The positions of the elemental matrices and vectors

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Unformatted text preview: ector. In the example e shown in Figure 5.5.3, the entry k11 is added to Row 4 and Column 4 of the global e sti ness matrix K. The entry k12 is added to Row 4 and Column 7 of K, etc. The assembly process avoids the explicit summations implied by (5.5.3) and yields A(V U ) = dT (K + M)c (5.5.9a) (V f ) = dT f (5.5.9b) where c = c1 c2 ::: c ] (5.5.9c) d = d1 d2 ::: d ] (5.5.9d) T N T N where K is the global sti ness matrix, M is the global mass matrix, f is the global load vector, and N is the dimension of the trial space (or the number of degrees of freedom). Imposing the Galerkin condition (5.5.1a) A(V U ) ; (V f ) = dT (K + M)c ; f ] = 0 8d 2 <N (5.5.10a) yields (K + M)c = f : (5.5.10b) 5.5.2 Essential and Neumann Boundary Conditions It's customary to ignore any essential boundary conditions during the assembly phase. Were boundary conditions not imposed, the matrix K + M would be singular. Essential boundary conditions constrain some of the ci, i = 1 2 ::: N , and they must be imposed before the algebraic system (5.5.10b) can be solved. In order to simplify the discussion, let us suppose that either M = 0 or that M has been added to K so that (5.5.10) may be written as d Kc ; f ] = 0 T Kc = f : 8d 2 <N (5.5.11a) (5.5.11b) 30 Mesh Generation and Assembly Global Local 4 1 1 03 7 2 1 0 8 3 1 08 1 0 1 0 1 0 1 0 7 1 40 1 0 2 e e e k11 k12 k13 e e e Ke = 4 k21 k22 k23 e e e k31 k32 k33 21 K= 6 6 6 6 6 6 6 6 6 6 6 6 4 1 0 1 0 2 1 10 234 56 7 3 2 f1e fe = 4 f2e f3e 5 89 e +k11 e e +k12 +k13 e +k21 e +k31 e e +k22 +k23 e e +k32 +k33 3 7 7 7 7 7 7 7 7 7 7 7 7 5 1 2 3 4 5 6 7 8 9 3 5 2 f 3 6 6 6 6 6 =6 6 6 6 6 4 7 7 7 7 7 7 7 7 7 7 5 +f1e +f2e +f3e Figure 5.5.3: Assembly of an element sti ness matrix and load vector into their global counterparts for a piecewise-linear polynomial approximation. The actual vertex indices are recorded and stored (top), the element sti ness matrix and load vector are calculated (center), and the indices are used to determine where to add the entries of the elemental matrix and vector into the global sti ness and mass matrix. 5.5. Element Matrices and Their Assembly 31 Essential boundary conditions may either constrain a single ci or impose constraints between several nodal variables. In the former case, we partition (5.5.11a) as d1 d2] K11 K12 K21 K22 c1 c2 f1 f2 ; =0 (5.5.12a...
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