Abstract
Large-scale natural frequency-related topology optimization problems pose a great challenge due to the high computational cost required by the iterative computation of the system eigenvalues and their sensitivities in each design iteration. This paper presents a new framework based on a multi-step relay strategy in conjunction with the idea of successive iteration of analysis and design (SIAD), to find high-quality solutions at affordable computational costs. The method starts from a relatively coarse finite element discretization, and then use gradually refined meshes to improve the design resolution. For a given level of mesh resolution, the method interleaves the eigenvalue solution routine with the optimization iterations to achieves sequential approximations of the eigenpairs along with the structural design evolution, thus avoiding the time-consuming eigenpair analysis in each optimization iteration. By sequentially solving the optimization problem and projecting intermediate designs and the corresponding approximate eigenmodes from a coarser mesh onto finer meshes, the proposed multi-step relay method can further substantially alleviate the computational burden and generate high-resolution boundaries in the final design. Numerical examples show that this method can be used to solve natural frequency maximization topology optimization problems with millions of degrees of freedom on a desktop workstation, and is much more efficient than the conventional double-loop method.
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References
Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidiscip Optim 48:1031–1055
Zargham S, Ward TA, Ramli R, Badruddin IA (2016) Topology optimization: a review for structural designs under vibration problems. Struct Multidiscip Optim 53:1157–1177
Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69:635–654
Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192:227–246
Allaire G, Jouve F, Toader A-M (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194:363–393
Huang X, Xie Y-M (2010) A further review of ESO type methods for topology optimization. Struct Multidiscip Optim 41:671–683
Huang X, Zuo ZH, Xie Y (2010) Evolutionary topological optimization of vibrating continuum structures for natural frequencies. Comput Struct 88:357–364
Vicente WM, Zuo ZH, Pavanello R, Calixto TKL, Picelli R, Xie YM (2016) Concurrent topology optimization for minimizing frequency responses of two-level hierarchical structures. Comput Methods Appl Mech Eng 301:116–136
Sanders C, Norato J, Walsh T, Aquino W (2020) An error-in-constitutive equations strategy for topology optimization for frequency-domain dynamics. Comput Methods Appl Mech Eng 372:113330
Liu H, Zhang WH, Gao T (2015) A comparative study of dynamic analysis methods for structural topology optimization under harmonic force excitations. Struct Multidiscip Optim 51:1321–1333
Liang X, Du JB (2019) Concurrent multi-scale and multi-material topological optimizationof vibro-acoustic structures. Comput Methods Appl Mech Eng 349:117–148
Díaz AR, Kikuchi N (1992) Solutions to shape and topology eigenvalue optimization problems using a homogenization method. Int J Numer Methods Eng 35:1487–1502
Ma ZD, Cheng HC, Kikuchi N (1994) Structural design for obtaining desired eigenfrequencies by using the topology and shape optimization method. Comput Syst Eng 5:77–89
Krog LA, Olhoff N (1999) Optimum topology and reinforcement design of disk and plate structures with multiple stiffness and eigenfrequency objectives. Comput Struct 72:535–563
Jensen JS, Pedersen NL (2006) On maximal eigenfrequency separation in two-material structures: the 1D and 2D scalar cases. J Sound Vib 289:967–986
Pedersen NL (2000) Maximization of eigenvalues using topology optimization. Struct Multidiscip Optim 20:2–11
Van Keulen F, Haftka R, Kim N (2005) Review of options for structural design sensitivity analysis. Part 1: linear systems. Comput Methods Appl Mech Eng 194:3213–3243
Du J, Olhoff N (2007) Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps. Struct Multidiscip Optim 34:91–110
Zhou P, Du J, Lu Z (2017) Topology optimization of freely vibrating continuum structures based on nonsmooth optimization. Struct Multidiscip Optim 56(3):603–618
Yoon GH, Donoso A, Bellido JC, Ruiz D (2020) Highly efficient general method for sensitivity analysis of eigenvectors with repeated eigenvalues without passing through adjacent eigenvectors. Int J Numer Methods Eng 121:4473–4492
Li Z, Shi T, Xia Q (2017) Eliminate localized eigenmodes in level set based topology optimization for the maximization of the first eigenfrequency of vibration. Adv Eng Softw 107:59–70
Yoon GH (2010) Maximizing the fundamental eigenfrequency of geometrically nonlinear structures by topology optimization based on element connectivity parameterization. Comput Struct 88:120–133
Gao W, Wang F, Sigmund O (2020) Systematic design of high-Q prestressed micro membrane resonators. Comput Methods Appl Mech Eng 361:112692
Wang X, Zhang P, Ludwick S, Belski E, To AC (2018) Natural frequency optimization of 3D printed variable-density honeycomb structure via a homogenization-based approach. Addit Manuf 20:189–198
He J, Kang Z (2018) Achieving directional propagation of elastic waves via topology optimization. Ultrasonics 82:1–10
Men H, Lee KY, Freund RM, Peraire J, Johnson SG (2014) Robust topology optimization of three-dimensional photonic-crystal band-gap structures. Opt Express 22:22632–22648
Li Y, Huang X, Meng F, Zhou S (2016) Evolutionary topological design for phononic band gap crystals. Struct Multidiscip Optim 54:595–617
Takezawa A, Yamamoto T, Zhang X, Yamakawa K, Nakano S, Kitamura M (2019) An objective function for the topology optimization of sound-absorbing materials. J Sound Vib 443:804–819
Liu T, Zhu JH, Zhang WH, Zhao H, Kong J, Gao T (2019) Integrated layout and topology optimization design of multi-component systems under harmonic base acceleration excitation. Struct Multidiscip Optim 59:1053–1073
Plocher J, Panesar A (2019) Review on design and structural optimisation in additive manufacturing: towards next-generation lightweight structures. Mater Des 183:108164
Aage N, Lazarov BS (2013) Parallel framework for topology optimization using the method of moving asymptotes. Struct Multidiscip Optim 47:493–505
Bathe KJ (2013) The subspace iteration method—revisited. Comput Struct 126:177–183
Kim TS, Kim JE, Kim YY (2004) Parallelized structural topology optimization for eigenvalue problems. Int J Solids Struct 41:2623–2641
Andreassen E, Ferrari F, Sigmund O, Diaz AR (2018) Frequency response as a surrogate eigenvalue problem in topology optimization. Int J Numer Methods Eng 113:1214–1229
Ferrari F, Lazarov BS, Sigmund O (2018) Eigenvalue topology optimization via efficient multilevel solution of the frequency response. Int J Numer Methods Eng 115:872–892
Ferrari F, Sigmund O (2020) Towards solving large-scale topology optimization problems with buckling constraints at the cost of linear analyses. Comput Methods Appl Mech Eng 363:112911
Kang Z, He J, Shi L, Miao Z (2020) A method using successive iteration of analysis and design for large-scale topology optimization considering eigenfrequencies. Comput Methods Appl Mech Eng 362:112847
Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50:2143–2158
Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190:3443–3459
Stolpe M, Svanberg K (2001) An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidiscip Optim 22:116–124
Aage N, Andreassen E, Lazarov BS (2015) Topology optimization using PETSc: an easy-to-use, fully parallel, open source topology optimization framework. Struct Multidiscip Optim 51:565–572
Squillacote AH, Ahrens J, Law C, Geveci B, Moreland K, King B (2007) The paraview guide. Kitware Inc, Clifton Park
Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24:359–373
Schaefer S, McPhail T, Warren J (2006) Image deformation using moving least squares. ACM Trans Gr 25:533–540
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The support of the National Natural Science Foundation of China (11872140) is gratefully acknowledged.
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Shi, L., Li, J., Liu, P. et al. A multi-step relay implementation of the successive iteration of analysis and design method for large-scale natural frequency-related topology optimization. Comput Mech 73, 403–418 (2024). https://doi.org/10.1007/s00466-023-02372-1
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DOI: https://doi.org/10.1007/s00466-023-02372-1