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Topology optimization for truss-like material distribution field with B-spline expression

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Abstract

Even through the topology optimal truss-like structure based on the truss-like material model is very close to its analytical solution, further post-processing is required to form a truss or a homogeneous isotropic perforated plate. In this paper, an optimal truss-like material distribution field is expressed by a B-spline function. For the convenience of processing, the optimal material distribution domain of a truss-like structures is firstly extended into a curved quadrilateral extension domain. Then, the relationship between the B-spline curve clusters and the distribution of truss-like material members is established by a sample points array. Combined with the distribution of materials, a topology structure that meets the design requirements is formed. Numerical examples with three models show that the B-spline function can describe the distribution field of truss-like material members efficiently, and the number of the truss-like material members can be actively controlled to meet different design needs. Comparison between the optimal results and them of the traditional analytical solution is made and results show that the topological structure is close to the analytical solution.

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References

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Bendsøe MP, Sigmund O (2003) Topology Optimization: Theory, Method and Applications. Springer-Verlag, Berlin

    MATH  Google Scholar 

  • Bendsøe MP, Díaz AR, Lipton R, Taylor JE (1995) Optimal design of material properties and material distribution for multiple loading conditions. Int J Numer Methods Eng 38:1149–1170

    Article  MathSciNet  Google Scholar 

  • Birker T (1995) Generalized Michell structures—exact least-weight truss layouts for combined stress and displacement constraints: Part II—Analytical solutions within a two-bar topology. Struct Multidiscip Optim 9:214–219

    Article  Google Scholar 

  • Bojczuk D, Mróz Z (1998) Optimal design of trusses with account for topology variation. Mech Struct Mach 26:21–40

    Article  Google Scholar 

  • Guedes JM, Taylor JE (1997) On the prediction of material properties and topology for optimal continuum structures. Struct Multidiscip Optim 14:193–199

    Article  Google Scholar 

  • Guo X, Zhang W, Zhong W (2014) Doing Topology Optimization Explicitly and Geometrically—A New Moving Morphable Components Based Framework. J Appl Mech 81:081009

    Article  Google Scholar 

  • Hörnlein HREM, Kovara M, Werner R (2001) Material optimization : bridging the gap between conceptual and preliminary design. Aerosp Sci Technol 5:541–554

    Article  Google Scholar 

  • Jin SY, Chen TF (2017) Lagrange Interpolation Method and Isoparametric Element Inverse Transformation Method Application in Thermosetting Coupling. Equipment Manufacturing Technology 07:232–236 (In Chinese)

    Google Scholar 

  • Lewiński T, Rozvany GIN (2007) Exact analytical solutions for some popular benchmark problems in topology optimization II: three-sided polygonal supports. Struct Multidiscip Optim 33:337–349

    Article  MathSciNet  Google Scholar 

  • Lewiński T, Rozvany GIN (2008a) Analytical benchmarks for topological optimization IV: square-shaped line support. Struct Multidiscip Optim 36:143–158

    Article  MathSciNet  Google Scholar 

  • Lewiński T, Rozvany GIN (2008b) Exact analytical solutions for some popular benchmark problems in topology optimization III: L-shaped domains. Struct Multidiscip Optim 35:165–174

    Article  MathSciNet  Google Scholar 

  • Li X, Zhou K (2013) Optimization method using Hermite elements based on truss-like material. Chin J Appl Mech 5:777–781 In Chinese

    Google Scholar 

  • Luo Z, Tong L, Kang Z (2009) A level set method for structural shape and topology optimization using radial basis functions. Comput Struct 87:425–434

    Article  Google Scholar 

  • Michell AGM (1904) The limit of economy of material in frame structures. Philos Mag 8:589–597

    Article  Google Scholar 

  • Mlejnek HP, Schirrmacher R (1993) An engineer's approach to optimal material distribution and shape finding. Comput Methods Appl Mech Eng 106:1–26

    Article  Google Scholar 

  • Munk DJ (2019) A bi-directional evolutionary structural optimization algorithm for mass minimization with multiple structural constraints. Int J Numer Methods Eng 118:93–120

    Google Scholar 

  • Norato JA, Bell BK, Tortorelli DA (2015) A geometry projection method for continuum-based topology optimization with discrete elements. Comput Methods Appl Mech Eng 293:306–327

    Article  MathSciNet  Google Scholar 

  • Rozvany GIN (1998) Exact analytical solutions for some popular benchmark problems in topology optimization. Struct Multidiscip Optim 15:42–48

    Article  Google Scholar 

  • Rozvany GIN, Birker T (1995) Generalized Michell structures—exact least-weight truss layouts for combined stress and displacement constraints: Part I—General theory for plane trusses. Struct Multidiscip Optim 9:178–188

    Article  Google Scholar 

  • Rozvany GIN, Querin OM, Gaspar Z, Pomezanski V (2003) Weight-increasing effect of topology simplification. Struct Multidiscip Optim 25:459–465

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Sokół T, Lewiński T (2011) Optimal design of a class of symmetric plane frameworks of least weight. Struct Multidiscip Optim 44:729–734

    Article  MathSciNet  Google Scholar 

  • Sui Y, Yang D (1998) A new method for structural topological optimization based on the concept of independent continuous variables and smooth model. Acta Mech Sinica 14:179–185

    Article  Google Scholar 

  • Taylor JE (1998) An energy model for the optimal design of linear continuum structures. Struct Multidiscip Optim 16:116–127

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Xie YM, Zuo ZH, Huang X, Black T, Felicetti P (2018) Application of topological optimisation technology to bridge design. Struct Eng Int 24:185–191

    Google Scholar 

  • Xu YP, Xiang Y, Liu QS, Lu AZ (2001) Research on numerical inverse isoparametric mapping interpolation and its application. Rock Soil Mech 02:226–228 (In Chinese)

    Google Scholar 

  • Yin L, Zhang F, Deng X, Wu P, Zeng H, Liu M (2019) Isogeometric Bi-Directional Evolutionary Structural Optimization. IEEE Access 7:91134–91145

    Article  Google Scholar 

  • Zhang W, Yang W, Zhou J, Li D, Guo X (2016a) Structural topology optimization through explicit boundary evolution. J Appl Mech 84:011011

    Article  Google Scholar 

  • Zhang W, Yuan J, Zhang J, Guo X (2016b) A new topology optimization approach based on Moving Morphable Components (MMC) and the ersatz material model. Struct Multidiscip Optim 53:1243–1260

    Article  MathSciNet  Google Scholar 

  • Zhang W, Li D, Yuan J, Song J, Guo X (2017a) A new three-dimensional topology optimization method based on moving morphable components (MMCs). Comput Mech 59:647–665

    Article  MathSciNet  Google Scholar 

  • Zhang W, Zhou J, Zhu Y, Guo X (2017b) Structural complexity control in topology optimization via moving morphable component (MMC) approach. Struct Multidiscip Optim 56:535–552

    Article  MathSciNet  Google Scholar 

  • Zhang W, Jiang S, Liu C, Li D, Kang P, Youn S-K, Guo X (2020) Stress-related topology optimization of shell structures using IGA/TSA-based Moving Morphable Void (MMV) approach. Comput Methods Appl Mech Eng 366:113036

    Article  MathSciNet  Google Scholar 

  • Zhou K, Hu Y (2002) A method of constructing michell truss using finite element method. Acta Mech Sinica 34:935–944 In Chinese

    Google Scholar 

  • Zhou K, Li X (2008) Topology optimization for minimum compliance under multiple loads based on continuous distribution of members. Struct Multidiscip Optim 37:49–56

    Article  Google Scholar 

  • Zuo ZH, Xie YM, Huang X (2012) An improved bi-directional evolutionary topology optimization method for frequencies. Int J Struct Stab Dyn 10:55–75

    Article  MathSciNet  Google Scholar 

  • Zuo ZH, Xie YM, Huang X (2016) Evolutionary topology optimization of structures with multiple displacement and frequency constraints. Adv Struct Eng 15:359–372

    Article  Google Scholar 

Download references

Acknowledgements

This work was financially supported by National Natural Science Foundation of China (11572131) and the Guiding Projects of Fujian Science and Technology Plan (2019H0012).

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National Natural Science Foundation of China (11572131) Guiding Projects of Fujian Science and Technology Plan (2019H0012).

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Authors and Affiliations

  1. College of Civil Engineering, Huaqiao University, Xiamen, 361021, Fujian, China

    Shunyi Shi & Kemin Zhou

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  1. Shunyi Shi
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  2. Kemin Zhou
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Correspondence to Kemin Zhou.

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The research results can be reproduced through the MATLAB code file in the attachment.

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Shi, S., Zhou, K. Topology optimization for truss-like material distribution field with B-spline expression. Struct Multidisc Optim 64, 2025–2043 (2021). https://doi.org/10.1007/s00158-021-02962-8

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  • DOI: https://doi.org/10.1007/s00158-021-02962-8

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