Semi-analytical solution of path-independent nonlinear finite element models

• Published in: Finite elements in analysis and design 2011-03, Vol.47 (3), p.281-287
• Main Author: Korelc, Jože
• Format: Article
• Language: English
• Subjects: Approximation; Automatic differentiation; Design engineering; Exact sciences and technology; Finite element method; Finite elements; Mathematical analysis; Mathematical models; Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); Nonlinearity; Numerical analysis. Scientific computation; Power series; Sciences and techniques of general use; Semi-analytical solution; Series expansion; Symbolic methods
• ISSN: 0168-874X; 1872-6925
• DOI: 10.1016/j.finel.2010.10.006

Within the semi-analytical solution of the finite element model, the results of the analysis are expressed as a function of one or several parameters of the problem. Severe growth of the size of resulting expressions limits the direct use of symbolic systems to the small number of input parameters left in a symbolic form and to the linear finite element formulations. One of the solutions to the problem of expression growth, followed also in this paper, is to construct an approximated symbolic solution of the finite element model in a form of multivariate series expansion. In order to obtain the semi-analytical solution of middle-scale path-independent nonlinear problems, the presented approach combines dual symbolic–numeric finite element environment with the run-time power series expansion at individual finite element level and a recursive algorithm for the multivariate power series expansion of the solution to the system of linear equations.