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\u2013Structure 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\u2013Temam projection scheme with semi\u2013implicit 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\u2013 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

A monolithic and a partitioned Reduced Basis Method for Fluid-Structure Interaction problems

The aim of this work is to present an overview about the combination of the Reduced Basis Method (RBM) with two different approaches for Fluid-Structure Interaction (FSI) problems, namely a monolithic and a partitioned approach. We provide the details of implementation of two reduction procedures, and we then apply them to the same test case of interest. We first implement a reduction technique that is based on a monolithic procedure where we solve the fluid and the solid problems all at once. We then present another reduction technique that is based on a partitioned (or segregated) procedure: the fluid and the solid problems are solved separately and then coupled using a fixed point strategy. The toy problem that we consider is based on the Turek-Hron benchmark test case, with a fluid Reynolds number Re = 100

Delay equations and characteristic roots: stability and more from a single curve

Delays appear always more frequently in applications, ranging, e.g., from population dynamics to automatic control, where the study of steady states is undoubt- edly of major concern. As many other dynamical systems, those generated by nonlinear delay equations usually obey the celebrated principle of linearized stability. Therefore, hyperbolic equilibria inherit the stability properties of the corresponding linearizations, the study of which relies on associated characteristic equations. The transcendence of the latter, due to the presence of the delay, leads to infinitely-many roots in the com- plex plane. Simple algebraic manipulations show, first, that all such roots belong to the intersection of two curves. Second, only one of these curves is crucial for stability, and relevant sufficient and/or necessary criteria can be easily derived from its analysis. Other aspects can be investigated under this framework and a link to the theory of modulus semigroups and monotone semiflows is also discussed

Projection based semi--implicit partitioned Reduced Basis Method for non parametrized and parametrized Fluid--Structure Interaction problems

We present a partitioned Model Order Reduction method for multiphysics problems, that is based on a semi-implicit treatment of the coupling conditions, and on a projection scheme. The proposed Reduced Order Method is based on the Proper Orthogonal Decomposition and on a Galerkin projection onto the reduced basis spaces; we aim of addressing both time-dependent and time-dependent, parametrized Fluid-Structure Interaction problems, where the fluid is incompressible and the structure is linear, elastic and two dimensional

Overcoming slowly decaying Kolmogorov n-width by transport maps: application to model order reduction of fluid dynamics and fluid--structure interaction problems

In this work we focus on reduced order modelling for problems for which the resulting reduced basis spaces show a slow decay of the Kolmogorov $n$-width, or, in practical calculations, its computational surrogate given by the magnitude of the eigenvalues returned by a proper orthogonal decomposition on the solution manifold. In particular, we employ an additional preprocessing during the offline phase of the reduced basis method, in order to obtain smaller reduced basis spaces. Such preprocessing is based on the composition of the snapshots with a transport map, that is a family of smooth and invertible mappings that map the physical domain of the problem into itself. Two test cases are considered: a fluid moving in a domain with deforming walls, and a fluid past a rotating cylinder. Comparison between the results of the novel offline stage and the standard one is presented.Comment: 26 pages, 11 figure

A Reduced Order Cut Finite Element method for geometrically parametrized steady and unsteady Navier-Stokes problems

We focus on steady and unsteady Navier-Stokes flow systems in a reduced-order modeling framework based on Proper Orthogonal Decomposition within a levelset geometry description and discretized by an unfitted mesh Finite Element Method. This work extends the approaches of [1 -3] to nonlinear CutFEM discretization. We construct and investigate a unified and geometry independent reduced basis which overcomes many barriers and complications of the past, that may occur whenever geometrical morphings are taking place. By employing a geometry independent reduced basis, we are able to avoid remeshing and transformation to reference configurations, and we are able to handle complex geometries. This combination of a fixed background mesh in a fixed extended background geometry with reduced order techniques appears beneficial and advantageous in many industrial and engineering applications, which could not be resolved efficiently in the past

Advances in Model Order Reduction for Fluid-Structure Interaction Problems

Contribution on the state of the art on reduced order modelling for FSI problems Digital Abstract: https://www.dropbox.com/s/uzdi1rlz9y7chrb/ADMOS_Rozza_short.mp4?dl=

A Monolithic and a Partitioned, Reduced Basis Method for Fluid–Structure Interaction Problems

The aim of this work is to present an overview about the combination of the Reduced Basis Method (RBM) with two different approaches for Fluid–Structure Interaction (FSI) problems, namely a monolithic and a partitioned approach. We provide the details of implementation of two reduction procedures, and we then apply them to the same test case of interest. We first implement a reduction technique that is based on a monolithic procedure where we solve the fluid and the solid problems all at once. We then present another reduction technique that is based on a partitioned (or segregated) procedure: the fluid and the solid problems are solved separately and then coupled using a fixed point strategy. The toy problem that we consider is based on the Turek–Hron benchmark test case, with a fluid Reynolds number Re=100

Projection Based Semi-Implicit Partitioned Reduced Basis Method for Fluid-Structure Interaction Problems

In this manuscript a POD-Galerkin based Reduced Order Model for unsteady Fluid-Structure Interaction problems is presented. The model is based on a partitioned algorithm, with semi-implicit treatment of the coupling conditions. A Chorin-Temam projection scheme is applied to the incompressible Navier-Stokes problem, and a Robin coupling condition is used for the coupling between the fluid and the solid. The coupled problem is based on an Arbitrary Lagrangian Eulerian formulation, and the Proper Orthogonal Decomposition procedure is used for the generation of the reduced basis. We extend existing works on a segregated Reduced Order Model for Fluid-Structure Interaction to unsteady problems that couple an incompressible, Newtonian fluid with a linear elastic solid, in two spatial dimensions. We consider three test cases to assess the overall capabilities of the method: an unsteady, non-parametrized problem, a problem that presents a geometrical parametrization of the solid domain, and finally, a problem where a parametrization of the solid's shear modulus is taken into account