Skip to main content
SpringerLink
Log in

Robust topology optimization with interval field model: on the spatially varied non-probabilistic uncertainty of material property, loading and geometry

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

In this work, we introduce the non-probabilistic uncertainties of spatial variation into the topology optimization problem. We investigate the material properties, geometry and loading uncertainties. A recently emerged interval field model is employed for modeling these spatially varied uncertainties. Based on the robust topology optimization framework, we propose an interval-field based perturbation analysis (IFPA) method for predicting the median and radius of structural compliance under uncertainty, and the sensitivity analysis is developed accordingly. Three numerical examples are presented, in which topology optimization problems with uncertainty aroused from material properties, loading and geometry are discussed separately. Comparing the results with those of Monte Carlo simulations, we illustrate the accuracy and efficiency of the IFPA in predicting structural compliance. The topology optimization results demonstrate the merit of emphasizing spatial dependence in topology optimization with non-probabilistic uncertainty.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

Data availability

The computational codes and numerical results presented in the paper are available from the authors upon reasonable request.

References

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

    MathSciNet  Google Scholar 

  2. Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Google Scholar 

  5. Guo X, Zhang W, Zhong W (2014) Doing topology optimization explicitly and geometrically—a new moving morphable components based framework. J Appl Mech 81(8):081009

  6. Aage N, Andreassen E, Lazarov BS, Sigmund O (2017) Giga-voxel computational morphogenesis for structural design. Nature 550:84–86

    Google Scholar 

  7. Zhu J-H, Zhang W-H, Xia L (2016) Topology optimization in aircraft and aerospace structures design. Arch Comput Methods Eng 23:595–622

    MathSciNet  Google Scholar 

  8. Helton JC, Johnson J, Oberkampf WL, Storlie CB (2007) A sampling-based computational strategy for the representation of epistemic uncertainty in model predictions with evidence theory. Comput Methods Appl Mech Eng 196:3980–3998

    MathSciNet  Google Scholar 

  9. Dantan J-Y, Gayton N, Qureshi AJ, Lemaire M, Etienne A (2013) Tolerance analysis approach based on the classification of uncertainty (aleatory/epistemic). Procedia CIRP 10:287–293

    Google Scholar 

  10. Ang AH-S, Tang WH (1984) Probability concepts in engineering planning and design, vol. 2: decision, risk, and reliability. Wiley, New York

    Google Scholar 

  11. Ghanem RG, Spanos PD (1991) Stochastic finite elements: a spectral approach. Springer

  12. Stefanou G (2009) The stochastic finite element method: past, present and future. Comput Methods Appl Mech Eng 198:1031–1051

    Google Scholar 

  13. Jaynes ET (1957) Information theory and statistical mechanics. Phys Rev 106:620

    MathSciNet  Google Scholar 

  14. Soize C (2001) Maximum entropy approach for modeling random uncertainties in transient elastodynamics. J Acoust Soc Am 109:1979–1996

    Google Scholar 

  15. Kruse, R., Meyer, K.D. (1988). Confidence Intervals for the Parameters of a Linguistic Random Variable. In: Kacprzyk, J., Fedrizzi, M. (eds) Combining Fuzzy Imprecision with Probabilistic Uncertainty in Decision Making. Lecture Notes in Economics and Mathematical Systems, vol 310. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46644-1_8

  16. Beer M, Ferson S, Kreinovich V (2013) Imprecise probabilities in engineering analyses. Mech Syst Signal Process 37:4–29

    Google Scholar 

  17. Moens D, Vandepitte D (2005) A survey of non-probabilistic uncertainty treatment in finite element analysis. Comput Methods Appl Mech Eng 194:1527–1555

    Google Scholar 

  18. Qiu Z, Elishakoff I (1998) Antioptimization of structures with large uncertain-but-non-random parameters via interval analysis. Comput Methods Appl Mech Eng 152:361–372

    Google Scholar 

  19. Ben-Haim Y, Elishakoff I (2013) Convex models of uncertainty in applied mechanics. Elsevier, Amsterdam

    Google Scholar 

  20. Zadeh LA (1984) Review of a mathematical theory of evidence. AI Mag 5:81–81

    Google Scholar 

  21. Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1:3–28

    MathSciNet  Google Scholar 

  22. Ben-Haim Y (1994) A non-probabilistic concept of reliability. Struct Saf 14:227–245

    Google Scholar 

  23. Ben-Haim Y, Elishakoff I (1995) Discussion on: a non-probabilistic concept of reliability. Struct Saf 17:195–199

    Google Scholar 

  24. Jiang C, Han X, Lu G, Liu J, Zhang Z, Bai Y (2011) Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique. Comput Methods Appl Mech Eng 200:2528–2546

    Google Scholar 

  25. Jiang C, Zhang Q, Han X, Qian Y (2014) A non-probabilistic structural reliability analysis method based on a multidimensional parallelepiped convex model. Acta Mech 225:383–395

    Google Scholar 

  26. Jiang C, Zhang Q, Han X, Liu J, Hu D (2015) Multidimensional parallelepiped model—a new type of non-probabilistic convex model for structural uncertainty analysis. Int J Numer Methods Eng 103:31–59

    MathSciNet  Google Scholar 

  27. Wang C, Matthies HG (2020) A modified parallelepiped model for non-probabilistic uncertainty quantification and propagation analysis. Comput Methods Appl Mech Eng 369:113209

    MathSciNet  Google Scholar 

  28. Moens D, Munck MD, Desmet W, Vandepitte D (2011) Numerical dynamic analysis of uncertain mechanical structures based on interval fields. IUTAM symposium on the vibration analysis of structures with uncertainties. Springer, Berlin, pp 71–83

    Google Scholar 

  29. Faes M, Moens D (2017) Identification and quantification of spatial interval uncertainty in numerical models. Comput Struct 192:16–33

    Google Scholar 

  30. van Mierlo C, Faes MG, Moens D (2021) Inhomogeneous interval fields based on scaled inverse distance weighting interpolation. Comput Methods Appl Mech Eng 373:113542

    MathSciNet  Google Scholar 

  31. Callens RR, Faes MG, Moens D (2021) Local explicit interval fields for non-stationary uncertainty modelling in finite element models. Comput Methods Appl Mech Eng 379:113735

    MathSciNet  Google Scholar 

  32. Sofi A (2015) Structural response variability under spatially dependent uncertainty: stochastic versus interval model. Probab Eng Mech 42:78–86

    Google Scholar 

  33. Wu D, Gao W (2017) Hybrid uncertain static analysis with random and interval fields. Comput Methods Appl Mech Eng 315:222–246

    MathSciNet  Google Scholar 

  34. Luo Y, Zhan J, Xing J, Kang Z (2019) Non-probabilistic uncertainty quantification and response analysis of structures with a bounded field model. Comput Methods Appl Mech Eng 347:663–678

    MathSciNet  Google Scholar 

  35. Zhan J, Luo Y, Zhang X, Kang Z (2020) A general assessment index for non-probabilistic reliability of structures with bounded field and parametric uncertainties. Comput Methods Appl Mech Eng 366:113046

    MathSciNet  Google Scholar 

  36. Ni BY, Jiang C (2020) Interval field model and interval finite element analysis. Comput Methods Appl Mech Eng 360:112713

  37. Jiang C, Ni B, Han X, Tao Y (2014) Non-probabilistic convex model process: a new method of time-variant uncertainty analysis and its application to structural dynamic reliability problems. Comput Methods Appl Mech Eng 268:656–676

    MathSciNet  Google Scholar 

  38. Hu H, Wu Y, Batou A, Ouyang H (2022) B-spline based interval field decomposition method. Comput Struct 272:106874

    Google Scholar 

  39. Kharmanda G, Olhoff N, Mohamed A, Lemaire M (2004) Reliability-based topology optimization. Struct Multidiscipl Optim 26:295–307

    Google Scholar 

  40. Luo Y, Kang Z, Luo Z, Li A (2009) Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model. Struct Multidiscipl Optim 39:297–310

    MathSciNet  Google Scholar 

  41. Meng Z, Guo L, Yıldız AR, Wang X (2022) Mixed reliability-oriented topology optimization for thermo-mechanical structures with multi-source uncertainties. Eng Comput 38:5489–5505

    Google Scholar 

  42. Ni B, Wang X, Lv T, Wang L, Li Z (2022) Non-probabilistic thermo-elastic reliability-based topology optimization (NTE-RBTO) of composite laminates with interval uncertainties. Eng Comput 38:5713–5732

    Google Scholar 

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

    MathSciNet  Google Scholar 

  44. Chen S, Chen W, Lee S (2010) Level set based robust shape and topology optimization under random field uncertainties. Struct Multidiscipl Optim 41:507–524

    MathSciNet  Google Scholar 

  45. Hoang V-N, Pham T, Ho D, Nguyen-Xuan H (2022) Robust multiscale design of incompressible multi-materials under loading uncertainties. Eng Comput 38:875–890

    Google Scholar 

  46. Enevoldsen I, Sørensen JD (1994) Reliability-based optimization in structural engineering. Struct Saf 15:169–196

    Google Scholar 

  47. Tu J, Choi KK, Park YH (1999) A new study on reliability-based design optimization. J Mech Des 121:557–564

    Google Scholar 

  48. De Gournay F, Allaire G, Jouve F (2008) Shape and topology optimization of the robust compliance via the level set method. ESAIM Control Optim Calculus Var 14:43–70

    MathSciNet  Google Scholar 

  49. Amir O, Sigmund O, Lazarov BS, Schevenels M (2012) Efficient reanalysis techniques for robust topology optimization. Comput Methods Appl Mech Eng 245:217–231

    MathSciNet  Google Scholar 

  50. Tootkaboni M, Asadpoure A, Guest JK (2012) Topology optimization of continuum structures under uncertainty—a polynomial chaos approach. Comput Methods Appl Mech Eng 201:263–275

    MathSciNet  Google Scholar 

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

    Google Scholar 

  52. Wu Y, Li E, He ZC, Lin XY, Jiang HX (2020) Robust concurrent topology optimization of structure and its composite material considering uncertainty with imprecise probability. Comput Methods Appl Mech Eng 364:112927

    MathSciNet  Google Scholar 

  53. da Silva GA, Cardoso EL, Beck AT (2020) Comparison of robust, reliability-based and non-probabilistic topology optimization under uncertain loads and stress constraints. Probab Eng Mech 59:103039

    Google Scholar 

  54. Zhan J, Luo Y (2019) Robust topology optimization of hinge-free compliant mechanisms with material uncertainties based on a non-probabilistic field model. Front Mech Eng 14:201–212

    Google Scholar 

  55. Luo Y, Zhan J (2020) Linear buckling topology optimization of reinforced thin-walled structures considering uncertain geometrical imperfections. Struct Multidiscipl Optim 62:3367–3382

    MathSciNet  Google Scholar 

  56. Wang L, Li Z, Ni B, Wang X, Chen W (2022) A robust topology optimization method considering bounded field parameters with uncertainties based on the variable time step parametric level-set method. Appl Math Model 107:441–463

    MathSciNet  Google Scholar 

  57. Van Trees HL (2004) Detection, estimation, and modulation theory, part I: detection, estimation, and linear modulation theory. Wiley, New York

    Google Scholar 

  58. Atkinson K, Han W (2009) Numerical solution of Fredholm integral equations of the second kind. Springer, New York, pp 473–549

    Google Scholar 

  59. Ni BY, Wu PG, Li JY, Jiang C (2020) A semi-analytical interval method for response bounds analysis of structures with spatially uncertain loads. Finite Elem Anal Des 182:103483

    MathSciNet  Google Scholar 

  60. Ghanem R (1999) The nonlinear Gaussian spectrum of log-normal stochastic processes and variables. J Appl Mech 66:964–973

    Google Scholar 

  61. Ghanem RG, Spanos PD (1991) Spectral stochastic finite-element formulation for reliability analysis. J Eng Mech 117:2351–2372

    Google Scholar 

  62. Torres AP, Warner JE, Aguiló MA, Guest JK (2021) Robust topology optimization under loading uncertainties via stochastic reduced order models. Int J Numer Methods Eng 122:5718–5743

    MathSciNet  Google Scholar 

  63. Zhang J, Xiao M, Li P, Gao L (2022) Quantile-based topology optimization under uncertainty using kriging metamodel. Comput Methods Appl Mech Eng 393:114690

    MathSciNet  Google Scholar 

  64. Zhou E, Wu Y, Lin X, Li Q, Xiang Y (2021) A normalization strategy for beso-based structural optimization and its application to frequency response suppression. Acta Mech 232:1307–1327

    MathSciNet  Google Scholar 

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

    Google Scholar 

  66. Olhoff N, Du J (2009) On topological design optimization of structures against vibration and noise emission. Springer Vienna, Vienna, pp 217–276

    Google Scholar 

  67. Neves M, Rodrigues H, Guedes J (1995) Generalized topology design of structures with a buckling load criterion. Struct Optim 10:71–78

    Google Scholar 

  68. Silva OM, Neves MM, Lenzi A (2019) A critical analysis of using the dynamic compliance as objective function in topology optimization of one-material structures considering steady-state forced vibration problems. J Sound Vib 444:1–20

    Google Scholar 

  69. Tortorelli DA, Michaleris P (1994) Design sensitivity analysis: overview and review. Inverse Probl Eng 1:71–105

    Google Scholar 

  70. Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190:3443–3459

    Google Scholar 

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

    Google Scholar 

  72. Svanberg K (1987) The method of moving asymptotes-a new method for structural optimization. Int J Numer Methods Eng 24:359–373

    MathSciNet  Google Scholar 

  73. Hu H, Wu Y, Batou A, Ouyang H (2023) Uncertainty propagation with b-spline based interval field decomposition method in boundary value problems. Appl Math Model (Accepted)

  74. Torii AJ (2019) Robust compliance-based topology optimization: a discussion on physical consistency. Comput Methods Appl Mech Eng 352:110–136

    MathSciNet  Google Scholar 

  75. Dunning PD, Kim HA, Mullineux G (2011) Introducing loading uncertainty in topology optimization. AIAA J 49:760–768

    Google Scholar 

  76. Ngoc NM, Hoang V-N, Lee D (2022) Concurrent topology optimization of coated structure for non-homogeneous materials under buckling criteria. Eng Comput 38:5635–5656

    Google Scholar 

  77. Banh TT, Lieu QX, Lee J, Kang J, Lee D (2023) A robust dynamic unified multi-material topology optimization method for functionally graded structures. Struct Multidiscipl Optim 66:75

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The work is supported by the Key Project of Chongqing Technological Innovation and Application Development Special Projects (CSTB2022TIAD-DEX0011), and partially supported by Leverhulme Trust Research Fellowship. Z. C He acknowledges the support from the National Natural Science Foundation of China (Grant No. U20A20285) and the Key Research and Development Program of Liuzhou (Grant No.2021AAA0111). We are grateful to Prof. Svanberg from KTH, Sweden for providing the code for the MMA algorithm. Yi Wu acknowledges the China Scholarship Council (CSC No. 201906130024).

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, University of Science and Technology Beijing, Beijing, 100083, People’s Republic of China

    Yi Wu

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

    Yi Wu, Jing Zheng & Z. C. He

  3. Univ Gustave Eiffel, CNRS, MSME UMR 8208, 77454, Marne-la-Vallée, France

    Yi Wu & Han Hu

  4. Department of Mechanical, Materials and Aerospace Engineering, School of Engineering, University of Liverpool, Liverpool, L69 7ZF, UK

    Han Hu

  5. School of Computing, Engineering and Digital Technologies, Teesside University, Middlesbrough, UK

    Yining Zhang & Eric Li

Authors
  1. Yi Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Han Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jing Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yining Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Eric Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Z. C. He
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Eric Li or Z. C. He.

Ethics declarations

Conflict of interest

The authors state that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, Y., Hu, H., Zheng, J. et al. Robust topology optimization with interval field model: on the spatially varied non-probabilistic uncertainty of material property, loading and geometry. Engineering with Computers 40, 1093–1109 (2024). https://doi.org/10.1007/s00366-023-01850-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-023-01850-7

Keywords

Search

Navigation