Advances in Computational Mathematics

, Volume 44, Issue 6, pp 1797–1820 | Cite as

Damping optimization of parameter dependent mechanical systems by rational interpolation

  • Zoran TomljanovićEmail author
  • Christopher Beattie
  • Serkan Gugercin


We consider an optimization problem related to semi-active damping of vibrating systems. The main problem is to determine the best damping matrix able to minimize influence of the input on the output of the system. We use a minimization criteria based on the \(\mathcal {H}_{2}\) system norm. The objective function is non-convex and the associated optimization problem typically requires a large number of objective function evaluations. We propose an optimization approach that calculates ‘interpolatory’ reduced order models, allowing for significant acceleration of the optimization process. In our approach, we use parametric model reduction (PMOR) based on the Iterative Rational Krylov Algorithm, which ensures good approximations relative to the \(\mathcal {H}_{2}\) system norm, aligning well with the underlying damping design objectives. For the parameter sampling that occurs within each PMOR cycle, we consider approaches with predetermined sampling and approaches using adaptive sampling, and each of these approaches may be combined with three possible strategies for internal reduction. In order to preserve important system properties, we maintain second-order structure, which through the use of modal coordinates, allows for very efficient implementation. The methodology proposed here provides a significant acceleration of the optimization process; the gain in efficiency is illustrated in numerical experiments.


Model reduction Interpolation Second-order systems Semi-active damping 

Mathematics Subject Classification (2010)

49J15 74P10 70Q05 41A05 


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Authors and Affiliations

  1. 1.Department of MathematicsJ.J. Strossmayer University of OsijekOsijekCroatia
  2. 2.Department of MathematicsVirginia Polytechnic Institute and State UniversityBlacksburgUSA

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