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Optimal design of wheel rim in elastic mechanics

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Abstract

In the context of shape optimization, this article proposes a numerical method for designing wheel rims as linear elastic structures. Our approach is based on a combination of the classical shape derivative and the Lagrangian method. The compliant shape derivative with respect to the domain variation has been computed by using Céa fast derivation method. With a novel utilization of the gradient shape optimization strategy, we present a numerical scheme for the optimal design of wheel rims. A remeshing technique is implemented to achieve the regularization of shapes. Design applications are examined on a general vehicle in various circumstances to illustrate the efficiency of the presented scheme. The proposed method applies to different wheel designs due to its simplicity and generic nature.

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References

  1. Wang, X., Zhang, X.: Simulation of dynamic cornering fatigue test of a steel passenger car wheel. Int. J. Fatigue 32(2), 434–442 (2010). https://doi.org/10.1016/j.ijfatigue.2009.09.006

    Article  Google Scholar 

  2. Zhu, Z., Zhu, Y., Wang, Q.: Fatigue mechanisms of wheel rim steel under off-axis loading. Mater. Sci. Eng. A 773, 138731 (2020). https://doi.org/10.1016/j.msea.2019.138731

    Article  Google Scholar 

  3. Zanchini, M., Longhi, D., Mantovani, S., Puglisi, F., Giacalone, M.: Fatigue and failure analysis of aluminium and composite automotive wheel rims: experimental and numerical investigation. Eng. Fail. Anal. 146, 107064 (2023). https://doi.org/10.1016/j.engfailanal.2023.107064

    Article  Google Scholar 

  4. Akbulut, H.: On optimization of a car rim using finite element method. Finite Elem. Anal. Des. 39(5–6), 433–443 (2003). https://doi.org/10.1016/s0168-874x(02)00091-4

    Article  Google Scholar 

  5. Schäfer, C., Finke, E.: Shape optimisation by design of experiments and finite element methods-an application of steel wheels. Struct. Multidiscip. Optim. 36(5), 477–491 (2007). https://doi.org/10.1007/s00158-007-0183-6

    Article  Google Scholar 

  6. Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71(2), 197–224 (1988). https://doi.org/10.1016/0045-7825(88)90086-2

    Article  MathSciNet  MATH  Google Scholar 

  7. Zuo, Z.H., Xie, Y.M., Huang, X.: Reinventing the wheel. J. Mech. Des. 133(2) (2011) https://doi.org/10.1115/1.4003411

  8. Xiao, D., Zhang, H., Liu, X., He, T., Shan, Y.: Novel steel wheel design based on multi-objective topology optimization. J. Mech. Sci. Technol. 28(3), 1007–1016 (2014). https://doi.org/10.1007/s12206-013-1174-8

    Article  Google Scholar 

  9. Ta, T.T.M.: Modélisation des problèmes bi-fluides par la méthode des lignes de niveaux et l’adaptation du maillage: Application à l’optimisation des formes. Ph.D. thesis, Univ. Pierre et Marie Curie (2015)

  10. Ta, T.T.M., Le, V.C., Pham, H.T.: Shape optimization for stokes flows using sensitivity analysis and finite element method. Appl. Numer. Math. 126, 160–179 (2018). https://doi.org/10.1016/j.apnum.2017.12.009

    Article  MathSciNet  MATH  Google Scholar 

  11. Le, V.C., Ta, T.T.M.: Constrained shape optimization problem in elastic mechanics. Comput. Appl. Math. 40(7), 23 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  12. Braess, D.: Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  13. Ciarlet, P.G.: Mathematical Elasticity: Three-Dimensional Elasticity. Society for Industrial and Applied Mathematics, Philadelphia (2021). https://doi.org/10.1137/1.9781611976786

  14. Allaire, G., Gournay, F., Jouve, F., Toader, A.M.: Structural optimization using topological and shape sensitivity via a level set method. Control. Cybern. 34(1), 59–80 (2005)

    MathSciNet  MATH  Google Scholar 

  15. Allaire, G., Dapogny, C., Frey, P.: Shape optimization with a level set based mesh evolution method. Comput. Methods Appl. Mech. Eng. 282, 22–53 (2014). https://doi.org/10.1016/j.cma.2014.08.028

    Article  MathSciNet  MATH  Google Scholar 

  16. Zeidler, E.: Nonlinear Functional Analysis and Its Applications. Springer, New York (1990)

    Book  MATH  Google Scholar 

  17. Allaire, G., Jouve, F., Toader, A.M.: Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194(1), 363–393 (2004). https://doi.org/10.1016/j.jcp.2003.09.032

    Article  MathSciNet  MATH  Google Scholar 

  18. Sigmund, O.: On the design of compliant mechanisms using topology optimization. Mech. Struct. Mach. 25(4), 493–524 (1997). https://doi.org/10.1080/08905459708945415

    Article  Google Scholar 

  19. Allaire, G., Jouve, F.: Optimal design of micro-mechanisms by the homogenization method. Rev. Européenne des Éléments Finis 11(2–4), 405–416 (2002). https://doi.org/10.3166/reef.11.405-416

    Article  MATH  Google Scholar 

  20. Allaire, G.: Conception Optimale de Structures vol. 58. Springer, Heidelberg (2006). https://doi.org/10.1007/978-3-540-36856-4

  21. Duysinx, P., Miegroet, L.V., Lemaire, E., Brüls, O., Bruyneel, M.: Topology and generalized shape optimization: Why stress constraints are so important? Int. J. Simul. Multi. Design Optim. 2(4), 253–258 (2008). https://doi.org/10.1051/ijsmdo/2008034

    Article  Google Scholar 

  22. Allaire, G., Jouve, F., Michailidis, G.: Molding direction constraints in structural optimization via a level-set method. In: Variational Analysis and Aerospace Engineering, pp. 1–39. Springer, Switzerland (2016). https://doi.org/10.1007/978-3-319-45680-5_1

  23. Allaire, G., Jouve, F., Michailidis, G.: Thickness control in structural optimization via a level set method. Struct. Multidiscip. Optim. 53(6), 1349–1382 (2016). https://doi.org/10.1007/s00158-016-1453-y

    Article  MathSciNet  Google Scholar 

  24. Hadamard, J.: Mémoire sur le problème d’analyse relatif à l’équilibre des plaque élastiques encastrées. Bull. Soc. Math. France, Technical report (1907)

  25. Sokolowski, J., Zolesio, J.-P.: Introduction to Shape Optimization; Shape Sensitivity Analysis. Series in Computational Mathematics, vol. 16. Springer, Heidelberg (1992). https://doi.org/10.1007/978-3-642-58106-9

  26. Murat, F., Simon, J.: Sur le contrôle par un domaine géométrique. Rr-76015, INRIA Rocquencourt (1976)

  27. Simon, J.: Differentiation with respect to the domain in boundary value problems. Numer. Funct. Anal. Optim. 2(7–8), 649–687 (1980). https://doi.org/10.1080/01630563.1980.10120631

    Article  MathSciNet  MATH  Google Scholar 

  28. Manzoni, A., Quarteroni, A., Salsa, S.: Optimal Control of Partial Differential Equations. Springer, Switzerland (2021). https://doi.org/10.1007/978-3-030-77226-0

  29. Azegami, H.: Shape Optimization Problems. Springer, Singapore (2020). https://doi.org/10.1007/978-981-15-7618-8

  30. Céa, J.: Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût. ESAIM: Math. Model. Numer. Anal. 20(3), 371–402 (1986) https://doi.org/10.1051/m2an/1986200303711

  31. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Grundlehren Der Mathematishen Wissenschaften 170. Springer, Heidelberg (1971)

  32. Pironneau, O.: Optimal Shape Design for Elliptic Systems. Springer, New York (1984). https://doi.org/10.1007/978-3-642-87722-3

  33. Allaire, G., Pantz, O.: Structural optimization with freefem++. Struct. Multidiscip. Optim. 32(3), 173–181 (2006). https://doi.org/10.1007/s00158-006-0017-y

    Article  MathSciNet  MATH  Google Scholar 

  34. Doǧan, G., Morin, P., Nochetto, R.H., Verani, M.: Discrete gradient flows for shape optimization and applications. Comput. Methods Appl. Mech. Eng. 196(37–40), 3898–3914 (2007). https://doi.org/10.1016/j.cma.2006.10.046

    Article  MathSciNet  MATH  Google Scholar 

  35. Allaire, G., Dapogny, C., Jouve, F.: Shape and topology optimization. In: Geometric Partial Differential Equations - Part II, pp. 1–132. Elsevier, Netherlands (2021). https://doi.org/10.1016/bs.hna.2020.10.004

  36. Schulz, V., Siebenborn, M., Welker, K.: Efficient pde constrained shape optimization based on steklov-poincaré type metrics (2015). https://doi.org/10.48550/ARXIV.1506.02244

  37. Heinkenschloss, M., Vicente, L.N.: Analysis of inexact trust-region SQP algorithms. SIAM J. Optim. 12(2), 283–302 (2002). https://doi.org/10.1137/s1052623499361543

    Article  MathSciNet  MATH  Google Scholar 

  38. Brockman, R.A.: Geometric sensitivity analysis with isoparametric finite elements. Commun. Appl. Numer. Methods 3(6), 495–499 (1987). https://doi.org/10.1002/cnm.1630030609

    Article  MATH  Google Scholar 

  39. Hecht, F., Pironneau, O., Hyaric, A.L., Ohtsuka, K.: Freefem++ manual. J. Numer. Math. 20 (2012)

  40. Hecht, F.: New development in freefem++. J. Numer. Math. 20(3–4), 251–265 (2012)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to thank the anonymous referees for their remarks which helped to significantly improve the paper. This work was supported by Vietnam Ministry of Education and Training under grant number B2024.BKA.18.

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  1. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Dai Co Viet, Hanoi, 11657, Vietnam

    Thi Thanh Mai Ta & Quang Huy Nguyen

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Ta, T.T.M., Nguyen, Q.H. Optimal design of wheel rim in elastic mechanics. Int J Adv Eng Sci Appl Math (2023). https://doi.org/10.1007/s12572-023-00362-3

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