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 nontrivial 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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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, Finite Element Method, Boundary value problem, Cj J, nite element

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