Skip to main content
SpringerLink
Log in

Topology optimization: a review for structural designs under vibration problems

  • REVIEW PAPER
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

Abstract

This article provides a comprehensive review of structural optimization employing topology methods for structures under vibration problems. Topology optimization allows creative and radical design modifications, compared to shape and size optimization techniques. Various works of structural topology optimization, which are subjected to vibration as the response function of the optimization process, are reviewed. Different types of calculus and numerical methods commonly used for solving structural topological optimization problems are briefly discussed. Moreover, different aspects of topology optimization related to vibration problems are explained. The articles reviewed are largely confined to linear systems that concern small vibration amplitudes. Accordingly, the works related to vibration topological optimization are classified according to the method employed (homogenization, evolutionary structural optimization, solid isotropic material with penalization, or level set). The reviewed works are tabulated according to their methodology, year, and the objective functions and applications of each work. Although the homogenization and evolutionary methods were common in the past, the solid isotropic material with penalization (SIMP) method is the most popular method applied in recent years. The advantages of the level set method show promise for future applications.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Achtziger W, Kocvara M (2007a) Structural topology optimization with eigenvalues. SIAM J Optim 18(4):1129–1164

    Article  MathSciNet  MATH  Google Scholar 

  • Achtziger W, Kočvara M (2007b) On the maximization of the fundamental eigenvalue in topology optimization. Struct Multidiscip Optim 34(3):181–195

    Article  MathSciNet  MATH  Google Scholar 

  • Akl W, El-Sabbagha A, Al-Mitanib K, Baz A (2009) Topology optimization of a plate coupled with acoustic cavity. Int J Solids Struct 46(10):2060–2074

    Article  MATH  Google Scholar 

  • Allaire G, Jouve F (2005) A level-set method for vibration and multiple loads structural optimization. Comput Methods Appl Mech Eng 194(30):3269–3290

    Article  MathSciNet  MATH  Google Scholar 

  • Allaire G, Aubry S, Jouve F (2001) Eigenfrequency optimization in optimal design. Comput Methods Appl Mech Eng 190(28):3565–3579

    Article  MathSciNet  MATH  Google Scholar 

  • Allaire G, Jouve F, Toader AM (2002) A level-set method for shape optimization. C R Math 334(12):1125–1130

    Article  MathSciNet  MATH  Google Scholar 

  • Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393

    Article  MathSciNet  MATH  Google Scholar 

  • Ananthasuresh GK, Kota S, Gianchandani Y (1994) A methodical approach to the design of compliant micromechanisms. In Solid-state sensor and actuator workshop. SC: IEEE 189–192

  • Arora JS (1993) Sequential linearization and quadratic programming techniques. Prog Astronaut Aeronaut 150:71–71

    Google Scholar 

  • Babuška I (1976) Solution of interface problems by homogenization I. SIAM J Math Anal 7(5):603–634

    Article  MathSciNet  MATH  Google Scholar 

  • Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Multidiscip Optim 1(4):193–202

    Article  Google Scholar 

  • Bendsøe MP (1995) Optimization of structural topology, shape and material. Springer Berlin Heidelberg, doi: 10.1007/978-3-662-03115-5

  • Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224

    Article  MathSciNet  MATH  Google Scholar 

  • Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9):635–654

    MATH  Google Scholar 

  • Bendsøe MP, Sigmund O (2004) Topology optimization: theory, methods, and applications, 2nd edn. Springer, Germany

    Book  MATH  Google Scholar 

  • Bendsøe MP, Diaz A, Kikuchi N (1993) Topology and generalized layout optimization of elastic structures. In: Bendsøe MP, Soares CAM (eds) Topology design of structures, NATO ASI Series 227. Kluwer Academic Publishers, Germany, pp 159–205

    Chapter  Google Scholar 

  • Bensoussan A, Lions JL, Papanicolaou G (1978) Asymptotic analysis for periodic structures. AMS Chelsea Publishing, American Mathematical Society

  • Bogomolny M (2010) Topology optimization for free vibrations using combined approximations. Int J Numer Methods Eng 82(5):617–636

    MATH  Google Scholar 

  • Bonkobara Y, Kondou T, Sakao D, Choi MS, Kuroki H, Sakamoto Y (2008) Topology optimization of a framed structure using generalized transfer stiffness coefficient method. Trans Jpn Soc Mech Eng A 74(741):1137–1144

    Article  Google Scholar 

  • Bruggi M, Venini P (2007) Topology optimization of incompressible media using mixed finite elements. Comput Methods Appl Mech Eng 196(33):3151–3164

    Article  MathSciNet  MATH  Google Scholar 

  • Bruggi M, Taliercio A (2012) Maximization of the fundamental eigenfrequency of micropolar solids through topology optimization. Struct Multidiscip Optim 46(4):549–560

    Article  MathSciNet  MATH  Google Scholar 

  • Choi JS, Yoo J (2008) Structural optimization of ferromagnetic materials based on the magnetic reluctivity for magnetic field problems. Comput Methods Appl Mech Eng 197(49):4193–4206

    Article  MATH  Google Scholar 

  • Choi SH, Kim SR, Park JY, Han SY (2007) Multi-objective optimization of the inner reinforcement for a vehicle’s hood considering static stiffness and natural frequency. Int J Automot Technol 8(3):337–342

    Google Scholar 

  • Chu DN (1997) Evolutionary structural optimization method for systems with stiffness and displacement constraints. Dissertation, Victoria University of Technology, Australia

  • Cioranescu D, Paulin JSJ (1979) Homogenization in open sets with holes. J Math Anal Appl 71(2):590–607

    Article  MathSciNet  MATH  Google Scholar 

  • Cui GY, Tai K, Wang BP (2002) Topology optimization for maximum natural frequency using simulated annealing and morphological representation. AIAA J 40(3):586–589

    Article  Google Scholar 

  • De Leon DM, De Souza CE, Fonseca JSO, Da Silva RGA (2012) Aeroelastic tailoring using fiber orientation and topology optimization. Struct Multidiscip Optim 46(5):663–677

    Article  MathSciNet  MATH  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(7):1487–1502

    Article  MathSciNet  MATH  Google Scholar 

  • Diaz AR, Mukherjee R (2006) A topology optimization problem in control of structures using modal disparity. J Mech Des 128(3):536–541

    Article  Google Scholar 

  • Dobbs MW, Felton LP (1969) Optimization of truss geometry. J Struct Div 95(10):2105–1118

    Google Scholar 

  • Dorn WS, Gomory RE, Greenberg HJ (1964) Automatic design of optimal structures. J Mech 3:25–52

    Google Scholar 

  • El-Sabbagh A, Akl W, Baz A (2008) Topology optimization of periodic Mindlin plates. Finite Elem Anal Des 44(8):439–449

    Article  MathSciNet  Google Scholar 

  • Fiacco AV, McCormick GP (1968) Nonlinear programming: sequential unconstrained minimization techniques. Society for Industrial and Applied Mathematics (SIAM), Philadelphia

    MATH  Google Scholar 

  • Fleury C (1979) Structural weight optimization by dual methods for convex programming. Int J Numer Methods Eng 14(12):1761–1783

    Article  MATH  Google Scholar 

  • Gholizadeh S, Salajegheh E, Torkzadeh P (2008) Structural optimization with frequency constraints by genetic algorithm using wavelet radial basis function neural network. J Sound Vib 312(1–2):316–331

    Article  Google Scholar 

  • Haftka RT, Gürdal Z (1992) Elements of structural optimization (Vol. 11), 3rd edn. Springer Science and Business Media

  • Halkjær S, Sigmund O, Jensen JS (2005) Inverse design of phononic crystals by topology optimization. Z Krist 220(9–10):895–905

    Google Scholar 

  • Haslinger J, Hillebrand A, Kärkkäinen T, Miettinen M (2002) Optimization of conducting structures by using the homogenization method. Struct Multidisci Optim 24(2):125–140

    Article  Google Scholar 

  • Hassani B, Hinton E (1998a) A review of homogenization and topology optimization I—homogenization theory for media with periodic structure. Comput Struct 69(6):707–717

    Article  MATH  Google Scholar 

  • Hassani B, Hinton E (1998b) A review of homogenization and topology opimization II—analytical and numerical solution of homogenization equations. Comput Struct 69(6):719–738

    Article  Google Scholar 

  • Hassani B, Hinton E (1998c) A review of homogenization and topology optimization III—topology optimization using optimality criteria. Comput Struct 69(6):739–756

    Article  MATH  Google Scholar 

  • He W, Bindel D, Govindjee S (2012) Topology optimization in micromechanical resonator design. Optim Eng 13(2):271–292

    Article  MathSciNet  MATH  Google Scholar 

  • Heyman J (1951) Plastic design of beams and frames for minimum material consumption. Quart Appl Math 8:373–381

    MathSciNet  MATH  Google Scholar 

  • Holland JH (1975) Adaptation in national and artificial systems. University of Michigan Press, Ann Arbor

    Google Scholar 

  • Huang X, Xie YM (2010a) Evolutionary topology optimization of continuum structures: methods and applications. Wiley

  • Huang X, Xie YM (2010b) A further review of ESO type methods for topology optimization. Struct Multidiscip Optim 41(5):671–683

    Article  Google Scholar 

  • Huang X, Zuo ZH, Xie YM (2010) Evolutionary topological optimization of vibrating continuum structures for natural frequencies. Comput Struct 88(5–6):357–364

    Article  Google Scholar 

  • Icerman LJ (1969) Optimal structural design for given dynamic deflection. Int J Solids Struct 5(5):473–490

    Article  Google Scholar 

  • Iga A, Nishiwaki S, Izui K, Yoshimura M (2009) Topology optimization for thermal conductors considering design-dependent effects, including heat conduction and convection. Int J Heat Mass Transf 52(11):2721–2732

    Article  MATH  Google Scholar 

  • Jensen JS (2009) Space-time topology optimization for one-dimensional wave propagation. Comput Methods Appl Mech Eng 198(5–8):705–715

    Article  MATH  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(4):967–986

    Article  Google Scholar 

  • Jianbin Du, Olhoff N (2005) Topology optimization of continuum structures with respect to simple and multiple eigenfrequencies. 6th World Congresses of Structural and Multidisciplinary Optimization, Rio de Janeiro, Brazil

  • Jianbin Du, Olhoff N (2007a) Minimization of sound radiation from vibrating bi-material structures using topology optimization. Struct Multidiscip Optim 33(4–5):305–321

    Article  MathSciNet  Google Scholar 

  • Jianbin Du, 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

    Article  MathSciNet  MATH  Google Scholar 

  • Jog CS (2002) Topology design of structures subjected to periodic loading. J Sound Vib 253(3):687–709

    Article  Google Scholar 

  • Jog CS, Haber RB, Bendsøe MP (1994) Topology design with optimized, self‐adaptive materials. Int J Numer Methods Eng 37(8):1323–1350

    Article  MathSciNet  MATH  Google Scholar 

  • Johnson E (1976) Disjoint design spaces in the optimization of harmonically excited structures. AIAA J 14(2):259–261

    Article  Google Scholar 

  • Jung EI, Park YS, Park KC (2005) Structural dynamics modification via reorientation of modification elements. Finite Elem Anal Des 42(1):50–70

    Article  Google Scholar 

  • Kalamkarov AL (1992) Composite and reinforced elements of construction. Wiley

  • Kang Z, Wang X, Wang R (2009) Topology optimization of space vehicle structures considering attitude control effort. Finite Elem Anal Des 45(6–7):431–438

    Article  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(1):51–67

    Article  MathSciNet  MATH  Google Scholar 

  • Khot NS (1985) Optimization of structures with multiple frequency constraints. Comput Struct 20(5):869–876

    Article  MATH  Google Scholar 

  • Kim TS, Kim YY (2000) Mac-based mode-tracking in structural topology optimization. Comput Struct 74(3):375–383

    Article  Google Scholar 

  • Kim TS, Kim JE, Kim YY (2004) Parallelized structural topology optimization for eigenvalue problems. Int J Solids Struct 41(9):2623–2641

    Article  MATH  Google Scholar 

  • Kim W, Song YH, Kim JE (2012) Topology optimization of actuator arms in hard disk drives for reducing bending resonance-induced off-tracks. Struct Multidiscip Optim 46(6):907–912

    Article  MathSciNet  Google Scholar 

  • Krog LA, Olhoff N (1999) Optimal topology and reinforcement design of disk and plate structures with multiple stiffness and eigenfrequency objectives. Comput Struct 72(4):535–563

    Article  MATH  Google Scholar 

  • Lau GK, Du H (2005) Topology optimization of head suspension assemblies using modal participation factor for mode tracking. Microsyst Technol 11(12):1243–1251

    Article  Google Scholar 

  • Lee S, Kwak BM (2008) Smooth boundary topology optimization for eigenvalue performance and its application to the design of a flexural stage. Eng Optim 40(3):271–285

    Article  MathSciNet  Google Scholar 

  • Lee JS, Kim JE, Kim YY (2007) Damage detection by the topology design formulation using modal parameters. Int J Numer Methods Eng 69(7):1480–1498

    Article  MATH  Google Scholar 

  • Lee D-K, Starossek U, Shin S-M (2010a) Optimized topology extraction of steel-framed DiaGrid structure for tall buildings. Int J Steel Struct 10(2):157–164

    Article  Google Scholar 

  • Lee D-K, Starossek U, Shin S-M (2010b) Topological optimized design considering dynamic problem with non-stochastic structural uncertainty. Struct Eng Mech 36(1):79–94

    Article  Google Scholar 

  • Shu L, Ma Z, Fang Z (2009) Topology-boundary optimization of coupled structural-acoustic systems. ASME 2009 Int Mech Eng Congr Expos 15:471–478

    Google Scholar 

  • Li Q, Steven GP, Querin OM, Xie YM (1999) Shape and topology design for heat conduction by evolutionary structural optimization. Int J Heat Mass Transf 42(17):3361–3371

    Article  MATH  Google Scholar 

  • Lian-shuan S, Huan-chun S, En-min F (2001) A method for topological optimization of structures with discrete variables under dynamic stress and displacement constraints. Appl Math Mech 22(7):781–787

    Article  Google Scholar 

  • Luo Z, Yang J, Chen L (2006) A new procedure for aerodynamic missile designs using topological optimization approach of continuum structures. Aerosp Sci Technol 10(5):364–373

    Article  MATH  Google Scholar 

  • Ma Z-D, Kikuchi N, Cheng HC, Hagiwara I (1992) Topology and shape optimization technique for structural dynamic problems. Recent advances in structural mechanics. ASME Press Vessels Piping Div Publ 248:133–143

    Google Scholar 

  • Ma Z-D, Kikuchi N, Hagiwara I (1993) Structural topology and shape optimization for a frequency response problem. Comput Mech 13(3):157–174

    Article  MathSciNet  MATH  Google Scholar 

  • Ma Z-D, Kikuchi N, Cheng H-C (1995a) Topology design for vibrating structures. Comput Methods Applied Mech Eng 121(1):259–280

    Article  MathSciNet  MATH  Google Scholar 

  • Ma Z-D, Kikuchi N, Cheng H-C, Hagiwara I (1995b) Topological optimization technique for free vibration problems. J Appl Mech 62(1):200–207

    Article  MathSciNet  MATH  Google Scholar 

  • Madeira JFA, Pina HL, Rodrigues HC (2010) GA topology optimization using random keys for tree encoding of structures. Struct Multidiscip Optim 40(1):227–240

    Article  MATH  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(5):597–628

    Article  MathSciNet  MATH  Google Scholar 

  • Maxwell G (1895) Scientific papers II: Cambridge University Press. 175–177

  • Michell AGM (1904) The limits of economy of material in frame-structures. London Edinburgh Dublin Philos Mag J of Sci 8(47):589–597

    Article  MATH  Google Scholar 

  • Molter, A, Fonseca JSO, Bottega V, Da Silveira OAA (2010) Simultaneous topology optimization and optimal control for vibration suppression in structural design. In 2nd International Conference on Engineering Optimization, Lisbon, Portugal

  • Murat F, Tartar L (1985) Optimality conditions and homogenization. Res Notes Math 127:1–8

    MathSciNet  MATH  Google Scholar 

  • Nagendra S, Haftka RT, Gurdal Z (1993) Design of blade stiffened composite panels by a genetic algorithm approach. In Proceedings of the 34th AlAA/ASME/AHS SDM Conference, La Jolla, USA. 2418–2436

  • Nandy AK, Jog CS (2012) Optimization of vibrating structures to reduce radiated noise. Struct Multidiscip Optim 45(5):717–728

    Article  MathSciNet  MATH  Google Scholar 

  • Neves MM, Rodrigues H, Guedes JM (1995) Generalized topology design of structures with a buckling load criterion. Struct Multidiscip Optim 10(2):71–78

    Article  Google Scholar 

  • Niemann H, Morlier J, Shahdin A, Gourinat Y (2010) Damage localization using experimental modal parameters and topology optimization. Mech Syst Signal Process 24(3):636–652

    Article  Google Scholar 

  • Nishiwaki S, Frecker MI, Min S, Kikuchi N (1998) Topology optimization of compliant mechanisms using the homogenization method. Int J Numer Methods Eng 42(3):535–559

    Article  MathSciNet  MATH  Google Scholar 

  • Nishiwaki S, Maeda Y, Izui K, Yoshimura M, Matsui K, Terada K (2007) Topology optimization of mechanical structures targeting vibration characteristics. J Environ Eng 2(3):480–492

    Article  Google Scholar 

  • Niu B, Yan J, Cheng G (2009) Optimum structure with homogenous optimum cellular material for maximum fundamental frequency. Struct Multidiscip Optim 39(2):115–132

    Article  Google Scholar 

  • Noilublao N, Bureerat S (2011) Simultaneous topology, shape and sizing optimisation of a three-dimensional slender truss tower using multiobjective evolutionary algorithms. Comput Struct 89(23–24):2531–2538

    Article  Google Scholar 

  • Ohsaki M, Fujisawa K, Katoh N, Kanno Y (1999) Semi-definite programming for topology optimization of trusses under multiple eigenvalue constraints. Comput Methods Appl Mech Eng 180(1):203–217

    Article  MATH  Google Scholar 

  • Olhoff N, Bendsøe MP, Rasmussen J (1991) On CAD-integrated structural topology and design optimization. Comput Methods Appl Mech Eng 89(1–3):259–279

    Article  MATH  Google Scholar 

  • Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys 79(1):12–49

    Article  MathSciNet  MATH  Google Scholar 

  • Ou JS, Kikuchi N (1996) Integrated optimal structural and vibration control design. Struct Multidiscip Optim 12(4):209–216

    Article  Google Scholar 

  • Pan J, Wang D-y (2006) Topology optimization of truss structure with fundamental frequency and frequency domain dynamic response constraints. Acta Mech Solida Sin 19(3):231–240

    Article  Google Scholar 

  • Papadrakakis M (1993) Solving large-scale problems in mechanics: the development and application of computational solution methods. Wiley, New York

    MATH  Google Scholar 

  • Papadrakakis M (1997) Parallel solution methods in computational mechanics. Wiley

  • Pedersen NL (2000) Maximization of eigenvalues using topology optimization. Struct Multidiscip Optim 20(1):2–11

    Article  Google Scholar 

  • Prager W, Shield RT (1968) Optimal design of multi-purpose structures. Int J Solids Struct 4(4):469–475

    Article  Google Scholar 

  • Prager W, Taylor JE (1968) Problems of optimal structural design. J Appl Mech 35(1):102–106

    Article  MATH  Google Scholar 

  • Prager W, Rozvany GIN (1977) Optimal layout of grillages. J Struct Mech 5(1):1–18

    Article  Google Scholar 

  • Qiao Z, Weihong Z, Jihong Z, Gao T (2012) Layout optimization of multi-component structures under static loads and random excitations. Eng Struct 43:120–128

    Article  Google Scholar 

  • Querin OM, Young V, Steven GP, Xie YM (2000) Computational efficiency and validation of bi-directional evolutionary structural optimisation. Comput Methods Appl Mech Eng 189(2):559–573

    Article  MATH  Google Scholar 

  • Rong JH, Xie YM, Yang XY, Liang QQ (2000) Topology optimization of structures under dynamic response constraints. J Sound Vib 234(2):177–189

    Article  Google Scholar 

  • Rosen JB (1960) The gradient projection method for nonlinear programming. Part I. Linear constraints. J Soc Ind Appl Math 8(1):181–217

    Article  MATH  Google Scholar 

  • Rozvany GIN (1981) Optimality criteria for grids, shells and arches. Optim Distributed Parameter Struct 1:112–151

    Article  Google Scholar 

  • Rozvany GIN (1989) Structural design via optimality criteria: the Prager approach to structural optimization. Kluwer Academic Publishers, Netherlands

    Book  MATH  Google Scholar 

  • Rozvany GIN (2009) A critical review of established methods of structural topology optimization. Struct Multidiscip Optim 37(3):217–237

    Article  MathSciNet  MATH  Google Scholar 

  • Rozvany GIN, Wang CM (1982) Extensions of Prager’s layout theory. In: Eshenauer H, Olhoff N (eds) Optimization methods in structural design. Procedures Euromech Colliquium, Siegen, pp 103–110

    Google Scholar 

  • Rozvany GIN, Querin OM (2002) Combining ESO with rigorous optimality criteria. Int J Vehicle Des 28(4):294–299

    Article  Google Scholar 

  • Rozvany GIN, Bendsøe MP, Kirsch U (1995) Layout optimization of structures. Appl Mech Rev 48(2):41–119

    Article  Google Scholar 

  • Rubio WM, Silva ECN, Paulino GH (2009) Toward optimal design of piezoelectric transducers based on multifunctional and smoothly graded hybrid material systems. J Intelligent Mater Syst Struct 20(14):1725–1746

    Article  Google Scholar 

  • Rubio WM, Paulino GH, Silva ECN (2011) Tailoring vibration mode shapes using topology optimization and functionally graded material concepts. Smart Mater Struct 20(2):025009

    Article  Google Scholar 

  • Sadek EA (1996) Minimum weight design of structures under frequency and frequency response constraints. Comput Struct 60(1):73–77

    Article  MathSciNet  MATH  Google Scholar 

  • Sánchez-Palencia E (1980) Non-homogeneous media and vibration theory. In lecture notes in physics, vol 127. Springer, Germany

    MATH  Google Scholar 

  • Schmit L (1960) Structural design by systematic synthesis. In Proceedings of the 2nd ASCE Conference on Electronic Computation, Pittsburgh PA 105–132

  • Sethian JA (1999) Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. Cambridge University Press

  • Sethian JA, Wiegmann A (2000) Structural boundary design via level set and immersed interface methods. J Comput Phys 163(2):489–528

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • Sigmund O Jensen JS (2003) Systematic design of phononic band–gap materials and structures by topology optimization. Philos Trans Royal Soc London Ser A: Math Phys Eng Sci 361:1001–1019

    Article  MathSciNet  MATH  Google Scholar 

  • Sigmund O (1994) Design of material structures using topology optimization. Danish center for applied mathematics and mechanics. Technical University of Denmark, Lyngby

    Google Scholar 

  • Sigmund O (1997) On the design of compliant mechanisms using topology optimization. J Struct Mech 25(4):493–524

    Google Scholar 

  • Sigmund O, Maute K (2003) Topology optimization approaches: a comparative review. Struct Multidiscip Optim 48(6):1031–1055

    Article  MathSciNet  Google Scholar 

  • Soto CA (1997) Structural topology optimization for tactile response improvement in the automotive industry. Proceeding of design optimization with applications in industry. Jt ASME ASCE SES Summer Meet 227:37–48

    Google Scholar 

  • Soto CA, Díaz AR (1993) On the modelling of ribbed plates for shape optimization. Struct Multidiscip Optim 6(3):175–188

    Article  Google Scholar 

  • Suzuki K, Kikuchi N (1991) A homogenization method for shape and topology optimization. Comput Methods Appl Mech Eng 93(3):291–318

    Article  MATH  Google Scholar 

  • Suzuki K, Kikuchi N (1992) Generalized layout optimization of three-dimensional shell structures. In Field DA, Komkov V (eds) Geometric Aspects of Industrial Design, SIAM 62–88

  • Suzuki K, Kikuchi N (1993) Layout optimization using the homogenization method. In: Rosvany GIN (ed) Optimization of large structural systems, Vol 231 NATO ASI Series. Springer, Netherlands, pp 157–175

    Chapter  Google Scholar 

  • Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373

    Article  MathSciNet  MATH  Google Scholar 

  • Swan CC, Arora JS (1997) Topology design of material layout in structured composites of high stiffness and strength. Struct Multidiscip Optim 13(1):45–59

    Article  Google Scholar 

  • Swan CC, Kosaka I (1997) Voigt-Reuss topology optimization for structures with linear elastic material behaviours. Int J Numer Methods Eng 40(16):3033–3057

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Tenek LH, Hagiwara I (1994) Eigenfrequency maximization of plates by optimization of topology using homogenization and mathematical programming. JSME Int J Ser C Dyn Control Robot Des Manuf 37(4):667–677

    Google Scholar 

  • Turner MJ (1967) Design of minimum mass structures with specified natural frequencies. AIAA J 5(3):406–412

    Article  MATH  Google Scholar 

  • Venkayya VB, Khot NS, Reddy VS (1968) Optimization of structures based on the study of strain energy distribution. DTIC Documen, USAF Flight Dynamics Lab, WPAFB, Dayton

    Google Scholar 

  • Wang L, Basu PK, Leiva JP (2004) Automobile body reinforcement by finite element optimization. Finite Elem Anal Des 40(8):879–893

    Article  Google Scholar 

  • 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 Multidiscip Optim 43(4):473–485

    Article  MathSciNet  MATH  Google Scholar 

  • Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49(5):885–896

    Article  Google Scholar 

  • Xie YM, Steven GP (1994a) Optimal design of multiple load case structures using an evolutionary procedure. Eng Comput 11(4):295–302

    Article  MATH  Google Scholar 

  • Xie YM, Steven GP (1994b) A simple approach to structural frequency optimization. Comput Struct 53(6):1487–1491

    Article  Google Scholar 

  • Xie YM, Steven GP (1996) Evolutionary structural optimization for dynamic problems. Comput Struct 58(6):1067–1073

    Article  MATH  Google Scholar 

  • Xie YM, Steven GP (1997) Basic evolutionary structural optimization. In: Steven GP, Xie YM (eds) Evolutionary structural optimization. Springer, London, pp 12–29

    Chapter  Google Scholar 

  • Xu B, Jiang J, Tong W, Wu K (2003) Topology group concept for truss topology optimization with frequency constraints. J Sound Vib 261(5):911–925

    Article  Google Scholar 

  • Yang RJ (1997) Multidiscipline topology optimization. Comput Struct 63(6):1205–1212

    Article  MATH  Google Scholar 

  • Yang RJ, Chuang CH (1994) Optimal topology design using linear programming. Comput Struct 52(2):265–275

    Article  MATH  Google Scholar 

  • Yang XY, Xei YM, Steven GP, Querin OM (1999a) Bidirectional evolutionary method for stiffness optimization. AIAA J 37(11):1483–1488

    Article  Google Scholar 

  • Yang XY, Xie YM, Steven GP, Querin OM (1999b) Topology optimization for frequencies using an evolutionary method. J Struct Eng 125(12):1432–1438

    Article  Google Scholar 

  • Yoon GH (2010a) Maximizing the fundamental eigenfrequency of geometrically nonlinear structures by topology optimization based on element connectivity parameterization. Comput Struct 88(1):120–133

    Article  Google Scholar 

  • Yoon GH (2010b) Structural topology optimization for frequency response problem using model reduction schemes. Comput Methods Appl Mech Eng 199(25):1744–1763

    Article  MathSciNet  MATH  Google Scholar 

  • Yoon GH, Sigmund O (2008) A monolithic approach for topology optimization of electrostatically actuated devices. Comput Methods Appl Mech Eng 197(45–48):4062–4075

    Article  MathSciNet  MATH  Google Scholar 

  • Yoon GH, Jensen JS, Sigmund O (2007) Topology optimization of acoustic–structure interaction problems using a mixed finite element formulation. Int J Numer Methods Eng 70(9):1049–1075

    Article  MathSciNet  MATH  Google Scholar 

  • Zarghamee M (1968) Frequency optimization. AIAA J 6:749–750

    Article  Google Scholar 

  • Zhao C, Steven GP, Xie YM (1997) Evolutionary natural frequency optimization of two-dimensional structures with additional non-structural lumped masses. Eng Comput 14(2):233–251

    Article  MATH  Google Scholar 

  • Zhao C, Steven GP, Xie YM (1998) A generalized evolutionary method for natural frequency optimization of membrane vibration problems in finite element analysis. Comput Struct 66(2):353–364

    Article  MATH  Google Scholar 

  • Zheng-Dong Ma ZD, Cheng H-C, Kikuchi N (1994) Structural design for obtaining desired eigenfrequencies by using the topology and shape optimization method. Comput Syst Eng 5(1):77–89

    Article  Google Scholar 

  • Zhou M, Rozvany GIN (1991) The COC algorithm, Part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech En 89(1):309–336

    Article  Google Scholar 

  • Jihong Z, Weihong Z (2006) Maximization of structural natural frequency with optimal support layout. Struct Multidiscip Optim 31(6):462–469

    Article  Google Scholar 

  • Jihong Z, Weihong Z, Ke-peng Q (2006) Investigation of localized modes in topology optimization of dynamic structures. ACTA Aeronauti Astronaut Sin Ser A B 27(4):619

    Google Scholar 

  • Jihong Z, Weihong Z, Ke-peng Q (2007) Bi-directional evolutionary topology optimization using element replaceable method. Comput Mech 40(1):97–109

    Article  Google Scholar 

  • Zifan F, Yang L, Daojia D, He K, Yi Z (2010) Research on topology optimization design for support structure. Zhongguo Jixie Gongcheng (China Mech Eng) 21(15):1836–1839

    Google Scholar 

  • Zoutendijk G (1960) Methods of feasible directions: a study in linear and non-linear programming. Elsevier, New York

    MATH  Google Scholar 

  • Zuo ZH, Xie YM, Huang X (2011) Optimal topological design of periodic structures for natural frequencies. J Struct Eng 137(10):1229–1240

    Article  Google Scholar 

Download references

Acknowledgments

This article was written under the auspices of the Center for Transportation Research at the Faculty of Engineering, University of Malaya. This research was funded under the High Impact Research MoE Grant: UM.C/625/1/HIR/MoE/ENG/53 (D000053-16001) from the Ministry of Education, Malaysia and the University of Malaya Research Grant: RG155-12AET.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, 50603, Kuala Lumpur, Malaysia

    Sajjad Zargham, Thomas Arthur Ward, Rahizar Ramli & Irfan Anjum Badruddin

Authors
  1. Sajjad Zargham
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas Arthur Ward
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rahizar Ramli
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Irfan Anjum Badruddin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Arthur Ward.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zargham, S., Ward, T.A., Ramli, R. et al. Topology optimization: a review for structural designs under vibration problems. Struct Multidisc Optim 53, 1157–1177 (2016). https://doi.org/10.1007/s00158-015-1370-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-015-1370-5

Keywords

Search

Navigation