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McGraw-Hill”) and protected by copyright and other state and federal laws. By opening and using this Manual the user agrees to the following restrictions, and if the recipient does not agree to these restrictions, the Manual should be promptly returned unopened to McGrawHill: This Manual is being provided only to authorized professors and instructors for use in preparing for the classes using the affiliated textbook. No other use or distribution of this Manual is permitted. This Manual may not be sold and may not be distributed to or used by any student or other third party.

10: An insulating wall is constructed of three homogeneous layers with conductivities k1 , k2 , and k3 in intimate contact (see Fig. 10). Under steadystate conditions, the temperatures of the media in contact at the left and right L and T R , respectively, and film surfaces of the wall are at ambient temperatures of T∞ ∞ coefficients βL and βR , respectively. Determine the temperatures when the ambient temperatures T0 and T5 and the (surface) are known. Assume that there is no internal heat generation and that the heat flow is one-dimensional (∂T /∂y = 0).

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. Alternatively, we identify the operator A of the problem to be A = −d2 /dx2 so that it does not include the unknown, λ (not consistent with the definition of the method). Then 0= Z 1 0 = = A(φi )R dx = n ½Z 1 X j=1 "Z Ã n 1 d2 φ d2 φ X i j j=1 n X j=1 0 dx2 0 ¾ A(φi ) [A(φj ) − λφj ] dx cj d2 φi + λ φj dx2 dx2 !

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


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