3.1 Introduction

In the solution of linear problems by a finite element method we always need to solve a set of simultaneous algebraic equations of the form

(3.1)

Provided the coefficient matrix is non-singular the solution to these equations is unique. In the solution of non-linear problems we will always obtain a set of algebraic equations; however, they generally will be non-linear. For example, in Chapter 2 we obtained the set (2.22) at each discrete time t _{n +1}. Here, we consider the generic problem which we indicate as

(3.2)

where u _{n +1} is the set of discretization parameters, f _{n+ 1} a vector which is independent of the parameters and P a vector dependent on the parameters. These equations may have multiple solutions [i.e. more than one set of u _{n +1} may satisfy Eq. (3.2)]. Thus, if a solution is achieved it may not necessarily be the solution sought. Physical insight into the nature of the problem and, usually, small-step incremental approaches from known solutions are essential to obtain realistic answers. Such increments are indeed always required if the problem is transient, if the constitutive law relating stress and strain is path dependent and/or if the load-displacement path has bifurcations or multiple branches at certain load levels.

The general problem should always starts from a nearby solution at

(3.3)

and often arises from changes in the forcing function f _n to

(3.4)

The determination of the change ? u _{n +1}