1. Introduction

The size of the models used for linear dynamic response calculations is becoming increasingly important, and the need for the precision of the results leads to finer and more expensive calculations. To significantly improve the calculation times of the responses of these systems to various requests, it is interesting to build a subspace of reduced dimensions that makes it possible to project the complete problem, and thus accelerate the various response calculations. These models of reduced components can be used either alone or in sub-structuring approaches, for linear analyses but also for certain non-linear analyses, such as the presence of shocks. This approach is also particularly interesting when it is necessary to take into account the internal dissipation devices specific to each structure.

The proposed method for enriching an initial subspace, built on the basis of the system’s own modes, is based on residue methods ([1,2]). This method makes it possible, on the basis of a solution that is close to the solution, to build vectors making it possible to improve the prediction for a given problem. This iterative method gives very good results with a limited number of iterations (in general, two iterations are sufficient).

The first iteration is generally carried out by calculating the terms classically called « static corrections », or « static modes ». This point is developed in the first section of this document. This technique is particularly suitable for the reduction of models in the presence of few external forces, even if they have a spatial distribution (fluctuating pressure, gravity, etc.). This approach also makes it possible to deal effectively with shock problems by adopting a penalization method (provided that we are interested in phenomena « far » from the shock zone). Important indications for controlling the quality of scale models built using this approach are also given in this section.

In the second part, we will discuss the particular cases of reduced models such as Craig&Bampton or McNeal, used for dynamic substructuring approaches. These two methods can be seen as the generalization of enrichments through static corrections. Techniques for limiting the size of these models will also be presented, in particular for the representation of interface behaviors.

In the third part, the concepts presented are extended to the cases of structures with internal forces such as in damped cases. We will mainly discuss viscous, hysteretic, and viscoelastic damping models. The first two approaches are in fact those usually used for dynamic studies. The case of damping associated with a viscoelastic law is interesting since this law introduces internal variables that are not present in the initial law of behavior of the material, but which can be taken into account through reduction. The particular case of the Rayleigh damping model, used frequently for seismic analyses, is not presented, since in these particular cases, the natural modes of the system also diagonalize the matrices associated with dissipation, and the base thus constructed does not require any particular enrichment, apart from those associated with external forces.

In the fourth part, using a study from the Code_Aster database (study No. 3185: Calculation of the complex modes of a sandwich plate including a visco-elastic material), we present the various techniques presented. These examples also make it possible to present the implementation of the methods in*Code_Aster*.