Evaluation of integrals for a ten-node isoparametric tetrahedral finite element

The use of isoparametric finite elemts in solving three-dimensional problems typically requires the numerical evaluation of a large number of integrals over individual element domains. The evaluation of these integrals by numerical quadrature, which is the traditional approach, can be computationally expensive. For certain problems the present study provides a more efficient method for evaluation of the needed integrals. For these problems some or all of the desired integrals can be evaluated as linear combinations of basic integrals whose integrands are either (i) products of shape (interpolation) functions or (ii) a derivative of a shape function times a product of one or more shape functions. Basic integrals of these two types (when written in terms of local coordinate systems) have integrands which are polynomial both in the variables of integration and in the nodal coordinates and, thus, can be expressed as linear combinations (with rational number coefficients) of a set of polynomial functions of the nodal coordinates. Group theoretic techniques can be employed to select appropriate sets of polynomial functions for use in these expansions and to reduce substantially the number of basic integrals which need to be explicitly evaluated.

The details for the approach have been worked out for a ten-node isoparametric tetrahedral element through the use of MACSYMA, a computer system for algebraic manipulation. The symmetry group for this element has order 24. The basic integrals of types (i) and (ii) are expressed as linear combinations of 20 and 26 terms, respectively. The special case of a straight-edged tetrahedral element with mid-edge nodes is also discussed.