Semi‐analytical integration of the 8‐node plane element stiffness matrix using symbolic computation

Abstract

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.

Elastic stiffness of straight‐sided triangular finite elements by analytical and numerical integration

“…Similar formulations for 4-node elements applied to steady seepage analysis have also been presented by Smith and Griffiths [14]. Videla et al [15,16] and Lozada et al [17] used the Derive and Maple packages to generate closed-form expressions for the integration of 4-and 8-node elastic plane strain FEs and reported significant CPU savings. Amberg et al [18] made extensive use of the Maple system to generate complete FE codes in Fortran for applications to research on welding, crystal growth, flow of viscoelastic liquids and more.…”

Semi‐analytical integration of the elastic stiffness matrix of an axisymmetric eight‐noded finite element

“…This work extend the formulation obtained by Lozada et al [11] to the integration of the eight-noded finite elements for axial symmetry.…”

“…We consider now the classification of terms presented by Lozada et al [11] for the eight-noded element, as shown in Table I: In the classification above, the symmetry of the stiffness matrix was considered. The eightnoded element was classified into 10 groups, considering the adjacency between DOF in the element nodes and the type of DOF (parallel or orthogonal).…”

“…Lozada et al [11] generalized Griffiths results and presented semianalytical formulas for the evaluation of the stiffness matrix of eight-noded finite elements in 2D elasticity. CPU times were improved about 37% when compared with numerical Gaussian integration.…”

Exact integration of the stiffness matrix of an 8‐node plane elastic finite element by symbolic computation

“…More recently, Lozada et al [23] have obtained semianalytical expressions to compute the stiffness matrix of an 8-noded plane elasticity superparametric finite element. The authors reported savings of 37% in CPU times.…”