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# The extra degrees of freedom associated with the

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Unformatted text preview: he approximation. Unknowns associated with interior shape functions are only coupled to unknowns on the element and can easily be eliminated by a variety of techniques. Considering the biquadratic polynomial in the form (4.3.3a), we might determine c3 3 so that the coe cient of the 4.4. Hierarchical Shape Functions 15 quartic term x2 y2 vanishes. Show how this may be done for a 2 2 square canonical element. Polynomials of this type have been called serendipity by Zienkiewicz 8]. In the next section, we shall see that they are also a part of the hierarchical family of approximations. The parameter c3 3 is said to be \constrained" since it is prescribed in advance and not determined as part of the Galerkin procedure. Plot or sketch shape functions associated with a vertex and a midside. 4.4 Hierarchical Shape Functions We have discussed the advantages of hierarchical bases relative to Lagrangian bases for one-dimensional problems in Section 2.5. Similar advantages apply in two and three dimensions. We'll again use the basis of Szabo and Babuska 7], but follow the construction procedure of Shephard et al. 6] and Dey et al. 5]. Hierarchical bases of degree p may be constructed for triangles and squares. Squares are the simpler of the two, so let us handle them rst. 4.4.1 Hierarchical Shape Functions on Squares We'll construct the basis on the canonical element f( )j ; 1 1g, indexing the vertices, edges, and interiors as described for the biquadratic approximation shown in Figure 4.3.1. The hierarchical polynomial of order p has a basis consisting of the following shape functions. Vertex shape functions. The four vertex shape functions are the bilinear functions (4.3.1c-e) Ni1j = Ni( )Nj ( ) i j=1 2 (4.4.1a) where N1 ( ) = 1 ; 2 N2 ( ) = 1 + : 2 (4.4.1b) The shape function N11 1 is shown in the upper left portion of Figure 4.4.1. Edge shape functions. For p 2, there are 4(p ; 1) shape functions associated with the midside nodes (3 1), (2 3), (3 2), and (1 3): N3k 1( N3k 2( N1k 3( N2k 3( ) ) ) ) = = = = N1( N2( N1( N2( )N k ( )N k ( )N k ( )N k ( ) ) ) ) k = 2 3 ::: p (4.4.2a) (4.4.2b) (4.4.2c) (4.4.2d) 16 Finite Element Approximation where N k ( ), k = 2 3 : : : p, are the one-dimensional hierarchical shape functions given by (2.5.8a) as r Z k ( ) = 2k ; 1 P ( )d : (4.4.2e) N 2 ;1 k;1 Edge shape functions N3k 1 are shown for k = 2 3 4, in Figure 4.4.1. The edge shape functions are the product of a linear function of the variable normal to the edge to which they are associated and a hierarchical polynomial of degree k in a variable on this edge. The linear function (Nj ( ), Nj ( ), j = 1 2) \blends" the edge function (N k ( ), N k ( )) onto the element so as to ensure continuity of the basis. Interior shape functions. For p 4, there are (p ; 2)(p ; 3)=2 internal shape functions associated with the centroid, Node (3 3). The rst internal shape function is the \bubble function" N34 30 0 = (1 ; 2)(1 ; 2): (4.4.3a) The remaining shape functions are products of N34 30 0 and the Legendre polynomials as N35 31 0 N35 30 1 N36 32 0 N36 31...
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