Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

IMPACT FACTOR 2017: 0.658

CiteScore 2017: 1.05

SCImago Journal Rank (SJR) 2017: 1.291
Source Normalized Impact per Paper (SNIP) 2017: 0.893

Mathematical Citation Quotient (MCQ) 2017: 0.76

See all formats and pricing
More options …
Volume 16, Issue 3


Computational Comparison of Surface Metrics for PDE Constrained Shape Optimization

Volker Schulz / Martin Siebenborn
Published Online: 2016-02-26 | DOI: https://doi.org/10.1515/cmam-2016-0009


We compare surface metrics for shape optimization problems with constraints, consisting mainly of partial differential equations (PDE), from a computational point of view. In particular, classical Laplace–Beltrami type metrics are compared with Steklov–Poincaré type metrics. The test problem is the minimization of energy dissipation of a body in a Stokes flow. We therefore set up a quasi-Newton method on appropriate shape manifolds together with an augmented Lagrangian framework, in order to enable a straightforward integration of geometric constraints for the shape. The comparison is focussed towards convergence behavior as well as effects on the mesh quality during shape optimization.

Keywords: PDE constrained shape optimization; optimization on Riemannian manifolds; Stokes problem

MSC 2010: 49Q10; 65K10; 65N30


  • [1]

    Absil P., Mahony R. and Sepulchre R., Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, 2008. Google Scholar

  • [2]

    Borzì A. and Schulz V., Computational Optimization of Systems Governed by Partial Differential Equations, SIAM Book Ser. Comput. Sci. Eng. 8, SIAM, Philadelphia, 2012. Google Scholar

  • [3]

    Conn A. R., Gould N. I. M. and Toint P. L., Lancelot, Springer, Berlin, 1992. Google Scholar

  • [4]

    Delfour M. C. and Zolésio J.-P., Shapes and Geometries: Analysis, Differential Calculus, and Optimization, Adv. Des. Control, SIAM, Philadelphia, 2001. Google Scholar

  • [5]

    Eppler K., Harbrecht H. and Schneider R., On convergence in elliptic shape optimization, SIAM J. Control Optim. 46 (2007), no. 1, 61–83. Web of ScienceGoogle Scholar

  • [6]

    Gangl P., Laurain A., Meftahi H. and Sturm K., Shape optimization of an electric motor subject to nonlinear magnetostatics, preprint 2015, http://arxiv.org/abs/1501.04752.

  • [7]

    Haack W., Geschoßformen kleinsten Wellenwiderstandes, Bericht der Lilienthal-Gesellschaft 136 (1941), no. 1, 14–28. Google Scholar

  • [8]

    Michor P. and Mumford D., Riemannian geometries on spaces of plane curves, J. Eur. Math. Soc. (JEMS) 8 (2006), 1–48. Google Scholar

  • [9]

    Mohammadi B. and Pironneau O., Applied Shape Optimization for Fluids, Num. Math. Sci. Comput., Clarendon Press, Oxford, 2001. Google Scholar

  • [10]

    Nägel A., Schulz V., Siebenborn M. and Wittum G., Scalable shape optimization methods for structured inverse modeling in 3D diffusive processes, Comput. Vis. Sci. 17 (2015), 79–88. Web of ScienceGoogle Scholar

  • [11]

    Paganini A., Approximative shape gradients for interface problems, technical report 2014-12, Seminar for Applied Mathematics, ETH Zürich, 2014. Google Scholar

  • [12]

    Pironneau O., On optimum profiles in stokes flow, J. Fluid Mech. 59 (1973), no. 1, 117–128. Google Scholar

  • [13]

    Ring W. and Wirth B., Optimization methods on Riemannian manifolds and their application to shape space, SIAM J. Optim. 22 (2012), 596–627. Google Scholar

  • [14]

    Schmidt S., Ilic C., Schulz V. and Gauger N., Three dimensional large scale aerodynamic shape optimization based on the shape calculus, AIAA J. 51 (2013), no. 11, 2615–2627. Web of ScienceGoogle Scholar

  • [15]

    Schulz V., A Riemannian view on shape optimization, Found. Comput. Math. 14 (2014), 483–501. Google Scholar

  • [16]

    Schulz V., Siebenborn M. and Welker K., A novel Steklov–Poincaré type metric for efficient PDE constrained optimization in shape spaces, preprint 2015, http://arxiv.org/abs/1506.02244v4.

  • [17]

    Schulz V., Siebenborn M. and Welker K., Structured inverse modeling in parabolic diffusion problems, SIAM J. Control Optim. 53 (2015), no. 6, 3319–3338. Web of ScienceGoogle Scholar

  • [18]

    Schulz V., Siebenborn M. and Welker K., Towards a Lagrange–Newton approach for PDE constrained shape optimization, New Trends in Shape Optimization, Internat. Ser. Numer. Math. 166, Birkhäuser, Cham (2015), 229–249. Google Scholar

  • [19]

    Sokolowski J. and Zolésio J., An introduction to shape optimization, Springer Ser. Comput. Math. 16, Springer, Berlin, 1992. Google Scholar

  • [20]

    Udawalpola R. and Berggren M., Optimization of an acoustic horn with respect to efficiency and directivity, Internat. J. Numer. Methods Engrg. 73 (2007), no. 11, 1571–1606. Google Scholar

About the article

Received: 2015-12-10

Revised: 2016-02-05

Accepted: 2016-02-10

Published Online: 2016-02-26

Published in Print: 2016-07-01

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: Schu804/12-1

This work has been partly supported by the Deutsche Forschungsgemeinschaft within the Priority program SPP 1648 “Software for Exascale Computing” under contract number Schu804/12-1. Furthermore, we acknowledge support by the Sino-German Science Center (grant id 1228) on the occasion of the Chinese-German Workshop on Computational and Applied Mathematics in Augsburg 2015.

Citation Information: Computational Methods in Applied Mathematics, Volume 16, Issue 3, Pages 485–496, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2016-0009.

Export Citation

© 2016 by De Gruyter.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

José A. Iglesias, Kevin Sturm, and Florian Wechsung
SIAM Journal on Scientific Computing, 2018, Volume 40, Number 6, Page A3807
Björn Führ, Volker Schulz, and Kathrin Welker
Vietnam Journal of Mathematics, 2018
Shengfeng Zhu and Zhiming Gao
Computer Methods in Applied Mechanics and Engineering, 2018
Alberto Paganini, Florian Wechsung, and Patrick E. Farrell
SIAM Journal on Scientific Computing, 2018, Volume 40, Number 4, Page A2356
Stephan Schmidt
SIAM Journal on Scientific Computing, 2018, Volume 40, Number 2, Page C210
Martin Siebenborn and Kathrin Welker
SIAM Journal on Scientific Computing, 2017, Volume 39, Number 6, Page B1156

Comments (0)

Please log in or register to comment.
Log in