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Fixed‐point iteration to nonlinear finite element analysis. Part I: Mathematical theory and background
Author(s) -
Zeid Ibrahim
Publication year - 1985
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620211107
Subject(s) - mathematics , nonlinear system , rate of convergence , fixed point iteration , newton's method , algebraic equation , finite element method , iterative method , mathematical analysis , fixed point , mathematical optimization , computer science , computer network , channel (broadcasting) , physics , quantum mechanics , thermodynamics
This paper sets the stage for the implementation of the Fixed‐Point Iteration (FPI) to nonlinear finite element analysis as an alternative to the existing Newton‐Raphson Method (NRM) or its derivatives. The superiority of the former method over the latter one is such that it enables one to obtain nonlinear structural static or dynamic responses without inverting the structural stiffness matrix. In the first part of the paper, a new convergence correction/acceleration factor has been developed for the FPI when applied to a single nonlinear algebraic equation. This new factor causes the iteration function of the equation under consideration to rotate about an axis that passes through one of its fixed points or roots. Using this observation, the slope of the iteration function can be adjusted in the neighbourhood of a specific root to ensure the convergence of the FPI. It is found that the optimum choice of the new factor corresponds to a zero slope, evaluated at the root, of the iteration function. The rate of convergence and the error estimate of this form of the FPI is developed and compared with the NRM. The equilibrium positions of a nonlinear loaded softening spring have been obtained by both methods as an illustrative numerical example to measure the effect of the new factor on the convergence rate. The second part of the paper extends the above concept to find the solution of a linear system of algebraic equations using the FPI. This leads to a better diagonal approximate inverse for the Jacobi iteration, or method of simultaneous displacements. If the elements of the solution vector of a specific system are all equal, the new Jacobi iteration becomes an exact method and the solution is obtained in one iteration. The concept is also extended to the Gauss‐Seidel iteration, or method of successive displacements. Systems involving symmetric as well as nonsymmetric coefficient matrices have been used as numerical examples and are presented. For future implementation to nonlinear finite element analysis, the active column or the skyline (or the non‐zero profile) FPI algorithms are developed for programming considerations.