A reduced‐order representation of the Poincaré–Steklov operator: an application to coupled multi‐physics problems

Abstract

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...

Localized model order reduction and domain decomposition methods for coupled heterogeneous systems

“…The physical parameters, $$$ {\rho}^s,E $$$ and $$$ \nu $$$ represent the density of the solid, the Young's modulus and Poisson's ratio, respectively. For this test we have used 37 $$$ {\rho}_s=1.0 $$$, $$$ E=1{0}^4 $$$, $$$ \nu =0.48 $$$. Equation (63) is split into two first‐order systems in time by defining $$$$ \boldsymbol{\xi} =\frac{\partial \boldsymbol{\eta}}{\partial t}.$$…”

“…-The functions 𝜑 j are also fixed. Inspired by similar works, 37 we choose them as the smallest eigenfunctions of the Laplace-Beltrami operator on the interface Γ, that is,…”

“…We now switch to a simple multi-physics problem, modeling the interaction between an incompressible flow governed by the unsteady Stokes equations and a porous medium described by the unsteady Darcy equation. 37 Although in a relatively simple context, this problem retains the main features of a multi-physics system, and is used to validate our technique with a more heterogeneous coupled system. We consider Ω = [0, 1] 2 , divided into two subdomains by a horizontal interface consider…”

“…The physical parameters, 𝜌 s , E and 𝜈 represent the density of the solid, the Young's modulus and Poisson's ratio, respectively. For this test we have used 37 The singular values of the local problems are reported in Figure 24 for each variable. The decay for the fluid problem, although exponential, is slower when compared to the previous test case.…”

“…The only condition on its value is the well posedness of (24), which is typically ensured if $$$ \lambda >0 $$$. The functions $$$ {\varphi}_j $$$ are also fixed. Inspired by similar works, 37 we choose them as the smallest eigenfunctions of the Laplace‐Beltrami operator on the interface $$$ \Gamma $$$, that is, $$$$ -\Delta {\varphi}_j={\omega}_j{\varphi}_j\kern.5em \mathrm{on}\kern.5em \Gamma, $$$$ with suitable boundary conditions. In order not to be restricted to a specific set of interface profiles, to increase the level of generality of the basis, and to have a larger insight on the local dynamics, we consider both homogeneous Dirichlet and homogeneous Neumann conditions.…”