1. Introduction

Thanks to the Finite Element Method, we are able to describe accurately the thermo-mechanical transformations that occur during processes like casting, forming and cutting. It is quite easy to construct a very complex finite element model. It is more difficult to forecast the state of the studied system during the process, by solving a time-dependant and non-linear thermo-mechanical problem. Generally a good knowledge of what are the main significant phenomena allows one to simplify the finite element model. Obviously, the simpler the finite element model is, the faster we forecast the state of the system.

An interesting approach to simplify a finite element model is to build a reduced-order model, thanks to an algorithm. Finite element modelling uses generally a large number of variables to describe the state of the studied system. The aim of model reduction is to define a transformation matrix to get few basis functions, from the Finite Element shape functions, to describe the spatially distributed state. In the framework of non-linear time-dependant problems we can distinguish an a posteriori approach based on Karhunen-Lo ve expansion [SIR 91] [PAR 01] and an a priori approach based on the Krylov subspace [KNO 95] [WEI 95] [SUN 01]. The first kind of approach is based on the knowledge of the evolution of the state variables of the system, the second one is not based on this knowledge. In the Karhunen-Lo ve...