A compendium of FEM integration formulas for symbolic work

Abstract

This paper presents a set of Mathematica modules that organizes numerical integration rules considered useful for finite element work. Seven regions are considered: line segments, triangles, quadrilaterals, tetrahedral, wedges, pyramids and hexahedra. Information can be returned in floating-point (numerical) form, or in exact symbolic form. The latter is useful for computer-algebra aided FEM work that carries along symbolic variables. A few quadrature rules were extracted from sources in the FEM and computational mathematics literature, and placed in symbolic form using Mathematica to generate own code. A larger class of formulas, previously known only numerically, were directly obtained through symbolic computations. Some unpublished non-product rules for pyramid regions were found and included in the collection. For certain regions: quadrilaterals, wedges and hexahedra, only product rules were included to economize programming. The collection embodies most FEM-useful formulas of low and moderate order for the seven regions noted above. Some gaps as regard region geometries and omission of non-product rules are noted in the conclusions. The collection may be used “as is” in support of symbolic FEM work thus avoiding contamination with floating arithmetic that precludes simplification. It can also be used as generator for low-level floating-point code modules in Fortran or C. Floating point accuracy can be selected arbitrarily. No similar modular collection applicable to a range of FEM work, whether symbolic or numeric, has been published before

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