Where the superscript 2 is used to identify

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Unformatted text preview: the form x( y( 3 3 ) = X X xij N 2 ( ) yij i j i=1 j =1 ) (5.4.4) where Ni2j ( ), i j = 1 2 3, is given by (5.4.3). This transformation produces an element in the (x y)-plane having curved (quadratic) edges as shown in Figure 5.4.3. An isoparametric approximation would be biquadratic 5.4. Coordinate Transformations y 19 η 3 11 00 3 11 00 11 00 6 11 00 11 00 11 00 11 00 1 11 00 5 11 00 11 00 11 002 11 00 11 00 11 00 11 00 6 11 00 5 11 00 11 00 11 00 11 00 11 00 11 00 4 x 1 11 00 ξ 11 00 4 2 Figure 5.4.4: Quadratic mapping of a triangle having one curved side. and have the form of (5.4.3). The interior node (3,3) is awkward and can be eliminated by using a second-order serendipity (cf. Problems 4.3.1) or hierarchical transformation (cf. Section 4.4). Example 5.4.3. The biquadratic transformation described in Example 5.4.2 is useful for discretizing domains having curved boundaries. With a similar goal, we describe a transformation for creating triangular elements having one curved and two straight sides (Figure 5.4.4). Let us approximate the curved boundary by a quadratic polynomial and map the element onto a canonical right triangle by the quadratic transformation x( y( 6 ) = X xi N 2 ( ) yi i i=1 ) (5.4.5a) where the quadratic Lagrange shape functions are (cf. Problem 4.2.1) Nj2 = 2 j ( N42 = 4 j ; 1=2) N52 = 4 12 j=1 2 3 23 N62 = 4 (5.4.5b) 31 (5.4.5c) and 1 =1; ; 2 = 3 =: (5.4.6) Equations (5.4.5) and (5.4.6) describe a general quadratic transformation. We have a more restricted situation with x4 = (x1 + x2 )=2 y4 = (y1 + y2)=2 x6 = (x1 + x3 )=2 y6 = (y1 + y3)=2: 20 Mesh Generation and Assembly This simpli es the transformation (5.4.5a) to x( y( ) = x1 N 2 + x2 N 2 + x3 N 2 + x5 N 2 ^ ^ ^ ^ ) y1 1 y2 2 y3 3 y5 5 (5.4.7a) where, upon use of (5.4.5) and (5.4.6), ^ N12 = N12 + (N42 + N62 )=2 = 1 = 1 ; ; (5.4.7b) ^ N22 = N22 + N42 =2 = (1 ; 2 ) (5.4.7c) ^ N32 = N32 + N62 =2 = (1 ; 2 ) (5.4.7d) ^ N52 = N52 = 4 : (5.4.7e) From these results, we see that the mappings on edges 1-2 ( = 0) and 1-3 ( = 0) are linear and are, respectively, given by x =...
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