Principles of the Method of Large Time Increments

Abstract

The large time increment method (acronym: LATIN) was introduced by Ladevèze [1985a, b]. It represents a break with classical incremental methods in the sense that it is not built on the notion of small increments; the interval of time studied, [0, T] does not have to be partitioned into small pieces. It is an iterative method that sometimes starts with a relative gross approximation (generally coming from an elastic analysis) for displacements, strains, and stresses at each point M belonging to the domain Ω and for all t belonging to [0, T]. At each iteration, an improvement is always made to these different quantities for all t ∈ [0, T] and for all M ∈ Ω. For the interval of study, [0, T] the method is built on three principles:

• P₁, separation of the difficulties—partition of the equations into two groups:

  • a group of equations local in space and time, possibly nonlinear

  • a group of linear equations, possibly global in the spatial variable.

• P₂,a two-step iterative approach where, at each iteration, one constructs, alternatively, a solution to the first group of equations and then a solution to the second group. The first problem is local in the spatial variable, perhaps nonlinear, and the second is linear but generally global

• P₃,use of an ad hoc space-time approximation based on mechanics for the treatment of the global problem defined on Ω × [0,T].