Adaptive computation of elliptic eigenvalue topology optimization with a phase-field approach
-
Jing Li
and
Shengfeng Zhu
Abstract
In this paper, we study adaptive approximations of an elliptic eigenvalue optimization problem in a phase-field setting by a conforming finite element method. An adaptive algorithm is proposed and implemented in several two-dimensional numerical examples for illustration of efficiency and accuracy. Theoretical findings consist in the vanishing limit of a subsequence of estimators and the convergence of the relevant subsequence of adaptively-generated solutions to a solution to the continuous optimality system.
-
Research ethics: Not applicable.
-
Informed consent: Not applicable.
-
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
-
Use of Large Language Models, AI and Machine Learning Tools: None declared.
-
Conflict of interest: The authors state no conflict of interest.
-
Research funding: We acknowledge the support by National Key Basic Research Program under grant 2022YFA1004402, National Natural Science Foundation of China under grants 12250013, 12261160361, 12271367, and 12471377, Science and Technology Commission of Shanghai Municipality through projects 20JC1413800, 22DZ2229014, 22ZR1421900, and 22ZR1445400, and General Research Fund (projects KF202318 and KF202468) from Shanghai Normal University.
-
Data availability: Not applicable.
References
[1] M. P. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods and Applications, Berlin, Springer, 2003.10.1007/978-3-662-05086-6Search in Google Scholar
[2] G. Allaire, S. Aubry, and F. Jouve, “Eigenfrequency optimization in optimal design,” Comput. Methods Appl. Mech. Eng., vol. 190, no. 28, pp. 3565–3579, 2001. https://doi.org/10.1016/s0045-7825(00)00284-x.Search in Google Scholar
[3] G. Allaire and F. Jouve, “A level-set method for vibration and multiple loads structural optimization,” Comput. Methods Appl. Mech. Eng., vol. 194, nos. 30–33, pp. 3269–3290, 2005. https://doi.org/10.1016/j.cma.2004.12.018.Search in Google Scholar
[4] G. Buttazzo and G. Dal Maso, “An existence result for a class of shape optimization problems,” Arch. Ration. Mech. Anal., vol. 122, no. 2, pp. 183–195, 1993. https://doi.org/10.1007/bf00378167.Search in Google Scholar
[5] H. Garcke, P. Hüttl, C. Kahle, P. Knopf, and T. Laux, “Phase-field methods for spectral shape and topology optimization,” ESAIM: COCV, vol. 29, no. 10, p. 57, 2023. https://doi.org/10.1051/cocv/2022090.Search in Google Scholar
[6] H. Garcke, P. Hüttl, and P. Knopf, “Shape and topology optimization including the eigenvalues of an elastic structure: a multi-phase-field approach,” Adv. Nonlinear Anal., vol. 11, no. 1, pp. 159–197, 2022. https://doi.org/10.1515/anona-2020-0183.Search in Google Scholar
[7] A. Henrot, “Extremum problems for eigenvalues of elliptic operators,” in Frontiers in Mathematics, Basel, Birkhäuser Verlag, 2006.10.1007/3-7643-7706-2Search in Google Scholar
[8] A. Henrot and M. Pierre, “Shape variation and optimization, a geometrical analysis,” in EMS Tracts in Mathematics, vol. 28, Zürich, European Mathematical Society (EMS), 2018.10.4171/178Search in Google Scholar
[9] N. L. Pedersen, “Maximization of eigenvalues using topology optimization,” Struct. Multidiscip. Optim., vol. 20, no. 1, pp. 2–11, 2000. https://doi.org/10.1007/s001580050130.Search in Google Scholar
[10] M. Qian, X. Hu, and S. Zhu, “A phase field method based on multi-level correction for eigenvalue topology optimization,” Comput. Methods Appl. Mech. Eng., vol. 401, no. Part B, p. 115646, 2022. https://doi.org/10.1016/j.cma.2022.115646.Search in Google Scholar
[11] J. Zhang, S. Zhu, C. Liu, and X. Shen, “A two-grid binary level set method for eigenvalue optimization,” J. Sci. Comput., vol. 89, no. 57, p. 21, 2021. https://doi.org/10.1007/s10915-021-01662-1.Search in Google Scholar
[12] B. Jin, J. Li, Y. Xu, and S. Zhu, “An adaptive phase-field method for structural topology optimization,” J. Comput. Phys., vol. 506, p. 112932, 2024. https://doi.org/10.1016/j.jcp.2024.112932.Search in Google Scholar
[13] S. Osher and F. Santosa, “Level set methods for optimization problems involving geometry and constraints. I. Frequencies of a two-density inhomogeneous drum,” J. Comput. Phys., vol. 171, no. 1, pp. 272–288, 2001. https://doi.org/10.1006/jcph.2001.6789.Search in Google Scholar
[14] S. Chanillo, D. Grieser, M. Imai, K. Kurata, and I. Ohnishi, “Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes,” Commun. Math. Phys., vol. 214, no. 2, pp. 315–337, 2000. https://doi.org/10.1007/pl00005534.Search in Google Scholar
[15] S. J. Cox and J. R. McLaughlin, “Extremal eigenvalue problems for composite membranes, I, II,” Appl. Math. Optim., vol. 22, no. 2, pp. 153–167, 169–187, 1990. https://doi.org/10.1007/bf01447325.Search in Google Scholar
[16] K. Liang, X. Lu, and J. Z. Yang, “Finite element approximation to the extremal eigenvalue problem for inhomogenous materials,” Numer. Math., vol. 130, no. 4, pp. 741–762, 2015. https://doi.org/10.1007/s00211-014-0678-1.Search in Google Scholar
[17] S. Zhu, Q. Wu, and C. Liu, “Variational piecewise constant level set methods for shape optimization of a two-density drum,” J. Comput. Phys., vol. 229, no. 13, pp. 5062–5089, 2010. https://doi.org/10.1016/j.jcp.2010.03.026.Search in Google Scholar
[18] Z. Zhang, K. Liang, and X. Cheng, “Greedy algorithms for eigenvalue optimization problems in shape design of two-density inhomogeneous materials,” Int. J. Comput. Math., vol. 88, no. 1, pp. 183–195, 2011. https://doi.org/10.1080/00207160903365891.Search in Google Scholar
[19] D. Bucur and G. Buttazzo, “Variational methods in shape optimization problems,” in Progress in Nonlinear Differential Equations and their Applications, vol. 65, Boston, Birkhäuser, 2005.10.1007/b137163Search in Google Scholar
[20] J. Sokolowski and J.-P. Zolesio, “Introduction to shape optimization: shape sensitivity analysis,” in Springer Series in Computational Mathematics, vol. 16, Berlin, Heidelberg, Springer-Verlag, 1992.10.1007/978-3-642-58106-9Search in Google Scholar
[21] L. He, C. Y. Kao, and S. Osher, “Incorporating topological derivatives into shape derivatives based level set methods,” J. Comput. Phys., vol. 225, no. 1, pp. 891–909, 2007. https://doi.org/10.1016/j.jcp.2007.01.003.Search in Google Scholar
[22] M. Burger and R. Stainko, “Phase-field relaxation of topology optimization with local stress constraints,” SIAM J. Control Optim., vol. 45, no. 4, pp. 1447–1466, 2006. https://doi.org/10.1137/05062723x.Search in Google Scholar
[23] M. Y. Wang and S. W. Zhou, “Phase field: a variational method for structural topology optimization,” Comput. Model. Eng. Sci., vol. 6, no. 6, pp. 547–566, 2004.Search in Google Scholar
[24] B. Bourdin and A. Chambolle, “Design-dependent loads in topology optimization,” ESAIM: Control Optim. Calc. Var., vol. 9, pp. 19–48, 2003. https://doi.org/10.1051/cocv:2002070.10.1051/cocv:2002070Search in Google Scholar
[25] B. Bourdin and A. Chambolle, “The phase-field method in optimal design,” in IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials, 2006, pp. 207–215.10.1007/1-4020-4752-5_21Search in Google Scholar
[26] J. F. Blowey and C. M. Elliott, “The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy Part I: mathematical analysis,” Eur. J. Appl. Math., vol. 2, no. 3, pp. 233–279, 1991. https://doi.org/10.1017/s095679250000053x.Search in Google Scholar
[27] A. Braides, “Γ-convergence for beginners,” in Oxford Lecture Series in Mathematics and its Applications, vol. 22, Oxford, Oxford University Press, 2002.Search in Google Scholar
[28] L. Modica, “The gradient theory of phase transitions and the minimal interface criterion,” Arch. Ration. Mech. Anal., vol. 98, no. 2, pp. 123–142, 1987. https://doi.org/10.1007/bf00251230.Search in Google Scholar
[29] M. Ainsworth and J. T. Oden, “A posteriori error estimation in finite element analysis,” in Pure and Applied Mathematics, New York, Wiley-Interscience, 2000.10.1002/9781118032824Search in Google Scholar
[30] R. Verfürth, A Posteriori Error Estimation Techniques for Finite Element Methods, Oxford, Oxford University Press, 2013.10.1093/acprof:oso/9780199679423.001.0001Search in Google Scholar
[31] J. M. Cascon, C. Kreuzer, R. H. Nochetto, and K. G. Siebert, “Quasi-optimal convergence rate for an adaptive finite element method,” SIAM J. Numer. Anal., vol. 46, no. 5, pp. 2524–2550, 2008. https://doi.org/10.1137/07069047x.Search in Google Scholar
[32] G. Gantner and D. Praetorius, “Plain convergence of adaptive algorithms without exploiting reliability and efficiency,” IMA J. Numer. Anal., vol. 42, no. 2, pp. 1434–1453, 2022. https://doi.org/10.1093/imanum/drab010.Search in Google Scholar
[33] E. M. Garau and P. Morin, “Convergence and quasi-optimality of adaptive FEM for Steklov eigenvalue problems,” IMA J. Numer. Anal., vol. 31, no. 3, pp. 914–946, 2011. https://doi.org/10.1093/imanum/drp055.Search in Google Scholar
[34] P. Morin, K. G. Siebert, and A. Veeser, “A basic convergence result for conforming adaptive finite elements,” Math. Models Methods Appl. Sci., vol. 18, no. 5, pp. 707–737, 2008. https://doi.org/10.1142/s0218202508002838.Search in Google Scholar
[35] R. H. Nochetto, K. G. Siebert, and A. Veeser, “Theory of adaptive finite element methods: an introduction,” in Multiscale, Nonlinear and Adaptive Approximation, R. A. DeVore, and A. Kunoth, Eds., New York, Springer, 2009, pp. 409–542.10.1007/978-3-642-03413-8_12Search in Google Scholar
[36] K. G. Siebert, “A convergence proof for adaptive finite elements without lower bounds,” IMA J. Numer. Anal., vol. 31, no. 3, pp. 947–970, 2011. https://doi.org/10.1093/imanum/drq001.Search in Google Scholar
[37] E. M. Garau, P. Morin, and C. Zuppa, “Convergence of adaptive finite element methods for eigenvalue problems,” Math. Models Methods Appl. Sci., vol. 19, no. 05, pp. 721–747, 2009. https://doi.org/10.1142/s0218202509003590.Search in Google Scholar
[38] R. Adams and J. Fournier, Sobolev Spaces, 2nd ed. Amsterdam, Elsevier Science/Academic Press, 2003.Search in Google Scholar
[39] I. Kossaczky, “A recursive approach to local mesh refinement in two and three dimensions,” J. Comput. Appl. Math., vol. 55, no. 3, pp. 275–288, 1994. https://doi.org/10.1016/0377-0427(94)90034-5.Search in Google Scholar
[40] C. Traxler, “An algorithm for adaptive mesh refinement in n dimensions,” Computing, vol. 59, no. 2, pp. 115–137, 1997. https://doi.org/10.1007/bf02684475.Search in Google Scholar
[41] A. Takezawa, S. Nishiwaki, and M. Kitamura, “Shape and topology optimization based on the phase field method and sensitivity analysis,” J. Comput. Phys., vol. 229, no. 7, pp. 2697–2718, 2010. https://doi.org/10.1016/j.jcp.2009.12.017.Search in Google Scholar
[42] B. Jin and Y. Xu, “Adaptive reconstruction for electrical impedance tomography with a piecewise constant conductivity,” Inverse Probl., vol. 36, no. 1, p. 014003, 2019. https://doi.org/10.1088/1361-6420/ab261e.Search in Google Scholar
[43] L. R. Scott and S. Zhang, “Finite element interpolation of nonsmooth functions satisfying boundary conditions,” Math. Comput., vol. 54, no. 190, pp. 483–493, 1990. https://doi.org/10.1090/s0025-5718-1990-1011446-7.Search in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
