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Semi‐analytical integration of the 8‐node plane element stiffness matrix using symbolic computation
Author(s) -
Lozada I. J.,
Osorio J. C.,
Griffiths D. V.,
Cerrolaza M.
Publication year - 2006
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20098
Subject(s) - numerical integration , gaussian quadrature , stiffness matrix , direct stiffness method , finite element method , mathematics , symbolic computation , quadrature (astronomy) , gaussian elimination , fortran , computation , matrix (chemical analysis) , direct integration of a beam , mathematical analysis , gaussian , computer science , algorithm , nyström method , structural engineering , integral equation , physics , materials science , electrical engineering , quantum mechanics , engineering , composite material , operating system
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. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006