An Intro. to the Finite Element Method [SOLUTIONS] by J. Reddy

By J. Reddy

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See Chapter 5 for additional details. The finite element model is obtained by substituting u(x) ≈ ueh (x) = n X uej φej (x) (7) Ke ue = Fe (8) j=1 into the weak form (3). We obtain [K e ]{ue } = {F e } or PROPRIETARY MATERIAL. c The McGraw-Hill Companies, Inc. ° All rights reserved. 34 AN INTRODUCTION TO THE FINITE ELEMENT METHOD where Z xb à dφi dφj ! 2: Construct the weak form and the finite element model of the differential equation − µ ¶ d du du =f a −b dx dx dx for 0 < x < L over a typical element Ωe = (xa , xb ).

Solution: The coefficient matrix is defined by e Kij = fie = Z xb à e dψ dψje i xa Z xb xa dx dx − ψie ψje ! dx (−x2 )ψie dx (1) The element coefficient matrix (for any element) is given by Eq. 00926 324 324 PROPRIETARY MATERIAL. c The McGraw-Hill Companies, Inc. ° All rights reserved. 13272 U4 Q32 (4) Since U1 = 0 and U4 = 0, the condensed equations are obtained by omitting the first and fourth row and column of the assembled equations. 0 The secondary variables can be computed using either the definition or from the element equations.

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