Domain decomposition methods based on proper generalized decomposition for parametric heterogeneous problems

Heterogeneous (or multi-physics) problems are very common in engineering and scientific applications. They typically arise when different phenomena occur in two or more subregions of the domain of interest such as, e.g., in the filtration of fluids through porous media in geophysical or industrial applications, in tissue perfusion in biomedicine, in the interactions between fluids and elastic structures. In such cases, at least two different sets of equations (e.g., incompressible fluid equations and elasticity equations) must be defined in each subregion and they must be suitably coupled into a global heterogeneous problem to correctly describe the physical system.

Solving these problems numerically is computationally demanding due to the need to accurately approximate all the different involved physical phenomena. The computational complexity increases even further when these problems must be solved several times for optimisation purposes as it occurs, e.g., in virtual design. Indeed, optimisation requires identifying the optimal values of several parameters used to describe various characteristics of the system such as geometrical features (e.g., the dimension of a structural element), material properties (e.g., the permeability of a porous medium) or process parameters (e.g., the inflow pressure in a filtering device). This is typically done by testing a large number of possible configurations, which dramatically increases the computational cost of numerical simulations and limits their practical applicability.

In this project, we will study a novel mathematical framework to make the numerical treatment of parametric heterogeneous problems more affordable by combining two mathematical methods: Domain Decomposition (DD) and Proper Generalized Decomposition (PGD).

The new method uses DD techniques to split multi-parametric heterogeneous problems into families of simpler subproblems of the same nature and with a reduced number of parameters. The solutions of these local subproblems can be computed by PGD that provides an efficient strategy to handle parameters of various nature in a unified manner. Finally, DD can 'compose' the local solutions to obtain the global 'general solution' of the original problem that accounts for all significant values of the parameters. Identifying effective and robust ways of 'composing' local solutions is not an easy task especially in the case of heterogeneous problems and it constitutes an open challenging research question in the PGD context that we address in this project.

We will lay the foundation of the DD-PGD method for heterogeneous problems and develop algorithms that will allow us to tackle the computational challenges encountered in the virtual design of multi-physics multi-parameter systems in various applications, e.g., membrane filtration processes.