# 6-18 - CHAP 6 Finite Elements for Plane Solids 331 18 A...

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CHAP 6 Finite Elements for Plane Solids 331 18. A quadrilateral element in the figure is mapped into the parent element. (a) A point P has a coordinate ( x, y ) = (½, y ) in the physical element and ( s, t ) = (−½, t ) in the parent element. Find y and t coordinates of the point using isoparametric mapping. (b) Calculate the Jacobian matrix at the center of the element. (c) Is the mapping valid? Explain your answer. Solution: (a) Using isoparametric interpolation of point P , 1 1 1 2 4 4 1 1 4 4 (1 )(1 ) 1 (1 )(1 ) 1 (1 )(1 ) 1 (1 )(1 ) 1 s t s t y s t s t (1) Substitute s = 1/2 in the first equation 1 3 1 1 1 4 2 4 2 2 (1 ) (1 ) t t The above equation can be solved for t = 0. Now substitute s = 1/2 and t = 0 into second equation, 1 1 1 1 3 4 2 4 2 8 2 y (b) From Eq. (1) with unknowns s and

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