The rst internal shape function is the bubble

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Unformatted text preview: 1 N36 30 2 = = = = = N34 30 0P1 ( N34 30 0P1 ( N34 30 0P2 ( N34 30 0P1 ( N34 30 0P2 ( ) ) ) )P1( ) ) :::: (4.4.3b) (4.4.3c) (4.4.3d) (4.4.3e) (4.4.3f) The superscripts k, , and , resectively, give the polynomial degree, the degree of P ( ), and the degree of P ( ). The rst six interior bubble shape functions N3k 3 , + = k ; 4, k = 4 5 6, are shown in Figure 4.4.2. These functions vanish on the element boundary to maintain continuity. On the canonical element, the interpolant U ( ) is written as the usual linear combination of shape functions U( p2 p 22 2 XX 1 1 X X k k X k k X X k )= ci j Ni j + c3 j N3 j + ci 3Ni 3] + c3 3 N3k 3 : i=1 j =1 k=2 j =1 i=1 k=4 + =k;4 (4.4.4) The notation is somewhat cumbersome but it is explicit. The rst summation identi es unknowns and shape functions associated with vertices. The two center summations identify edge unknowns and shape functions for polynomial orders 2 to p. And, the third summation identi es the interior unknowns and shape functions of orders 4 to p. 4.4. Hierarchical Shape Functions 17 0 1 −0.1 0.8 −0.2 0.6 −0.3 −0.4 0.4 −0.5 0.2 −0.6 0 −1 1 −0.5 −0.7 1 0.5 0 0.5 0 0.5 1 0 −0.5 1 0.5 0 −0.5 −1 −0.5 −1 0.4 −1 0.25 0.3 0.2 0.2 0.15 0.1 0.1 0.05 0 0 −0.1 −0.05 −0.2 −0.1 −0.3 −0.15 −0.4 1 −0.2 1 0.5 1 0.5 0 0.5 1 0.5 0 0 −0.5 0 −0.5 −0.5 −1 −0.5 −1 −1 −1 Figure 4.4.1: Hierarchical vertex and edge shape functions for k = 1 (upper left), k = 2 (upper right), k = 3 (lower left), and k = 4 (lower right). Summations are understood to be zero when their initial index exceeds the nal index. A degree p approximation has 4 + 4(p ; 1)+ + (p ; 2)+(p ; 3)+=2 unknowns and shape functions, where q+ = max(q 0). This function is listed in Table 4.4.1 for p ranging from 1 to 8. For large values of p there are O(p2) internal shape functions and O(p) edge functions. 4.4.2 Hierarchical Shape Functions on Triangles We'll express the hierarchical shape functions for triangular elements in terms of triangular coordinates, indexing the vertices as 1, 2, and 3 the edges as 4, 5, and 6 and the centroid as 7 (Figure 4.4.3). The basis consists of the following shape functions. Vertex Shape functions. The three vertex shape functions are the linear barycentric coordinates (4.2.7) Ni1 ( 1 2 3) = i i = 1 2 3: (4.4.5) 18 Finite Element Approximation 1 0.4 0.3 0.8 0.2 0.1 0.6 0 0.4 −0.1 −0.2 0.2 −0.3 0 −1 1 −0.5 −0.4 −1 0.5 0 1 −0.5 0 0.5 0.5 0 −0.5 1 0 0.5 −1 −0.5 1 0.4 0.2 0.3 −1 0.1 0.2 0 0.1 −0.1 0 −0.2 −0.1 −0.3 −0.2 −0.4 −0.3 −0.4 −1 1 −0.5 −0.5 −1 0.5 0 1 −0.5 0 0.5 0.5 0 −0.5 1 0 0.5 −1 −0.5 1 0.15 0.2 0.1 −1 0.1 0.05 0 0 −0.1 −0.05 −0.2 −0.1 −0.3 −0.15 −0.4 −0.2 −1 1 −0.5 0.5 0 0 0.5 −0.5 1 −1 −0.5 −1 1 −0.5 0.5 0 0 0.5 −0.5 1 −1 Figure 4.4.2: Hierarchical interior shape functions N34 30 0 , N35 31 0 (top), N35 30 1 , N36 32 0 (middle), and N36 31 1 , N36 30 2 (bottom). 4.4. Hierarchical Shape Functions 19 p Square Triangle Dimension Dimension 1 4 3 2 8 6 3 12 10 4 17 15 5 23 21 6 30 28 7 38 36 8 4...
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