A General Nonlinear Relaxation Iteration Technique for Solving Nonlinear Problems in Mechanics

Abstract

The nonlinear relaxation method, an iterative approach used in conjunction with finite-difference approximations, is illustrated via the solution to a very simple problem. Subsequently, the method is used to solve three geometrically nonlinear problems in mechanics: finite bending of a circular thin walled tube, the large deflection membrane response of a spherical cap, and finite deformations of a uniformly loaded circular membrane. Formulations for the three problems are quite different but this difference does not inhibit the use of the nonlinear relaxation technique. Solutions were obtained in approximately one man day per problem including the total time devoted to examining, planning, programming, debugging, etc. Solutions compare very favorably with results found elsewhere in the literature. The essential and important advantages of the nonlinear relaxation technique are (a) versatility and ease of application, (b) efficiency with respect to people and computer time utilized, (c) insensitivity to starting values as far as convergence is concerned, and (d) simplicity of logic that makes it a trivial task to learn how to employ it.