

1  

= [22, 1] E F − [12, 1] E F
2
a  
[22, 2] F G [12, 2] F G
 

1 1 0
 1 1 
 r uu · r vv − r uv · r uv 2 E u F u − E v 2 E v 2 G u 

2
1  
1 1
= F v − G u E F E v E F ,
2 2 − 2
a  
 1 F G 1 
2 G v 2 G u F G
again using simple properties of determinants and eqns (3.51). Moreover, it is straight-
forward to verify that
1
1
r uu · r vv − r uv · r uv =− E vv + F uv − G uu . (3.133)
2 2
Hence we obtain the equation
 1 1 1 1
 F uv − E vv − G uu 2 E u F u − E v

2
2
2
1  1
κ G = F v − G u E F
2 2 2
(EG − F ) 
 1
G v F G
2
0 E v
1 1 
2 2 G u 


1
2
− E v E F . (3.134)

1 
2 G u F G
Exercise 15. Verify that when the co-ordinate curves are orthogonal (so that F = 0),
this formula may be expressed in the simpler form
√ √

1 ∂ 1 ∂ G ∂ 1 ∂ E
κ G =−√ √ + √ . (3.135)
EG ∂u E ∂u ∂v G ∂v
The significance of the Gauss-Codazzi eqns (3.127) and (3.130) may be appre-
ciated by referring to the fundamental existence theorem for surfaces (not proved
α
here), which states that given two quadratic differential forms a αβ du du β and
β
α
β
α
b αβ du du , such that a αβ du du is positive definite and that the six coefficients
a 11 ,a 12 ,a 22 ,b 11, b 12 ,b 22 satisfy the Gauss-Codazzi equations, then there exists a sur-
face, uniquely determined apart from its precise position in space, whose first and
second fundamental forms are given respectively by these forms.
3.9 Div, grad, and the Beltrami operator on surfaces
The purpose of generating grids is normally to solve a physical problem represented by
a partial differential equation, the so-called hosted equation. In this section we present