Reduction techniques for PDEs built upon Reduced Basis and Domain Decomposition Methods with applications to hemodynamics

This thesis focuses on the development and validation of a reduced order technique for cardiovascular simulations. The method is based on the combined use of the Reduced Basis method and a Domain Decomposition approach and can be seen as a particular implementation of the Reduced Basis Element method. Our contributions include the application to the unsteady three-dimensional Navier--Stokes equations, the introduction of a reduced coupling between subdomains, and the reconstruction of arteries with deformed elementary building blocks. The technique is divided into two main stages: the offline and the online phases. In the offline phase, we define a library of reference building blocks (e.g., tubes and bifurcations) and associate with each of these a set of Reduced Basis functions for velocity and pressure. The set of Reduced Basis functions is obtained by Proper Orthogonal Decomposition of a large number of flow solutions called snapshots; this step is expensive in terms of computational time. In the online phase, the artery of interest is geometrically approximated as a composition of subdomains, which are obtained from the parametrized deformation of the aforementioned building blocks. The local solution in each subdomain is then found as a linear combination of the Reduced Basis functions defined in the corresponding building block. The strategy to couple the local solutions is of utmost importance. In this thesis, we devise a nonconforming method for the coupling of Partial Differential Equations that takes advantage of the definition of a small number of Lagrange multiplier basis functions on the interfaces. We show that this strategy allows us to preserve the h-convergence properties of the discretization method of choice for the primal variable even when a small number of Lagrange multiplier basis functions is employed. Moreover, we test the flexibility of the approach in scenarios in which different discretization algorithms are employed in the subdomains, and we also use it in a fluid-structure interaction benchmark. The introduction of the Lagrange multipliers, however, is associated with stability problems deriving from the saddle-point structure of the global system. In our Reduced Order Model, the stability is recovered by means of supremizers enrichment.

In our numerical simulations, we specifically focus on the effects of the Reduced Basis and geometrical approximations on the quality of the results. We show that the Reduced Order Model performs similarly to the corresponding high-fidelity one in terms of accuracy. Compared to other popular models for cardiovascular simulations (namely 1D models), it also allows us to compute a local reconstruction of the Wall-Shear Stress on the vessel wall. The speedup with respect to the Finite Element method is substantial (at least one order of magnitude), although the current implementation presents bottlenecks that are addressed in depth throughout the thesis.