Advertisement

Structural and Multidisciplinary Optimization

, Volume 57, Issue 3, pp 1061–1078 | Cite as

Structural topology optimization under harmonic base acceleration excitations

  • Ji-Hong Zhu
  • Fei He
  • Tao Liu
  • Wei-Hong Zhang
  • Qinglin Liu
  • Chong Yang
RESEARCH PAPER

Abstract

This work is focused on the structural topology optimization methods related to dynamic responses under harmonic base acceleration excitations. The uniform acceleration input model is chosen to be the input form of base excitations. In the dynamic response analysis, we propose using the large mass method (LMM) in which artificial large mass values are attributed to each driven nodal degree of freedom (DOF), which can thus transform the base acceleration excitations into force excitations. Mode displacement method (MDM) and mode acceleration method (MAM) are then used to calculate the harmonic responses and the design sensitivities due to their balances between computing efficiency and accuracy especially when frequency bands are taken into account. A density based topology optimization method of minimizing dynamic responses is then formulated based on the integration of LMM and MDM or MAM. Moreover, some particular appearances such as the precision of response analysis using MDM or MAM, and the duplicated frequencies are briefly discussed. Numerical examples are finally tested to verify the accuracy of the proposed schemes in dynamic response analysis and the quality of the optimized design in improving dynamic performances.

Keywords

Topology optimization Dynamic response Harmonic base acceleration excitations Large mass method Mode displacement method Mode acceleration method 

Notes

Acknowledgements

The authors would like to thank Prof Jianbin Du from Tsinghua University for valuable discussions. This work is supported by National Natural Science Foundation of China (11722219, 11432011, 11620101002), National Key Research and Development Program (2017YFB1102800), Key Research and Development Program of Shaanxi (2017KW-ZD-11 & 2017ZDXM-GY-059).

References

  1. Allahdadian S, Boroomand B (2016) Topology optimization of planar frames under seismic loads induced by actual and artificial earthquake records. Eng Struct 115:140–154CrossRefGoogle Scholar
  2. Allahdadian S, Boroomand B et al (2012) Towards optimal design of bracing system of multi-story structures under harmonic base excitation through a topology optimization scheme. Finite Elem Anal Des 61:60–74MathSciNetCrossRefGoogle Scholar
  3. Alvin KF (1997) Efficient computation of eigenvector sensitivities for structural dynamics. AIAA J 35(11):1760–1766CrossRefzbMATHGoogle Scholar
  4. Bathe KJ, Wilson EL (1972) Stability and accuracy analysis of direct integration methods. Earthq Eng Struct Dyn 1(3):283–291CrossRefGoogle Scholar
  5. Bathe K, Wilson EL (1976) Numerical methods in finite element analysis. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  6. Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224MathSciNetCrossRefzbMATHGoogle Scholar
  7. Besselink B, Tabak U et al (2013) A comparison of model reduction techniques from structural dynamics, numerical mathematics and systems and control. J Sound Vib 332(19):4403–4422CrossRefGoogle Scholar
  8. Bi K, Hao H et al (2013) Seismic response of a concrete filled steel tubular arch bridge to spatially varying ground motions including local site effect. Adv Struct Eng 16(10):1799–1817CrossRefGoogle Scholar
  9. Cai J, Xia Q, Luo Y, Zhang L, Wang MY (2015) A variable-width harmonic probe for multifrequency atomic force microscopy. Appl Phys Lett 106(7)Google Scholar
  10. Chen M, Ali A (1998) An efficient and robust integration technique for applied random vibration analysis. Comput Struct 66(6):785–798CrossRefzbMATHGoogle Scholar
  11. Chen W, Huang G (2010) Seismic wave passage effect on dynamic response of submerged floating tunnels. Procedia Engineering 4:217–224CrossRefGoogle Scholar
  12. Clough RW, Penzien J (1975) Dynamics of structures. McGraw-Hill Inc., USzbMATHGoogle Scholar
  13. Cornwell RE, Craig RR et al (1983) On the application of the mode-acceleration method to structural engineering problems. Earthq Eng Struct Dyn 11(5):679–688CrossRefGoogle Scholar
  14. Craig RR Jr (1981) Structural dynamics an introduction to computer methods. Wiley, New YorkGoogle Scholar
  15. Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49(1):1–38MathSciNetCrossRefGoogle Scholar
  16. Díaaz AR, Kikuchi N (1992) Solutions to shape and topology eigenvalue optimization problems using a homogenization method. Int J Numer Methods Eng 35(7):1487–1502MathSciNetCrossRefzbMATHGoogle Scholar
  17. Du J (2015) Structure optimization and its application in dynamic and acoustic design. Tsinghua University Press, BeijingGoogle Scholar
  18. 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(2):91–110MathSciNetCrossRefzbMATHGoogle Scholar
  19. Gupta AK (1992). Response spectrum method in seismic analysis and design of structures. Blackwell, CambridgeGoogle Scholar
  20. Haftka RT, Starnes JH (1976) Applications of a quadratic extended interior for structural optimization penalty function. AIAA J 14:718–724CrossRefzbMATHGoogle Scholar
  21. Hao X, Li M, et al. (2010). Optimal design of strap-down inertial navigation support under random loads. Information and Automation (ICIA), 2010 I.E. International Conference on, IEEEGoogle Scholar
  22. Jog CS (2002) Topology design of structures subjected to periodic loading. J Sound Vib 253(3):687–709CrossRefGoogle Scholar
  23. Kim Y, Jhung MJ (2014) A study on large mass method for dynamic problem of multiple degree-of-freedom system excited by ground acceleration time history. J Mech Sci Technol 28(1):25–41CrossRefGoogle Scholar
  24. Kiureghian AD, Neuenhofer A (1992). Response spectrum method for multi-support seismic excitations. Earthq Eng Struct Dyn 21(8):713-740CrossRefGoogle Scholar
  25. Leger P, Ide IM et al (1990) Multiple-support seismic analysis of large structures. Comput Struct 36(6):1153–1158CrossRefGoogle Scholar
  26. Lin Z, Gea HC et al (2011) Design of piezoelectric energy harvesting devices subjected to broadband random vibrations by applying topology optimization. Acta Mech Sinica 27(5):730–737MathSciNetCrossRefzbMATHGoogle Scholar
  27. Liu M, Gorman DG (1995) Formulation of Rayleigh damping and its extensions. Comput Struct 57(57):277–285CrossRefzbMATHGoogle Scholar
  28. Liu T, Zhu JH et al (2016) A MAC based excitation frequency increasing method for 4 structural topology optimization under harmonic excitations. Int J Simul Multisci Des Optim.  https://doi.org/10.1051/smdo/2016012
  29. Liu H., Zhang W et al. (2015). A comparative study of dynamic analysis methods for structural topology optimization under harmonic force excitations. Struct Multidiscip Optim, 51(6):1321–1333Google Scholar
  30. Ma J, Li Y (2013). Analysis of Traveling Wave Effect on Half-Through CFST Arch Bridge by Large Mass Method. Key Engineering Materials, Trans Tech PublGoogle Scholar
  31. Ma Z, Kikuchi N et al (1995) Topological design for vibrating structures. Comput Methods Appl Mech Eng 121(1):259–280MathSciNetCrossRefzbMATHGoogle Scholar
  32. Mills-Curran WC (1988) Calculation of eigenvector derivatives for structures with repeated eigenvalues. AIAA J 26:867–871CrossRefzbMATHGoogle Scholar
  33. Ni C, Yan J, Cheng G, Guo X (2014) Integrated size and topology optimization of skeletal structures with exact frequency constraints. Struct Multidiscip Optim 50(1):113–128CrossRefGoogle Scholar
  34. Ojalvo IU (1987) Efficient computation of mode-shape derivatives for large dynamic systems. AIAA J 25(10):1386–1390CrossRefzbMATHGoogle Scholar
  35. Olhoff N, Du J (2005). Topological design of continuum structures subjected to forced vibration. Proceedings of 6th world congresses of structural and multidisciplinary optimization, Rio de Janeiro, BrazilGoogle Scholar
  36. Olhoff N, Du J (2014). Structural Topology Optimization with Respect to Eigenfrequencies of Vibration. In: Rozvany GIN, Lewiński T (ed) Topology Optimization in Structural and Continuum Mechanics. Springer, Vienna, pp 275–297Google Scholar
  37. 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(5):1113–1141MathSciNetCrossRefGoogle Scholar
  38. Olhoff N, Niu B (2015) 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):1–22MathSciNetGoogle Scholar
  39. Olhoff N, Niu B et al (2012) Optimum design of band-gap beam structures. Int J Solids Struct 49(22):3158–3169CrossRefGoogle Scholar
  40. Pedersen NL (2000) Maximization of eigenvalues using topology optimization. Struct Multidiscip Optim 20(1):2–11CrossRefGoogle Scholar
  41. Prasad B, Haftka RT (1979) A cubic extended interior penalty function for structural optimization. Int J Numer Methods Eng 14(8):1107–1126CrossRefzbMATHGoogle Scholar
  42. Rong J, Jiang J et al (2003) Optimal dynamic design of structures under earthquake environment excitation. J Vibration Eng 16(1):46–51Google Scholar
  43. Seyranian AP, Lund E et al (1994) Multiple eigenvalues in structural optimization problems. Struct Optim 8(4):207–227CrossRefGoogle Scholar
  44. Shu L, Wang MY et al (2011) Level set based structural topology optimization for minimizing frequency response. J Sound Vib 330(24):5820–5834CrossRefGoogle Scholar
  45. Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidiscip Optim 48(6):1031–1055MathSciNetCrossRefGoogle Scholar
  46. Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16(1):68–75CrossRefGoogle Scholar
  47. Stolpe M, Svanberg K (2001) An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidiscip Optim 22(2):116–124CrossRefGoogle Scholar
  48. Svanberg K (1995). A globally convergent version of MMA without linesearch. Proceedings of the first world congress of structural and multidisciplinary optimization, Goslar, GermanyGoogle Scholar
  49. Tcherniak D (2002) Topology optimization of resonating structures using SIMP method. Int J Numer Methods Eng 54(11):1605–1622CrossRefzbMATHGoogle Scholar
  50. Vicente WM, Zuo ZH et al (2016) Concurrent topology optimization for minimizing frequency responses of two-level hierarchical structures. Comput Methods Appl Mech Eng 301:116–136MathSciNetCrossRefGoogle Scholar
  51. Wang D, Zhang WH, Jiang JS (2005) What are the repeated frequencies? J Sound Vib 281:1186–1194CrossRefGoogle Scholar
  52. Wang B, Zhou Y, Zhou YM et al (2017) Diverse competitive Design for Topology Optimization. Struct Multidiscip Optim.  https://doi.org/10.1007/s00158-017-1762-9
  53. Yoon GH (2010) Structural topology optimization for frequency response problem using model reduction schemes. Comput Methods Appl Mech Eng 199(25):1744–1763MathSciNetCrossRefzbMATHGoogle Scholar
  54. Zhang MY (2007) Study on dynamic re-analysis of complex structures. PhD thesis, Fudan University, ChinaGoogle Scholar
  55. Zhang Q (2012). Topology and layout optimization design of component structures under random excitation, Northwestern Polytechnical University. Ph.DGoogle Scholar
  56. Zhang N, Xia H et al (2010) Dynamic analysis of a train-bridge system under multi-support seismic excitations. J Mech Sci Technol 24(11):2181–2188CrossRefGoogle Scholar
  57. Zhu JH, Zhang WH (2006) Maximization of structural natural frequency with optimal support layout. Struct Multidiscip Optim 31(6):462–469CrossRefGoogle Scholar
  58. Zhu JH, Beckers P et al (2010) On the multi-component layout design with inertial force. J Comput Appl Math 234(7):2222–2230CrossRefzbMATHGoogle Scholar
  59. Zhu JH, Zhang WH, Xia L (2016) Topology optimization in aircraft and aerospace structures design. Arch Comput Method E 23(4):595–622MathSciNetCrossRefzbMATHGoogle Scholar
  60. Zhu JH, Guo WJ, Zhang WH, Liu T (2017) Integrated layout and topology optimization design of multi-frame and multi-component fuselage structure systems. Struct Multidiscip Optim 56(1):21–45MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.State IJR Center of Aerospace Design and Additive ManufacturingNorthwestern Polytechnical UniversityXianChina
  2. 2.MIIT Lab of Metal Additive Manufacturing and Innovative DesignNorthwestern Polytechnical UniversityXianChina
  3. 3.Institute of Intelligence Material and Structure, Unmanned System TechnologiesNorthwestern Polytechnical UniversityXianChina
  4. 4.Institute of Systems EngineeringChina Academy of Engineering PhysicsMianyangChina

Personalised recommendations