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Calcolo
Erschienen in: Calcolo 1/2018

01.03.2018

An inexact Newton method for solving complementarity problems in hydrodynamic lubrication

verfasst von: Francesco Mezzadri, Emanuele Galligani

Erschienen in: Calcolo | Ausgabe 1/2018

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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.
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Fußnoten
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.
 
Literatur
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)MathSciNetCrossRefMATH
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)MathSciNetCrossRef
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)MathSciNetCrossRefMATH
6.
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)CrossRef
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)CrossRef
10.
Bernstein, D.S.: Matrix Mathematics: Theory, Facts and Formulas, 2nd edn. Princeton University Press, Princeton (2009)MATH
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)CrossRef
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)MathSciNetCrossRefMATH
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
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)MathSciNetCrossRefMATH
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)MathSciNetCrossRefMATH
17.
Cottle, R.W., Dantzig, G.B.: Complementarity pivot theory of mathematical programming. Linear Algebra Appl. 1, 103–125 (1968)MathSciNetCrossRefMATH
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)MathSciNetCrossRefMATH
21.
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)MathSciNetCrossRefMATH
23.
Durazzi, C.: On the Newton interior-point method for nonlinear programming problems. J. Optim. Theory Appl. 104, 73–90 (2000)MathSciNetCrossRefMATH
24.
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)MathSciNetCrossRefMATH
26.
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)MathSciNetCrossRefMATH
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.
31.
Kelley, C.: Iterative Methods for Linear and Nonlinear Equations. Frontiers in Applied Mathematics. SIAM, Philadelphia (1995)CrossRef
32.
Khonsari, M., Booser, E.: Applied Tribology: Bearing, Design and Lubrication, 2nd edn. Wiley, Chichester (2008)CrossRef
33.
Kostreva, M.: Elasto-hydrodynamic lubrication: a nonlinear complementarity problem. Int. J. Numer. Meth. Fl 4, 377–397 (1984)CrossRefMATH
34.
Lakshmikantham, V., Trigiante, D.: Theory of Difference Equations: Numerical Methods and Applications, 2nd edn. Marcel Dekker Inc., New York (2002)CrossRefMATH
35.
Laratta, A., Menchi, O.: Approssimazione della soluzione di una disequazione variazionale. Applicazione ad un problema di frontiera libera. Calcolo 11, 243–267 (1974)MathSciNetCrossRefMATH
36.
37.
McAllister, G., Rohde, S.: An optimization problem in hydrodynamic lubrication theory. Appl. Math. Optim. 2, 223–235 (1976)MathSciNetCrossRef
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)MathSciNetCrossRefMATH
39.
40.
Oh, K., Goenka, P.: The elastohydrodynamic solution of journal bearings under dynamic loading. J. Tribol. 107(3), 389–395 (1985)CrossRef
41.
Rheinboldt, W.: Methods for Solving Systems of Nonlinear Equations, 2nd edn. SIAM, Philadelphia (1998)CrossRefMATH
42.
Siefert, C., De Sturler, E.: Preconditioners for generalized saddle-point problems. SIAM J. Numer. Anal. 44(3), 1275–1296 (2006)MathSciNetCrossRefMATH
43.
Süli, E., Mayers, D.: An Introduction to Numerical Analysis. Cambridge University Press, Cambridge (2003)CrossRefMATH
44.
45.
Woloszynski, T., Podsiadlo, P., Stachowiak, G.W.: Efficient solution to the cavitation problem in hydrodynamic lubrication. Tribol. Lett. 58, 1–11 (2015)CrossRef
Metadaten
Titel
An inexact Newton method for solving complementarity problems in hydrodynamic lubrication
verfasst von
Francesco Mezzadri
Emanuele Galligani
Publikationsdatum
01.03.2018
Verlag
Springer Milan
Erschienen in
Calcolo / Ausgabe 1/2018
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-018-0244-9

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