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Structural and Multidisciplinary Optimization

, Volume 59, Issue 4, pp 1143–1162 | Cite as

Smooth size design for the natural frequencies of curved Timoshenko beams using isogeometric analysis

  • Hongliang Liu
  • Dixiong YangEmail author
  • Xuan Wang
  • Yutian Wang
  • Chen Liu
  • Zhen-Pei Wang
Research Paper
  • 177 Downloads

Abstract

Curved thick beams with smoothly variable cross-section size are very common in practical engineering problems. Designing the variable size based on the traditional finite element methods often leads to non-smooth solutions. To guarantee the smoothness of size distribution, an isogeometric analysis (IGA)-based design approach is proposed in this work to optimize the cross-section size of curved Timoshenko beams for natural frequencies. Due to the geometric exactness and high-order continuity of IGA, there is high accuracy of natural frequency prediction and sensitivity analysis for design optimization. It is found that small numbers of design variables lead to parameterization-dependent solutions, while large numbers of design variables induce design fluctuation. To avoid the undesirable design fluctuation, a stability transformation method-based K-S aggregation constraint scheme is proposed to regularize the size distribution and also achieve the stable convergence of optimal solutions. Multiple design studies including the deterministic and reliability-based optimization problems are performed to demonstrate the applicability and effectiveness of the proposed approach.

Keywords

Curved Timoshenko beam Isogeometric analysis Smooth size design Natural frequency optimization K-S aggregation constraint 

Notes

Acknowledgments

The authors are grateful to Professor Krister Svanberg for providing his MMA code and the anonymous reviewers for their insightful suggestions and comments on the early version of this paper.

Funding information

This study was financially supported by the National Natural Science Foundation of China (Grant Nos. 11302041 and 11772079) and the Open Foundation of State Key Laboratory on Disaster Reduction in Civil Engineering (Grant No. SLDRCE17-03).

References

  1. Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, GermanyzbMATHGoogle Scholar
  2. Cheng G, Liu X (2011) Discussion on symmetry of optimum topology design. Struct Multidiscip Optim 44(5):713–717MathSciNetCrossRefzbMATHGoogle Scholar
  3. Chidamparam P, Leissa AW (1993) Vibrations of planar curved beams, rings, and arches. Appl Mech Rev 46:467–483CrossRefGoogle Scholar
  4. Cho S, Ha SH (2009) Isogeometric shape design optimization: exact geometry and enhanced sensitivity. Struct Multidiscip Optim 38(1):53–70MathSciNetCrossRefzbMATHGoogle Scholar
  5. Choi MJ, Cho S (2018) Constrained isogeometric design optimization of lattice structures on curved surfaces: computation of design velocity field. Struct Multidiscip Optim 58(1):17–34MathSciNetCrossRefGoogle Scholar
  6. Choi MJ, Yoon M, Cho S (2016) Isogeometric configuration design sensitivity analysis of finite deformation curved beam structures using Jaumann strain formulation. Comput Methods Appl Mech Eng 309:41–73MathSciNetCrossRefGoogle Scholar
  7. Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward the integration of CAD and FEA. Wiley, SingaporeCrossRefzbMATHGoogle Scholar
  8. Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49(1):1–38MathSciNetCrossRefGoogle Scholar
  9. 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
  10. ESA, 1991. Hermes cutaway. http://www.esa.int/spaceinimages/ Images/2003/07/Hermes cutaway, accessed: 2017-10-30
  11. Eisenberger M, Efraim E (2001) In-plane vibrations of shear deformable curved beams. Int J Numer Methods Eng 52:1221–1234CrossRefzbMATHGoogle Scholar
  12. Guo X, Ni C, Cheng G, Du Z (2012) Some symmetry results for optimal solutions in structural optimization. Struct Multidiscip Optim 46(5):631–645MathSciNetCrossRefzbMATHGoogle Scholar
  13. Haftka RT, Gürdal Z (2012) Elements of structural optimization. Springer Science & Business MediaGoogle Scholar
  14. Hao P, Wang B, Li G, Meng Z, Wang L (2015) Hybrid framework for reliability-based design optimization of imperfect stiffened shells. AIAA J 53(10):2878–2889CrossRefGoogle Scholar
  15. Hao P, Wang B, Tian K, Li G, Du KF, Niu F (2016) Efficient optimization of cylindrical stiffened shells with reinforced cutouts by curvilinear stiffeners. AIAA J 54(4):1350–1363CrossRefGoogle Scholar
  16. Herrema AJ, Wiese NM, Darling CN, Ganapathsubramanian B, Krishnamurthy A, Hsu MC (2017) A framework for parametric design optimization using isogeometric analysis. Comput Methods Appl Mech Eng 316:944–965MathSciNetCrossRefGoogle Scholar
  17. Hosseini SF, Moetakef-Imani B, Hadidi-Moud S, Hassani B (2018) Pre-bent shape design of full free-form curved beams using isogeometric method and semi-analytical sensitivity analysis. Struct Multidiscip Optim.  https://doi.org/10.1007/s00158-018-2041-0
  18. Huang X, Zuo ZH, Xie YM (2010) Evolutionary topological optimization of vibrating continuum structures for natural frequencies. Comput Struct 88(5–6):357–364CrossRefGoogle Scholar
  19. Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195MathSciNetCrossRefzbMATHGoogle Scholar
  20. Kamat MP, Simitses GJ (1973) Optimal beam frequencies by the finite element displacement method. Int J Solids Struct 9:415–429CrossRefzbMATHGoogle Scholar
  21. Karihaloo BL, Niordson FI (1973) Optimal design of vibrating cantilevers. J Optim Theory Appl 11(6):638–654.Google Scholar
  22. Katsikadelis JT, Tsiatas GC (2006) Regulating the vibratory motion of beams using shape optimization. J Sound Vib 292:390–401CrossRefGoogle Scholar
  23. Kennedy GJ, Hicken JE (2015) Improved constraint-aggregation methods. Comput Methods Appl Mech Eng 289:332–354MathSciNetCrossRefzbMATHGoogle Scholar
  24. Kiendl J, Schmidt R, Wüchner R, Bletzinger KU (2014) Isogeometric shape optimization of shells using semi-analytical sensitivity analysis and sensitivity weighting. Comput Methods Appl Mech Eng 274:148–167MathSciNetCrossRefzbMATHGoogle Scholar
  25. Liu L, Pal S, Xie H (2012) Mems mirrors based on a curved concentric electrothermal actuator. Sensors Actuators A Phys 188:349–358CrossRefGoogle Scholar
  26. Liu HL, Zhu XF, Yang DX (2016) Isogeometric method based in-plane and out-of-plane free vibration analysis for Timoshenko curved beams. Struct Eng Mech 59(3):503–526CrossRefGoogle Scholar
  27. Ma ZD, Kikuchi N, Hagiwara I (1993) Structural topology and shape optimization for a frequency response problem. Comput Mech 13(3):157–174MathSciNetCrossRefzbMATHGoogle Scholar
  28. Ma HM, Gao XL, Reddy JN (2008) A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J Mech Phys Solids 56(12):3379–3391MathSciNetCrossRefzbMATHGoogle Scholar
  29. Manh ND, Evgrafov A, Gersborg AR, Gravesen J (2011) Isogeometric shape optimization of vibrating membranes. Comput Methods Appl Mech Eng 200(13):1343–1353MathSciNetCrossRefzbMATHGoogle Scholar
  30. Nagy AP, Abdalla MM, Gürdal Z (2010) Isogeometric sizing and shape optimisation of beam structures. Comput Methods Appl Mech Eng 199:1216–1230MathSciNetCrossRefzbMATHGoogle Scholar
  31. Nagy AP, Abdalla MM, Gürdal Z (2011) Isogeometric design of elastic arches for maximum fundamental frequency. Struct Multidiscip Optim 43(1):135–149MathSciNetCrossRefzbMATHGoogle Scholar
  32. Nagy AP, IJsselmuiden ST, Abdalla MM (2013) Isogeometric design of anisotropic shells: optimal form and material distribution. Comput Methods Appl Mech Eng 264:145–162MathSciNetCrossRefzbMATHGoogle Scholar
  33. Nguyen VP, Anitescu C, Bordas SPA, Rabczuk T (2015) Isogeometric analysis: an overview and computer implementation aspects. Math Comput Simul 117:89–116MathSciNetCrossRefGoogle Scholar
  34. Niordson FI (1965) On the optimal design of a vibrating beam. Q Appl Math 23(1):47–53MathSciNetCrossRefGoogle Scholar
  35. Olhoff N (1976) Optimization of vibrating beams with respect to higher order natural frequencies. J Struct Mech 4(1):87–122CrossRefGoogle Scholar
  36. Olhoff N (1977) Maximizing higher order eigenfrequencies of beams with constraints on the design geometry. J Struct Mech 5(2):107–134MathSciNetCrossRefGoogle Scholar
  37. Olhoff N, Plaut RH (1983) Bimodal optimization of vibrating shallow arches. Int J Solids Struct 19(6):553–570CrossRefzbMATHGoogle Scholar
  38. Park BU, Seo YD, Sigmund O, Youn SK (2013) Shape optimization of the stokes flow problem based on isogeometric analysis. Struct Multidiscip Optim 48(5):965–977MathSciNetCrossRefGoogle Scholar
  39. Piegl L, Tiller W (2012) The NURBS book. Springer Science & Business MediaGoogle Scholar
  40. Plaut RH, Olhoff N (1983) Optimal forms of shallow arches with respect to vibration and stability. J Struct Mech 11(1):81–100CrossRefGoogle Scholar
  41. Qian X (2010) Full analytical sensitivities in NURBS based isogeometric shape optimization. Comput Methods Appl Mech Eng 199:2059–2071MathSciNetCrossRefzbMATHGoogle Scholar
  42. Qian X, Sigmund O (2011) Isogeometric shape optimization of photonic crystals via coons patches. Comput Methods Appl Mech Eng 200(25):2237–2255MathSciNetCrossRefzbMATHGoogle Scholar
  43. Raveendranath P, Singh G, Rao GV (2001) A three-noded shear-flexible curved beam element based on coupled displacement field interpolations. Int J Numer Methods Eng 51(1):85–101CrossRefzbMATHGoogle Scholar
  44. Rozvany GIN, Zhou M (1991) The COC algorithm, part I: cross-section optimization or sizing. Comput Methods Appl Mech Eng 89(1–3):281–308CrossRefGoogle Scholar
  45. Schmelcher P, Diakonos FK (1997) Detecting unstable periodic orbits of chaotic dynamical systems. Phys Rev Lett 78(25):4733–4736CrossRefGoogle Scholar
  46. Shojaee S, Izadpanah E, Valizadeh N, Kiendl J (2012) Free vibration analysis of thin plates by using a NURBS-based isogeometric approach. Finite Elem Anal Des 61:23–34MathSciNetCrossRefGoogle Scholar
  47. Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373MathSciNetCrossRefzbMATHGoogle Scholar
  48. Taheri AH, Hassani B (2014) Simultaneous isogeometrical shape and material design of functionally graded structures for optimal eigenfrequencies. Comput Methods Appl Mech Eng 277:46–80MathSciNetCrossRefzbMATHGoogle Scholar
  49. Takewaki I (1996) Optimal frequency design of tower structures via an approximation concept. Comput Struct 58(3):445–452CrossRefGoogle Scholar
  50. Thomas J, Abbas BAH (1975) Finite element model for dynamic analysis of Timoshenko beam. J Sound Vib 41(3):291–299CrossRefGoogle Scholar
  51. Verbart A, Langelaar M, Keulen FV (2017) A unified aggregation and relaxation approach for stress-constrained topology optimization. Struct Multidiscip Optim 55(2):663–679MathSciNetCrossRefGoogle Scholar
  52. Wang ZP, Kumar D (2017) On the numerical implementation of continuous adjoint sensitivity for transient heat conduction problems using an isogeometric approach. Struct Multidiscip Optim 56(2):487–500MathSciNetCrossRefGoogle Scholar
  53. Wang ZP, Turteltaub S (2015) Isogeometric shape optimization for quasi-static processes. Int J Numer Methods Eng 104(5):347–371MathSciNetCrossRefzbMATHGoogle Scholar
  54. Wang ZP, Poh LH, Dirrenberger J, Zhu Y, Forest S (2017a) Isogeometric shape optimization of smoothed petal auxetic structures via computational periodic homogenization. Comput Methods Appl Mech Eng 323:250–271MathSciNetCrossRefGoogle Scholar
  55. Wang ZP, Turteltaub S, Abdalla MM (2017b) Shape optimization and optimal control for transient heat conduction problems using an isogeometric approach. Comput Struct 185:59–74CrossRefGoogle Scholar
  56. Wu YT, Wirsching PH (1987) New algorithm for structural reliability estimation. J Eng Mech 113(9):1319–1336CrossRefGoogle Scholar
  57. Yang DX, Yang PX (2010) Numerical instabilities and convergence control for convex approximation methods. Nonlinear Dyn 61:605–622MathSciNetCrossRefzbMATHGoogle Scholar
  58. Yang DX, Yi P (2009) Chaos control of performance measure approach for evaluation of probabilistic constraints. Struct Multidiscip Optim 38(1):83–92CrossRefGoogle Scholar
  59. Yang F, Sedaghati R, Esmailzadeh E (2008) Free in-plane vibration of general curved beams using finite element method. J Sound Vib 318(4):850–867CrossRefGoogle Scholar
  60. Yoon M, Ha SH, Cho S (2013) Isogeometric shape design optimization of heat conduction problems. Int J Heat Mass Transf 62:272–285CrossRefGoogle Scholar
  61. Zhou M, Rozvany G (1991) The COC algorithm, part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89(1–3):309–336CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, International Research Center for Computational MechanicsDalian University of TechnologyDalianPeople’s Republic of China
  2. 2.Department of Civil & Environmental EngineeringNational University of SingaporeSingaporeSingapore

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