Stefan Burgard

Passivity preserving parametric model-order reduction for non-affine parameters

Parametric model-order reduction (pMOR) has become a well-established technology for analysing large-scale systems with multiple parameters. However, the treatment of non-affine parameters is still posing significant challenges, because projection-based order-reduction methods cannot be applied directly. A common remedy is to establish affine parameter-dependencies approximately, but present extraction methods do not take important system properties, such as passivity, into account. This article proposes a new order-reduction approach that preserves passivity, reciprocity and causality and applies to a wide class of linear time-invariant (LTI) systems. We present the theory of the suggested method and demonstrate its practical usefulness by numerical examples taken from computational electromagnetics.

A Posteriori Error Bounds for Krylov-Based Fast Frequency Sweeps of Finite-Element Systems

Projection-based model reduction is a well-established methodology for computing fast frequency sweeps of finite-element (FE) approximations to passive microwave structures. This contribution presents a novel provable error bound for moment-matching reduced-order models of lossless systems. It improves over existing methods by increasing the accuracy of the estimate and by reducing numerical costs. Numerical studies demonstrate the benefits of the suggested approach.

A novel parametric model order reduction approach with applications to geometrically parameterized microwave devices

Purpose – The goal is to derive a numerical method for computing parametric reduced-order models (PROMs) from finite-element (FE) models of microwave structures that feature geometrical parameters. Design/methodology/approach – First, a parameter-dependent FE mesh is constructed by a topology-preserving mesh-morphing algorithm. Then, multivariate polynomial interpolation is employed to achieve explicit geometrical parameterization of all FE matrices. Finally, a PROM based on parameter-dependent projection matrices is constructed by means of interpolation and state transformation techniques. Findings – The resulting PROMs are of low dimension and fast to evaluate. Moreover, the method features high rates of convergence, and the number of FE solutions required for constructing the PROM is small. The accuracy of the PROM is only limited by that of the underlying FE model and can be controlled by varying the PROM dimension. Research limitations/implications – Since the method uses topology-preserving mesh-mor...

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Methods of parametric order reduction are very appealing for solving parameter-dependent models at the fields level, because they provide fast simulations and low systematic error. This paper presents a self-adaptive framework for computing reduced-order models featuring affine and non-affine parameters. It is based on a hypercube partitioning of the domain of non-affine parameters and employs non-uniform grid refinement, controlled by a suitable error indicator. Compared with state-of-the-art entire-domain methods, the proposed sub-domain approach achieves significant improvements in memory consumption and computer run time.

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Numerical simulation methods at the fields level, such as the finite-element method, are highly accurate but computationally expensive. In the context of mathematical optimization, this implies that the cost function, which needs to be evaluated a large number of times, is very expensive to compute. To overcome this shortcoming, this paper proposes to employ parametric reduced-order models for computing the cost function and, if required, its gradients. They introduce low systematic error, require little memory, and allow for hundreds to thousands of model evaluations per second. The great utility of the suggested approach in both deterministic and stochastic optimization methods is demonstrated by a numerical example, featuring 11 geometric parameters.

Adaptive Sub-Domain Framework for Parametric Order Reduction

Abstract Since typical finite-element systems are of high dimension, the analysis of parameter-dependent microwave structures over broad frequency bands tends to be very time-consuming. This issue is addressed by parametric order reduction, which provides a systematic methodology for constructing surrogate models that are cheap to evaluate and feature low and controllable levels of error. This article presents an order reduction technique for finite-element models that depends on the operating frequency and features explicit and implicit parameters for material properties and shape, respectively. It uses polynomial interpolation to resolve implicit parameter dependencies and employs parameter-dependent bases defined on sub-domains of parameter space. The resulting reduced-order models are of very small dimension and preserve the structure and frequency dependency of the original finite-element model. Numerical results demonstrate that the proposed method reduces solution times by several orders of magnitude compared to the underlying finite-element model at very low error.

Fast Shape Optimization of Microwave Devices Based on Parametric Reduced-Order Models

Purpose – The purpose of this paper is to determine the broadband frequency response of the impedance matrix of wireless power transfer (WPT) systems comprising litz wire coils. Design/methodology/approach – A finite-element (FE)-based method is proposed which treats the microstructure of litz wires by an auxiliary cell problem. In the macroscopic model, litz wires are represented by a block with a homogeneous, artificial material whose properties are derived from the cell problem. As the frequency characteristics of the material closely resemble a Debye relaxation, it is possible to convert the macroscopic model to polynomial form, which enables the application of model reduction techniques of moment-matching type. Findings – FE-based model-order reduction using litz wire homogenization provides an efficient approach to the broadband analysis of WPT systems. The error of the reduced-order model (ROM) is comparable to that of the underlying original model and can be controlled by varying the ROM dimension...

Reduced-Order Models of Finite-Element Systems Featuring Shape and Material Parameters

The finite-element time-domain simulation of nonlinear eddy-current problems requires the iterative solution of a large, sparse system of equations at every time-step. Model-order reduction is a powerful tool for reducing the computational effort for this task. In this paper, an adaptive order-reduction methodology with error control is proposed. In contrast to previous approaches, it treats the nonlinearity without simplification, by rewriting the original equations as a quadratic-bilinear system.

Litz wire homogenization by finite elements and model-order reduction

Abstract Established order reduction methods for nonlinear descriptor systems project the nonlinear system onto a linear submanifold. This may lead to reduced models of large scale which provide little computational speed-up. To overcome this deficiency, the proposed method replaces the construction of a global projection matrix by an interpolation of locally reduced models. The present paper gives the underlying theory and demonstrates the accuracy and efficiency of the suggested approach by a finite element model of an eddy current problem.