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Layout optimization of continuum structures considering the probabilistic and fuzzy directional uncertainty of applied loads based on the cloud model

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Abstract

This paper reports an efficient approach for uncertain topology optimization in which the uncertain optimization problem is equivalent to that of solving a deterministic topology optimization problem with multiple load cases. Probabilistic and fuzzy property of the directional uncertainty of the applied loads is considered in the topology optimization; the cloud model is employed to describe that property which can also take the correlations of the probability and fuzziness into account. Convergent and mesh-independent bi-directional evolutionary structural optimization (BESO) algorithms are utilized to obtain the final optimal solution. The proposed method is suitable for linear elastic problems with uncertain applied loads, subject to volume constraint. Several numerical examples are presented to demonstrate the capability and effectiveness of the proposed approach. In-depth discussions are also given on the effects of considering the probability and fuzziness of the directions of the applied loads on the final layout.

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References

  • Bae HR, Grandhi RV, Canfield RA (2004) An approximation approach for uncertainty quantification using evidence theory. Reliab Eng Syst Saf 86(3):215–225

    Article  Google Scholar 

  • 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  Google Scholar 

  • Dunning PD, Kim HA (2013) Robust topology optimization: minimization of expected and variance of compliance. AIAA J 51(11):2656–2664

    Article  Google Scholar 

  • Duysinx P, Bendsøe MP (1998) Topology optimization of continuum structures with local stress constraints. Int J Numer Methods Eng 43(8):1453–1478

    Article  MATH  Google Scholar 

  • Guest JK (2015) Optimizing the layout of discrete objects in structures and materials: a projection-based topology optimization approach. Comput Methods Appl Mech Eng 283:330–351

    Article  Google Scholar 

  • Guest JK, Igusa T (2008) Structural optimization under uncertain loads and nodal locations. Comput Methods Appl Mech Eng 198(1):116–124

    Article  MathSciNet  MATH  Google Scholar 

  • Gunawan S, Papalambros PY (2006) A bayesian approach to reliability-based optimization with incomplete information. J Mech Des 128(4):909–918

    Article  Google Scholar 

  • Guo X, Bai W, Zhang W, Gao X (2009) Confidence structural robust design and optimization under stiffness and load uncertainties. Comput Methods Appl Mech Eng 198(41):3378–3399

    Article  MathSciNet  MATH  Google Scholar 

  • Huang X, Xie YM (2007) Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elem Anal Des 43(14):1039–1049

    Article  Google Scholar 

  • Huang X, Xie YM (2009) Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials. Comput Mech 43(3):393–401

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

  • Huang X, Xie YM (2011) Evolutionary topology optimization of continuum structures including design-dependent self-weight loads. Finite Elem Anal Des 47(8):942–948

    Article  Google Scholar 

  • Jung HS, Cho S (2004) Reliability-based topology optimization of geometrically nonlinear structures with loading and material uncertainties. Finite Elem Anal Des 41(3):311–331

    Article  MathSciNet  Google Scholar 

  • Kanno Y, Takewaki I (2006) Sequential semidefinite program for maximum robustness design of structures under load uncertainty. J Optim Theory Appl 130(2):265–287

    Article  MathSciNet  MATH  Google Scholar 

  • Kaya N, Karen İ, Öztürk F (2010) Re-design of a failed clutch fork using topology and shape optimisation by the response surface method. Mater Des 31(6):3008–3014

    Article  Google Scholar 

  • Kharmanda G, Olhoff N, Mohamed A et al (2004) Reliability-based topology optimization. Struct Multidiscip Optim 26(5):295–307

    Article  Google Scholar 

  • Leary M, Merli L, Torti F, Mazur M, Brandt M (2014) Optimal topology for additive manufacture: a method for enabling additive manufacture of support-free optimal structures. Mater Des 63:678–690

    Article  Google Scholar 

  • Li JP (2015) Truss topology optimization using an improved species-conserving genetic algorithm. Eng Optim 47(1):107–128

    Article  MathSciNet  Google Scholar 

  • Luhandjula M (2004) Optimisation under hybrid uncertainty. Fuzzy Sets Syst 146(2):187–203

    Article  MathSciNet  MATH  Google Scholar 

  • Luo Y, Kang Z, Luo Z et al (2009) Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model. Struct Multidiscip Optim 39(3):297–310

    Article  MathSciNet  Google Scholar 

  • Querin O, Steven G, Xie YM (1998) Evolutionary structural optimisation (ESO) using a bidirectional algorithm. Eng Comput 15(8):1031–1048

    Article  MATH  Google Scholar 

  • Rozvany G (1996) Difficulties in truss topology optimization with stress, local buckling and system stability constraints. Struct Optimization 11(3–4):213–217

    Article  Google Scholar 

  • Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optimization 16(1):68–75

    Article  Google Scholar 

  • Silva M, Tortorelli DA, Norato JA et al (2010) Component and system reliability-based topology optimization using a single-loop method. Struct Multidiscip Optim 41(1):87–106

    Article  MathSciNet  MATH  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 

  • Takezawa A, Nii S, Kitamura M, Kogiso N (2011) Topology optimization for worst load conditions based on the eigenvalue analysis of an aggregated linear system. Comput Methods Appl Mech Eng 200(25):2268–2281

    Article  MathSciNet  MATH  Google Scholar 

  • Vicente W, Picelli R, Pavanello R, Xie Y (2015) Topology optimization of frequency responses of fluid–structure interaction systems. Finite Elem Anal Des 98:1–13

    Article  Google Scholar 

  • Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1):227–246

    Article  MATH  Google Scholar 

  • Wang Z, Huang HZ, Li Y, Pang Y, Xiao NC (2012) An approach to system reliability analysis with fuzzy random variables. Mech Mach Theory 52:35–46

    Article  Google Scholar 

  • Wang G, Xu C, Li D (2014) Generic normal cloud model. Inf Sci 280:1–15

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Xie YM, Steven G (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. Springer, London

    Book  Google Scholar 

  • Zhao J, Wang C (2014a) Robust structural topology optimization under random field loading uncertainty. Struct Multidiscip Optim 50(3):517–522

    Article  MathSciNet  Google Scholar 

  • Zhao J, Wang C (2014b) Robust topology optimization of structures under loading uncertainty. AIAA J 52(2):398–407

    Article  Google Scholar 

  • Zhao J, Wang C (2014c) Robust topology optimization under loading uncertainty based on linear elastic theory and orthogonal diagonalization of symmetric matrices. Comput Methods Appl Mech Eng 273:204–218

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  • Zuo ZH, Xie YM (2015) A simple and compact Python code for complex 3D topology optimization. Adv Eng Softw 85:1–11

    Article  Google Scholar 

  • Zuo ZH, Huang X, Rong JH, Xie YM (2013) Multi-scale design of composite materials and structures for maximum natural frequencies. Mater Des 51:1023–1034

    Article  Google Scholar 

Download references

Acknowledgments

This work was supported jointly by the National Science Fund for Distinguished Young Scholars in China (No. 11225212), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20120161130001), the Science Fund of the State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body of China (No. 71275003), the Open Research Fund of the State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body (41215001).

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Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, People’s Republic of China

    Jie Liu & Guilin Wen

  2. Centre for Innovative Structures and Materials, School of Civil, Environmental and Chemical Engineering, RMIT University, GPO Box 2476, Melbourne, 3001, Australia

    Yi Min Xie

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  1. Jie Liu
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  2. Guilin Wen
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  3. Yi Min Xie
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Correspondence to Guilin Wen.

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Liu, J., Wen, G. & Xie, Y.M. Layout optimization of continuum structures considering the probabilistic and fuzzy directional uncertainty of applied loads based on the cloud model. Struct Multidisc Optim 53, 81–100 (2016). https://doi.org/10.1007/s00158-015-1334-9

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  • DOI: https://doi.org/10.1007/s00158-015-1334-9

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