53 assembly of an element sti ness matrix and load

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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 right-hand 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 right-hand 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...
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