Localized Reduced Basis Methods for Time Harmonic Maxwell’s Equations

Abstract

WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTERLocalized model order reduction methods have attracted significant attention during the last years. They have favorable parallelization properties and promise to perform well on cloud architectures, which become more and more commonplace. We introduced ArbiLoMod [1], a localized reduced basis method targeted at the important use case of changing problem definition, wherein the changes are of local nature. This is a common situation in simulation software used by engineers optimizing a CAD model. An especially interesting application is the simulation of electromagnetic fields in printed circuit boards, which is necessary to design high frequency electronics. The simulation of the electromagnetic fields can be done by solving the time-harmonic Maxwell's equations, which results in a parameterized, inf-sup stable problem which has to be solved for many parameters. In this multi-query setting, the reduced basis method can perform well. Experiments have shown two dimensional time-harmonic Maxwell's to be amenable to localized model reduction [2]. However, Galerkin projection of an inf-sup stable problem is not guaranteed to be stable. Existing stabilization methods for the reduced basis method involve global computations and are thus not applicable in a localized setting. Replacing the Galerkin projection with the minimization of a localized a posteriori error estimator provides a stable reduction for inf-sup stable projects which retains all the advantageous properties of localized model order reduction. It allows for an offline-online decomposition and requires no global computations in the unreduced space.

Dictionary-based model reduction for state estimation

“…A common approach to compute the residual norm efficiently is to expand offline r(Ua, •) 2 U as r(Ua; ξ) 2 U = a T M 1 (ξ)a − 2a T M 2 (ξ) + M 3 (ξ), ∀ξ ∈ P, where M 1 : P → R (K+m)×(K+m) , M 2 : P → R (K+m) and M 3 : P → R all admit affine representations, with respectively m 2 B , m B m f and m 2 f affine terms. Although this expression is theoretically exact, in practice the high number of affine terms makes this approach more sensitive to round-off errors, as pointed out in [8,2,7], and the associated offline cost scales as…”

“…which is prohibitive since we aim for large K. Another approach proposed [7] is to compute an orthonormal basis of the space in which the residual lies. This approach is more stable than the previous one, but comes with a similar prohibitive offline cost.…”

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

“…Localized model reduction of multi-component systems has been investigated in a number of works. 7 A related approach is the Reduced basis, Domain decomposition, Finite elements (RDF) method, 8 which constructs local surrogates using a parametrization of the interface data with Lagrange or Fourier basis functions and a greedy algorithm. This method shares many features with our approach, but the use of the full model in a few regions of the computational domain and the application to a single steady-state diffusion problem limit its generalization potential.…”

“…Localized model reduction of multi‐component systems has been investigated in a number of works 7 . A related approach is the Reduced basis, Domain decomposition, Finite elements (RDF) method, 8 which constructs local surrogates using a parametrization of the interface data with Lagrange or Fourier basis functions and a greedy algorithm.…”

Multiscale modeling of linear elastic heterogeneous structures via localized model order reduction

“…To alleviate these shortcomings, methods combining multiscale methods, domain decomposition and model order reduction were developed. Approaches of this kind are known as localized model order reduction methods , and an extensive review is given by Buhr et al 35 The main idea is the construction of local reduced spaces on subdomains, that is, parts of the global domain, which are then coupled (either in a conforming or non‐conforming way) to obtain a global approximation.…”