The adjoint for an algebraic finite element
Computers & Mathematics with Applications
Surface- and volume-based techniques for shape modeling and analysis
SIGGRAPH Asia 2013 Courses
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Polynomials suffice as finite element basis functions for triangles, parallelograms, and some other elements of little practical importance. Rational basis functions extend the range of allowed elements to the much wider class of well-set algebraic elements, where well-set is a convexity type constraint. The extension field from R(x,y) to R(x,y,x^2+y^2) removes this quadrilateral constraint as described in Chapter 8 of [E.L. Wachspress, A Rational Finite Element Basis, Academic Press, 1975]. The basis function construction described there is clarified here, first for concave quadrilaterals and then for concave polygons. Its application is enhanced by the GADJ algorithm [G. Dasgupta, E.L. Wachspress, The adjoint for an algebraic finite element, Computers and Mathematics with Applications, doi:10.1016/j.camwa.2004.03.021] for finding the denominator polynomial common to all the basis functions.