# Even so lagrangian interpolants are simple to

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Unformatted text preview: ge polynomials. Multi-dimensional polynomials formed in this manner are called \tensor-product" approximations. we'll proceed by constructing polynomial shape functions on canonical 2 2 square elements and mapping these elements to an arbitrary quadrilateral elements. We describe a simple bilinear mapping here and postpone more complex mappings to Chapter 5. We consider the canonical 2 2 square f( )j ; 1 1g shown in Figure 4.3.1. For simplicity, the vertices of the element have been indexed with a double subscript as (1 1), (2 1), (1 2), and (2 2). At times it will be convenient to index the vertex coordinats as 1 = ;1, 2 = 1, 1 = ;1, and 2 = 1. With nodes at each vertex, we construct a bilinear Lagrangian polynomial U ( ) whose restriction to the canonical 4.3. Lagrange Shape Functions on Rectangles y 11 1,2 2,2 1,2 3,2 2,2 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 1,3 11 00 3,3 11 00 2,3 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 1,1 2,1 11 00 1,1 11 00 3,1 2,1 x Figure 4.3.1: Node indexing for canonical square elements with bilinear (left) and biquadratic (right) polynomial shape functions. element has the form U( ) = c1 1N1 1 ( ) + c2 1N2 1( ) + c2 2 N2 2( ) + c1 2 N1 2 ( As with Lagrangian polynomials on triangles, the shape function Ni j ( Ni j ( Once again, U ( functions k l) k l) = i k j l (4.3.1a) ) satis es (4.3.1b) = ck l however, now Ni j is the product of one-dimensional hat Ni j ( with k l = 1 2: ): ) = Ni( )Nj ( ) N1 ( ) = 1 ; 2 1+ N2 ( ) = 2 (4.3.1c) (4.3.1d) 1: ;1 (4.3.1e) Similar formulas apply to Nj ( ), j = 1 2, with replaced by and i replaced by j . The shape function N1 1 is shown in Figure 4.3.2. By examination of either this gure or (4.3.1c-e), we see that Ni j ( ) is a bilinear function of the form Ni j ( ) = a1 + a2 + a3 + a4 ;1 1: (4.3.2) The shape function is linear along the two edges containing node (i j ) and it vanishes along the two opposite edges. A basis may be constructed by uniting shape functions on elements sharing a node. The piecewise bilinear basis functions i j when Node (i j ) is at the intersection of four 12 Finite Element Approximation 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 −1 1 −0.5 0.5 0 0.5 0 −0.5 1 1 −0.5 0 0.5 0 −1 0 0.5 −1 −0.5 1 −1 Figure 4.3.2: Bilinear shape function N1 1 on the ;1 1] ;1 1] canonical square element (left) and bilinear basis function at the intersection of four square elements (right). square elements is shown in Figure 4.3.2. Since each shape function is a linear polynomial along element edges, the basis will be continuous on a grid of square (or rectangular) elements. The restriction to a square (or rectangular) grid is critical and the approximation would not be continuous on an arbitrary mesh of quadrilateral elements. To construct biquadratic shape functions on the canonical square, we introduce 9 nodes: (1,1), (2,1), (2,2), and (1,2) at the vertices (3,1), (2,3), (3,2), and (1,3) at midsides and (3,3) at the center (Figure 4.3.1). The restriction of the interpolant U to this element has the form 33 XX U( ) = ci j Ni j ( ) (4.3.3a) i=1 j =1 where the shape functions Ni j , i j = 1 2 3, are products of the...
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