An inexact Newton method for solving complementarity problems in hydrodynamic lubrication

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.

Splitting Methods for a Class of Horizontal Linear Complementarity Problems

“…5 and 6, we then present and solve several numerical examples, ranging from random test problems to HLCPs of practical interest. We thus validate the procedures, demonstrate their efficiency and also perform a comparison with an IP method recently introduced for solving HLCPs in hydrodynamic lubrication [11]. Finally, the conclusions of this work are summarized in Sect.…”

“…Then, we consider HLCPs arising in mechanical engineering to model cavitation (which is the formation of gaseous bubbles) in hydrodynamic lubrication. The formulation of the problem in complementarity form can be found in [15], while we follow [11] for the HLCP formulation arising from finite difference discretizations of the differential problem. The complementarity variables represent here pressure and density and are thus denoted by p and r (instead of x and y).…”

“…We solve the four test problems presented in [11] and identify them by P1, P2, P3 and P4, respectively. These problems, whose solution is known, encompass various cases which can occur.…”

Some Unconstrained Optimization Methods

“…Some interesting applications of Newton, modified Newton, inexact Newton, and quasi-Newton methods can be found, for example, in [73][74][75][76][77][78][79][80][81][82][83], etc.…”

Modulus‐based synchronous multisplitting methods for solving horizontal linear complementarity problems on parallel computers

“…In particular, the HLCP( A , B , q ) consists in finding a pair of vectors such that with . Solution techniques for these problems include reduction to LCP, 29,30 interior point methods, 31,32 homotopy approaches, 33 neural networks, 34 and projected splitting methods 35 . Recently, HLCPs have been reformulated as an implicit fixed‐point system and solved by generalizations of modulus‐based matrix splitting methods, 28 demonstrating the effectiveness of such strategies for solving HLCPs.…”