A new paradigm in numerical simulation – Separated variables decomposition for a priori model reduction – SIM-DREAM

Many problems in science and engineering remain today intractable, despite the impressive progresses in computer science and the computational resources today available, because their numerical complexity is simply unimaginable. Among the models that remain today intractable, we can distinguish two main families:
- The first family of models consists of engineering models usually encountered in computational mechanics, defined in large and complex 3D geometries, involving many multiphysics couplings, many scales (in space and time), strong non-linearities, and whose transient simulation needs extremely small time steps. Despite the maturity of such models, as soon as the loading becomes complex (e.g. cyclic loadings, uncertainties …), the material involves uncertainties, heterogeneities in the microscopic scale, multiphysics couplings, or as soon as one focuses in parametric analysis … the model becomes simply unsolvable.
- The other family of challenging models concerns those models defined in highly dimensional spaces. For example, this kind of models appears naturally in the modeling of the structure and properties of materials at the finest scales. These models exhibit the redoubtable curse of dimensionality when usual mesh-based discretization techniques are applied.

For alleviating the difficulties related to the first family of models, many authors considered the use of proper orthogonal decomposition (POD) based techniques. However these techniques are far to be optimal, because the reduced basis are only optimal when constructed “a posteriori”, being only approximated when constructed “a priori”. Obviously, the optimal alternative lies in the simultaneous construction of both the solution and the associated reduced basis for expressing such a solution.

The partners of this project (P. Ladeveze, F. Chinesta, A. Ammar and A. Nouy) proposed some years ago a novel efficient technique able to circumvent the challenging issues just described. They called this technique: Proper Generalized Decomposition (PGD). It is based on a separated representation of the unknown solution.

We would like to recall that using these approaches, we have reduced the computing time related to the solutions of problems belonging to the first family of models (defined above) in several orders of magnitude (millions in some cases) and on the other hand we solved successfully highly multidimensional models never until now solved because they were considered suffering of the irremediable curse of dimensionality.

Because the novelty and the youngness of the PGD method, many aspects have not been addressed. The aim of this proposal is to push back the limits of this method. Definitively, these developments could lead to a real change of paradigm in computational mechanics. Imagine the possibility of solving efficiently any parameterized (including parameterized geometries), multi-scale and multi-physics model. Inverse identification and optimization would be a direct post-treatment. This proposal groups the “inventors” of the PGD method, all of them being highly recognized within the international community of computational mechanics. The partners are developing actively and independently these methods for many years. This proposal aims at grouping together researchers from the different partners in order to facilitate and accelerate the developments of new ideas within the context of PGD methods.

This project could have a significant impact from both fundamental and applicative points of view. The solution of the complex systems encountered in high-tech applications (aeronautic and space among many others) are waiting for new proposals able to solve challenging models without using impressive computer resources. Alternatives other than the increase of computational resources and the speed-up of standard techniques merit to be considered seriously.