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31 the restriction of the interpolant u to this

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Unformatted text preview: one-dimensional quadratic polynomial Lagrange shape functions Ni j ( ) = Ni( )Nj ( ) i j=1 2 3 (4.3.3b) with (cf. Section 2.4) N1 ( ) = ; (1 ; )=2 (4.3.3c) N2 ( ) = (1 + )=2 (4.3.3d) N3 ( ) = (1 ; 2) ;1 1: (4.3.3e) Shape functions for a vertex, an edge, and the centroid are shown in Figure 4.3.3. Using (4.3.3b-e), we see that shape functions are biquadratic polynomials of the form Ni j ( ) = a1 + a2 + a3 + a4 2 + a5 + a6 2 + a7 2 + a8 2 + a9 2 2 : (4.3.4) 4.3. Lagrange Shape Functions on Rectangles 1.2 13 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −0.2 −1 −0.2 −1 1 −0.5 0.5 0 1 −0.5 0.5 0 0.5 0 0 −0.5 1 0.5 −0.5 −1 1 −1 1 0.8 0.6 0.4 0.2 0 −1 1 −0.5 0.5 0 0 0.5 −0.5 1 −1 Figure 4.3.3: Biquadratic shape functions associated with a vertex (left), an edge (right), and the centroid (bottom). Although (4.3.4) contains some cubic and quartic monomial terms, interpolation accuracy is determined by the highest-degree complete polynomial that can be represented exactly, which, in this case, is a quadratic polynomial. Higher-order shape functions are constructed in similar fashion. 4.3.1 Bilinear Coordinate Transformations Shape functions on the canonical square elements may be mapped to arbitrary quadrilaterals by a variety of transformations (cf. Chapter 5). The simplest of these is a picewise-bilinear function that uses the same shape functions (4.3.1d,e) as the nite element solution (4.3.1a). Thus, consider a mapping of the canonical 2 2 square S to a quadrilateral Q having vertices at (xi j yi j ), i j = 1 2, in the physical (x y)-plane (Figure 4.3.4) using a bilinear transformation written in terms of (4.3.1d,e) as 14 Finite Element Approximation y 2,2 (x 22 22) ,y 1 0 1 0 (x 12,y12 ) η 1,2 1 0 1 0 1,2 1 0 1 0 2,2 1 0 1 0 1 ξ 1 02,1 1 0 1 0 1 0 1,1 (x (x 21,y21 ) ,y ) 11 11 x 1 1,1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 2,1 1 1 0 1 0 1 Figure 4.3.4: Bilinear mapping of the canonical square to a quadrilateral. 22 ) = X X xij N ( ) (4.3.5) ) yij i j i=1 j =1 where Ni j ( ) is given by (4.3.1b). The transformation is linear on each edge of the element. In particular, transforming the edge = ;1 to the physical edge (x11 y11 - (x21 y21) yields x = x11 1 ; + x21 1 + ;1 1: y y11 y21 2 2 As varies from -1 to 1, x and y vary linearly from (x11 y11) to (x21 y21). The locations of the vertices (1,2) and (2,2) have no e ect on the transformation. This ensures that a continuous approximation in the ( )-plane will remain continuous when mapped to the (x y)-plane. We have to ensure that the mapping is invertible and we'll show in Chapter 5 that this is the case when Q is convex. x( y( Problems 1. As noted, interpolation errors of the biquadratic approximation (4.3.3) are the same order as for a quadratic approximation on a triangle. Thus, for example, the L2 error in interpolating a smooth function u(x y) by a piecewise biquadratic function U (x y) is O(h3), where h is the length of the longest edge of an element. The extra degrees of freedom associated with the cubic and quartic terms do not generally improve the order of accuracy. Hence, we might try to eliminate some shape functions and reduce the complexity of t...
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