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# 410a 7410b the four second order hierarchical shape

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Unformatted text preview: η (1,2) (3,2) 11 00 11 00 (2,2) 11 00 11 00 11 00 11 00 (1,3) (2,3) 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 (1,1) ξ 11 00 11 00 (3,1) (2,1) Figure 7.4.2: Nodal placement for bilinear and hierarchical biquadratic shape functions on a canonical 2 2 square element. Node indexing is given in Figure 7.4.2 The restriction of a piecewise bilinear nite element solution U to the square canonical element is U( )= 2 2 XX i=1 j =1 1 c1 Nij ( ij ): (7.4.12) Using either (7.4.8) or (7.4.9), the restriction of the error approximation E to the canonical element is the second-order hierarchical function E( )= 2 2 XX i=1 j =1 cij Nij ( 2 1 )+ 2 X i=1 dN( 2 i3 2 i3 )+ 2 X j =1 d2j N32j ( 3 ): (7.4.13) The local problems (7.4.8) or (7.4.9) are transformed to the canonical element and solved for the eight unknowns, c2 , i j = 1 2, d23, i = 1 2, d2j , j = 1 2, using the test functions ij i 3 k , i j = 1 2 3, k = 1 2. V = Nij Several simpli cations and variations are possible. One of these may be called verte...
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