Skip to main content
SpringerLink
Log in

An inexact Newton method for solving complementarity problems in hydrodynamic lubrication

  • Published:
Calcolo Aims and scope Submit manuscript

Abstract

We present an iterative procedure based on a damped inexact Newton iteration for solving linear complementarity problems. We introduce the method in the framework of a popular problem arising in mechanical engineering: the analysis of cavitation in lubricated contacts. In this context, we show how the perturbation and the damping parameter are chosen in our method and we prove the global convergence of the entire procedure. A Fortran implementation of the method is finally analyzed. First, we validate the procedure and analyze all its components, performing also a comparison with a recently proposed technique based on the Fischer–Burmeister–Newton iteration. Then, we solve a 2D problem and provide some insights on an efficient implementation of the method exploiting routines of the Lapack and of the PETSc packages for the solution of inner linear systems.

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

Similar content being viewed by others

Notes

  1. If we replace \(\varphi ^{(k)}(\alpha )\) and \(\psi ^{(k)}(\alpha )\) with their expressions in (28) and (29), it is easy to note that the condition \(\varphi ^{(k)}(\alpha ) \ge 0\) forces the iterates far from \((\,{{\varvec{p}}}, {{\varvec{r}}})\ge 0\), while \(\psi ^{(k)}(\alpha ) \ge 0\) makes the sequence \(\{ {{{\varvec{p}}}^{(k)}(\alpha )}^T {{\varvec{r}}}^{(k)}(\alpha ) \}\) to converge to zero slower than \(\{ \Vert {{\varvec{F}}}_1(\,{{\varvec{p}}}^{(k)}(\alpha ), {{\varvec{r}}}^{(k)}(\alpha )) \Vert \}\).

  2. This reduction, introduced in [5], consists in reducing the value of \(\alpha _k\) computed by the feasibility conditions multiplying it by a factor \(\hat{\theta }\) defined as

    $$\begin{aligned} \hat{\theta } = {\left\{ \begin{array}{ll} \max (0.8, 1-100(\,{{\varvec{p}}}^{(k)^T} {{\varvec{r}}}^{(k)})) &{} \text{ if } \alpha _k = 1 \\ \max (0.8, \min (0.9995,1-100(\,{{\varvec{p}}}^{(k)^T} {{\varvec{r}}}^{(k)}))) &{} \text{ if } \alpha _k < 1 \end{array}\right. } \end{aligned}$$

  3. The use of PETSc allows also to easily change solver and it provides a parallel implementation of the linear solvers through MPI. For better reproducibility, all the result here reported have however been obtained by running the programs sequentially on a single core.

References

  1. Almqvist, A., Wall, P.: Modelling cavitation in (elasto)hydrodynamic lubrication. In: Darji, P.H. (ed.) Advances in Tribology (2016)

  2. Amestoy, P.R., Duff, I.S., Koster, J., L’Excellent, J.Y.: A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix. Anal. A 23(1), 15–41 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Amestoy, P.R., Guermouche, A., L’Excellent, J.Y., Pralet, S.: Hybrid scheduling for the parallel solution of linear systems. Parallel Comput. 32(2), 136–156 (2006)

    Article  MathSciNet  Google Scholar 

  4. Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.: LAPACK Users’ Guide, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia, PA (1999)

  5. Argaez, M., Tapia, R., Velazquez, L.: Numerical comparisons of path-following strategies for a primal-dual interior-point method for nonlinear programming. J. Optim. Theory App. 114, 255–272 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Armijo, L.: Minimization of functions having Lipschitz-continuous first partial derivatives. Pac. J. Math. 16, 1–3 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  7. Balay, S., Gropp, W.D., McInnes, L.C., Smith, B.F.: Efficient management of parallelism in object oriented numerical software libraries. In: Bruaset, A.M., Langtangen, H.P., Arge, E. (eds.) Modern Software Tools in Scientific Computing, pp. 163–202. Birkhäuser Press, Cambridge (1997)

    Chapter  Google Scholar 

  8. Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Rupp, K., Smith, B.F., Zampini, S., Zhang, H., Zhang, H.: PETSc users manual. Tech. Rep. ANL-95/11 - Revision 3.8, Argonne National Laboratory, (2017). http://www.mcs.anl.gov/petsc

  9. Benzi, M., Wathen, A.J.: Some preconditioning techniques for saddle point problems. In: Schilders, W.H.A., van der Vorst, H.A., Rommes, J. (eds.) Model Order Reduction: Theory, Research Aspects and Applications, vol. 13, pp. 195–211. Springer, Berlin (2008)

    Chapter  Google Scholar 

  10. Bernstein, D.S.: Matrix Mathematics: Theory, Facts and Formulas, 2nd edn. Princeton University Press, Princeton (2009)

    MATH  Google Scholar 

  11. Bertocchi, L., Dini, D., Giacopini, M., Fowell, M., Baldini, A.: Fluid film lubrication in the presence of cavitation: a mass-conserving two-dimensional formulation for compressible, piezoviscous and non-Newtonian fluids. Tribol. Int. 67, 61–71 (2013)

    Article  Google Scholar 

  12. Bonettini, S., Galligani, E., Ruggiero, V.: Inner solvers for interior point methods for large scale nonlinear programming. Comput. Optim. Appl. 37, 1–34 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bonneau, D., Hajjam, M.: Modélisation de la rupture et de la reformation des films lubrifiants dans les contacts élastohydrodynamiques. Eur. J. Comput. Mech. 10, 679–704 (2001)

    MATH  Google Scholar 

  14. Capriz, G., Cimatti, G.: Free boundary problems in the theory of hydrodynamic lubrication: A survey. Tech. Rep. Nota Informatica C81-7, Istituto di Matematica, University of Pisa (1981)

  15. Chapiro, G., Gutierrez, A.E., Herskovits, J., Mazorche, S.R., Pereira, W.S.: Numerical solution of a class of moving boundary problems with a nonlinear complementarity approach. J. Optim. Theory Appl. 168(2), 534–550 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cimatti, G., Menchi, O.: On the numerical solution of a variational inequality connected with the hydrodynamica lubrication of a complete journal bearing. Calcolo 15, 249–258 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cottle, R.W., Dantzig, G.B.: Complementarity pivot theory of mathematical programming. Linear Algebra Appl. 1, 103–125 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Classics in Applied Mathematics, SIAM (2009)

  19. Cryer, C.: The numerical solution of a degenerate variational inequality. In: Parter, S.V. (ed.) Numerical Methods for Partial Differential Equations (1979)

  20. Cryer, C., Dempster, A.: Equivalence of linear complementarity problems and linear programs in vector lattice Hilbert spaces. SIAM J. Control Optim. 18(1), 76–90 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dembo, R., Eisenstat, S., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal. 19, 400–408 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  22. Demmel, J.W., Eisenstat, S.C., Gilbert, J.R., Li, X.S., Liu, J.W.H.: A supernodal approach to sparse partial pivoting. SIAM J. Matrix Anal. A 20(3), 720–755 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Durazzi, C.: On the Newton interior-point method for nonlinear programming problems. J. Optim. Theory Appl. 104, 73–90 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Eisenstat, S., Walker, H.: Globally convergent inexact Newton methods. SIAM J. Optim. 4, 393–422 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  25. El-Bakri, A., Tapia, R., Tsuchiya, T., Zhang, Y.: On the formulation and theory of the Newton interior point method for nonlinear programming. J. Optim. Theory Appl. 89, 507–541 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Fischer, A.: A special Newton-type optimization method. Optimization 24, 269–284 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  27. Fokkema, D., Sleijpen, G., Van der Vost, H.: Accelerated inexact Newton schemes for large systems of nonlinear equations. SIAM J. Sci. Comput. 19, 657–674 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Galligani, E.: Analysis of the convergence of an inexact Newton method for solving Karush-Kuhn-Tucker systems. In: Atti del Seminario Matematico e Fisico dell’Università di Modena e Reggio Emilia LII, pp. 331–368 (2004)

  29. Giacopini, M., Fowell, M., Dini, D., Strozzi, A.: A mass-conserving complementarity formulation to study lubricant films in the presence of cavitation. J. Tribol. 132(4), 041702 (2010)

  30. Horn, R., Johnson, C.: Matrix Analysis. Cambridge University Press, Cambridge (1985)

    Book  MATH  Google Scholar 

  31. Kelley, C.: Iterative Methods for Linear and Nonlinear Equations. Frontiers in Applied Mathematics. SIAM, Philadelphia (1995)

    Book  Google Scholar 

  32. Khonsari, M., Booser, E.: Applied Tribology: Bearing, Design and Lubrication, 2nd edn. Wiley, Chichester (2008)

    Book  Google Scholar 

  33. Kostreva, M.: Elasto-hydrodynamic lubrication: a nonlinear complementarity problem. Int. J. Numer. Meth. Fl 4, 377–397 (1984)

    Article  MATH  Google Scholar 

  34. Lakshmikantham, V., Trigiante, D.: Theory of Difference Equations: Numerical Methods and Applications, 2nd edn. Marcel Dekker Inc., New York (2002)

    Book  MATH  Google Scholar 

  35. Laratta, A., Menchi, O.: Approssimazione della soluzione di una disequazione variazionale. Applicazione ad un problema di frontiera libera. Calcolo 11, 243–267 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  36. Li, X., Demmel, J., Gilbert, J., Grigori, L., Shao, M., Yamazaki, I.: SuperLU Users’ Guide. Tech. Rep. LBNL-44289, Lawrence Berkeley National Laboratory, http://crd.lbl.gov/~xiaoye/SuperLU/ (1999)

  37. McAllister, G., Rohde, S.: An optimization problem in hydrodynamic lubrication theory. Appl. Math. Optim. 2, 223–235 (1976)

    Article  MathSciNet  Google Scholar 

  38. Morini, B., Simoncini, V., Tani, M.: A comparison of reduced and unreduced kkt systems arising from interior point methods. Comput. Optim. Appl. 68(1), 1–27 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  39. Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York (1999)

    Book  MATH  Google Scholar 

  40. Oh, K., Goenka, P.: The elastohydrodynamic solution of journal bearings under dynamic loading. J. Tribol. 107(3), 389–395 (1985)

    Article  Google Scholar 

  41. Rheinboldt, W.: Methods for Solving Systems of Nonlinear Equations, 2nd edn. SIAM, Philadelphia (1998)

    Book  MATH  Google Scholar 

  42. Siefert, C., De Sturler, E.: Preconditioners for generalized saddle-point problems. SIAM J. Numer. Anal. 44(3), 1275–1296 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  43. Süli, E., Mayers, D.: An Introduction to Numerical Analysis. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  44. Varga, R.: Matrix Iterative Analysis, 2nd edn. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  45. Woloszynski, T., Podsiadlo, P., Stachowiak, G.W.: Efficient solution to the cavitation problem in hydrodynamic lubrication. Tribol. Lett. 58, 1–11 (2015)

    Article  Google Scholar 

Download references

Acknowledgements

The authors desire to thank the anonymous referees for their valuable comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Engineering “Enzo Ferrari”, University of Modena and Reggio Emilia, via P Vivarelli 10/1, building 26, 41125, Modena, Italy

    Francesco Mezzadri & Emanuele Galligani

Authors
  1. Francesco Mezzadri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Emanuele Galligani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francesco Mezzadri.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mezzadri, F., Galligani, E. An inexact Newton method for solving complementarity problems in hydrodynamic lubrication. Calcolo 55, 1 (2018). https://doi.org/10.1007/s10092-018-0244-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10092-018-0244-9

Keywords

Mathematics Subject Classification

Search

Navigation