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Thus let u0 u x y and v0 v x y

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Unformatted text preview: .4b) where g1e = p(x( g2e = p(x( g3e = p(x( ) y( ) y( ) y( 2 2 )) x + y ] )) xx )) 2 x + yy (5.5.4c) (5.5.4d) 2 + y ]: (5.5.4e) The integrand of (5.5.4b) might appear to be polynomial for constant p and a polynomial mapping however, this is not the case. In Section 4.6, we showed that the inverse coordinate mapping satis es y x y x : (5.5.5) x= y=; x=; y= det(Je) det(Je) det(Je) det(Je) The functions gie, i = 1 2 3, are proportional to 1= det(Je)]2 thus, the integrand of (5.5.4b) is a rational function unless, of course, det(Je) is a constant. Let us write U0 and V0 in the form U0 ( ) = cT N( e ) = N( )T ce V0 ( ) = dT N( e ) = N( )T de (5.5.6) where the vectors ce and de contain the elemental parameters and N( ) is a vector containing the elemental shape functions. Example 5.5.1. For a linear polynomial on the canonical right 45 triangular element having vertices numbered 1 to 3 as shown in Figure 5.5.1, 2 3 2 3 ce 1 1; ; 4 ce 2 5 4 5: ce = N( ) = ce 3 The actual vertex indices, shown as i, j , abd k, are mapped to the canonical indices 1, 2, and 3. Example 5.5.2. The treatment of hierarchical polynomials is more involved because there can be more than one parameter per node. Consider the case of a cubic hierarchical 5.5. Element Matrices and Their Assembly 27 1 0 1 0k 1 0 η 1 0k, 3 1 0 1 0 e 1 0 1 0 1 0 0 1 0 1 0 1 0 i 1 0 1 0 1 0 j 1 0 1 0 1 0 i, 1 ξ j, 2 Figure 5.5.1: Linear transformation of a triangular element e to a canonical right 45 triangle. function on a triangle. Translating the basis construction of Section 4.4 to the canonical element, we obtain an approximation of the form (5.5.6) with c = c 1 c 2 ::: c 10] T e e e e N = N1 ( ) N2( ) ::: N10( )]: T The basis has ten shape functions per element (cf. (4.4.5-9)), which are ordered as N1 ( )= 1 =1; ; p N2 ( )= 2 = p N3 ( )= 3 = p N4 ( ) = ; 6 1 2 N5 ( ) = ; 6 2 3 N6( ) = ; 6 3 p p N7( ) = ; 10 1 2(2 ; 1) N8( ) = ; 10 2 3(2 ; 1) 1 p N9 ( ) = ; 10 1 3(1 ; 2 ) N10( ) = 1 2 3: With this ordering, the rst three shape functions are associa...
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