Application of optimisation-based domain--decomposition reduced order models to parameter-dependent fluid dynamics and multiphysics problems

With the increase in the potential of high-performance computing in the last years, there is an immense necessity for numerical methods and approximation techniques that can perform real-time simulations of Partial Differential Equations (PDEs). The applications range from naval to aeronautical, to biomedical engineering ones, to name a few. There exist many techniques that might help in achieving such a goal, among which reduced–order models (ROMs) and domain–decomposition (DD) algorithms, namely those employed in this manuscript. The DD methodology is a very efficient tool in the framework of PDEs. Any DD algorithm is constructed by an effective splitting of the domain of interest into different subdomains (overlapping or not), and the original problem is then restricted to each of these subdomains with some coupling conditions on the intersections of the subdomains. The coupling conditions may be very different, they depend on the physical meaning of the problem at hand, and they must render a certain degree of continuity among these subdomains. These methods are extremely important for multiphysics problems when efficient subcomponent numerical codes are already available, or when we do not have direct access to the numerical algorithms for some parts of the systems. Model-order reduction methods are another set of methods mentioned before, which are extremely useful when dealing with real-time simulations or multi-query tasks. These methods are successfully employed in the settings of non-stationary and/or parameter-dependent PDEs. ROMs are extremely effective thanks to the splitting of the computational effort into two stages: the offline stage, which contains the most expensive part of the computations, and the online stage, which allows performing fast computational queries using structures that are pre-computed in the offline stage. This thesis aims to introduce a framework where both aforementioned techniques, namely DD algorithms and ROMs are combined in order to achieve better performance of numerical simulations. We choose to model the DD using an optimisation approach to ensure the coupling of the interface conditions among subdomains. Starting from the domain decomposition approach, we derive an optimal control problem, for which we present the convergence analysis. The snapshots for the high–fidelity model are obtained with the Finite Element discretisation, and the model order reduction is then proposed both in terms of time and/or physical parameters, with a standard Proper Orthogonal Decomposition (POD)-Galerkin projection or with non--intrusive methods, such as POD-neural network (NN). The methodology has been tested on a couple of Computational Fluid Dynamics (CFD) benchmark problems. The final aim of the thesis is to produce a fully-segregated method for multiphysics problems using the aforementioned techniques. We have managed successfully to build a model for a non-stationary Fluid-Structure Interaction (FSI) problem. The resulting numerical method shows an extremely important feature - it is stable under the assumption of the “added mass” effect, which causes instabilities of many partitioned approaches to FSI problems. It has been evidenced by the numerical experiments of the model presented for a two-dimensional haemodynamics benchmark FSI problem.