Explicit integration of the stiffness matrix of a four-noded-plane-elasticity finite element

Abstract

SUMMARYThe paper deals with the symbolic integration of a 4-noded isoparametric finite element for plane elasticity. An efficient approach to generate explicit formulas for computing the elementary stiffness matrix is discussed. The procedure is based on the use of the Derive symbolic manipulation code as well as in a posteriori manipulation of the expressions obtained. The accuracy of the results is tested in extremely distorted and geometrically ill-conditioned elements. Three practical engineering models are presented and the accuracy of the results is discussed. A computer time comparison between both numerical and symbolic integration approaches is also included, showing that relevant CPU savings are obtained when applying symbolic integration.

Analytic Integration of the Stiffness Matrix of a 8-Noded Plane Cuadrilateral Subparametric Finite Element

“…Se recomienda revisar el trabajo donde se discuten y presentan este tipo de integrales (Griffiths 2002). Videla et al (1996) presenta unas fórmulas analíticas de integración para los términos de la matriz de rigidez de un elemento finito plano de cuatro nodos isoparamétrico, reportando mejoras de un 50% en los tiempos de cálculo al comparar la integración numérica con la integración analítica, además de mejorar la precisión en el cálculo de elementos distorsionados. El autor utilizó el programa DERIVE-1989 para obtener las expresiones analíticas de la matriz de rigidez.…”

A finite element elasticity programming in Mathematica software

“…Choi and Nomura presented an application of Wolfram Mathematica to solve a two‐dimensional elasticity problems. Additionally, other works have been presented in which closed‐form integration of the stiffness matrix were obtained . Recently, closed‐form of elastic constants in frame‐like periodic cellular solids have been analytically derived using symbolic objetc‐oriented finite element program in MATLAB [14,16].…”

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

“…Similar formulations for 4-node elements applied to steady seepage analysis have also been presented by Griffiths and Smith [16]. Videla et al [17] used the Derive package to generate closed-form analytical expressions for the integration of a 4-node elastic plane strain finite element and showed important CPU savings. The formulas yielded by the symbolic software were then optimized to avoid unnecessary calculations.…”