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This assignment solution was submitted to Amar Sharma for Finite Element Method course at Aligarh Muslim University. It includes: Quadratic, Interpolation, Function, One-dimensional, Element, Nodes, Coordinates, Matrices, Row, Operation
Typology: Exercises
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The quadratic interpolation function of a one dimensional element with three nodes is given by ɸ(x)=a 1 +a 2 x+a 3 x^2 If the x coordinates of the nodes 1, 2 and 3 are 1, 3 and 5 inches respectively. Find the matrices: [ɳ], [ɳ]-1 and [N]. Solution: As we know [ɳ] = ɳT^ = [1 x x^2 ] ɳ 1 T^ = [1 1 1] ɳ 2 T^ = [1 3 9] ɳ 3 T^ = [1 5 25]
we can find [ɳ]-1 by row operation, [ɳ]-1^ = [N]= ɳT[ɳ]- [1 x x2][ɳ]-1^ = [(1.875-x+0.125x^2 ); (-1.25+1.5x-0.25x^2 ); (0.375-0.5x+0.125x^2 )] N1 = 1.875-x+0.125x^2 N2 = -1.25+1.5x-0.25x^2 N3 = 0.375-0.5x+0.125x^2
Determine the shape function of a 3-D simple element: ɸ = a 1 +a 2 x+a 3 y+a 4 z ɳT= { 1 x y z } [N] = [Ni Nj Nk Nl] [NT] = { Ni Nj Nk Nl} Ni [N] = Nj Nk Nl Ni ʃNNtds = ʃʃʃvNNTdV = ʃʃʃv Nj [Ni Nj Nk Nl] dV Nk Nl
