Abstract
This work deals with the topological design of vibrating continuum structures. The vibration of continuum structure is excited by time-harmonic external mechanical loading with prescribed frequency and amplitude. In comparison with well-known compliance minimization in static topology optimization, various objective functions are proposed in literature to minimize the response of vibrating structures, such as power flow, vibration transmission, and dynamic compliances, etc. Even for the dynamic compliance, different definitions are found in literature, which have quite different formulations and influences on the optimization results. The aim of this paper is to provide a comparison of these different objective functions and propose reference forms of objective functions for design optimization of vibration problems. Analytical solutions for two degrees of freedom system and topological design of plane structures in numerical examples are compared using different optimization formulations for given various excitation frequencies. The results are obtained by the finite element method and gradient based optimization using analytical sensitivity analysis. The optimized topologies and vibration response of the optimized structures are presented. The influence of excitation frequencies and the eigenfrequencies of the structure are discussed in the numerical examples.
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References
Allaire G (2002) Shape optimization by the homogenization method. Springer Verlag, New York
Allaire G, Aubry S, Jouve F (2001) Eigenfrequency optimization in optimal design. Comput Methods Appl Mech Eng 190(28):3565–3579
Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1(4):193–202
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224
Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9–10):635–654
Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer-Verlag, Berlin
Cheng G, Olhoff N (1981) An investigation concerning optimal design of solid elastic plates. Int J Solids Struct 17(3):305–323
Cherkaev A (2000) Variational methods for structural optimization. Springer Verlag, New York
Cremer L, Heckl M, Petersson BAT (2005) Structure-borne sound. Springer, Berlin
Diaz AR, Kikuchi N (1992) Solutions to shape and topology eigenvalue optimization problems using a homogenization method. Int J Numer Methods Eng 35(7):1487–1502
Du J, Olhoff N (2007a) Minimization of sound radiation from vibrating bi-material structures using topology optimization. Struct Multidiscip Optim 33(4–5):305–321
Du JB, Olhoff N (2007b) Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps. Struct Multidiscip Optim 34(2):91–110
Eschenauer HA, Olhoff N (2001) Topology optimization of continuum structures: a review. Appl Mech Rev 54(4):331–389
Gardonio P, Elliott SJ (1998) Driving point and transfer mobility matrices for thin plates excited in flexure. In: ISVR technical report no 277. University of, Southampton
Goyder HGD, White RG (1980) Vibrational power flow from machines into built-up structures, part I: introduction and approximate analyses of beam and plate-like foundations. J Sound Vib 68(1):59–75
Guo X, Cheng G-D (2010) Recent development in structural design and optimization. Acta Mech Sinica 26(6):807–823
Jensen JS (2007) Topology optimization of dynamics problems with Pade approximants. Int J Numer Methods Eng 72(13):1605–1630
Jensen JS (2011) Waves and vibrations in inhomogeneous structures - bandgaps and optimal designs. Technical University of Denmark, Lyngby
Jog CS (2002) Topology design of stpuctures subjected to periodic loading. J Sound Vib 253(3):687–709
Jung J, Hyun J, Goo S, Wang S (2015) An efficient design sensitivity analysis using element energies for topology optimization of a frequency response problem. Comput Methods Appl Mech Eng 296:196–210
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(1):51–67
Krog LA, Olhoff N (1999) Optimum topology and reinforcement design of disk and plate structures with multiple stiffness and eigenfrequency objectives. Comput Struct 72(4):535–563
Larsen AA, Laksafoss B, Jensen JS, Sigmund O (2009) Topological material layout in plates for vibration suppression and wave propagation control. Struct Multidiscip Optim 37(6):585–594
Le C, Bruns TE, Tortorelli DA (2012) Material microstructure optimization for linear elastodynamic energy wave management. J Mech Phys Solids 60(2):351–378
Liu H, Zhang W, Gao T (2015) A comparative study of dynamic analysis methods for structural topology optimization under harmonic force excitations. Struct Multidiscip Optim 51(6):1321–1333
Ma ZD, Kikuchi N, Hagiwara I (1993) Structural topology and shape optimization for a frequency-response problem. Comput Mech 13(3):157–174
Ma ZD, Kikuchi N, Cheng HC (1995) Topological Design for Vibrating Structures. Comput Methods Appl Mech Eng 121(1–4):259–280
Min S, Kikuchi N, Park YC, Kim S, Chang S (1999) Optimal topology design of structures under dynamic loads. Struct Optim 17(2–3):208–218
Niu B, Yan J, Cheng G (2009) Optimum structure with homogeneous optimum cellular material for maximum fundamental frequency. Struct Multidiscip Optim 39(2):115–132
Olhoff N, Du JB (2006). Topological design optimization of vibrating structures. Cjk-Osm 4: the fourth China-Japan-Korea joint symposium on optimization of structural and mechanical systems. Kunming, China, November 6-9, 2006 G. D. Cheng, S. T. Liu and X. Guo (eds): 509–514
Olhoff N, Du J (2014). Topological design for minimum dynamic compliance of structures under forced vibration. Topology Optimization in Structural and Continuum Mechanics, 325-339. G. Rozvany and T. Lewiński. Springer, Heidelberg
Olhoff N, Du J (2016) Generalized incremental frequency method for topological designof 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(5):1113–1141
Olhoff N, Niu B (2016) Minimizing the vibrational response of a lightweight building by topology and volume optimization of a base plate for excitatory machinery. Struct Multidiscip Optim 53(3):567–588
Olhoff N, Niu B, Cheng G (2012) Optimum design of band-gap beam structures. Int J Solids Struct 49(22):3158–3169
Pedersen NL (2000) Maximization of eigenvalues using topology optimization. Struct Multidiscip Optim 20(1):2–11
Rivin EI (2003) Passive vibration isolation. ASME Press, New York
Rivin EI (2010) Handbook on stiffness and damping in mechanical design. ASME Press, New York
Rozvany GIN, Lewiński T (2014) Topology optimization in structural and continuum mechanics. Springer, Heidelberg
Rozvany GIN, Zhou M, Birker T (1992) Generalized shape optimization without homogenization. Struct Optim 4(3–4):250–252
Shu L, Wang MY, Fang Z, Ma Z, Wei P (2011) Level set based structural topology optimization for minimizing frequency response. J Sound Vib 330(24):5820–5834
Sigmund O (1994) Design of material structures using topology optimization. Technical University of Denmark, Lyngby
Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33(4–5):401–424
Sigmund O, Jensen JS (2003) Systematic design of phononic band-gap materials and structures by topology optimization. Philos Trans R Soc A Math Phys Eng Sci 361(1806):1001–1019
Sigmund O, Maute K (2013) Topology optimization approaches - A comparative review. Struct Multidiscip Optim:1–25
Sun HL, Chen HB, Zhang K, Zhang PQ (2008) Research on performance indices of vibration isolation system. Appl Acoust 69(9):789–795
Svanberg K (1987) The method of moving asymptotes - a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373
Tartar L (2000) An introduction to the homogenization method in optimal design in optimal shape design (Troia, 1998). In: Cellina A, Ornelas A (eds) Lecture notes in mathematics, vol 1740. Springer, Berlin, pp 47–156
Tavakoli R (2016) Optimal design of multiphase composites under elastodynamic loading. Comput Methods Appl Mech Eng 300:265–293
Tcherniak D (2002) Topology optimization of resonating structures using SIMP method. Int J Numer Methods Eng 54(11):1605–1622
Tenek LH, Hagiwara I (1994) Eigenfrequency maximization of plates by optimization of topology using homogenization and mathematical-programming. JSME Int J Ser C 37(4):667–677
Tortorelli DA, Michaleris P (1994) Design sensitivity analysis: overview and review. Inverse Prob Eng 1(1):71–105
Wang J, Mak CM (2013) An indicator for the assessment of isolation performance of transient vibration. J Vib Control 19(16):2459–2468
Yi G, Youn BD (2016) A comprehensive survey on topology optimization of phononic crystals. Struct Multidiscip Optim 54(5):1315–1344
Yoon GH (2010) Structural topology optimization for frequency response problem using model reduction schemes. Comput Methods Appl Mech Eng 199(25–28):1744–1763
Zargham S, Ward TA, Ramli R, Badruddin IA (2016) Topology optimization: a review for structural designs under vibration problems. Struct Multidiscip Optim 53(6):1157–1177
Zhao J, Wang C (2016) Dynamic response topology optimization in the time domain using model reduction method. Struct Multidiscip Optim 53(1):101–114
Acknowledgements
This work was partially supported by the National Natural Science Foundation of China (no. 51505064, 51675082, 51790172, 51621064), and Natural Science Foundation of Liaoning Province (no. 2015020154), and Fundamental Research Funds for the Central Universities (DUT17ZD207). These supports are gratefully appreciated.
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Niu, B., He, X., Shan, Y. et al. On objective functions of minimizing the vibration response of continuum structures subjected to external harmonic excitation. Struct Multidisc Optim 57, 2291–2307 (2018). https://doi.org/10.1007/s00158-017-1859-1
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DOI: https://doi.org/10.1007/s00158-017-1859-1