prob_06_02_01 - Problem 6.2-1 For the element in Fig....

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Problem 6.2-1 For the element in Fig. 6.2-1a, let x = [1 ξ η ξη ] [a 1 a 2 a 3 a 4 ] T . (a) Write matrix [A] in the relation [x 1 x 2 x 3 x 4 ] T = [A] [a 1 a 2 a 3 a 4 ] T . (b) By inspection of Eqs. 6.2-3, write [A] -1 in the relation x = [1 ξ η ] [A] -1 [x 1 x 2 x 3 x 4 ] T (c) Check answers to parts (a) and (b) by seeing if [A] [A] -1 = [ I ] Solution: (a) Applying the interpolation equation: x1 ξηξ η () a 1 a 2 a 3 a 4 = at each node gives: x 1 x 2 x 3 x 4 A a 1 a 2 a 3 a 4 = A 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 := (b) since a 1 a 2 a 3 a 4 A 1 x 1 x 2 x 3 x 4 = then η a 1 a 2 a 3 a 4 = becomes η A 1 x 1 x 2 x 3 x 4 = From Eq. 6.2-3 N 1 1 4 1 ξ 1 η = 1 4 1 ξ η ξη + = 1 4 1 η 11 1 1 T = N 2 1 4 1 ξ + 1 η = 1 4 1 ξ + η ξη = 1 4 1 η 11 1 1 T = N 3 1 4 1 ξ + 1 η + = 1 4 1 ξ + η + ξη + = 1 4 1 η 1111 T = N 4 1 4 1 ξ 1 η +
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This note was uploaded on 01/11/2011 for the course MAE 5020 taught by Professor Folkman during the Fall '10 term at Utah State University.

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