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Topology optimization of phononic crystals with uncertainties

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Abstract

Topology optimization of phononic crystals (PnCs) is generally based on deterministic models without considering effects of inherent uncertainties existed in PnCs. However, uncertainties presented in PnCs may significantly affect band gap characteristics. To address this, an interval Chebyshev surrogate model-based heuristic algorithm is proposed for topology optimization of PnCs with uncertainties. Firstly, the interval model is introduced to handle the uncertainties, and then the interval Chebyshev surrogate model (ICSM), in which the improved fast plane wave expansion method (IFPWEM) is used to calculate the integral points to construct the ICSM, is introduced for band structure analysis with uncertainties efficiently. After that, the sample data, which is randomly generated by the Monte Carlo method (MCM), is applied to the ICSM for predicting the interval bounds of the band structures. Finally, topology optimization of PnCs is conducted to generate the widest band gaps with uncertainties included by utilizing the genetic algorithm (GA) and the ICSM. Numerical results show the effectiveness and efficiency of the proposed method which has promising prospects in a range of engineering applications.

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References

  • Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New York

    MATH  Google Scholar 

  • Antonio MA, Jorge CZ, Sergio CA et al (2014) Multi-objective adaptive evolutionary strategy for tuning compilations. Neurocomputing 123:381–389

    Article  Google Scholar 

  • Babaee S, Wang P, Bertoldi K (2015) Three-dimensional adaptive soft phononic crystals. J Appl Phys 117(24):244903

    Article  Google Scholar 

  • Bilal OR, Hussein MI (2011a) Ultrawide phononic band gap for combined in-plane and out-of-plane waves. Phys Rev E 84(6):065701

    Article  Google Scholar 

  • Bilal OR, Hussein MI (2011b) Optimization of phononic crystals for the simultaneous attenuation of out-of-plane and in-plane waves. Proceeding of the ASME 2011 International mechanical Enginnering congress and exposition Novenber 11–17, 2011, Denver, Colorado

  • Chen SK, Chen W (2011) A new level-set based approach to shape and topology optimization under geometric uncertainty. Struct Multidiscip Optim 44(1):1–18

    Article  MATH  MathSciNet  Google Scholar 

  • Chen YY, Zhu R, Barnhart MV, Huang GL (2016) Enhanced flexural wave sensing by adaptive gradient-index metamaterials. Scientific Reports 6:35048

    Article  Google Scholar 

  • da Silva GA, Cardoso EL (2016) Topology optimization of continuum structures subjected to uncertainties in material properties. Int J Numer Meth Eng 106:192–212

    Article  MATH  MathSciNet  Google Scholar 

  • Deckers E, Claeys C, Atak O, Groby JP, Dazel O, Desmet WA (2016) Wave based method to predict the absorption, reflection and transmission coefficient of two-dimensional rigid frame porous structures with periodic inclusions. J Comput Phys 312:115–138

    Article  MATH  MathSciNet  Google Scholar 

  • Diaz AR, Haddow AG, Ma L (2005) Design of band-gap grid structures. Struct Multidiscip Optim 29(6):418–431

    Article  Google Scholar 

  • Dong HW, Su XX, Wang YS, Zhang CZ (2014a) Topology optimization of two-dimensional asymmetrical phononic crystals. Phys Lett A 378(4):434–441

    Article  Google Scholar 

  • Dong HW, Su XX, Wang YS, Zhang CZ (2014b) Topological optimization of two-dimensional phononic crystals based on the finite element method and genetic algorithm. Struct Multidiscip Optim 50(4):593–604

    Article  MathSciNet  Google Scholar 

  • Dyn N, Levin D, Rippa S (1986) Numerical procedures for surface fitting of scattered data by radial basis functions. SIAM J Sci Stat Comput 7:639–659

    Article  MATH  Google Scholar 

  • Gao W, Kessissoglou NJ (2007) Dynamic response analysis of stochastic truss structures under non-stationary random excitation using the random factor method. Comput Methods Appl Mech Eng 196(25–28):2765–2773

    Article  MATH  Google Scholar 

  • Gao W, Wu D, Song C, Tin-Loi F, Li X (2011) Hybrid probabilistic interval analysis of bar structures with uncertainty using a mixed perturbation Monte-Carlo method. Finite Elem Anal Des 47(7):643–652

    Article  MathSciNet  Google Scholar 

  • Gazalet J, Dupont S, Kastelik JC et al (2013) A tutorial survey on waves propagating in periodic media: electronic, photonic and phononic crystals. Perception of the Bloch theorem in both real and Fourier domains. Wave Motion 50(3):619–654

    Article  MathSciNet  Google Scholar 

  • Gazonas GA, Weile DS, Wildman R, Mohan A (2006) Genetic algorithm optimization of phononic bandgap structures. Int J Solids Struct 43(18–19):5851–5866

    Article  MATH  Google Scholar 

  • Guest JK, Igusa T (2008) Structural optimization under uncertain loads and nodal locations. Comput Methods Appl Mech Eng 198(1):116–124

    Article  MATH  MathSciNet  Google Scholar 

  • Herath MT, Natarajan S, Prusty BG, John NS (2014) Smoothed finite element and genetic algorithm based optimization for shape adaptive composite marine propellers. Compos Struct 109:189–197

    Article  Google Scholar 

  • Hou Z, Fu X, Liu Y (2006) Singularity of the Bloch theorem in the fluid/solid phononic crystal. Phys Rev B 73(2):024304

    Article  Google Scholar 

  • Hu W, Azarm S, Almansoori A (2013) New approximation assisted multi-objective collaborative robust optimization (new AA-McRO) under interval uncertainty. Struct Multidiscip Optim 47(1):19–35

    Article  MATH  MathSciNet  Google Scholar 

  • Hussein MI, Hamza K, Hulbert GM, Scott RA, Saitou K (2006) Multiobjective evolutionary optimization of periodic layered materials for desired wave dispersion characteristics. Struct Multidiscip Optim 31(1):60–75

    Article  Google Scholar 

  • Hussein MI, Hamza K, Hulbert GM, Saitou K (2007) Optimal synthesis of 2D phononic crystals for broadband frequency isolation. Waves in Random Complex Media 17(4):491–510

    Article  MATH  MathSciNet  Google Scholar 

  • Jin R, Chen W, Simpson TW (2001) Comparative studies of metamodelling techniques under multiple modelling criteria. Struct Multidiscip Optim 23(1):1–13

    Article  Google Scholar 

  • Kang Z, Luo YJ (2010) Reliability-based structural optimization with probability and convex set hybrid models. Struct Multidiscip Optim 42(1):89–102

    Article  Google Scholar 

  • Khelif A, Houjaa AC, Benchabane S, Djafari-Rouhani B, Laude V (2004) Guiding and bending of acoustic waves in highly confined phononic crystal waveguides. Appl Phys Lett 84(22):4400–4402

    Article  Google Scholar 

  • Lazarov BS, Schevenels M, Sigmund O (2012) Topology optimization with geometric uncertainties by perturbation techniques. Int J Numer Meth Eng 90(11):1321–1336

    Article  MATH  Google Scholar 

  • Li YF, Huang X, Meng F, Zhou S (2016a) Evolutionary topological design for phononic band gap crystals. Struct Multidiscip Optim 54(3):595–617

    Article  MathSciNet  Google Scholar 

  • Li YF, Huang X, Zhou S (2016b) Topological design of cellular phononic band gap crystals. Materials 9(3):186

    Article  Google Scholar 

  • Lin PT (2016) An efficient method of solving design optimization problems with arbitrary random distributions in gradient-based design spaces. Struct Multidiscip Optim 54(6):1653–1670

    Article  MathSciNet  Google Scholar 

  • Liu ZF, Wu B, He CF (2014) Band-gap optimization of two-dimensional phononic crystals based on genetic algorithm and FPWE. Waves Random Complex Media 24(3):286–305

    Article  MathSciNet  Google Scholar 

  • Luo Y, Kang Z, Li A (2009) Structural reliability assessment based on probability and convex set mixed model. Comput Struct 87:1408–1415

    Article  Google Scholar 

  • Mansouri I, Ozbakkaloglu T, Kisi O, Xie TY (2016) Predicting behavior of FRP-confined concrete using neuro fuzzy, neural network, multivariate adaptive regression splines and M5 model tree techniques. Mater Struct 49(10):4319–4334

    Article  Google Scholar 

  • Men H, Lee KYK, Freund RM et al (2014) Robust topology optimization of three-dimensional photonic-crystal band-gap structures. Opt Express 22(19):22632–22648

    Article  Google Scholar 

  • Myers RH, Montgomery D (1995) Response surface methodology: process and product optimization using designed experiments. John Wiley and Sons Inc., Toronto

    MATH  Google Scholar 

  • Nguyen TH, Song J, Paulino GH (2011) Single-loop system reliabilitybased topology optimization considering statistical dependence between limit-states. Struct Multidiscip Optim 44(5):593–611

    Article  MATH  MathSciNet  Google Scholar 

  • Oskooi A, Mutapcic A, Noda S et al (2012) Robust optimization of adiabatic tapers for coupling to slow-light photonic-crystal waveguides. Opt Express 20(19):21558–21575

    Article  Google Scholar 

  • Park JH, Ma PS, Kim YY (2015) Design of phononic crystals for selfcollimation of elastic waves using topology optimization method. Struct Multidiscip Optim 51(6):1199–1209

    Article  MathSciNet  Google Scholar 

  • Popa BI, Zigoneanu L, Cummer SA (2011) Experimental acoustic ground cloak in air. Phys Rev Lett 106(25):253901

    Article  Google Scholar 

  • Rupp CJ, Evgrafov A, Maute K, Dunn ML (2007) Design of phononic materials/structures for surface wave devices using topology optimization. Struct Multidiscip Optim 34(2):111–121

    Article  MATH  MathSciNet  Google Scholar 

  • Shan S, Wang GG (2010) Metamodeling for high dimensional simulation-based design problems. J Mech Des 132:051001–051011

    Article  Google Scholar 

  • Sigmund O, Jensen JS (2003) Systematic design of phononic band-gap materials and structures by topology optimization. Phil Trans R Soc Lond A 361(1806):1001–1019

  • Sigmund O (2001) Microstructural design of elastic band-gap structures. Proceeding of the 4th World Congress of Structural and Multidisciplinary Optimization, June 4–8, 2001, Dalian, China

  • Silva M, Tortorelli DA, Norato JA, Ha C, Bae HR (2010) Component and system reliability-based topology optimization using a single-loop method. Struct Multidiscip Optim 41(1):87–106

    Article  MATH  MathSciNet  Google Scholar 

  • Simpson TW, Mauery TM, Korte JJ, Mistree F (2001) Kriging models for global approximation in simulation-based multidisciplinary design optimization. AIAA J 39:2233–2241

    Article  Google Scholar 

  • Su R, Wang X, Gui L, Fan Z (2011) Multi-objective topology and sizing optimization of truss structures based on adaptive multi-island search strategy. Struct Multidiscip Optim 43(2):275–286

    Article  Google Scholar 

  • Tootkaboni M, Asadpoure A, Guest JK (2012) Topology optimization of continuum structures under uncertainty – a polynomial chaos approach. Comput Methods Appl Mech Eng 201-204:263–275

    Article  MATH  MathSciNet  Google Scholar 

  • Vasseur JO, Matar OB, Robillard JF, Hladky-Hennion AC, Deymier PA (2011) Band structures tenability of bulk 2D phononic crystals made of magneto-elastic materials. AIP Adv 1:041904

    Article  Google Scholar 

  • Wang C, Qiu Z (2015a) Hybrid uncertain analysis for temperature field prediction with random, fuzzy and interval parameters. Int J Therm Sci 98:124–134

    Article  Google Scholar 

  • Wang C, Qiu Z (2015b) Improved numerical prediction and reliability-based optimization of transient heat conduction problem with interval parameters. Struct Multidiscip Optim 51(1):113–123

    Article  Google Scholar 

  • Wang GG, Shan S (2007) Review of metamodeling techniques in support of engineering design optimization. J Mech Des 129:370–380

    Article  Google Scholar 

  • Wang YZ, Li FM, Huang WH, Wang YS (2008) The propagation and localization of Rayleigh waves in disordered piezoelectric phononic crystals. J Mech Phys Solids 56:1578–1590

    Article  MATH  Google Scholar 

  • Wang F, Jensen JS, Sigmund O (2011) Robust topology optimization of photonic crystal waveguides with tailored dispersion properties. JOSA B 28:387–397

    Article  Google Scholar 

  • Wang F, Li J, Liu S et al (2014) An improved adaptive genetic algorithm for image segmentation and vision alignment used in microelectronic bonding. IEEE-ASME Trans Mechatron 19(3):916–923

    Article  Google Scholar 

  • Wang C, Qiu Z, Yang Y (2016) Collocation methods for uncertain heat convection-diffusion problem with interval input parameters. Int J Therm Sci 107:230–236

    Article  Google Scholar 

  • Warmuth F, Körner C (2015) Phononic band gaps in 2D quadratic and 3D cubic cellular structures. Materials 8(12):8327–8337

    Article  Google Scholar 

  • Wei X, Wu YZ, Chen LP (2012) A new sequential optimal sampling method for radial basis functions. Appl Math Comput 218:9635–9646

    MATH  MathSciNet  Google Scholar 

  • Wen XS, Wen JH, Yu DL et al (2009) Phononic crystals. National Defense Industry Press, Beijing

    MATH  Google Scholar 

  • Wu J, Luo Z, Zhang Y, Zhang N, Chen L (2013) Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions. Int J Numer Meth Eng 95(7):608–630

    Article  MATH  MathSciNet  Google Scholar 

  • Wu J, Luo Z, Zhang N, Zhang Y (2015) A new interval uncertain optimization method for structures using Chebyshev surrogate models. Comput Struct 146:185–196

    Article  Google Scholar 

  • Wu J, Gao J, Luo Z, Brown T (2016a) Robust topology optimization for structures under interval uncertainty. Adv Eng Softw 99:36–48

    Article  Google Scholar 

  • Wu J, Luo Z, Zheng J, Jiang C (2016b) Incremental modeling of a new high-order polynomial surrogate model. Appl Math Model 40:4681–4699

    Article  MathSciNet  Google Scholar 

  • Xi ZM, Youn BD, Jung BC, Yoon JT (2015) Random field modeling with insufficient field data for probability analysis and design. Struct Multidiscip Optim 51(3):599–611

    Article  Google Scholar 

  • Xia B, Dai H, Yu D (2016) Symmetry-broken metamaterial for blocking, cloaking, and supertunneling of sound in a subwavelength scale. Appl Phys Lett 108(25):251902

    Article  Google Scholar 

  • Xie LX, Xia BZ, Liu J, Huang GL, Lei JR (2017) An improved fast plane wave expansion method for topology optimization of phononic crystals. Int J Mech Sci 120:171–181

    Article  Google Scholar 

  • Yang XF, Liu YS, Gao Y, Zhang YS, Gao ZZ (2015) An active learning kriging model for hybrid reliability analysis with both random and interval variables. Struct Multidiscip Optim 51(5):1003–1016

    Article  MathSciNet  Google Scholar 

  • Yi G, Youn BD (2016) A comprehensive survey on topology optimization of phononic crystals. Struct Multidiscip Optim 54(5):1315–1344

    Article  MathSciNet  Google Scholar 

  • Yoo D, Lee I (2014) Sampling-based approach for design optimization in the presence of interval variables. Struct Multidiscip Optim 49(2):253–266

    Article  MathSciNet  Google Scholar 

  • Yoo S, Kim YJ, Kim YY (2012) Hybrid phononic crystals for broad-band frequency noise control by sound blocking and localization. J Acoust Soc Am 132(5):411–416

    Article  Google Scholar 

  • Zhao ZH, Han X, Jiang C, Zhou XX (2010) A nonlinear interval-based optimization method with local-densifying approximation technique. Struct Multidiscip Optim 42(4):559–573

    Article  Google Scholar 

  • Zheng LY, Wu Y, Ni X, Chen ZG, Lu MH, Chen YF (2014) Acoustic cloaking by a near-zero-index phononic crystal. Appl Phys Lett 104(16):161904

    Article  Google Scholar 

  • Zhong HL, Wu FG, Yao LN (2006) Application of genetic algorithm in optimization of band gap of two-dimensional phononic crystals. Acta Phys Sin-Ch Ed 55(1):275–280

    Google Scholar 

  • Zhou Q, Shao X, Jiang P et al (2015) A deterministic robust optimisation method under interval uncertainty based on the reverse model. J Eng Des 26(10–12):416–444

    Article  Google Scholar 

  • Zhou Q, Shao X, Jiang P et al (2016) An active learning variable-fidelity metamodelling approach based on ensemble of metamodels and objective-oriented sequential sampling. J Eng Des 27(4–6):205–231

    Article  Google Scholar 

  • Zhou Q, Jiang P, Shao X et al (2017a) A variable fidelity information fusion method based on radial basis function. Adv Eng Inform 32:26–39

    Article  Google Scholar 

  • Zhou H, Zhou Q, Liu C, Zhou T (2017b) A kriging metamodel-assisted robust optimization method based on a reverse model. Eng Optim. doi:10.1080/0305215X.2017.1307355

  • Zhu R, Huang GL, Huang HH, Sun CT (2011) Experimental and numerical study of guided wave propagation in a thin metamaterial plate. Phys Lett A 375(30):2863–2867

    Article  Google Scholar 

  • Zi B, Zhou B (2016) A modified hybrid uncertain analysis method for dynamic response field of the LSOAAC with random and interval parameters. J Sound Vib 374:111–137

    Article  Google Scholar 

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Acknowledgements

The paper is supported by National Natural Science Foundation of China (No.11402083, 11572121), Independent Research Project of State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body in Hunan University (No.51375002), and Heavy-duty Tractor Intelligent Manufacturing Technology Research and System Development (No.2016YFD0701105).

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Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, Hunan, 410082, China

    Longxiang Xie, Baizhan Xia, Jirong Lei & Jian Liu

  2. Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO, 65211, USA

    Guoliang Huang

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  1. Longxiang Xie
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  2. Baizhan Xia
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  3. Guoliang Huang
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  4. Jirong Lei
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  5. Jian Liu
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Correspondence to Jian Liu.

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Xie, L., Xia, B., Huang, G. et al. Topology optimization of phononic crystals with uncertainties. Struct Multidisc Optim 56, 1319–1339 (2017). https://doi.org/10.1007/s00158-017-1723-3

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  • DOI: https://doi.org/10.1007/s00158-017-1723-3

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