We extend the hierarchical model reduction procedure previously introduced in
This work investigates a model reduction method applied to coupled multi-physics systems. The case in which a system of interest interacts with an external system is considered. An approximation of the Poincaré-Steklov operator is computed by simulating, in an offline phase, the external problem when the inputs are the Laplace-Beltrami eigenfunctions defined at the interface. In the online phase, only the reduced representation of the operator is needed to account for the influence of the external problem on the main system. An online basis enrichment is proposed in order to guarantee a precise reduced-order computation. Several test cases are proposed on different fluid -structure couplings.the Poincaré-Steklov operator the one associated with a generic linear PDE, even if historically this name refers to the case in which the secondary system is described by a Laplace equation (see [2] for a first analysis of the problem).The need to set up efficient solvers and to decouple the solution of the problems in interaction is related to the ability to solve the problem at the interface. The use of the Poincaré-Steklov operator as a preconditioner in fluid-structure interaction iterations was investigated in [3]. An efficient nonlinear coupling strategy was devised in [4] to set up an uncertainty quantification method applied to networks of coupled systems. Unfortunately, the problem at the interface is in general not sparse and ill-conditioned [1]. To tackle this issue, several strategies were proposed in the literature. They can be broadly divided into two classes: 'local' and spectral approximations. A local approximation of the Poincaré-Steklov operator consists of solving one or more external problems (P 2 in the present work) in a strip localized around the interface. Such a method was proposed, for example, in [5], for applications in hydrology. A similar procedure, based on a two-scale method, was presented in [6]: a local problem in a strip localized around the interface is solved, and then, thanks to the residual, a global correction is computed. A different strategy consists of approximating the leading part of the action of the Poincaré-Steklov operator through a spectral decomposition. Such an approach was proposed in [7,8] in the case of elliptic problems, and a multiscale version was proposed in [9] for applications in heterogeneous media. An approximation of the Poincaré-Steklov operator via a Padé expansion was detailed in [10] for the study of the vibrations in fluid-structure couplings. In [11], the Poincaré-Steklov operator is computed in the context of the wave propagation in elastodynamics by considering a family of smooth functions at the interface and by solving the problem P 2 by taking these functions as inputs. In the recent work [12], a compressed sensing approach is proposed to retrieve the discretized Poincaré-Steklov operator for coupled Helmholtz problems. The method consists of probing, randomly, the matrix associated with the Poincaré-Steklov operator, by selecting inputs from a kerne...
A simplified fluid-structure model for arterial flow.Application to retinal hemodynamics. AbstractWe propose a simplified fluid-structure interaction model for applications in hemodynamics. This work focuses on simulating the blood flow in arteries, but it could be useful in other situations where the wall displacement is small. As in other approaches presented in the literature, our simplified model mainly consists of a fluid problem on a fixed domain, with Robin-like boundary conditions and a first order transpiration. Its main novelty is the presence of fibers in the solid. As an interesting numerical side effect, the presence of fibers makes the model less sensitive than others to strong variations or inaccuracies in the curvatures of the wall. An application to retinal hemodynamics is investigated.
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