Abstract
In the field of model order reduction for frequency response problems, the minimal rational interpolation (MRI) method has been shown to be quite effective. However, in some cases, numerical instabilities may arise when applying MRI to build a surrogate model over a large frequency range, spanning several orders of magnitude. We propose a strategy to overcome these instabilities, replacing an unstable global MRI surrogate with a union of stable local rational models. The partitioning of the frequency range into local frequency sub-ranges is performed automatically and adaptively, and is complemented by a (greedy) adaptive selection of the sampled frequencies over each sub-range. We verify the effectiveness of our proposed method with two numerical examples.
Highlights
The numerical simulation of frequency-domain dynamical systems is crucial in many engineering applications, spanning from the analysis of the natural modes of elastic structures to the frequency response of electrical circuits
To alleviate the computational burden, in the field of model order reduction (MOR), it is customary to build a surrogate model for the quantity of interest (QoI) and use such surrogate instead of the original model when performing the frequency response analysis
The precise way in which the surrogate is built can vary: e.g., the reduced basis method (RBM) performs a Galerkin projection of (1) onto a subspace of CnS, whereas, in the Loewner framework and with the vector fitting (VF) method, one constructs a rational approximation by fitting the samples. We focus on the latter class of MOR approaches, commonly referred to as non-intrusive, since they only rely on samples of the QoI, and not on the specific structure of the original system (1)
Summary
The numerical simulation of frequency-domain dynamical systems is crucial in many engineering applications, spanning from the analysis of the natural modes of elastic structures to the frequency response of electrical circuits. 1.1 The model reduction endeavor Assume that our target is the frequency response analysis of a quantity of interest (QoI) associated with system (1) (most often, the state x or the linear output y) To this aim, we have to evaluate the QoI at many values of the frequency, located within a “frequency range of interest” Z ⊂ C, often an interval on the imaginary axis. We can identify the main drawback of MRI in its difficulty in dealing with very large frequency ranges, namely, with frequencies spanning several orders of magnitude (this is a situation of practical interest, e.g., when making Bode diagrams) In such cases, numerical instabilities may arise, hindering the construction of a stable surrogate, as we will discuss more in detail in Sect.
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