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Concurrent optimization design of axial shape and cross-sectional topology for beam structures

  • Ji Liu
  • Quhao Li
  • Shutian LiuEmail author
  • Liyong Tong
Research Paper
  • 43 Downloads

Abstract

This paper presents a new method for simultaneously optimizing both axial shape and cross-sectional topology of a non-uniform beam with geometrically similar cross-sections. One challenge in this concurrent shape and topology optimization is the high cost associated with calculating sectional properties of different cross-sections. In resolving this, a mapping function is proposed to calculate the sectional properties of all other cross-sections by using those of one reference cross-section. Two types of design variables are introduced, and they are the mapping parameter for each beam element for defining varying axial shape and the density for each cross-sectional element for describing the topology of the reference cross-section. Three objective functions considered are minimum compliance, maximum fundamental frequency, and the maximum gap between two adjacent frequencies, and the associated sensitivity analyses and algorithms are also developed. Several numerical examples involving a rectangular and an airfoil cross-sections are presented to illustrate the capability and efficiency of the present method.

Keywords

Concurrent optimization Non-uniform beam Mapping function Beam shape Cross-sectional topology 

Notes

Funding information

The authors gratefully acknowledge the financial support to this work by the National Natural Foundation of China (Grant Nos. 11332004, 11572073, and 11802164), the 111 Project (B14013), the Fundamental Research Funds for the Central Universities of China (DUT18ZD103), the Fundamental Research Funds of Shandong University (31360078614014), and the Australian Research Council via Discovery-Project Grants (DP140104408).

References

  1. Banichuk N, Ragnedda F, Serra M (2002) Optimum shapes of bar cross-sections. Struct Multidiscip Optim 23(3):222–232Google Scholar
  2. Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224MathSciNetzbMATHGoogle Scholar
  3. Bendsoe MP, Sigmund O (2003) Topology optimization: theory, methods, and applications. Springer, BerlinGoogle Scholar
  4. Blasques JP (2014) Multi-material topology optimization of laminated composite beams with eigenfrequency constraints. Compos Struct 111:45–55Google Scholar
  5. Blasques JP, Stolpe M (2012) Multi-material topology optimization of laminated composite beam cross sections. Compos Struct 94(11):3278–3289Google Scholar
  6. Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158MathSciNetzbMATHGoogle Scholar
  7. Cheng G, Yu N, Olhoff N (2015) Optimum design of thermally loaded beam-columns for maximum vibration frequency or buckling temperature. Int J Solids Struct 66:20–34Google Scholar
  8. Dems K (1980) Multiparameter shape optimization of elastic bars in torsion. Int J Numer Methods Eng 15(10):1517–1539MathSciNetzbMATHGoogle Scholar
  9. Donoso A, Sigmund O (2004) Topology optimization of multiple physics problems modelled by Poisson’s equation. Latin Am J Solids Struct 1(2):169–184Google Scholar
  10. 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):91–110MathSciNetzbMATHGoogle Scholar
  11. Fukada Y et al (2018) Response of shape optimization of thin-walled curved beam and rib formation from unstable structure growth in optimization. Struct Multidiscip Optim 58(4):1769–1782.Google Scholar
  12. Giavotto V et al (1983) Anisotropic beam theory and applications. Comput Struct 16(1–4):403–413zbMATHGoogle Scholar
  13. Haftka RT, Gürdal Z (2012) Elements of structural optimization, vol 11. Springer Science & Business Media, BerlinzbMATHGoogle Scholar
  14. Karihaloo BL, Hemp W (1983) Minimum-weight thin-walled cylinders of given torsional and flexural rigidity. J Appl Mech 50(4a):892–894Google Scholar
  15. Karihaloo B, Hemp W (1987) Optimum sections for given torsional and flexural rigidity. Proc R Soc Lond A 409(1836):67–77zbMATHGoogle Scholar
  16. Karihaloo B, Parbery R (1979) The optimal design of beam-columns. Int J Solids Struct 15(11):855–859MathSciNetzbMATHGoogle Scholar
  17. Karihaloo B, Parbery R (1980) Optimal design of beam-columns subjected to concentrated moments. Eng Optim 5(1):59–65Google Scholar
  18. Kim YY, Kim TS (2000) Topology optimization of beam cross sections. Int J Solids Struct 37(3):477–493zbMATHGoogle Scholar
  19. Kim TS, Kim YY (2002) Multiobjective topology optimization of a beam under torsion and distortion. AIAA J 40(2):376–381Google Scholar
  20. Kreisselmeier G, Steinhauser R (1979) Systematic control design by optimizing a vector performance index. In: IFAC Symp. Computer Aided Design of Control Systems, ZurichGoogle Scholar
  21. Liu S, An X, Jia H (2008) Topology optimization of beam cross-section considering warping deformation. Struct Multidiscip Optim 35(5):403–411Google Scholar
  22. Nguyen H-D et al (2018) Finite prism method based topology optimization of beam cross section for buckling load maximization. Struct Multidiscip Optim 57(1):55–70MathSciNetGoogle Scholar
  23. Niordson FI (1965) On the optimal design of a vibrating beam. Q Appl Math 23(1):47–53MathSciNetGoogle Scholar
  24. Olhoff N (1976) Optimization of vibrating beams with respect to higher order natural frequencies. J Struct Mech 4(1):87–122MathSciNetGoogle Scholar
  25. Olhoff N (1977) Maximizing higher order eigenfrequencies of beams with constraints on the design geometry. J Struct Mech 5(2):107–134MathSciNetGoogle Scholar
  26. Olhoff N, Parbery R (1984) Designing vibrating beams and rotating shafts for maximum difference between adjacent natural frequencies. Int J Solids Struct 20(1):63–75zbMATHGoogle Scholar
  27. Olhoff N, Niu B, Cheng G (2012) Optimum design of band-gap beam structures. Int J Solids Struct 49(22):3158–3169Google Scholar
  28. Parbery R, Karihaloo B (1977) Minimum-weight design of hollow cylinders for given lower bounds on torsional and flexural rigidities. Int J Solids Struct 13(12):1271–1280zbMATHGoogle Scholar
  29. Parbery R, Karihaloo B (1980) Minimum-weight design of thin-walled cylinders subject to flexural and torsional stiffness constraints. J Appl Mech 47(1):106–110zbMATHGoogle Scholar
  30. Pedersen P, Pedersen NL (2009) Analytical optimal designs for long and short statically determinate beam structures. Struct Multidiscip Optim 39(4):343–357MathSciNetzbMATHGoogle Scholar
  31. Qin H et al (2018) Two-level multiple cross-sectional shape optimization of automotive body frame with exact static and dynamic stiffness constraints. Struct Multidiscip Optim 58(5):2309–2323.Google Scholar
  32. Raspanti C, Bandoni J, Biegler L (2000) New strategies for flexibility analysis and design under uncertainty. Comput Chem Eng 24(9):2193–2209Google Scholar
  33. Seyranian AP, Lund E, Olhoff N (1994) Multiple eigenvalues in structural optimization problems. Struct Optim 8(4):207–227Google Scholar
  34. Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33(4–5):401–424Google Scholar
  35. Soares CM et al (1984) Optimization of the geometry of shafts using boundary elements. J Mech Transm Autom Des 106(2):199–202Google Scholar
  36. Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373MathSciNetzbMATHGoogle Scholar
  37. Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1):227–246MathSciNetzbMATHGoogle Scholar
  38. Wittrick W (1962) Rates of change of eigenvalues, with reference to buckling and vibration problems. Aeronaut J 66(621):590–591Google Scholar
  39. Xia L et al (2018) Bi-directional evolutionary structural optimization on advanced structures and materials: a comprehensive review. Arch Comput Methods Eng 25(2):437–478MathSciNetzbMATHGoogle Scholar
  40. Xingsi L (1992) An entropy-based aggregate method for minimax optimization. Eng Optim 18(4):277–285Google Scholar
  41. Zhu J-H, Zhang W-H, Xia L (2016) Topology optimization in aircraft and aerospace structures design. Arch Comput Methods Eng 23(4):595–622MathSciNetzbMATHGoogle Scholar
  42. Zuo WJ, Bai JT (2016) Cross-sectional shape design and optimization of automotive body with stamping constraints. Int J Automot Technol 17(6):1003–1011Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Structural Analysis for Industrial EquipmentDalian University of TechnologyDalianChina
  2. 2.School of Mechanical EngineeringShandong UniversityJinanChina
  3. 3.School of Aerospace, Mechanical and Mechatronic EngineeringThe University of SydneySydneyAustralia

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