Comparison of Computer Algebra Systems for Closed Form Stiffness Matrix Generation

Abstract

Computer algebra systems (CAS) have been advantageously employed to generate closed form expressions for finite elements. The advantages relate to the time improvements or savings realized by employing closed form generated expressions as compared to numerical integration. However as the element order increases, the size of the closed form generated expressions become unmanageable causing the source code files to possibly become unusable due to their size. One approach to reducing the size of the source files is to take advantage of the utilities found in CAS to identify common expressions or sub-expressions. In this manuscript we present on-going research by comparing two widely used CAS, Mathematica and Maple, as they relate to identifying common expressions in low order tetrahedral finite element stiffness matrices generated in symbolic form, associated time savings and possible issues. The results indicate that the use of CAS could be advantageously employed to identify common subexpressions through pattern matching to further reduce the size of the generated source files and realize time improvements during execution of the source codes. In addition, the developed procedures could be easily applied to higher order elements with much larger number of entries of closed form expressions where even more savings could be realized.