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Computational Comparison of Surface Metrics for PDE Constrained Shape Optimization

  • Volker Schulz EMAIL logo and Martin Siebenborn
Published/Copyright: February 26, 2016

Abstract

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.

MSC 2010: 49Q10; 65K10; 65N30

Award Identifier / Grant number: Schu804/12-1

Funding statement: 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.

References

[1] Absil P., Mahony R. and Sepulchre R., Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, 2008. 10.1515/9781400830244Search in 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. 10.1137/1.9781611972054Search in Google Scholar

[3] Conn A. R., Gould N. I. M. and Toint P. L., Lancelot, Springer, Berlin, 1992. 10.1007/978-3-662-12211-2Search in Google Scholar

[4] Delfour M. C. and Zolésio J.-P., Shapes and Geometries: Analysis, Differential Calculus, and Optimization, Adv. Des. Control, SIAM, Philadelphia, 2001. Search in 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. 10.1137/05062679XSearch in Google 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. 10.1137/15100477XSearch in Google Scholar

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

[8] Michor P. and Mumford D., Riemannian geometries on spaces of plane curves, J. Eur. Math. Soc. (JEMS) 8 (2006), 1–48. 10.4171/JEMS/37Search in Google Scholar

[9] Mohammadi B. and Pironneau O., Applied Shape Optimization for Fluids, Num. Math. Sci. Comput., Clarendon Press, Oxford, 2001. Search in 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. 10.1007/s00791-015-0248-9Search in Google Scholar

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

[12] Pironneau O., On optimum profiles in stokes flow, J. Fluid Mech. 59 (1973), no. 1, 117–128. 10.1017/S002211207300145XSearch in 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. 10.1137/11082885XSearch in 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. 10.2514/6.2011-3718Search in Google Scholar

[15] Schulz V., A Riemannian view on shape optimization, Found. Comput. Math. 14 (2014), 483–501. 10.1007/s10208-014-9200-5Search in 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. Search in Google Scholar

[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. 10.1137/140985883Search in Google 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. 10.1007/978-3-319-17563-8_10Search in Google Scholar

[19] Sokolowski J. and Zolésio J., An introduction to shape optimization, Springer Ser. Comput. Math. 16, Springer, Berlin, 1992. 10.1007/978-3-642-58106-9Search in 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. 10.1002/nme.2132Search in Google Scholar

Received: 2015-12-10
Revised: 2016-2-5
Accepted: 2016-2-10
Published Online: 2016-2-26
Published in Print: 2016-7-1

© 2016 by De Gruyter

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