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A Shape Optimization Algorithm for Interface Identification Allowing Topological Changes

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Abstract

In this work, we investigate a combination of classical optimization techniques from optimal control and a rounding strategy based on shape optimization for interface identification for problems constrained by partial differential equations. The goal is to identify the location of pollution sources in a fluid flow represented by a control that is either active or inactive. We use a relaxation of the binary problem on a coarse grid as initial guess for the shape optimization with higher resolution. The result is a computationally cheap method, where large shape deformations do not have to be performed. We demonstrate that our algorithm is, moreover, able to change the topology of the initial guess.

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References

  1. Cheney, M., Isaacson, D., Newell, J.: Electrical impedance tomography. SIAM Rev. 41(1), 85–101 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Hintermüller, M., Laurain, A.: Electrical impedance tomography: from topology to shape. Control Cybern. 37(4), 913–933 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Allaire, G., Jouve, F., Toader, A.: Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194(1), 363–393 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Schmidt, S., Wadbro, E., Berggren, M.: Large-scale three-dimensional acoustic horn optimization. SIAM J. Sci. Comput. 38(6), B917–B940 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Mohammadi, B., Pironneau, O.: Applied Shape Optimization for Fluids. Oxford University Press, Oxford (2001)

    MATH  Google Scholar 

  6. Jameson, A.: Aerodynamic Shape Optimization Using the Adjoint Method. Lectures at the Von Karman Institute, Brussels (2003)

    Google Scholar 

  7. Giles, M.B., Pierce, N.A.: An introduction to the adjoint approach to design. Flow Turbul. Combust. 65(3–4), 393–415 (2000)

    Article  MATH  Google Scholar 

  8. Allaire, G.: Shape Optimization by the Homogenization Method, vol. 146. Springer, Berlin (2012)

    MATH  Google Scholar 

  9. Gangl, P., Langer, U., Laurain, A., Meftahi, H., Sturm, K.: Shape optimization of an electric motor subject to nonlinear magnetostatics. SIAM J. Sci. Comput. 37(6), B1002–B1025 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Schulz, V., Siebenborn, M.: Computational comparison of surface metrics for PDE constrained shape optimization. Comput. Methods Appl. Math. 16, 485–496 (2016). https://doi.org/10.1515/cmam-2016-0009

    Article  MathSciNet  MATH  Google Scholar 

  11. Schulz, V., Siebenborn, M., Welker, K.: Efficient PDE constrained shape optimization based on Steklov–Poincaré-type metrics. SIAM J. Optim. 26(4), 2800–2819 (2016). https://doi.org/10.1137/15M1029369

    Article  MathSciNet  MATH  Google Scholar 

  12. Laurain, A., Sturm, K.: Distributed shape derivative via averaged adjoint method and applications. ESAIM Math. Model. Numer. Anal. 50(4), 1241–1267 (2016). https://doi.org/10.1051/m2an/2015075

    Article  MathSciNet  MATH  Google Scholar 

  13. Hante, F.M., Sager, S.: Relaxation methods for mixed-integer optimal control of partial differential equations. Comput. Optim. Appl. 55(1), 197–225 (2013). https://doi.org/10.1007/s10589-012-9518-3

    Article  MathSciNet  MATH  Google Scholar 

  14. Tröltzsch, F.: Optimal Control of Partial Differential Equations. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2010). https://doi.org/10.1090/gsm/112

  15. Hintermüller, M., Ito, K., Kunisch, K.: The primal–dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13(3), 865–888 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Weiser, M., Gänzler, T., Schiela, A.: A control reduced primal interior point method for a class of control constrained optimal control problems. Comput. Optim. Appl. 41(1), 127–145 (2008). https://doi.org/10.1007/s10589-007-9088-y

    Article  MathSciNet  MATH  Google Scholar 

  17. Ulbrich, M.: Semismooth Newton methods for operator equations in function spaces. SIAM J. Optim. 13(3), 805–841 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Herzog, R., Kunisch, K.: Algorithms for PDE-constrained optimization. GAMM-Mitteilungen 33(2), 163–176 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bergounioux, M., Kunisch, K.: Augmented Lagrangian techniques for elliptic state constrained optimal control problems. SIAM J. Control Optim. 35(5), 1524–1543 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sokolowski, J., Zolesio, J.-P.: Introduction to Shape Optimization. Springer Series in Computational Mathematics, vol. 16, pp. 5–12. Springer, Berlin, Heidelberg (1992)

  21. Delfour, M., Zolésio, J.P.: Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization. Advances in Design and Control, vol. 22, 2nd edn. SIAM, Philadelphia (2001)

    MATH  Google Scholar 

  22. Correa, R., Seeger, A.: Directional derivative of a minmax function. Nonlinear Anal. Theory Methods Appl. 9(1), 13–22 (1985)

    Article  MATH  Google Scholar 

  23. Haslinger, J., Mäkinen, R.: Introduction to Shape Optimization: Theory, Approximation, and Computation. Advances in Design and Control, vol. 7. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  24. Welker, K.: Efficient PDE constrained shape optimization in shape spaces. Ph.D. Thesis, Universität Trier (2016)

  25. Evans, L.: Partial Differential Equations. American Mathematical Society, Providence (1993)

    MATH  Google Scholar 

  26. Nittka, R.: Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains. J. Differ. Equ. 251(4–5), 860–880 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46(1), 131–150 (1999)

    Article  MATH  Google Scholar 

  28. Geuzaine, C., Remacle, J.F.: Gmsh: a 3-d finite element mesh generator with built-in pre-and post-processing facilities. Int. J. Numer. Methods Eng. 79(11), 1309–1331 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work has been partly supported by the German Research Foundation (DFG) within the priority program SPP 1648 “Software for Exascale Computing” under Contract No. Schu804/12-1 and the research training group 2126 “Algorithmic Optimization.”

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  1. Universität Hamburg, Bundesstraße 55, 20146, Hamburg, Germany

    Martin Siebenborn

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  1. Martin Siebenborn
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Correspondence to Martin Siebenborn.

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Communicated by Zenon Mróz.

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Siebenborn, M. A Shape Optimization Algorithm for Interface Identification Allowing Topological Changes. J Optim Theory Appl 177, 306–328 (2018). https://doi.org/10.1007/s10957-018-1279-4

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  • DOI: https://doi.org/10.1007/s10957-018-1279-4

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