Stable reduced order modeling of piezoelectric energy harvesting modules using implicit Schur complement

Hu, Shan; Yuan, Chengdong; Castagnotto, Alessandro; Lohmann, Boris; Bouhedma, Sofiane; Hohlfeld, Dennis; Bechtold, Tamara

doi:10.1016/j.microrel.2018.03.026

Cited by 14 publications

(21 citation statements)

References 24 publications

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“…In the previous research [43][44][45], the stability of the reduced system (2) could not be guaranteed by the conventional MOR methods. Several stabilization approaches have been introduced and mathematically proven in [46][47][48]. In this work, we apply the efficient approach 'Schur after MOR' to generate a stable ROM of the piezoelectric energy harvester model.…”

Section: Model Order Reductionmentioning

confidence: 99%

System-Level Model and Simulation of a Frequency-Tunable Vibration Energy Harvester

et al. 2020

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In this paper, we present a macroscale multiresonant vibration-based energy harvester. The device features frequency tunability through magnetostatic actuation on the resonator. The magnetic tuning scheme uses external magnets on linear stages. The system-level model demonstrates autonomous adaptation of resonance frequency to the dominant ambient frequencies. The harvester is designed such that its two fundamental modes appear in the range of (50,100) Hz which is a typical frequency range for vibrations found in industrial applications. The dual-frequency characteristics of the proposed design together with the frequency agility result in an increased operative harvesting frequency range. In order to allow a time-efficient simulation of the model, a reduced order model has been derived from a finite element model. A tuning control algorithm based on maximum-voltage tracking has been implemented in the model. The device was characterized experimentally to deliver a power output of 500 µW at an excitation level of 0.5 g at the respected frequencies of 63.3 and 76.4 Hz. In a design optimization effort, an improved geometry has been derived. It yields more close resonance frequencies and optimized performance.

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Section: Model Order Reductionmentioning

confidence: 99%

System-Level Model and Simulation of a Frequency-Tunable Vibration Energy Harvester

et al. 2020

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“…Based on these two models, we first recapture the stabilization methods: “MOR after Schur,” “Schur after MOR” and “multiphysics structure preserving MOR” in Section 3. The necessary conditions sufficient for stable second-order DAEs from study by Salimbahrami (2005) are then introduced, as it is needed for the proof of stability preservation, similar to works by Hu et al (2018a) and Hu et al (2018b). Finally, we conclude with the mathematical proof that “multiphysics structure preserving MOR” stabilizes the reduced piezoelectric model.…”

Section: Introductionmentioning

confidence: 98%

“…This kind of structure-preserving or index-aware MOR method preserves the original block structure of the system (Freund, 2004; Banagaaya, 2014; Benner et al , 2016). Past year, based on the “MOR after Schur” method and the work in Castagnotto et al (2015), Yuan et al (2018) and Hu et al (2018a) introduced the “MOR after implicit Schur” method, which further speeds up the MOR process. Further, in studies by Hu et al (2018a) and Hu et al (2018b), the stability preservation of “MOR after Schur,” “MOR after implicit Schur” and “Schur after MOR” were proven mathematically.…”

Section: Introductionmentioning

confidence: 99%

“…Past year, based on the “MOR after Schur” method and the work in Castagnotto et al (2015), Yuan et al (2018) and Hu et al (2018a) introduced the “MOR after implicit Schur” method, which further speeds up the MOR process. Further, in studies by Hu et al (2018a) and Hu et al (2018b), the stability preservation of “MOR after Schur,” “MOR after implicit Schur” and “Schur after MOR” were proven mathematically. Proceeding with their work, in this paper we provide the “last piece of the jigsaw.” We demonstrate the stability preservation proof of the “multiphysics structure preserving MOR” method mathematically.…”

Section: Introductionmentioning

confidence: 99%

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Stable compact modeling of piezoelectric energy harvester devices

Yuan

Bechtold

2020

COMPEL

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Purpose Based on the framework of Krylov subspace-based model order reduction (MOR), compact models of the piezoelectric energy harvester devices can be generated. However, the stability of reduced piezoelectric model often cannot be preserved. In previous research studies, “MOR after Schur,” “Schur after MOR” and “multiphysics structure preserving MOR” methods have proven successful in obtaining stable reduced piezoelectric energy harvester models. Though the stability preservation of “MOR after Schur” and “Schur after MOR” methods has already been mathematically proven, the “multiphysics structure preserving MOR” method was not. This paper aims to provide the missing mathematical proof of “multiphysics structure preserving MOR.” Design/methodology/approach Piezoelectric energy harvesters can be represented by system of differential-algebraic equations obtained by the finite element method. According to the block structure of its system matrices, “MOR after Schur” and “Schur after MOR” both perform Schur complement transformations either before or after the MOR process. For the “multiphysics structure preserving MOR” method, the original block structure of the system matrices is preserved during MOR. Findings This contribution shows that, in comparison to “MOR after Schur” and “Schur after MOR” methods, “multiphysics structure preserving MOR” method performs the Schur complement transformation implicitly, and therefore, stabilizes the reduced piezoelectric model. Originality/value The stability preservation of the reduced piezoelectric energy harvester model obtained through “multiphysics structure preserving MOR” method is proven mathematically and further validated by numerical experiments on two different piezoelectric energy harvester devices.

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“…Today, the successful applications of model order reduction (MOR) in mechanical engineering deal with different system and problem classes from different physical domains, like: -structural and multibody dynamics, modeled by linear or nonlinear differential equations [18,17,21,67,69,35,34,53,3,31,91,90,5,104,19,49,47,95,110]; -fluid dynamics, including fluid-structure interaction and aerodynamics [93,26,89,88,83,2,31]; -thermo-mechanical, thermo-fluid, thermo-acoustic, and thermo-electrical systems [68,12,14,11,97,75,54,46].…”

Section: Introductionmentioning

confidence: 99%