HEVICO · Virtual control methods for coupling heterogeneous problems

Virtual control methods for coupling heterogeneous problems

The Marie Curie CIG project HeViCo has studied a new mathematical and numerical technique based on optimal control theory to couple heterogeneous problems described by partial differential equations. Heterogeneous or multi-physics problems arise in many practical applications, e.g., whenever different physical phenomena must to be taken into account in two or more sub-regions of the computational domain, or when simplified models are locally used to reduce the overall computational complexity. Starting from the well-consolidated framework provided by domain decomposition methods for multi-physics problems and from optimal control for partial differential equations, this project has studied a particular class of techniques, named Interface Control Domain Decomposition (ICDD) methods, to handle the intrinsic heterogeneity of the problems of our interest. The first part of the project was devoted to the study of ICDD for elliptic problems and for the Stokes equations. On the basis of the encouraging results obtained in this phase, the second part has mainly focused on the Stokes-Darcy problem, a model of great interest for many important applications. In fact, being used to describe filtration processes through porous media, it can be applied to a vast range of contexts such as membrane filtration, geophysics (e.g., filtration of water through the soil), and biomedicine (e.g., filtration of blood through human tissues). The ICDD method allows to couple the Stokes and the Darcy equations differently from the most common approach based on the so-called Beavers-Joseph condition. In particular, ICDD exploits a transition region between the free fluid and the porous medium in analogous way to the meso-scopic averaged methods. However, one of the advantages of ICDD is that no further modelling of the transition region is required so that the difficulties of experimentally estimating several physical parameters are overcome. The new approach has been analysed theoretically and it has been compared with more standard methods available in the literature. Numerical results have shown that the velocity and pressure of the fluid estimated by ICDD are in very good agreement with those provided by other more consolidated techniques. Moreover, this method has proved to be robust and much easier to implement than algorithms based on non-overlapping decompositions of the computational domain. Summarising, the contribution of this project has been twofold. On the one hand we have provided new solution algorithms for heterogeneous problems such as the coupling between advection-diffusion and pure advection equations, and the Stokes-Darcy problem. On the other hand, the analysis that we have carried out provides a solid basis for the application of ICDD to the more general context of heterogeneous problems. In particular, this methodology could be further extended to the case of geometrically multi-scale problems, or to the coupling of macro- (e.g., Navier-Stokes or Darcy equations), meso- (e.g., Lattice-Boltzmann) and micro-scopic models.

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