A priori hyperreduction method: an adaptive approach

Model reduction methods are usually based on preliminary computations to build the shape function of the reduced order model (ROM) before the computation of the reduced state variables. They are a posteriori approaches. Most of the time these preliminary computations are as complex as the simulation which we want to simplify by the ROM. The reduction method we propose avoids such preliminary computations. It is an a priori approach based on the analysis of some state evolutions, such that all the state evolutions needed to perform the model reduction are described by an approximate ROM. The ROM and the state evolution are simultaneously improved by the method, thanks to an adaptive strategy. Obviously, an initial set of known shape functions can be used to define the ROM to adapt. But it is not necessary. The adaptive procedure includes extensions of the subspace spanned by the shape functions of the ROM and selections of the most relevant shape functions in order to represent the state evolution. The hyperreduction is achieved by selecting a part of the integration points of the finite element model to forecast the evolution of the reduced state variables. Hence both the number of degrees of freedom and the number of integration points are reduced. To perform the adaptive procedure, different computational strategies can be developed. In this paper, we propose an incremental algorithm involving adaptive periods. During these adaptive periods the incremental computation is restarted until a quality criterion is satisfied. This approach is compatible with classical formulations of the equations. (C) 2004 Elsevier Inc. All rights reserved.