Enhanced solution control for physically and geometrically non-linear problems. Part I - the subplane control approach

Geometrically or physically non-linear problems are often characterized by the presence of critical points with snapping behaviour in the structural response. These structural or material instabilities usually lead to ine ciency of standard numerical solution techniques. Special numerical procedures are therefore required to pass critical points. This paper presents a solution technique which is based on a constraint equation that is de ned on a subplane of the degrees-of-freedom (dof ʹs) hyperspace or a hyperspace constructed from speci c functions of the degrees-of-freedom. This uni ed approach includes many existing methods which have been proposed by various authors. The entire computational process is driven from only one control function which is either a function of a number of degrees-of-freedom (local subplane method) or a single automatically weighted function that incorporates all dof ʹs directly or indirectly (weighted subplane method). The control function is generally computed in many points of the structure, which can be related to the nite element discretization. Each point corresponds to one subplane. In the local subplane method, the subplane with the control function that drives the load adaptation is selected automatically during the deformation process. Part I of this two-part series of papers fully elaborates the proposed solution strategy, including a fully automatic load control, i.e. load estimation, adaptation and correction. Part II presents a comparative analysis in which several choices for the control function in the subplane method are confronted with classical update algorithms. The comparison is carried out by means of a number of geometrically and physically non-linear examples. General conclusions are drawn with respect to the e ciency and applicability of the subplane solution control method for the numerical analysis of engineering problems. Copyright ? 1999 John Wiley & Sons Ltd.