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Calcolo
Published in: Calcolo 1/2018

01-03-2018

An inexact Newton method for solving complementarity problems in hydrodynamic lubrication

Authors: Francesco Mezzadri, Emanuele Galligani

Published in: Calcolo | Issue 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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Footnotes
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.
 
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Metadata
Title
An inexact Newton method for solving complementarity problems in hydrodynamic lubrication
Authors
Francesco Mezzadri
Emanuele Galligani
Publication date
01-03-2018
Publisher
Springer Milan
Published in
Calcolo / Issue 1/2018
Print ISSN: 0008-0624
Electronic ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-018-0244-9

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