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315b the vector lj is called the element load vector

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Unformatted text preview: s by making the mesh uniform with hj = h = 1=N , j = 1 2 : : : N , and summing AS , AM , and (V f )j separately. Thus, summing (1.3.12) jj N X j =1 AS = j N X p dj 1 dj ] h j =1 ; 1 ;1 cj 1 : cj ;1 ; 1 The rst and last contributions have to be modi ed because of the boundary conditions which, as noted, prescribe c0 = cN = d0 = dN = 0. Thus, N X p p AS = d1 ] h 1] c1] + d1 d2] h j j =1 p dN 1 ] h 1 ;1 c1 + c2 ;1 1 1 ;1 cN 2 + d ] p 1] c ]: N1 ;1 1 cN 1 h N1 Although this form of the summation can be readily evaluated, it obscures the need for the matrices and complicates implementation issues. Thus, at the risk of further complexity, we'll expand each matrix and vector to dimension N ; 1 and write the summation as + dN ; 2 N X k=1 ; ; AS = d1 d2 j ; ; ; 2 1 6 p6 dN 1 ] h 6 6 4 ; 32 76 76 76 74 5 3 c1 c2 7 7 ... 7 5 cN 1 ; 14 + d1 d2 + 2 1 6 ;1 p6 dN 1] h 6 6 4 + d1 d2 + d1 d2 32 76 76 76 74 5 ;1 1 ; 2 6 p6 dN 1] h 6 6 4 3 c1 c2 7 7 ... 7 5 cN 32 76 76 76 7 1 ;1 5 4 ; 2 6 p6 dN 1] h 6 6 4 Introduction ;1 32 76 76 76 74 5 ; 1 ; 1 c1 c2 ... 3 c1 c2 7 7 ... 7 5 cN 3 7 7 7 5 1 ; cN 1 1 Zero elements of the matrices have not...
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