A Rational Finite Element Basis by Wachspress

By Wachspress

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The role of pyramids and wedges in patchwork approximation and some of the geometric implications is described by Synge (1957). Some of the definitions and symbols introduced in this chapter are new. 5 . 30 RATIONAL FINITE ELEMENT BASIS have been implicit in the finite element literature, but they have not previously been brought so sharply into focus. Recognition of the importance of these properties is a starting point for construction of finite element basis functions. Isoparametric coordinates are an ingenious alternative for a useful class of elements.

Similarly, It is thus shown that the discontinuous limit functions of the quadrilateral wedges may be combined to yield the continuous linear basis functions for the limiting triangle. Areal coordinates are a degenerate form of rational quadrilateral wedges. 4 AN EXAMPLE OF QUADRILATERAL WEDGES By way of illustration, we determine the wedges for a sample quadrilateral. Referring to Fig. 7, we have (4;l) = y, (1;2) = (2y - 3 x ) / m , - 8 y ) / m , (3;4) = (4 - 2~ (2;3) = (5 + 2~ Q1 = (20 + 8~ 17y)/m, - - y)/&‘, and the rational basis functions for degree one approximation over the quadrilateral are: 39 THE QUADRILATERAL Fig.

13 . The 3 - c o n : a c t u a l a n d model. MATI ON The local coordinate system is completely defined by the location of the six 3-con or eight 4-con nodes. These coordinates are p , q, r for the 3-con and c , n for the 4-conl as illustrated in Figs. 15. For Fig. 14: w1 = W3 = r(2r-1) ~ ( 2 p - 1 ) ~W2 = q(2q-11, W 5 = 4qr, w4 = 4Pql W6 = 4rp . P P= Fig. 14. I s o p a r a m e t r i c c o o r d i n a t e s for a 3-con ( p + q + r = 1). For Fig. 1 (1+5+ll) - ( l + n ) (1+5)(1-5-n) 4 * 4 w2 w4 26 = - = - ( 1 + 5 ) ( 1 7 )( l - C + r l ) ( l + r l ) (1-5) (l+<-Tl) 4 4 RATIONAL FINITE ELEMENT BASIS Fig.

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