36 since using 134b u x0 c0 and u xn cn

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Unformatted text preview: cN = 0. Thus, the representations (1.3.5) or (1.3.6) are identical however, (1.3.6) would be useful with non-trivial boundary data. 3. The restriction of the nite element solution (1.3.5) or (1.3.6) to the element xj 1 xj ] is the linear function ; U (x) = cj 1 ; j 1 ; (x) + cj j (x) x 2 xj 1 xj ] ; (1.3.7) 1.3. A Simple Finite Element Problem since j 1 ; and 11 are the only nonzero basis elements on xj j ; 1 xj ] (Figure 1.3.2). Using Galerkin's method in the form (1.2.12c), we have to solve N1 X ; ck A( k=1 j k) = ( j f) j = 1 2 : : : N ; 1: (1.3.8) Equation (1.3.8) can be evaluated in a straightforward manner by substituting replacing k and j using (1.3.4) and evaluating the strain energy and L2 inner product according to (1.3.2b,c). This development is illustrated in several texts (e.g., 9], Section 1.2). We'll take a slightly more complex path to the solution in order to focus on the computer implementation of the nite element method. Thus, write (1.2.12a) as the summation of contributions from each element N X j =1 N 8V 2 S0 Aj (V U ) ; (V f )j ] = 0 (1.3.9a) where Aj (V U )...
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