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Unformatted text preview: face of the cube that are uniquely determined by values at their four vertices on that face. Once again, this ensures that shape functions and U are C 0 functions on a uniform grid of cubes or rectangular parallelepipeds. Since each shape function is the product of one-dimensional linear polynomials, the interpolant is a trilinear function of the form U( ) = a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 : Other approximations and transformations follow their two-dimensional counterparts. For example, tri-quadratic shape functions on the canonical cube are constructed by placing 27 nodes at the vertices, midsides, midfaces, and centroid of the element (Figure 4.5.4). The shape function associated with Node (i j k) is given by (4.5.7a) with Ni ( ) given by (4.3.3b-d). 4.4. Three-Dimensional Shape Functions 27 4.5.3 Hierarchical Approximations As with the two-dimensional hierarchical approximations described in Section 4.4, we use Szabo and Babuska's 7] shape function with the representation of Shephard et al. 6]. The basis for a tetrahedral or a canonical cube begins with the vertex functions (4.5.1) or (4.5.7), respectively. As noted in Section 4.4, higher-order shape functions are written as products Nik (x y z) = k ( k of an entity function ) i( ) (4.5.8) and a blending function i. The entity function is de ned on a mesh entity (vertex, edge, face, or element) and varies with the degree k of the approximation. It does not depend on the shapes of higher-dimensional entities. The blending function distributes the entity function over higher-dimensional entities. It depends on the shapes of the higher-dimensional entities but not on k. The entity functions that are used to construct shape functions for cubic and tetrahedral elements follow. Edge functions for both cubes and tetrahedra are given by (4.4.6c) and (4.4.2e) as p Z k ( ) = 2(2k ; 1) Pk;1( )d k2 (4.5.9a) 1; 2 ;1 where 2 ;1 1] is a coordinate on the edge. The rst four edge functions are presented in (4.4.8). Face functions for squares are given by (4.4.3) divided by the square face blending function (4.4.3a) k Here, ( ( ) = P ( )P ( ) + =k;4 k 4: (4.5.9b) ) are canonical coordinates on the face. The rst six square face functions are 400 = 1 510 = 501 = 620 = 3 2;1 2 2;1 611 = 602 = 3 2: Face functions for triangles are given by (4.4.9) divided the triangular face blending function (4.4.9a) k ( 1 2 3) = P ( 2 ; 1)P (2 3 ; 1) + =k;3 k 3: (4.5.9c) 28 Finite Element Approximation As with square faces, ( 1 2 3) form a canonical coordinate system on the face. The rst six triangular face functions are 300 = 1 401 = 2 511 = ( 3;1 2 ; 1 )(2 3 ; 1) 410 = 520 = 2; 1 3( 2 ; 1)2 ; 1 2 2 5 0 2 = 3(2 3 ; 1) ; 1 : 2 Now, let's turn to the blending functions. The tetrahedral element blending function for an edge is ij ( 1 2 3 4 ) = i j (4.5.10a) when the edge is directed from Vertex i to Vertex j . Using either Figure 4.5.2 or Figure 4.5.3 as references, we see that the blending function ensures that the shape function vani...
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