Solution Methods for Nonlinear Evolution Problems

Nearly all the methods one encounters in mechanics for solving nonlinear evolution problems are incremental methods. The load, or rather the interval of time [0, T] considered, is decomposed into a succession of (generally) small intervals. The history of different quantities being known until the present instant t, one studies a new interval of time [ t, t + Δt] where Δt is the increment. The problem is then to determine the history on the interval [t, t + Δt]. In assuming, for example, a linear history on [t, t + Δt] that is, a history that depends only on values at the instant t + Δt we are led to a classical nonlinear problem where the time does not enter. This problem, which determines various quantities at time t + Δt is generally treated by a Newton-type method. We note that the only mechanics property used is the principle of causality.