Unformatted text preview: ) where the essential boundary conditions are c2 = 2: (5.5.12b) Recall (Chapters 2 and 3), that the test function V should vanish on @
sponding to (5.5.12b) d2 = 0: E thus, corre(5.5.12c) The second \block" of equations in (5.5.12a) should never have been generated and,
actually, we should have been solving d1 K11 c1 + K12c2 ; f1 ] = d1 K11c1 + K12
T T 2 ; f1 ] = 0: (5.5.13a) Imposing the Galerkin condition that (5.5.13a) vanish for all d1, K11 c1 = f1 ; K12 2: (5.5.13b) Partitioning (5.5.11) need not be done explicitly as in (5.5.11). It can be done implicitly without rearranging equations. Consider the original system (5.5.11b)
2 k11
6.
.
6.
6
6 kj 1
6.
4.
.
kN 1 k1j
...
kjj
...
kN j k1N
...
kjN
...
kN N 32 c1
76 .
.
76 .
76
7 6 cj
76 .
54 .
.
cN 3 2 f1 3
7 6.7
.7
7 6.7
76
7 = 6 fj 7 :
7 6.7
5 4.5
.
fN (5.5.14) Suppose that one boundary condition speci es cj = j , then the j th equation (row) of
the system is deleted, cj is replaced by the boundary condition, and the coe cients of cj
are moved to the righthand side to obtain
2
6
6
6
6
6
6
6
4 k11
... kj;1 1
kj+1 1
...
kN 1 k1 j;1
...
kj ;1 j ;1
kj+1 j;1
...
kN j;1 k1 j+1
...
kj;1 j+1
kj+1 j+1
...
kN j+1 k1N
... kj;1 N
kj+1 N
...
kN N 32
76
76
76
76
76
76
76
54 c1
... cj;1
cj+1
...
cN 3 2 f1 ; k1 j j 3
...
76
7
76
7
76
7
7 6 fj ;1 ; kj ;1 j j 7
7=6
7:
7 6 fj +1 ; kj +1 j j 7
76
7
...
54
5
fN ; kN j j 32 Mesh Generation and Assembly When the algebraic system is large, the cost of moving data when rows and columns
are removed from the system may outweigh the cost of solving a larger algebraic system.
In this case, the boundary condition cj = j can be inserted as the j th equation of
(5.5.14). Although not necessary, the j th column is usually moved to the righthand
side to preserve symmetry. The resulting larger problem is
2 k11
6.
.
6.
6
60
6.
4.
.
kN 1 0
...
1
...
0 k1N
...
0
...
kN N 32 c1
76 .
.
76 .
76
7 6 cj
76 .
54 .
.
cN 3 2 76
76
76
7=6
76
54 f1 ; k1 j
... j j ...
fN ; kN j 3
7
7
7
7:
7
5 j The treatment of essential boundary conditions that impose constraints among...
View
Full
Document
This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

Click to edit the document details