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Sequentially optimal sensor placement in thermoelastic models for real time applications

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Abstract

Optimal sensor placement problems are considered for a thermoelastic solid body. The main motivation is the real-time capable prediction of the displacement of the tool center point (TCP) in a machine tool by temperature measurements. A reduced order model based on proper orthogonal decomposition is used to describe the temperature field. The quality of the TCP displacement estimation is measured in terms of the associated covariance matrix. Based on this criterion, a sequential placement algorithm is described which stops when a certain prediction quality is reached. Numerical tests are provided.

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Notes

  1. \(L^2(\varOmega )\) is the Lebesgue space of square integrable functions, and \(H^1(\varOmega )^3\) stands for the Sobolev space of vector valued square integrable functions whose weak first-order derivatives are also square integrable.

  2. To be precise, (2.19) is valid only for \(i \ge n_{{\text {POD}}}\ge 3\), i.e., one needs at least as many sensors as one has output quantities.

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Acknowledgments

This work was supported by a DFG grant within the http://transregio96.deSpecial Research Program SFB/ Transregio 96 (Thermo-energetische Gestaltung von Werkzeugmaschinen), which is gratefully acknowledged.

The authors wish to thank Dipl.-Ing. Carsten Zwingenberger (Fraunhofer Institute for Machine Tools and Forming Technology, Chemnitz) for providing the CAD geometry and mesh of the machine column.

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  1. Faculty of Mathematics, Technische Universität Chemnitz, 09107, Chemnitz, Germany

    Roland Herzog & Ilka Riedel

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  1. Roland Herzog
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  2. Ilka Riedel
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Correspondence to Ilka Riedel.

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Herzog, R., Riedel, I. Sequentially optimal sensor placement in thermoelastic models for real time applications. Optim Eng 16, 737–766 (2015). https://doi.org/10.1007/s11081-015-9275-0

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