Abstract
Topology optimization of eigenfrequencies has significant applications in science, engineering, and industry. Eigenvalue problems as constraints of optimization with partial differential equations are solved repeatedly during optimization and design process. The nonlinearity of the eigenvalue problem leads to expensive numerical solvers and thus requires huge computational costs for the whole optimization process. In this paper, we propose a simple yet efficient linearization approach and use a phase field method for topology optimization of eigenvalue problems with applications in two models: vibrating structures and photonic crystals. More specifically, the eigenvalue problem is replaced by a linear source problem every few optimization steps for saving computational costs. Numerical evidence suggests first-order accuracy of approximate eigenvalues and eigenfunctions with respect to the time step and mesh size. Numerical examples are presented to illustrate the effectiveness and efficiency of the algorithms.
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References
Allaire G (2005) A level-set method for vibration and multiple loads structural optimization. Comput Methods Appl Mech Eng 194(30–33):3269–3290. https://doi.org/10.1016/j.cma.2004.12.018
Allaire G, Aubry S, Jouve F (2001) Eigenfrequency optimization in optimal design. Comput Methods Appl Mech Eng 190(28):3565–3579. https://doi.org/10.1016/s0045-7825(00)00284-x
Andkjær J, Sigmund O (2011) Topology optimized low-contrast all-dielectric optical cloak. Appl Phys Lett 98(2):021,112. https://doi.org/10.1063/1.3540687
Bendsøe MP, Sigmund O (2004) Topology optimization. Springer, Berlin. https://doi.org/10.1007/978-3-662-05086-6
Brenner SC, Scott LR (2008) The mathematical theory of finite element methods, texts in applied mathematics, vol 15. Springer, New York. https://doi.org/10.1007/978-0-387-75934-0
Chanillo S, Grieser D, Imai M, Kurata K, Ohnishi I (2000) Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes. Commun Math Phys 214(2):315–337. https://doi.org/10.1007/pl00005534
Chen W, Chou CS, Kao CY (2016) Minimizing eigenvalues for inhomogeneous rods and plates. J Sci Comput 69(3):983–1013. 10/gr55sm
Cheng X, Yang J (2013) Maximizing band gaps in two-dimensional photonic crystals in square lattices. J Opt Soc Am A 30(11):2314. 10/gr55sn
Cox SJ (1991) The two phase drum with the deepest bass note. Jpn J Indust Appl Math 8(3):345–355. https://doi.org/10.1007/bf03167141
de Gournay F (2006) Velocity extension for the level-set method and multiple eigenvalues in shape optimization. SIAM J Control Optim 45(1):343–367. https://doi.org/10.1137/050624108
Dobson DC, Cox SJ (1999) Maximizing band gaps in two-dimensional photonic crystals. SIAM J Appl Math 59(6):2108–2120. https://doi.org/10.1137/s0036139998338455
Du J, Olhoff N (2007) Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps. Struct Multidisc Optim 34(6):545–545. https://doi.org/10.1007/s00158-007-0167-6
Ferrari F, Sigmund O (2020) Towards solving large-scale topology optimization problems with buckling constraints at the cost of linear analyses. Comput Methods in Appl Mech Eng 363:112911
Guo X, Zhang W, Zhong W (2014) Doing topology optimization explicitly and geometrically—a new moving morphable components based framework. J Appl Mech 81(8):081,009. https://doi.org/10.1115/1.4027609
Haug EJ, Rousselet B (1980) Design sensitivity analysis in structural mechanics. II. Eigenvalue variations. J Struct Mech 8(2):161–186. https://doi.org/10.1080/03601218008907358
He L, Kao CY, Osher S (2007) Incorporating topological derivatives into shape derivatives based level set methods. J Comput Phys 225(1):891–909. https://doi.org/10.1016/j.jcp.2007.01.003
Hecht F (2012) New development in freefem++. J Numer Math. https://doi.org/10.1515/jnum-2012-0013
Joannopoulos JD, Johnson SG, Winn JN, Meade RD (2011) Photonic crystals: molding the flow of light. In: Photonic crystals. Princeton University Press, Princeton
Kao CY, Osher S, Yablonovitch E (2005) Maximizing band gaps in two-dimensional photonic crystals by using level set methods. Appl Phys B 81(2–3):235–244. https://doi.org/10.1007/s00340-005-1877-3
Leader M, Chin TW, Kennedy G (2019) High-resolution topology optimization with stress and natural frequency constraints. AIAA J 57:1–17. https://doi.org/10.2514/1.J057777
Ma ZD, Kikuchi N, Cheng HC (1995) Topological design for vibrating structures. Comput Methods Appl Mech Eng 121(1–4):259–280. https://doi.org/10.1016/0045-7825(94)00714-x
Novotny AA, Sokołowski J (2013) Topological derivatives in shape optimization. Springer, Berlin, Interaction of mechanics and mathematics. https://doi.org/10.1007/978-3-642-35245-4
Osher SJ, Santosa F (2001) Level set methods for optimization problems involving geometry and constraints. J Comput Phys 171(1):272–288. https://doi.org/10.1006/jcph.2001.6789
Qian M, Zhu S (2022) A level set method for Laplacian eigenvalue optimization subject to geometric constraints. Comput Optim Appl 82(2):499–524. https://doi.org/10.1007/s10589-022-00371-1
Qian M, Hu X, Zhu S (2022) A phase field method based on multi-level correction for eigenvalue topology optimization. Comput Methods Appl Mech Eng 401(115):646. https://doi.org/10.1016/j.cma.2022.115646
Sigmund O, Hougaard K (2008) Geometric Properties of optimal photonic crystals. Phys Rev Lett 100(15):153,904. https://doi.org/10.1103/physrevlett.100.153904
Sokolowski J, Zolesio JP (1992) Introduction to shape optimization, Springer series in computational mathematics, vol 16. Springer, Berlin Heidelberg. https://doi.org/10.1007/978-3-642-58106-9
Sun J, Tian Q, Hu H, Pedersen NL (2019) Topology optimization for eigenfrequencies of a rotating thin plate via moving morphable components. J Sound Vib 448:83–107. https://doi.org/10.1016/j.jsv.2019.01.054
Takezawa A, Kitamura M (2014) Phase field method to optimize dielectric devices for electromagnetic wave propagation. J Comput Phys 257:216–240. https://doi.org/10.1016/j.jcp.2013.09.051
Takezawa A, Nishiwaki S, Kitamura M (2010) Shape and topology optimization based on the phase field method and sensitivity analysis. J Comput Phys 229(7):2697–2718. https://doi.org/10.1016/j.jcp.2009.12.017
Warren JA, Kobayashi R, Lobkovsky AE, Craig Carter W (2003) Extending phase field models of solidification to polycrystalline materials. Acta Mater 51(20):6035–6058. https://doi.org/10.1016/s1359-6454(03)00388-4
Wu S, Hu X, Zhu S (2018) A multi-mesh finite element method for phase-field based photonic band structure optimization. J Comput Phys 357:324–337. https://doi.org/10.1016/j.jcp.2017.12.031
Xia Q, Shi T, Wang MY (2011) A level set based shape and topology optimization method for maximizing the simple or repeated first eigenvalue of structure vibration. Struct Multidisc Optim 43(4):473–485. https://doi.org/10.1007/s00158-010-0595-6
Xu J, Zhou A (1999) A two-grid discretization scheme for eigenvalue problems. Math Comp 70(233):17–25. https://doi.org/10.1090/s0025-5718-99-01180-1
Zargham S, Ward TA, Ramli R, Badruddin IA (2016) Topology optimization: a review for structural designs under vibration problems. Struct Multidisc Optim 53(6):1157–1177. https://doi.org/10.1007/s00158-015-1370-5
Zhang Z, Chen W (2018) An approach for maximizing the smallest eigenfrequency of structure vibration based on piecewise constant level set method. J Comput Phys 361:377–390. https://doi.org/10.1016/j.jcp.2018.01.050
Zhang W, Yuan J, Zhang J, Guo X (2016) A new topology optimization approach based on Moving Morphable Components (MMC) and the ersatz material model. Struct Multidisc Optim 53(6):1243–1260. https://doi.org/10.1007/s00158-015-1372-3
Zhang J, Zhu S, Liu C, Shen X (2021) A two-grid binary level set method for eigenvalue optimization. J Sci Comput 89(3):57. https://doi.org/10.1007/s10915-021-01662-1
Zhu S, Wu Q, Liu C (2010) Variational piecewise constant level set methods for shape optimization of a two-density drum. J Comput Phys 229(13):5062–5089. https://doi.org/10.1016/j.jcp.2010.03.026
Funding
This work was supported in part by the Key Technologies Research and Development Program (Grant No. 2022YFA1004402), the National Natural Science Foundation of China (Grant No. 12071149), the Science and Technology Commission of Shanghai Municipality (No. 22ZR1421900 and No. 22DZ2229014), and the Natural Science Foundation of Chongqing, China (Grant No. CSTB2023NSCQ-MSX1079).
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Hu, X., Qian, M. & Zhu, S. Accelerating a phase field method by linearization for eigenfrequency topology optimization. Struct Multidisc Optim 66, 242 (2023). https://doi.org/10.1007/s00158-023-03692-9
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DOI: https://doi.org/10.1007/s00158-023-03692-9