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# Therefore it su ces to check that detje has the same

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Unformatted text preview: ; (x12 ; x11 )(y21 ; y11): The cross product formula for two-component vectors indicates that det(Je(;1 ;1)) = h1h2 sin 12 where h1 , h2, and 12 are the lengths of two adjacent sides and the angle between them (Figure 5.4.1). Similar formulas apply at the other vertices. Therefore, det(Je) will not vanish if and only if ij < at each vertex, i.e., if and only if the quadrilateral is convex. Polynomial shape functions and bases are constructed on the canonical element as described in Chapter 4. For example, the restriction of a bilinear (isoparametric) trial function to the canonical element would have the form U( )= 2 2 XX i=1 j =1 ci j Ni j ( ): A subparametric approximation might, for example, use a piecewise-bilinear coordinate transformation (5.4.2) with a piecewise-biquadratic trial function. Let us illustrate this using the element node numbering of Section 4.3 as shown in Figure 5.4.2. Using (4.3.3), the restriction of the piecewise-biquadratic polynomial trial function to the canonical element is U( )= 3 3 XX i=1 j =1 ci j Ni2j ( ) (5.4.3a) 18 Mesh Generation and Assembly 1 0 1 0 1,2 1 0 1 0 1 0 1,3 1 02,2 1 0 1 0 3,2 1 0 y 3,2 1 0 1 0 1 0 2,2 1 0 1 0 1 0 3,3 1 0 1 0 1 0 2,3 1 0 1 0 1 0 1,1 1 0 1 0 1 0 3,1 1 0 1 0 1 0 1,30 1 1 0 1 02,1 1 0 1 0 1 0 1 03,1 1 0 1 0 1 0 1,1 η 1 0 1 0 1 02,3 1 0 1 0 1 0 1 03,3 1 0 1 0 1 0 1 0 1,2 1 0 1 0 1 0 ξ 1 2,10 1 0 1 0 x Figure 5.4.2: Bilinear mapping to a unit square with a biquadratic trial function. 3,2 1 0 1 0 y 1 02,2 1 0 1,2 11 00 11 00 1,3 1 02,1 1 0 1 0 1 0 1 03,1 1,3 1 0 1 0 1,1 2,2 3,3 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 2,3 1 0 1 0 3,3 1 0 1 0 3,2 1 0 1 0 1,2 1 0 1 0 η 1,1 3,1 1 0 1 0 1 0 1 0 2,3 2,1 ξ 1 0 1 0 x Figure 5.4.3: Biquadratic mapping of the unit square to a curvilinear element. where the superscript 2 is used to identify biquadratic shape functions Ni2j ( with ) = Ni2( )Nj2 ( ) i j=1 2 3 (5.4.3b) 8 <; (1 ; )=2 if i = 1 Ni2 ( ) = : (1 + )=2 if i = 2 : (5.4.3c) 2 1; if i = 3 Example 5.4.2. A biquadratic transformation of the canonical square has...
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