Computational Mechanics

, Volume 63, Issue 5, pp 853–868 | Cite as

Geometric element parameterization and parametric model order reduction in finite element based shape optimization

  • Benjamin Fröhlich
  • Jan Gade
  • Florian Geiger
  • Manfred Bischoff
  • Peter EberhardEmail author
Original Paper


This contribution proposes a new approach to derive geometrically parameterized, reduced order finite element models. An element formulation for geometrically parameterized finite elements is suggested. The parameterized elements are used to derive models with a parameterized geometry where the parameterized system matrices are expressed in an affine representation. Parametric model order reduction can then be efficiently used to reduce the full order parameterized model to a reduced order parameterized model. The approach shows two beneficial features. First, design studies and shape optimizations can be conducted with parameterized reduced order model of much lower dimension compared to the parameterized, full order model. Second, it is possible to compute sensitivities analytically, and therefore, to avoid the computation of finite differences gradients. The approach is illustrated with two numerical examples. The first example includes a detailed error analysis. The second example is a shape optimization example of an adaptive structure.


Parametric model order reduction Shape optimization Reduced order modeling Moment matching 



The authors gratefully thank the German Research Foundation (DFG) for the support of this research work within the collaborative research centre SFB/CRC 1244, “Adaptive Skins and Structures for the Built Environment of Tomorrow” with the projects A04 and B01 and as well as the project DFG EB 195/11-2 “Model Order Reduction for Elastic Multibody Systems with Moving Interactions”.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Aliyev N, Benner P, Mengi E, Schwerdtner P, Voigt M (2017) Large-scale computation of \(L_\infty \)-norms by a Greedy subspace method. SIAM J Matrix Anal Appl 38(4):1496–1516MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ammar A, Huerta A, Chinesta F, Cueto E, Leygue A (2014) Parametric solutions involving geometry: a step towards efficient shape optimization. Comput Methods Appl Mech Eng 268:178–193MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Amsallem D, Farhat C (2008) Interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J 46(7):1803–1813CrossRefGoogle Scholar
  4. 4.
    Antoulas A (2005) Approximation of large-scale dynamical systems. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  5. 5.
    Baumann M (2016) Parametrische Modellreduktion in elastischen Mehrkörpersystemen. Dissertation, Schriften aus dem Institut für Technische und Numerische Mechanik der Universität Stuttgart, vol 43. Shaker, Aachen (in German) Google Scholar
  6. 6.
    Baur U, Beattie C, Benner P, Gugercin S (2011) Interpolatory projection methods for parameterized model reduction. SIAM J Sci Comput 33(5):2489–2518MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Baur U, Benner P (2009) Modellreduktion für parametrisierte Systeme durch balanciertes Abschneiden und Interpolation. at-Automatisierungstechnik 57(8):411–419CrossRefGoogle Scholar
  8. 8.
    Baur U, Benner P, Haasdonk B, Himpe C, Martini I, Ohlberger M (2017) Comparison of methods for parametric model order reduction of time-dependent problems. In: Benner P, Cohen A, Ohlberger M, Willcox K (eds) Model reduction and approximation theory and algorithms. SIAM, PhiladelphiaGoogle Scholar
  9. 9.
    Benner P, Gugercin S, Willcox K (2015) A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev 57(4):483–531MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bletzinger KU (2014) A consistent frame for sensitivity filtering and the vertex assigned morphing of optimal shape. Struct Multidiscip Optim 49(6):873–895MathSciNetCrossRefGoogle Scholar
  11. 11.
    Daniel L, Siong O, Chay L, Lee K, White J (2004) A multiparameter moment-matching model-reduction approach for generating geometrically parameterized interconnect performance models. IEEE Trans Comput Aided Des Integr Circuits Syst 23(5):678–693CrossRefGoogle Scholar
  12. 12.
    Fehr J (2011) Automated and error-controlled model reduction in elastic multibody systems. Dissertation, Schriften aus dem Institut für Technische und Numerische Mechanik der Universität Stuttgart, vol 21. Shaker, AachenGoogle Scholar
  13. 13.
    Fehr J, Grunert D, Holzwarth P, Fröhlich B, Walker N, Eberhard P (2017) Morembs—a model order reduction package for elastic multibody systems and beyond. In: Keiper W, Milde A, Volkwein S (eds) Reduced-order modeling (ROM) for simulation and optimization. Powerful algorithms as key enablers for scientific computing.
  14. 14.
    Feng L, Rudnyi E, Korvink J (2005) Preserving the film coefficient as a parameter in the compact thermal model for fast electrothermal simulation. IEEE Trans Comput Aided Des Integr Circuits Syst 24(12):1838–1847CrossRefGoogle Scholar
  15. 15.
    Fischer M, Eberhard P (2014) Application of parametric model reduction with matrix interpolation for simulation of moving loads in elastic multibody systems. Adv Comput Math 41(5):1049–1072MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Grimme E (1997) Krylov projection methods for model reduction. Dissertation, University of Illinois at Urbana-ChampaignGoogle Scholar
  17. 17.
    Huerta A, Nadal E, Chinesta F (2017) Proper generalized decomposition solutions within a domain decomposition strategy. Int J Numer Methods Eng 113(13):1972–1994MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lang N, Saak J, Benner P (2014) Model order reduction for systems with moving loads. at-Automatisierungstechnik 62(7):512–522CrossRefGoogle Scholar
  19. 19.
    Lehner M (2007) Modellreduktion in elastischen Mehrkörpersystemen. Dissertation, Schriften aus dem Institut für Technische und Numerische Mechanik der Universität Stuttgart, vol 10. Shaker, Aachen (in German) Google Scholar
  20. 20.
    Nocedal J, Wright SJ (2006) Numerical optimization. Springer series in operations research. Springer, BerlinGoogle Scholar
  21. 21.
    Panzer H, Mohring J, Eid R, Lohmann B (2010) Parametric model order reduction by matrix interpolation. at-Automatisierungstechnik 58(8):475–484CrossRefGoogle Scholar
  22. 22.
    Salimbahrami SB (2005) Structure preserving order reduction of large scale second order models. Dissertation, Technische Universität MünchenGoogle Scholar
  23. 23.
    Stavropoulou E, Hojjat M, Bletzinger KU (2014) In-plane mesh regularization for node-based shape optimization problems. Comput Methods Appl Mech Eng 275:39–54MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Teuffel P (2004) Entwerfen Adaptiver Strukturen. Dissertation, Universität Stuttgart (in German) Google Scholar
  25. 25.
    Walker N, Fröhlich B, Eberhard P (2018) Model order reduction for parameter dependent substructured systems using krylov subspaces. In: Proceedings of the 9th Vienna conference on mathematical modelling (MATHMOD), Vienna, AustriaGoogle Scholar
  26. 26.
    Yoo EJ (2010) Parametric model order reduction for structural analysis and control. Dissertation, Technische Universität MünchenGoogle Scholar
  27. 27.
    Zienkiewicz OC (1971) The finite element method. McGraw-Hill, LondonzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Engineering and Computational MechanicsUniversity of StuttgartStuttgartGermany
  2. 2.Institute for Structural MechanicsUniversity of StuttgartStuttgartGermany

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