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Contact Optimization Problems for Stationary and Sliding Conditions

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Mathematical Modeling and Optimization of Complex Structures

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 40))

Abstract

The contact stress distribution is frequently not regular. It may contain singularities reducing the lifetime of machine elements. In order to eliminate such stress singularities, the application of contact pressure control is recommended in the contact conditions. In the paper, several classes of optimization problems are formulated for stationary and sliding contacts. Further, they are illustrated by specific examples. The relation to wear process is made as a natural way to attain the steady state contact profile satisfying the optimality conditions corresponding to minimization of the wear dissipation rate. It is assumed that the displacements and strains are small and the materials of the contacting bodies are elastic.

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References

  1. N.V. Banichuk, P.J. Neittaanmäki, Structural Optimization with Uncertainties. Solid Mechanics and Its Applications, vol. 162 (Springer, Berlin, 2010)

    Google Scholar 

  2. N.V. Banichuk, Introduction to Optimization of Structures (Springer, London, 2011). Reprint of the original 1st edn. (1990)

    Google Scholar 

  3. Z. Mróz, Optimal design of structures of composite materials. Int. J. Solids Struct. 6(7), 859–870 (1970)

    Article  Google Scholar 

  4. Z. Mróz, Design sensitivity of critical loads and vibration frequencies of nonlinear structures, in Optimization of Large Structural Systems, vol. I, of NATO ASI Series, Series E: Applied Sciences, vol. 321, ed. by G.I.N. Rozvany (Kluwer Academic Publishers, Dordrecht, 1993), pp. 455–476

    Google Scholar 

  5. Z. Mróz, D. Bojczuk, Shape and topology sensitivity analysis and its application to structural design. Arch. Appl. Mech. 82(10–11), 1541–1555 (2012)

    Article  MATH  Google Scholar 

  6. Z. Mróz, R.T. Haftka, Design sensitivity analysis of non-linear structures in regular and critical states. Int. J. Solids Struct. 31(15), 2071–2098 (1994)

    Article  MATH  Google Scholar 

  7. Z. Mróz, T. Lekszycki, Optimal support reaction in elastic frame structures. Comput. Struct. 14(3–4), 179–185 (1981)

    Article  MATH  Google Scholar 

  8. J. Haslinger, P.P. Neittaanmäki, Finite Element Approximation for Optimal Shape Design: Theory and Applications (Wiley, London, 1988)

    MATH  Google Scholar 

  9. R.L. Benedict, J.E. Taylor, Optimal design for elastic bodies in contact, in Optimization of Distributed Parameter Structures, vol. II, ed. by E.J. Haug, J. Cea (Sijthoff and Noordhoff, Alphen aan den Rijn, 1981), pp. 1553–1599

    Chapter  Google Scholar 

  10. N. Kikuchi, J.E. Taylor, Shape optimization for unilateral elastic contact problems, in Numerical Methods in Coupled Problems (University College, Swansea, 1981), pp. 430–441

    Google Scholar 

  11. A. Klarbring, On the problem of optimizing contact force distributions. J. Optim. Theory Appl. 74(1), 131–150 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Klarbring, J. Haslinger, On almost constant contact stress distributions by shape optimization. Struct. Multidisc. Optim. 5(4), 213–216 (1993)

    Article  Google Scholar 

  13. T.F. Conry, A. Seireg, A mathematical programming method for design of elastic bodies in contact. J. Appl. Mech. 38(2), 387–392 (1971)

    Article  Google Scholar 

  14. J. Oda, J. Sakamoto, K. San, A method for producing a uniform contact stress distribution in composite bodies with interface. Struct. Multidisc. Optim. 3(1), 23–28 (1991)

    Article  Google Scholar 

  15. I. Páczelt, B. Herpai, Some remarks on the solution of contact problems of elastic shells. Archiwum Budowy Maszyn 24, 197–202 (1977)

    Google Scholar 

  16. J. Petersson, Behaviourally constrained contact force optimization. Struct. Multidisc. Optim. 9(3–4), 189–193 (1995)

    Article  Google Scholar 

  17. I. Páczelt, Some new developments in contact pressure optimization. Eng. Trans. 43(1–2), 297–312 (1995)

    Google Scholar 

  18. I. Páczelt, Iterative methods for solution of contact optimization problems. Arch. Mech. 52(4–5), 685–711 (2000)

    MATH  Google Scholar 

  19. I. Páczelt, A. Baksa, Examination of contact optimization and wearing problems. J. Comput. Appl. Mech. 3(1), 61–84 (2002)

    MATH  Google Scholar 

  20. I. Páczelt, T. Szabó, Optimal shape design for contact problems. Struct. Multidisc. Optim. 7(1–2), 66–75 (1994)

    Article  Google Scholar 

  21. B. Szabó, I. Babuska, Finite Element Analysis (Wiley, New York, 1991)

    MATH  Google Scholar 

  22. D. Hilding, A. Klarbring, J. Petersson, Optimization of structures in unilateral contact. Appl. Mech. Rev. 52(4), 139–160 (1999)

    Article  Google Scholar 

  23. P. Beremlijski, J. Haslinger, M. Kočvara, R. Kučera, J.V. Outrata, Shape optimization in three-dimensional contact problems with coulomb friction. SIAM J. Optim. 20(1), 416–444 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Czekanski, S.A. Meguid, Solution of dynamic frictional contact problems using nondifferentiable optimization. Internat. J. Mech. Sci. 43(6), 1369–1386 (2001)

    Article  MATH  Google Scholar 

  25. D. Hilding, The equilibrium state of a structure subject to frictional contact. Eur. J. Mech. A Solids 19(6), 1029–1040 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. D. Hilding, A. Klarbring, Optimization of structures in frictional contact. Comput. Methods Appl. Mech. Eng. 205–208, 83–90 (2012)

    Article  MathSciNet  Google Scholar 

  27. Y. Kanno, J.A.C. Martins, Arc-length method for frictional contact problems using mathematical programming with complementarity constraints. J. Optim. Theory Appl. 131(1), 89–113 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. N.H. Kim, K.K. Choi, J.S. Chen, Shape design sensitivity analysis and optimization of elasto-plasticity with frictional contact. AIAA J. 38(9), 1742–1753 (2000)

    Article  Google Scholar 

  29. A. Klarbring, A. Mikelič, M. Shillor, Optimal shape design in contact problems with normal compliance and friction. Appl. Math. Lett. 5(2), 51–55 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  30. G.E. Stavroulakis, Optimal prestress of structures with frictional unilateral contact interfaces. Arch. Appl. Mech. 66(1–2), 71–81 (1995)

    Article  MATH  Google Scholar 

  31. W. Li, Q. Li, G.P. Steven, Y.M. Xie, An evolutionary approach to elastic contact optimization of frame structures. Finite Elem. Anal. Des. 40(1), 61–81 (2003)

    Article  Google Scholar 

  32. A. Myslinski, Level set method for optimization of contact problems. Eng. Anal. Bound. Elem. 32(11), 986–994 (2008)

    Article  MATH  Google Scholar 

  33. W. Li, Q. Li, G.P. Steven, Y.M. Xie, An evolutionary shape optimization for elastic contact problems subject to multiple load cases. Comput. Methods Appl. Mech. Eng. 194(30–33), 3394–3415 (2005)

    Article  MATH  Google Scholar 

  34. N.V. Banichuk, S.Y. Ivanova, Shape optimization in contact problems of the theory of elasticity with incomplete external loading data. J. Appl. Math. Mech. 73(6), 696–704 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. N.V. Banichuk, Optimization of the contact pressure in the problem of the interaction of a punch and an elastic medium. J. Appl. Math. Mech. 74(3), 334–340 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. B. Desmorat, Structural rigidity optimization with frictionless unilateral contact. Int. J. Solids Struct. 44(3–4), 1132–1144 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. D. Hilding, B. Torstenfelt, A. Klarbring, A computational methodology for shape optimization of structures in frictionless contact. Comput. Methods Appl. Mech. Eng. 190(31), 4043–4060 (2001)

    Article  MATH  Google Scholar 

  38. F.F. Mahmoud, A.A. El-Sharkawy, K.M. Hassan, Contour design for contact stress minimization by interior penalty method. Appl. Math. Model. 13(10), 596–600 (1989)

    Article  MATH  Google Scholar 

  39. Y. Tada, S. Nishihara, Optimum shape design of contact surface with finite element method. Adv. Eng. Softw. 18(2), 75–85 (1993)

    Article  Google Scholar 

  40. S. Šimunović, S. Saigal, Contact surface optimization using boundary element method. Comput. Struct. 56(5), 745–750 (1995)

    Article  MATH  Google Scholar 

  41. H. Ou, B. Lu, Z.S. Cui, C. Lin, A direct shape optimization approach for contact problems with boundary stress concentration. J. Mech. Sci. Tech. 27(9), 2751–2759 (2013)

    Article  Google Scholar 

  42. M. Ignesti, A. Innocenti, L. Marini, E. Meli, A. Rindi, P. Toni, Wheel profile optimization on railway vehicles from the wear viewpoint. Int. J. Non-Linear Mech. 53, 41–54 (2013)

    Article  Google Scholar 

  43. R. Smallwood, J.C. Sinclair, K.J. Sawley, An optimization technique to minimize rail contact stresses. Wear 144(1–2), 373–384 (1991)

    Article  Google Scholar 

  44. S. Zakharov, I. Goryacheva, V. Bogdanov, D. Pogorelov, I. Zharov, V. Yazykov, E. Torskaya, S. Soshenkov, Problems with wheel and rail profiles selection and optimization. Wear 265(9–10), 1266–1272 (2008)

    Article  Google Scholar 

  45. P.C. Sui, S. Anderle, Optimization of contact pressure profile for performance improvement of a rotary elastomeric seal operating in abrasive drilling environment. Wear 271(9–10), 2466–2470 (2011)

    Article  Google Scholar 

  46. I. Páczelt, Z. Mróz, Optimal shapes of contact interfaces due to sliding wear in the steady relative motion. Int. J. Solids Struct. 44(3–4), 895–925 (2007)

    Article  MATH  Google Scholar 

  47. I. Páczelt, Z. Mróz, On the analysis of steady-state sliding wear processes. Trib. Int. 42(2), 275–283 (2009)

    Article  Google Scholar 

  48. I. Páczelt, Z. Mróz, Numerical analysis of steady thermo-elastic wear regimes induced by translating and rotating punches. Comput. Struct. 89(23–24), 2495–2521 (2011)

    Article  Google Scholar 

  49. I.G. Goryacheva, Contact Mechanics in Tribology (Kluwer Academic Publishers, Dordrecht, 1998)

    Book  MATH  Google Scholar 

  50. I. Páczelt, Some optimization problems of contact bodies within the linear theory of elasticity, in Variational methods in the Mechanics of Solids (Evanston, IL, 1978), ed. by S. Nemat-Nasser (Pergamon Press, Oxford, 1980), pp. 349–356

    Google Scholar 

  51. I. Páczelt, A. Baksa, T. Szabó, Product design using a contact-optimization technique. Strojniški Vestnik J. Mech. Eng. 53(7–8), 442–461 (2007)

    Google Scholar 

  52. J.J. Kalker, Three-dimensional Elastic Bodies in Rolling Contact (Kluwer Academic Publishers, Dordrecht, 1990)

    Book  MATH  Google Scholar 

  53. I. Páczelt, Z. Mróz, On steady wear states for monotonic relative sliding of contacting bodies. Key Eng. Mater. 618, 49–71 (2014)

    Article  Google Scholar 

  54. I. Páczelt, Z. Mróz, Analysis of thermo-mechanical wear problems for reciprocal punch sliding. Adv. Eng. Softw. 80, 139–155 (2015)

    Article  Google Scholar 

  55. I. Páczelt, Z. Mróz, Solution of wear problems for monotonic and periodic sliding with p-version of finite element method. Comput. Methods Appl. Mech. Eng. 249–252, 75–103 (2012)

    Article  Google Scholar 

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Acknowledgments

The present research was partially supported by the Hungarian Academy of Sciences, by grants OTKA K67825 and K115701, within the program TÁMOP 4.2.1.B-10/2/KONV-2010-0001.

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Authors and Affiliations

  1. Institute of Applied Mechanics, University of Miskolc, 3515, Miskolc-egyetemváros, Hungary

    István Páczelt & Attila Baksa

  2. Institute of Fundamental Technological Research, A. Pawińskiego 5B, 02-106, Warsaw, Poland

    Zenon Mróz

Authors
  1. István Páczelt
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  2. Attila Baksa
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  3. Zenon Mróz
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Correspondence to István Páczelt .

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Editors and Affiliations

  1. Mathematical Information Technology, University of Jyväskylä, Jyväskylä, Finland

    Pekka Neittaanmäki

  2. Mathematical Information Technology, University of Jyväskylä, Jyväskylä, Finland

    Sergey Repin

  3. Mathematical Information Technology, University of Jyväskylä, Jyväskylä, Finland

    Tero Tuovinen

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Páczelt, I., Baksa, A., Mróz, Z. (2016). Contact Optimization Problems for Stationary and Sliding Conditions. In: Neittaanmäki, P., Repin, S., Tuovinen, T. (eds) Mathematical Modeling and Optimization of Complex Structures. Computational Methods in Applied Sciences, vol 40. Springer, Cham. https://doi.org/10.1007/978-3-319-23564-6_16

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  • DOI: https://doi.org/10.1007/978-3-319-23564-6_16

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  • Online ISBN: 978-3-319-23564-6

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