4  Application of the BSEMs to the Test Problem

To demonstrate the direct and indirect boundary and shell element methods, the test problem with the domain of the unit square and with a discontinuity between ([1/2],[1/2]) and ([1/2],1) is introduced. The boundary conditions are such that j(p) = 1 for 0 < p1 < [1/2] and p2 = 1, j(p) = -1 for [1/2] < p1 < 1 and p2 = 1 and v(p)=0 on the remainder of the boundary. The shell condition is such that n(p)=V(p)=0 for all points on the shell. The test problem is illustrated in figure 3.


The discrete forms of the integral operators are computed through the use of a subroutine described in Kirkup (1992). These numerical integrations are computed to sufficient accuracy so the error does not contribute significantly to the overall error in the integral equation methods.

4.1  Direct Boundary and Shell Element Method

The following linear system of equations follows from approximations (10)-(12) and (17),

é
ê
ê
ê
ê
ê
ê
ë
(MSS + 1
2
 ISS )
- LSS
0S G
- MS G
DaSS
DbSS
0S G
0S G
MGS
- LGS
IGG
-MGG
NGS
-MtGS
0GG
-NGG
 ù
ú
ú
ú
ú
ú
ú
û 		é
ê
ê
ê
ê
ê
ê
ê
ë
^
j
 
S 
^
v
 
S 
^
F
 
G 
^
d
 
G 
ù
ú
ú
ú
ú
ú
ú
ú
û 		= é
ê
ê
ê
ê
ê
ë
0S
gS
0G
0G
 ù
ú
ú
ú
ú
ú
û
(18)
On solution the approximations [^(j)]S, [^(v)]S, [^(F)]G, [^(d)]S to jS, vS, FG, dS are obtained. The discrete form of equation (4) is then used to compute the solution in the domain D.

q
Table 1: Solution via the direct BSEM
h p = (0.25,0.25) p = (0.25,0.5) p = (0.25,0.75)
[1/2] 0.225927 0.447135 0.721443
[1/4] 0.200476 0.396208 0.692895
[1/8] 0.188335 0.373434 0.677145
[1/16] 0.182496 0.362574 0.669262
[1/32] 0.179660 0.357271 0.665348
[1/64] 0.178268 0.354654 0.663403
[1/128] 0.177576 0.353348 0.662427
Results from the application of the method to the test problem are given in table 1. The solution is sought at the points (0.25,0.25), (0.25,0.5), and (0.25, 0.75) in the domain with 9 (8 boundary and 1 shell), 18 (16 + 2), 36 (32 + 4), 72 (64 + 8), 144 (128 + 16), 288 (256 + 32) and 576 (512 + 64) uniform elements. Thus the element lengths h are [1/2], [1/4], [1/8], [1/16], [1/32], [1/64] and [1/128] respectively.

4.2  Indirect Boundary and Shell Element Method

The following linear system of equations follows from equations (13)-(17),

é
ê
ê
ê
ê
ê
ê
ê
ë
ISS
0SS
- LSS
0S G
- MS G
0SS
ISS
-(MtSS + 1
2
 ISS)
0S G
- NS G
DaSS
DbSS
0SS
0S G
0S G
0GS
0GS
- LGS
IGG
- MGG
0GS
0GS
- MtGS
0GG
- NGG
ù
ú
ú
ú
ú
ú
ú
ú
û 		é
ê
ê
ê
ê
ê
ê
ê
ê
ê
ë
^
j
 
S 
^
v
 
S 
^
s
 
S 
^
F
 
G 
^
d
 
G 
ù
ú
ú
ú
ú
ú
ú
ú
ú
ú
û 		= é
ê
ê
ê
ê
ê
ê
ë
0S
0S
gS
0G
0G
 ù
ú
ú
ú
ú
ú
ú
û
(19)
On solution the approximations [^(j)]S, [^(v)]S, [^(s)]S, [^(F)]G, [^(d)]S to jS, vS, sS, FG, dS are obtained. The discrete form of equation (9) is then used to compute the solution in the domain D. The results were obtained under the conditions described in section 4.1 but using the indirect method.


Table 2: Solution via the indirect BSEM
h p = (0.25,0.25) p = (0.25,0.5) p = (0.25,0.75)
[1/2] 0.207509 0.412761 0.672304
[1/4] 0.195115 0.387571 0.682660
[1/8] 0.186049 0.369937 0.672389
[1/16] 0.181445 0.361050 0.667046
[1/32] 0.179140 0.356563 0.664308
[1/64] 0.177995 0.354308 0.662904
[1/128] 0.177427 0.353173 0.662181