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J. Murin, G. Meschke, P. Helnwein, H.A. Mang:
"A Study on Solution Algorithms for Geometrically and Physically Nonlinear 1D Finite Elements";
in: "Proceedings of the International Conference on Numerical Methods in Continuum Mechanics", University Zilina, Stará Lesná, Slowakei, 1994, 118 - 127.

This paper is concerned with a 1D study of different algotirhms for the solution of the Euler-Lagrange equations in the context of the geometrically and physically nonlinear theory. In particular, two solution procedures are suggested, which employ information from the previous load step via a scalar parameter \al. A simple algorithm for the determination of this parameter is suggested. One of the proposed algorithms aims at reducing the error in the first iteration step. The subsequent iterations are then based on the consistent linearization of the Euler-Lagrange equations. An alternative method, which completely circumvents inversions of the stiffness matrix, yields quadratic convergence for the 1-D problem under consideration with significantly reduced computer costs compared to the classical Newton iteration scheme.

Keywords: fem, solution, nonlinear, algorithms, 1D, Finite Elements, geometrically, physically, stiffness matrix