Keywords: time multiscale, proper generalized decomposition, separated representations, transient models.

Models encountered in computational mechanics could involve many time scales. When these time scales cannot be separated one must solve the evolution model in the entire time interval by using the finest time step that the model implies. In some cases the solution procedure becomes cumbersome because the extremely large number of time steps needed for integrating the evolution model in the whole time interval. In this work, we consider an alternative approach that lies in separating the time axis (one dimensional in nature) in a multidimensional time space. This is an usual procedure for computational homogenization of multiscale problems. But it requires a neat separation of scales so as to be able to obtain a proper description of the governing equation in terms of both scales. In this work we consider the situation in which there is not such a neat separation of scales. The motivation comes only from computational requirements of extremely small time steps.

The time-multiscale approach mentioned before results in a multi-dimensional problem whose meshing can be prohibitive, depending on the number of chosen dimensions. To circumvent the resulting curse of dimensionality the proper generalized decomposition (PGD) method is applied, thus allowing for a fast solution with significant computing time savings with respect to a standard incremental integration. PGD methods assume an approximation of the field of interest in terms of a finite sum of separable functions, thus allowing to treat very efficiently models defined in highly-dimensional spaces. Such spaces can not be meshed by traditional finite element techniques due to the exponential growth of the number of degrees of freedom with the number of dimensions (i.e. the so-called curse of dimensionality). The method can be viewed as a generalization of the LATIN method, and also shares some similarities with classical techniques like Hartree-Fock approaches for quantum mechanics. The main difference lies in the fact that the functions that build the approximation are constructed on the fly by the method itself.

An analysis is then made on the techniques for imposing continuity in the time variable. An approach based upon penalty techniques has revealed to be very efficient, simple to code, and gives excellent results in terms of number of iterations needed for convergence. Results for academic problems are reported in the paper.

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