Reduced Basis Methods for the Solution of Parametrized PDEs in Repetitive and Complex Networks with Application to CFD

The objective of this work is to develop a numerical framework to perform rapid and reliable simulations for solving parametric problems in domains represented by networks and to extend the classical reduced basis method. Aimed at this scope, we propose two original methodological approaches for the approximation of partial differential equations in domains made up by repetitive parametrized geometries where topological features are recurrent: the reduced basis hybrid method (RBHM) and the reduced basis-domain decomposition-finite element (RDF) method. The common paradigm of these methods is the consideration that the blocks composing the computational domain are topologically similar to a few reference shapes. On the latter, we compute representative solutions, corresponding to the same governing partial differential equations, but for different values of some parameters of interest and representing, for example, the deformation of the blocks. A new desired solution for a new deformed domain is recovered as projection on the reduced spaces built by the previously precomputed solutions and the continuity of the solution across subdomain interfaces is guaranteed by suitable coupling conditions. The different choices for the reduced spaces and coupling conditions adopted characterize one method with respect to the other one. The geometrical parametrization of the considered domains, by transfinite maps, induces non-affine parameter dependence: an empirical interpolation technique is used to recover an approximate affine parameter dependence and a sub–sequent offline/online decomposition of the reduced basis procedure. This computational decomposition yields a considerable reduction of the problem complexity. Results computed on some combinations of 2D and 3D geometries, representing cardiovascular networks, show the flexibility and the advantages of the proposed methods in terms of reduced computational costs and complexities. The computational time with these new approaches is, in general, much reduced with respect to a classical finite element method on the whole domain but also only marginally slower than a classical reduced basis approach on the whole domain. However, these approaches decrease drastically the offline time to pre-compute the reduced basis by splitting the total number of parameters characterizing the problem into smaller subsets for each reference block, moreover they allow to considerably increase the geometrical flexibility and versatility.

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