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Unformatted text preview: Chapter 3 1 Chapter 3 Higher-Order Parametric Elements 1. Shape Functions in Natural Coordinates Shape functions may be expressed directly in terms of the global coordinates x , y (an example is the constant strain triangle of Section 2, Chapter 2), or in natural coordinates. Natural coordinates (section 4, Chapter 2) offers the following advantages: They are dimensionless and independent of the orientation and position of the element in space. They are pure interpolating functions that can readily be applied to different physical quantities such as displacements, position, temperature, material variation, thickness, .etc.. They make it possible to develop higher-order isoparametric elements with curved edges. (0,1) (1,0) 1 = =0 =0 1 2 3 1 2 3 (0,0) (a) Parent element (b) Actual element + 1 x y 4 (-1,1) 2 (1,-1) 1 (-1,-1) 3 (1,1) 5 6 7 8 1 , 1 (c) Parent element (d) Actual element 1 2 3 4 5 6 7 8 x y Chapter 3 2 Figure 1.1 Parent elements and actual elements Natural coordinates and are defined for a simple parent element which may have an intrinsic shape and a specified number of nodes as shown in Fig.1.1. The shape functions interpolate the nodal quantities i=1,2,n to give ( , ) at an arbitrary point in the element: ( , ) = N 1 ( , ) 1 + + N n ( , ) n = i = 1, n N i ( , ) i = N 1 n n 1 (1.1) where , = natural coordinates N i = shape functions for node i n = number of element's nodes = vector of nodal quantities of interest. The mapping between the parent element and the actual element is done by the transformation: x = i = 1, n N i ( , ) x i y = i = 1, n N i ( , ) y i (1.2) where x i , y i are the global coordinates of the actual element in space. This transformation maps the parent's node i to the actual element's node i , and the parent's generic point ( , ) to the point ( x , y ) in the actual element. Similarly, if a physical quantity of interest is defined at the actual nodes (in the real element) as T n 1 , then its value at a point ( x , y ) is interpolated by T ( x , y ) = N ( ( x , y ), ( x , y )) T n 1 (1.3) provided that the transformation in Eq.1.2 is a one-to-one mapping that yields one unique pair of ( , ) for each unique pair of ( x , y ). 1.1 Polynomial Basis for Shape Functions Shape functions in Eq.1.1 may be derived starting from a polynomial expression: ( , ) = a 1 + a 2 + a 3 + a 4 + a n = [1 ] n 1 a 1 n = P ( , ) a (1.4) where the coefficients a i = 1,2,.. n are to be found in terms of the nodal quantities i = 1,2,.. n ....
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This note was uploaded on 02/08/2010 for the course MECHANICAL 6371 taught by Professor Ha during the Winter '10 term at Concordia Canada.
- Winter '10