The semi-analytical integration of an 8-node plane strain finite element stiffness matrix is presented in this work. The element is assumed to be super-parametric, having straight sides. Before carrying out the integration, the integral expressions are classified into several groups, thus avoiding duplication of calculations. Symbolic manipulation and integration is used to obtain the basic formulae to evaluate the stiffness matrix. Then, the resulting expressions are postprocessed, optimized, and simplified in order to reduce the computation time. Maple symbolic-manipulation software was used to generate the closed expressions and to develop the corresponding Fortran code. Comparisons between semi-analytical integration and numerical integration were made. It was demonstrated that semi-analytical integration required less CPU time than conventional numerical integration (using Gaussian-Legendre quadrature) to obtain the stiffness matrix.
The finite element method (FEM) is a numerical method for approximate solution of partial differential equations with appropriate boundary conditions. This work describes a methodology for generating the elastic stiffness matrix of an axisymmetric eight-noded finite element with the help of Computer Algebra Systems. The approach is described as "semi analytical" because the formulation mimics the steps taken using Gaussian numerical integration techniques. The semianalytical subroutines developed herein run 50% faster than the conventional Gaussian integration approach. The routines, which are made publically available for download, 1 should help FEM researchers and engineers by providing significant reductions of CPU times when dealing with large finite element models.
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