Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

A level set method for Laplacian eigenvalue optimization subject to geometric constraints

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

We consider to solve numerically the shape optimization problems of Dirichlet Laplace eigenvalues subject to volume and perimeter constraints. By combining a level set method with the relaxation approach, the algorithm can perform shape and topological changes on a fixed grid. We use the volume expressions of Eulerian derivatives in shape gradient descent algorithms. Finite element methods are used for discretizations. Two and three-dimensional numerical examples are presented to illustrate the effectiveness of the algorithms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

Data availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. Buttazzo, G., Belhachmi, Z., Bucur, D., Sac-Epee, J.M.: Shape optimization problems for eigenvalues of elliptic operators. ZAMM Z. Angew. Math. Mech. 86(3), 171–184 (2006)

    Article  MathSciNet  Google Scholar 

  2. Dobson, D.C., Cox, S.J.: Maximizing band gaps in two-dimensional photonic crystals. SIAM J. Appl. Math. 59, 2108–2120 (1999)

    Article  MathSciNet  Google Scholar 

  3. Pedersen, N.L.: Maximization of eigenvalues using topology optimization. Struct. Multidiscip. Optim. 20, 2–11 (2000)

    Article  Google Scholar 

  4. Bucur, D., Buttazzo, G.: Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations 65, Birkhuser (2005)

  5. Lamboley, J., Laurain, A., Nadin, G., Privat, Y.: Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions. Calc. Var. Partial Differ. Equations 55(6) (2016)

  6. Antunes, P.R.S., Freitas, P.: Numerical optimization of low eigenvalues of the Dirichlet and Neumann Laplacians. J. Optim. Theory Appl. 154(1), 235–257 (2012)

    Article  MathSciNet  Google Scholar 

  7. Sokołowski, J., Zolesio, J.-P.: Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer, Heidelberg (1992)

    Book  Google Scholar 

  8. Allaire, G., Aubry, S., Jouve, F.: Eigenfrequency optimization in optimal design. Comput. Methods Appl. Mech. Eng. 190(28), 3565–3579 (2001)

    Article  MathSciNet  Google Scholar 

  9. Du, Q., Lin, F.: Numerical approximations of a norm-preserving gradient flow and applications to an optimal partition problem. Nonlinearity 22(1), 67–83 (2009)

    Article  MathSciNet  Google Scholar 

  10. Allaire, G., Jouve, F.: A level-set method for vibration and multiple loads structural optimization. Comput. Methods Appl. Mech. Eng. 194(30–33), 3269–3290 (2005)

    Article  MathSciNet  Google Scholar 

  11. Osher, S., Santosa, F.: Level set methods for optimization problems involving geometry and constraints I. frequencies of a two-density inhomogeneous drum. J. Comput. Phys. 171(1), 272–288 (2001)

  12. Osher, S., Sethian, J.A: Front propagation with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12-49 (1988)

  13. Laurain, A., Sturm, K.: Distributed shape derivative via averaged adjoint method and applications. ESAIM Math. Model. Numer. Anal. 50(4), 1241–1267 (2016)

    Article  MathSciNet  Google Scholar 

  14. Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators. Birkhäuser Verlag (2006)

  15. Laurain, A.: Distributed and boundary expressions of first and second order shape derivatives in nonsmooth domains. J. Math. Pures Appl. 134(9), 328–368 (2020)

    Article  MathSciNet  Google Scholar 

  16. Hiptmair, R., Paganini, A., Sargheini, S.: Comparison of approximate shape gradients. BIT Numer. Math. 55, 459–485 (2015)

    Article  MathSciNet  Google Scholar 

  17. Zhu, S.: Effective shape optimization of Laplace eigenvalue problems using domain expressions of Eulerian derivatives. J. Optim. Theory and Appl. 176, 17–34 (2018)

    Article  MathSciNet  Google Scholar 

  18. Zhu, S., Hu, X., Liao, Q.: Convergence analysis of Galerkin finite element approximations to shape gradients in eigenvalue optimization. BIT Numer. Math. 60, 853–878 (2020)

    Article  MathSciNet  Google Scholar 

  19. Zhu, S., Hu, X., Wu, Q.: On accuracy of approximate boundary and distributed H1 shape gradient flows for eigenvalue optimization. J. Comput. Appl. Math. 365 (2020)

  20. Oudet, E.: Numerical minimization of eigenmodes of a membrane with respect to the domain. ESAIM Control Optim. Calc. Var. 10(3), 315–330 (2004)

    Article  MathSciNet  Google Scholar 

  21. Bogosel, B., Oudet, E.: Qualitative and numerical analysis of a spectral problem with perimeter constraint. SIAM J. Control Optim. 54(1), 317–340 (2016)

    Article  MathSciNet  Google Scholar 

  22. Zhu, S., Hu, X., Wu, Q.: A level set method for shape optimization in semilinear elliptic problems. J. Comput. Phys. 355(15), 104–120 (2018)

    Article  MathSciNet  Google Scholar 

  23. Laurain, A.: A level set-based structural optimization code using FEniCS. Struct. Multidiscip. Optim. 58, 1311–1334 (2018)

    Article  MathSciNet  Google Scholar 

  24. Bucur, D.: Minimization of the \(k\)-th eigenvalue of the Dirichlet Laplacian. Arch. Ration. Mech. Anal. 206(3), 1073–1083 (2012)

    Article  MathSciNet  Google Scholar 

  25. Mazzoleni, D., Pratelli, A.: Existence of minimizers for spectral problems. J. Math. Pures Appl. 100(3), 433–453 (2013)

    Article  MathSciNet  Google Scholar 

  26. De Philippis, G., Velichkov, B.: Existence and regularity of minimizers for some spectral functionals with perimeter constraint. Appl. Math. Optim. 69, 199–231 (2014)

    Article  MathSciNet  Google Scholar 

  27. de Gournay, F.: Velocity extension for the level-set method and multiple eigenvalues in shape optimization. SIAM J. Control Optim. 45(1), 343–367 (2006)

    Article  MathSciNet  Google Scholar 

  28. Hecht, F.: New development in FreeFem++. J. Numer. Math. 20(3–4), 251–265 (2012)

    MathSciNet  MATH  Google Scholar 

  29. Antunes, P.R.S., Oudet, E.: Numerical Results for Extremal Problem for Eigenvalues of the Laplacian. Shape Optimization and Spectral Theory, pp. 398–411. De Gruyter Open, Warsaw (2017)

  30. Bogosel, B.: Optimisation de formes et problèmes spectraux. PhD thesis, Grenoble Alpes (2015)

Download references

Acknowledgements

The work was supported by the National Natural Science Foundation of China (grant 12071149) and the Science and Technology Commission of Shanghai Municipality (Grant Numbers 19ZR1414100, 18dz2271000, and 21JC1402500). The authors thank Xindi Hu in East China Normal University for many helpful comments in numerical experiments.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, East China Normal University, Shanghai, 200241, China

    Meizhi Qian & Shengfeng Zhu

Authors
  1. Meizhi Qian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shengfeng Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shengfeng Zhu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qian, M., Zhu, S. A level set method for Laplacian eigenvalue optimization subject to geometric constraints. Comput Optim Appl 82, 499–524 (2022). https://doi.org/10.1007/s10589-022-00371-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-022-00371-1

Keywords

Search

Navigation