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Topology optimization of vibrating structures with frequency band constraints

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Abstract

Engineering structures usually operate in some specific frequency bands. An effective way to avoid resonance is to shift the structure’s natural frequencies out of these frequency bands. However, in the optimization procedure, which frequency orders will fall into these bands are not known a priori. This makes it difficult to use the existing frequency constraint formulations, which require prescribed orders. For solving this issue, a novel formulation of the frequency band constraint based on a modified Heaviside function is proposed in this paper. The new formulation is continuous and differentiable; thus, the sensitivity of the constraint function can be derived and used in a gradient-based optimization method. Topology optimization for maximizing the structural fundamental frequency while circumventing the natural frequencies located in the working frequency bands is studied. For eliminating the frequently happened numerical problems in the natural frequency topology optimization process, including mode switching, checkerboard phenomena, and gray elements, the “bound formulation” and “robust formulation” are applied. Three numerical examples, including 2D and 3D problems, are solved by the proposed method. Frequency band gaps of the optimized results are obtained by considering the frequency band constraints, which validates the effectiveness of the developed method.

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References

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

    MathSciNet  MATH  Google Scholar 

  • Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202

    Google Scholar 

  • Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224

    MathSciNet  MATH  Google Scholar 

  • Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69:635–654

    MATH  Google Scholar 

  • Bendsøe MP, Olhoff N, Taylor JE (1983) A variational formulation for multicriteria structural optimization. J Struct Mech 11:523–544

    Google Scholar 

  • Díaaz AR, Kikuchi N (1992) Solutions to shape and topology eigenvalue optimization problems using a homogenization method. Int J Numer Methods Eng 35:1487–1502

    MathSciNet  MATH  Google Scholar 

  • 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

    MathSciNet  MATH  Google Scholar 

  • Guo X, Zhang W, Zhong W (2014) Doing topology optimization explicitly and geometrically—a new moving morphable components based framework. J Appl Mech 81:081009–081012

    Google Scholar 

  • 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

    Google Scholar 

  • Kang Z, Zhang X, Jiang S, Cheng G (2012) On topology optimization of damping layer in shell structures under harmonic excitations. Struct Multidiscip Optim 46:51–67

    MathSciNet  MATH  Google Scholar 

  • 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

    MathSciNet  MATH  Google Scholar 

  • 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

    MATH  Google Scholar 

  • Leader MK, Chin TW, Kennedy GJ (2019) High-resolution topology optimization with stress and natural frequency constraints. AIAA J 57:3562–3578

    Google Scholar 

  • Lee WS, Youn SK (2004) Topology optimization of rubber isolators considering static and dynamic behaviours. Struct Multidiscip Optim 27:284–294

    Google Scholar 

  • 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

    Google Scholar 

  • Li Q, Xu R, Liu J, Liu S, Zhang S (2019) Topology optimization design of multi-scale structures with alterable microstructural length-width ratios. Compos Struct 230:111454

    Google Scholar 

  • Liu S, Hu R, Li Q, Zhou P, Dong Z, Kang R (2014) Topology optimization-based lightweight primary mirror design of a large-aperture space telescope. Appl Opt 53:8318–8325

    Google Scholar 

  • Liu T, Zhu J, He F, Zhao H, Liu Q, Yang C (2017) A MAC based excitation frequency increasing method for structural topology optimization under harmonic excitations. Int J Simul Multidiscip Des Optim 8:1–10

    Google Scholar 

  • Liu J, Li QH, Liu ST, Tong LY (2018) Dynamic topology optimization design of rotating beam cross-section with gyroscopic effects. Struct Multidiscip Optim 58:1467–1487

    MathSciNet  Google Scholar 

  • Ma Z, Kikuchi N, Hagiwara I (1993) Structural topology and shape optimization for a frequency response problem. Comput Mech 13:157–174

    MathSciNet  MATH  Google Scholar 

  • Ma Z, Cheng H, Kikuchi N (1994) Structural design for obtaining desired eigenfrequencies by using the topology and shape optimization method. Comput Syst Eng 5:77–89

    Google Scholar 

  • Maeda Y, Nishiwaki S, Izui K, Yoshimura M, Matsui K, Terada K (2006) Structural topology optimization of vibrating structures with specified eigenfrequencies and eigenmode shapes. Int J Numer Methods Eng 67:597–628

    MathSciNet  MATH  Google Scholar 

  • Nakshatrala PB, Tortorelli DA (2015) Topology optimization for effective energy propagation in rate-independent elastoplastic material systems. Comput Methods Appl Mech Eng 295:305–326

    MathSciNet  MATH  Google Scholar 

  • Niu B, He X, Shan Y, Yang R (2018) On objective functions of minimizing the vibration response of continuum structures subjected to external harmonic excitation. Struct Multidiscip Optim 57:2291–2307

    MathSciNet  Google Scholar 

  • Noh JY, Yoon GH (2012) Topology optimization of piezoelectric energy harvesting devices considering static and harmonic dynamic loads. Adv Eng Softw 53:45–60

    Google Scholar 

  • Olhoff N (1989) Multicriterion structural optimization via bound formulation and mathematical programming. Struct Multidiscip Optim 1:11–17

    Google Scholar 

  • Olhoff N, Du J (2014) Topological design for minimum dynamic compliance of structures under forced vibration. In: Topology optimization in structural and continuum mechanics. Springer, Vienna, pp. 325–339

  • Olhoff N, Du J (2016) Generalized incremental frequency method for topological design of continuum structures for minimum dynamic compliance subject to forced vibration at a prescribed low or high value of the excitation frequency. Struct Multidiscip Optim 54:1113–1141

    MathSciNet  Google Scholar 

  • Overton ML (1988) On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J Matrix Anal Appl 9:256–268

    MathSciNet  MATH  Google Scholar 

  • Pedersen NL (2000) Maximization of eigenvalues using topology optimization. Struct Multidiscip Optim 20:2–11

    Google Scholar 

  • Pierson BL (1972) A survey of optimal structural design under dynamic constraints. Int J Numer Methods Eng 4:491–499. https://doi.org/10.1002/nme.1620040404

    Article  Google Scholar 

  • Seiranyan AP (1987) Multiple eigenvalues in optimization problems. J Appl Math Mech 51:272–275

    MathSciNet  MATH  Google Scholar 

  • Sethian JA, Wiegmann A (2000) Structural boundary design via level set and immersed interface methods. J Comput Phys 163:489–528

    MathSciNet  MATH  Google Scholar 

  • Seyranian AP, Lund E, Olhoff N (1994) Multiple eigenvalues in structural optimization problems. Struct Optim 8:207–227

    Google Scholar 

  • Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33:401–424

    Google Scholar 

  • Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidiscip Optim 48:1031–1055

    MathSciNet  Google Scholar 

  • Sun W, Zhao Y, Li J, Zhang L, Gao H (2012) Active suspension control with frequency band constraints and actuator input delay. IEEE Trans Ind Electron 59:530–537. https://doi.org/10.1109/TIE.2011.2134057

    Article  Google Scholar 

  • Tcherniak D (2002) Topology optimization of resonating structures using SIMP method. Int J Numer Methods Eng 54:1605–1622

    MATH  Google Scholar 

  • Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192:227–246

    MathSciNet  MATH  Google Scholar 

  • Wang D, Zhang W, Jiang J (2004) Truss optimization on shape and sizing with frequency constraints. AIAA J 42:622–630

    Google Scholar 

  • Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767–784

  • Wang B, Zhou Y, Tian K, Wang G (2020) Novel implementation of extrusion constraint in topology optimization by Helmholtz-type anisotropic filter. Struct Multidiscip Optim 1–10

  • Xie Y, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49:885–896

    Google Scholar 

  • Yi G, Youn BD (2016) A comprehensive survey on topology optimization of phononic crystals. Struct Multidiscip Optim 54:1315–1344

    MathSciNet  Google Scholar 

  • Yoon GH (2010) Structural topology optimization for frequency response problem using model reduction schemes. Comput Methods Appl Mech Eng 199:1744–1763

    MathSciNet  MATH  Google Scholar 

  • 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

    MathSciNet  Google Scholar 

  • Zhang X, Kang Z (2013) Topology optimization of damping layers for minimizing sound radiation of shell structures. J Sound Vib 332:2500–2519

    Google Scholar 

  • Zhang W, Zhou L (2018) Topology optimization of self-supporting structures with polygon features for additive manufacturing. Comput Methods Appl Mech Eng 334:56–78

    MathSciNet  MATH  Google Scholar 

  • Zhou M, Rozvany G (1991) The COC algorithm, Part II: Topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89:309–336

    Google Scholar 

  • Zhu J-H, Zhang W-H, Xia L (2016) Topology optimization in aircraft and aerospace structures design. Arch Comput Methods Eng 23(4): 595–622

Download references

Funding

This work is funded by the National Natural Science Foundation of China (grant nos.11802164, U1808215), Shandong Provincial Natural Science Foundation (grant no. ZR2019BEE005), Key Research & Development Program of Shandong Province (no. 2019GGX104101), and the project funded by China Postdoctoral Science Foundation.

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Authors and Affiliations

  1. Key Laboratory of High Efficiency and Clean Mechanical Manufacture of MOE, School of Mechanical Engineering, Shandong University, Jinan, 250061, People’s Republic of China

    Quhao Li & Qiangbo Wu

  2. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Quhao Li, Jingjie He & Shutian Liu

  3. Institute of Systems Engineering, China Academy of Engineering Physics, Mianyang, Sichuan, 621900, People’s Republic of China

    Ji Liu

Authors
  1. Quhao Li
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  2. Qiangbo Wu
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  3. Ji Liu
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  4. Jingjie He
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  5. Shutian Liu
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Correspondence to Shutian Liu.

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Replication of results

Details on the numerical implementation for the replication of the results have been provided in Section 4, with the pseudocode and the optimization parameters. The design problems, mesh size, and boundary conditions are given in Section 5. If the information provided in the paper is not enough, we sincerely welcome scientists or interested parties to contact us for further explanation.

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Li, Q., Wu, Q., Liu, J. et al. Topology optimization of vibrating structures with frequency band constraints. Struct Multidisc Optim 63, 1203–1218 (2021). https://doi.org/10.1007/s00158-020-02753-7

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  • DOI: https://doi.org/10.1007/s00158-020-02753-7

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