Topology optimization of elastic contact problems using B-spline parameterization

Abstract

This work extends B-spline parameterization method to topology optimization of elastic contact problems. Unlike the traditional density-based method, design variables directly refer to the control parameters of the B-spline. A continuous pseudo-density field representing the material distribution over the concerned design domain is constructed by means of B-spline parameterization and then discretized onto the finite element (FE) mesh. The threshold projection is further introduced to regularize the B-spline pseudo-density field for the reduction of gray areas related to the local support property of B-spline. 2D and 3D frictionless and frictional problems are solved to demonstrate the effectiveness of the proposed method. Results are also compared with those obtained by the traditional density-based method. It is shown that the B-spline parameterization is independent of the FE model and suitable to deal with contact problems of inherent contact nonlinearity. The optimized configuration with refined details and smoothed boundaries can be obtained.

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References

  1. Alberdi R, Zhang GD, Li L, Khandelwal K (2018) A unified framework for nonlinear path-dependent sensitivity analysis in topology optimization. Int J Numer Methods Eng 115:1–56

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

  3. Bendsoe MP, Sigmund O (2013) Topology optimization: theory, methods, and applications. Springer Science & Business Media,

    Google Scholar 

  4. Benedict RL (1982) Maximum stiffness design for elastic bodies in contact. J Mech Des 104:825–830

    Google Scholar 

  5. Bennett JA, Botkin ME (1985) Structural shape optimization with geometric description and adaptive mesh refinement. AIAA J 23:458–464

    Article  Google Scholar 

  6. Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50:2143–2158

    MathSciNet  MATH  Article  Google Scholar 

  7. Bruggi M, Duysinx P (2013) A stress-based approach to the optimal design of structures with unilateral behavior of material or supports. Struct Multidiscip Optim 48:311–326

    MathSciNet  Article  Google Scholar 

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

    MATH  Article  Google Scholar 

  9. Buhl T, Pedersen CBW, Sigmund O (2000) Stiffness design of geometrically nonlinear structures using topology optimization. Struct Multidiscip Optim 19:93–104

    Article  Google Scholar 

  10. Costa G, Montemurro M, Pailhès J (2018) A 2D topology optimisation algorithm in NURBS framework with geometric constraints. Int J Mech Mater Des 14:669–696

    Article  Google Scholar 

  11. Desmorat B (2007) Structural rigidity optimization with frictionless unilateral contact. Int J Solids Struct 44:1132–1144

    MathSciNet  MATH  Article  Google Scholar 

  12. Fancello E (2006) Topology optimization for minimum mass design considering local failure constraints and contact boundary conditions. Struct Multidiscip Optim 32:229–240

    MathSciNet  MATH  Article  Google Scholar 

  13. Gao J, Gao L, Luo Z, Li PG (2019) Isogeometric topology optimization for continuum structures using density distribution function. Int J Numer Methods Eng 119:991–1017

    MathSciNet  Article  Google Scholar 

  14. Guest JK, Prevost JH, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61:238–254

    MathSciNet  MATH  Article  Google Scholar 

  15. Haug EJ, Kwak BM (1978) Contact stress minimization by contour design. Int J Numer Methods Eng 12:917–930

    MATH  Article  Google Scholar 

  16. Jeong GE, Youn SK, Park KC (2018) Topology optimization of deformable bodies with dissimilar interfaces. Comput Struct 198:1–11

    Article  Google Scholar 

  17. Kang Z, Wang YQ (2011) Structural topology optimization based on non-local Shepard interpolation of density field. Comput Methods Appl Mech Eng 200:3515–3525

    MathSciNet  MATH  Article  Google Scholar 

  18. Kawamoto A, Matsumori T, Yamasaki S, Nomura T, Kondoh T, Nishiwaki S (2011) Heaviside projection based topology optimization by a PDE-filtered scalar function. Struct Multidiscip Optim 44:19–24

    MATH  Article  Google Scholar 

  19. Klarbring A, Petersson J, Rönnqvist M (1995) Truss topology optimization including unilateral contact. J Optim Theory Appl 87:1–31

    MathSciNet  MATH  Article  Google Scholar 

  20. Kristiansen H, Poulios K, Aage N (2020) Topology optimization for compliance and contact pressure distribution in structural problems with friction. Comput Methods Appl Mech Eng 364:112915

    MathSciNet  MATH  Article  Google Scholar 

  21. Lawry M, Maute K (2015) Level set topology optimization of problems with sliding contact interfaces. Struct Multidiscip Optim 52:1107–1119

    MathSciNet  Article  Google Scholar 

  22. Lawry M, Maute K (2018) Level set shape and topology optimization of finite strain bilateral contact problems. Int J Numer Methods Eng 113:1340–1369

    MathSciNet  Article  Google Scholar 

  23. Li W, Li Q, Steven GP, Xie YM (2003) An evolutionary approach to elastic contact optimization of frame structures. Finite Elem Anal Des 40:61–81

    Article  Google Scholar 

  24. Li Y, Zhu JH, Wang FW, Zhang WH, Sigmund O (2019) Shape preserving design of geometrically nonlinear structures using topology optimization. Struct Multidiscip Optim 59:1033–1051

    MathSciNet  Article  Google Scholar 

  25. Lopes CG, dos Santos RB, Novotny AA, Sokolowski J (2017) Asymptotic analysis of variational inequalities with applications to optimum design in elasticity. Asymptot Anal 102:227–242

    MathSciNet  MATH  Article  Google Scholar 

  26. Luo YJ, Li M, Kang Z (2016) Topology optimization of hyperelastic structures with frictionless contact supports. Int J Solids Struct 81:373–382

    Article  Google Scholar 

  27. Ma YH, Chen XQ, Zuo WJ (2020) Equivalent static displacements method for contact force optimization. Struct Multidiscip Optim:1–14

  28. Mankame ND, Ananthasuresh GK (2004) Topology optimization for synthesis of contact-aided compliant mechanisms using regularized contact modeling. Comput Struct 82:1267–1290

    Article  Google Scholar 

  29. Meng L et al (2019) From topology optimization design to additive manufacturing: today’s success and tomorrow’s roadmap. Arch Comput Methods Eng 27:805–830

    Article  Google Scholar 

  30. Michaleris P, Tortorelli DA, Vidal CA (1994) Tangent operators and design sensitivity formulations for transient non-linear coupled problems with applications to elastoplasticity. Int J Numer Methods Eng 37:2471–2499

    MATH  Article  Google Scholar 

  31. Mulaik SA (2009) Foundations of factor analysis. CRC press

  32. Myśliński A (2015) Piecewise constant level set method for topology optimization of unilateral contact problems. Adv Eng Softw 80:25–32

    Article  Google Scholar 

  33. Myśliński A, Wróblewski M (2017) Structural optimization of contact problems using Cahn-Hilliard model. Comput Struct 180:52–59

    Article  Google Scholar 

  34. Niu C, Zhang WH, Gao T (2019) Topology optimization of continuum structures for the uniformity of contact pressures. Struct Multidiscip Optim 60:185–210

    MathSciNet  Article  Google Scholar 

  35. Niu C, Zhang WH, Gao T (2020) Topology optimization of elastic contact problems with friction using efficient adjoint sensitivity analysis with load increment reduction. Comput Struct 238:106296

    Article  Google Scholar 

  36. Oden JT, Pires EB (1983) Nonlocal and nonlinear friction laws and variational principles for contact problems in elasticity. J Appl Mech 50:67–76

    MathSciNet  MATH  Article  Google Scholar 

  37. Paczelt I, Baksa A, Mroz Z (2016) Contact optimization problems for stationary and sliding conditions vol 40. Mathematical Modeling and Optimization of Complex Structures. Springer-Verlag Berlin, Berlin

  38. Petersson J, Patriksson M (1997) Topology optimization of sheets in contact by a subgradient method. Int J Numer Methods Eng 40:1295–1321

    MathSciNet  MATH  Article  Google Scholar 

  39. Piegl L, Tiller W (2012) The NURBS book. Springer Science & Business Media

    Google Scholar 

  40. Qian XP (2013) Topology optimization in B-spline space. Comput Methods Appl Mech Eng 265:15–35

    MathSciNet  MATH  Article  Google Scholar 

  41. Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33:401–424

    Article  Google Scholar 

  42. Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidiscip Optim 48:1031–1055

    MathSciNet  Article  Google Scholar 

  43. Sokół T, Rozvany GIN (2013) Exact truss topology optimization for external loads and friction forces. Struct Multidiscip Optim 48:853–857

    MathSciNet  Article  Google Scholar 

  44. Stromberg N (2010) Topology optimization of structures with manufacturing and unilateral contact constraints by minimizing an adjustable compliance-volume product. Struct Multidiscip Optim 42:341–350

    MathSciNet  MATH  Article  Google Scholar 

  45. Strömberg N (2010) Topology optimization of two linear elastic bodies in unilateral contact. In: Proc. of the 2nd Int Conf on Engineering Optimization, Lisbon, Portugal, 2010

  46. Stromberg N (2013) The influence of sliding friction on optimal topologies. In: Stavroulakis GE (ed) Recent advances in contact mechanics, vol 56. Lecture Notes in Applied and Computational Mechanics. pp 327-336

  47. Stromberg N, Klarbring A (2010) Topology optimization of structures in unilateral contact. Struct Multidiscip Optim 41:57–64

    MathSciNet  MATH  Article  Google Scholar 

  48. Svanberg K (1995) A globally convergent version of MMA without linesearch. In: Proceedings of the first world congress of structural and multidisciplinary optimization, 1995. Goslar, Germany, pp 9–16

    Google Scholar 

  49. Wang MM, Qian XP (2015) Efficient filtering in topology optimization via b-splines. J Mech Des 137

  50. Wang FW, Lazarov BS, Sigmund O (2010) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43:767–784

    MATH  Article  Google Scholar 

  51. Wang J, Zhu JH, Hou J, Wang C, Zhang WH (2020) Lightweight design of a bolt-flange sealing structure based on topology optimization. Struct Multidiscip Optim

  52. Wriggers P (2006) Computational contact mechanics, 2nd edn. Springer, Berlin

    Google Scholar 

  53. Xia L, Fritzen F, Breitkopf P (2017) Evolutionary topology optimization of elastoplastic structures. Struct Multidiscip Optim 55:569–581

    MathSciNet  Article  Google Scholar 

  54. Xu SL, Cai YW, Cheng GD (2010) Volume preserving nonlinear density filter based on heaviside functions. Struct Multidiscip Optim 41:495–505

    MathSciNet  MATH  Article  Google Scholar 

  55. Xu Z, Zhang WH, Gao T, Zhu JH (2020) A B-spline multi-parameterization method for multi-material topology optimization of thermoelastic structures. Struct Multidiscip Optim 61:923–942

    MathSciNet  Article  Google Scholar 

  56. Yang RJ, Chen CJ (1996) Stress-based topology optimization. Struct Optim 12:98–105

    Article  Google Scholar 

  57. Zhang WH, Niu C (2018) A linear relaxation model for shape optimization of constrained contact force problem. Comput Struct 200:53–67

    Article  Google Scholar 

  58. Zhang WH, Zhou Y, Zhu JH (2017) A comprehensive study of feature definitions with solids and voids for topology optimization. Comput Methods Appl Mech Eng 325:289–313

    MathSciNet  MATH  Article  Google Scholar 

Download references

Funding

This work is supported by the National Key Research and Development Program of China (2017YFB1102800), National Natural Science Foundation of China (12032018, 11620101002), and China Postdoctoral Science Foundation (2020M673520).

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Correspondence to Weihong Zhang.

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Li, J., Zhang, W., Niu, C. et al. Topology optimization of elastic contact problems using B-spline parameterization. Struct Multidisc Optim (2021). https://doi.org/10.1007/s00158-020-02837-4

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Keywords

  • Topology optimization
  • B-spline parameterization
  • Contact problems