Analytical Evaluation of the Integral of Any Order Polynomials on Tetrahedral Regions

Abstract

This paper presents an analytical method to set out the integral of any polynomial function f(x, y,z) on a tetrahedral region T by using its four vertexes. The method uses a coordinate transformation which involves the four vertexes of the tetrahedron, whose Jacobian is simple. The last integral is not difficult to solve given that recurrence formula is very simple, furthermore we have developed an algorithm which can evaluate the integral when integrating function is generated by several multiplications of polynomials without necessity of develop the products. This method can be used in finite element method because the most functions involved in this method are polynomial ones. The method here presented is faster than Gauss-Legendre quadrature or n order if the amount of monomials present on f(x, y,z) is least than n3