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Extra resources for A Rational Finite Element Basis
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. Analysis by Ciarlet and Raviart (1972 a,b) provides a theoretical basis for application.
The homogeneous coordinates (k,m,n) linear in x and y which assume the values (a,O,O), (O,b,O) and (O,O,c) at the vertices of a triangle, where a, b, and c are arbitrary, are called the "barycentric" coordinates of the triangle. The point (ga,gb,gc) for any g # 0 is identioal to the point (a,b,c), this being a characteristic of any set of homogeneous coordinates. Barycentric coordinates normalized to k+m+n = 1 are called "areal" coordinates. These are the values of the triangle wedges: (k,m,n) = (W1,w2,W,) .
Property ( 4 ) implies that both (2;3) and (3;4) must appear as factors in any wedge for vertex 1. Introduction of other factors will only increase the degree of variation on the adjacent sides. 5 exists for the quadilateral, except for the special case of a parallelogram. 2 RATIONAL WEDGES Polynomials have many properties that are beneficial in numerical application. They are easily evaluated, continuous, may be differentiated and integrated readily any number of times, and there are many results in approximation theory concerning polynomials.
A Rational Finite Element Basis by Wachspress