On the application of the Reduced Basis Method to Fluid-Structure Interaction problems

With this thesis the author aims at giving an extensive overview on the application of the Reduced Basis Method to Fluid–Structure Interaction (FSI) problems. The work exposed is divided into three main research directions: the First two methods presented are based on a standard Finite Element discretization of the problem of interest, whereas the third method presented differs from the other two because it is based on an embedded Finite Element discretization. In this way the author wants to show the advantages of pursuing a model order reduction with either a standard Finite Element method or with a Cut Finite Element method, depending on the particular problem of interest: throughout the Chapters it will be shown that a reduction method based on a classical Finite Element discretization is well suited for multiphysics problems where the geometry of the domain does not change significantly; on the contrary, a Cut Finite Element approach shows its full potentiality in situations where, for example, the structure undergoes a large deformation. The algorithms presented in this thesis are: a partitioned (or segregated) Reduced Basis Method that is based on a Chorin–Temam projection scheme with semi–implicit coupling of the solid and the fluid problem, a Reduced Basis Method enriched with a preprocessing of the snapshots during the offline phase, and lastly a Reduced Order Method in a Cut Finite Element framework. According to the approach adopted to adress the particular problem of interest, the thesis proposes a modification and an improvement of the Reduced Basis Method in order to obtain a complete model order reduction procedure. Several test cases are considered throughout the work: a toy problem that describes the deformation of two leaflets under the influence of the jet of a fluid; a Fluid– Structure Interaction problem whose solution exhibits a transport dominated behaviour, and, in addition, some Computational Fluid Dynamics toy problems, also in the case of parameter dependence. For each one of the test cases considered, first there is an introduction to the problem formulation, and then the proposed model order reduction procedure follows.