Extending the convergence domain of the Newton-Raphson method in structural analysis

• Published in: Computers & structures Vol. 9; no. 3; pp. 265 - 272
• Main Author: Schmidt, William F.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.01.1978
• ISSN: 0045-7949; 1879-2243
• DOI: 10.1016/0045-7949(78)90109-8

This paper contains a procedure for continuing the solution of nonlinear structural analysis problems past limit and branch points when using the Newlon-Raphson or the modified Newton-Raphson methods in conjunction with incremental loading. The singular or ill conditioned Jacobian which arises at or in the vicinity of critical load points is eliminated by adding positive values to select diagonal elements of the Jacobian. A technique for determining which elements to modify together with a method for efficiently determining the optimum magnitude of the modification for minimum solution error are developed. The effectiveness of the procedure is illustrated by a number of example problems of hyperelastic bodies at finite strain which experience either snap-buckling, bifurcation, or a soft load-displacement response followed by stiffening. In addition a few cases are presented to illustrate the sensitivity of the particular problems to imperfections.