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

Computer algebra systems (CAS) are powerful tools for obtaining analytical expressions for many engineering applications in both academic and industrial environments. CAS have been used in this paper to generate exact expressions for the stiffness matrix of an 8‐node plane elastic finite element. The Maple software system was used to identify six basic formulas from which all the terms of the stiffness matrix could be obtained. The formulas are functions of the Cartesian coordinates of the corner nodes of the element, and elastic parameters Young's modulus and Poisson's ratio. Many algebraic manipulations were performed on the formulas to optimize their efficiency. The redaction in CPU time using the exact expressions as opposed to the classical Gauss–Legendre numerical integration approach was over 50%. In an additional study of accuracy, it was shown that the numerical approach could lead to quite significant errors as compared with the exact approach, especially as element distortion was increased.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007