A reduced order model for domain decompositions with non-conforming interfaces

TL;DR

A reduced order modeling strategy for two-way Dirichlet-Neumann parametric coupled problems solved with domain-decomposition (DD) sub-structuring methods that is numerically verified on both steady and unsteady coupled problems, in the case of non-conforming FE interfaces.

Abstract

In this paper, we propose a reduced-order modeling strategy for two-way Dirichlet-Neumann parametric coupled problems solved with domain-decomposition (DD) sub-structuring methods. We split the original coupled differential problem into two sub-problems with Dirichlet and Neumann interface conditions, respectively. After discretization by, e.g., the finite element method, the full-order model (FOM) is solved by Dirichlet-Neumann iterations between the two sub-problems until interface convergence is reached. We then apply the reduced basis (RB) method to obtain a low-dimensional representation of the solution of each sub-problem. Furthermore, we apply the discrete empirical interpolation method (DEIM) at the interface level to achieve a fully reduced-order representation of the DD techniques implemented. To deal with non-conforming FE interface discretizations, we employ the INTERNODES method combined with the interface DEIM reduction. The reduced-order model (ROM) is then solved by sub-iterating between the two reduced-order sub-problems until the convergence of the approximated high-fidelity interface solutions. The ROM scheme is numerically verified on both steady and unsteady coupled problems, in the case of non-conforming FE interfaces.

Figures (28)

• Figure 1: Schematic representation of the two discretizations of the domain $\Omega$ used to compute the FOM snapshots (left and center) and the discretization of the domain $\Omega$ used to compute the ROM snapshots (right).
• Figure 2: Schematic representation of the reduced order Dirichlet--Neumann domain decomposition algorithm.
• Figure 3: Geometrical interpretation of the interface reduction with the DEIM. The blue points represent the magic points on $\Gamma_2$ and the values of $\mathbf{u}_{\Gamma_2}^k$ at these magic points, the orange lines represent the piecewise constant interpolating function $\tilde{\mathbf{u}}_{\Gamma_2}^k$, while the red crosses are the points on $\Gamma_1$ corresponding to the magic points of $\Gamma_2$ and the values of $\mathbf{u}_{\Gamma_1}^{k+1}$ at these points.
• Figure 4: Test#1. From left to right: Half of the slave domain, half of the master domain, lateral view, and cross--section of the two half subdomains. In green the interface $\Gamma$.
• Figure 5: Test#1. Slave solution FOM (top), ROM (center) solutions, and absolute error (bottom) for three different vectors of testing parameters.

Theorems & Definitions (14)

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