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All rights reserved. 792/a2 . 779/a2 . 22: Find the first two eigenvalues associated with the differential equation d2 u − 2 = λu, 0 < x < 1 dx u(0) = 0, u(1) + u0 (1) = 0 Use the least squares method. Use the operator definition to be A = −(d2 /dx2 ) to avoid increasing the degree of the characteristic polynomial for λ. Solution: For this problem, the choice of the operator A is crucial. Ã ¸ A(φi )A(φj ) dx cj d2 φj + λφj dx2 Ã ! # dx cj d2 φj d2 φj d2 φi + λ φ + φj i dx2 dx2 dx2 ! 2 + λ φi φj # ) dx cj (1) which is a quadratic (matrix) eigenvalue problem, and it is more difficult (but not impossible) to solve.

4a) c The McGraw-Hill Companies, Inc. ° All rights reserved. 2: Develop the weighted-residual finite element model (not weakform finite element model) of the following pair of equations: − d2 w0 M = 0, − 2 dx EI − d2 M =q dx2 (1) Assume the following approximations of the form w0 (x) ≈ 4 X (1) ∆i ϕi (x), i=1 M (x) ≈ 4 X (2) Λi ϕi (x), (2) i=1 The finite element equations should be in the form 0= 0= m X j=1 m X j=1 PROPRIETARY MATERIAL. 11 e Kij ∆j + 21 e Kij ∆j + n X j=1 n X j=1 12 e Kij Λj − Fi1 (3a) 22 e Kij Λj − Fi2 (3b) c The McGraw-Hill Companies, Inc.

SInce none of the UI are specified, the condensed equations are the same as the assembled equations. However, the coefficient matrix of the assembled equations is singular and the solution can be determined by specifying one of the UI . 10132 which coincides with the exact solution at the nodes. 14, and both problems have the same solution gradient, du/dx, as indicated by the exact solutions of the two problems. PROPRIETARY MATERIAL. c The McGraw-Hill Companies, Inc. ° All rights reserved. PROPRIETARY AND CONFIDENTIAL This Manual is the proprietary property of The McGraw-Hill Companies, Inc.

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