6.2  Shell Element Method

In this section collocation is applied to derive the discrete form of the integral equations and a method for the general solution of such methods is developed. The shell(s) are divided into N panels: DG1, DG2, ..., DGN. The shell functions d(p), n(p), F(p), V(p), a(p), b(p), f(p), A(p), B(p), F(p) are approximated by a constant on each panel. Hence the function d(p)   (p Î G) can be represented by the vector

d = [ d(p1),d(p2), ...,d(pN)]T, where p1, p2,..., pN are representative points on the respective panels DG1, DG2, ..., DGN. The other shell functions are discretised in a similar way, represented by the vectors , F, V, a, b, f, A, B, F.

The discrete shell conditions (91), (92) can be written in the form

Da d + Db n = f
(96)
and

DA F + DB V = F .
(97)

The discrete equivalent of integral equations (94) and (95) are as follows:

F » f* +M d - L n ,

V » V* + N d - Mt n
Alternatively, we may write

^
F
 
 = f* +M ^
d
 
 - L ^
n
 
  ,
(98)

^
V
 
 = V* + N ^
d
 
 - Mt ^
n
 
(99)
as the exact relationship between approximate values corresponding to the approximate relationship between exact values above.

6.2.1  Linear System of Equations

In order to progress with the numerical solution method, the problem must be posed in the form that can be solved by subroutine GLS, as outline in Appendix 1. Equations (98) and (99) can be used to develop the following system

é
ê
ë
-M
I
N
0
 ù
ú
û 		é
ê
ë 		^
d
F
 ù
ú
û 		= é
ê
ë
- L
0
Mt
I
 ù
ú
û 		é
ê
ë 		^
n
V
 ù
ú
û 		+ é
ê
ë
f*
V*
 ù
ú
û
(100)

The shell condition can be written in the form

é
ë
a
A
 ù
û 		T
 
 é
ë
d
F
 ù
û 		+ é
ë
b
B
 ù
û 		T
 
 é
ë
F
V
 ù
û 		= é
ë
f
F
 ù
û
(101)


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