Three Fixed Point Strategies For The Unilateral Contact Between Two Membranes

The unilateral contact problem between two membranes has a wide range of appli-cations in the field of elasticity,such as the obstacle problem and the Signorini problem.This problem is not only nonlinear,but also involves relation between the displacements and interaction force of two elastic membranes,which make the problem much difficult for study of the problem.Usually,the contact between two membranes is formulated as variational inequalities for the properties of the solution.Following the work,we use the saddle point problem based on Lagrangian and augmented Lagrangian functions,and obtain three numerical methods:the Uzawa method,the semi-smooth Newton method and the augmented Lagrangian method.For the unilateral contact problem between two membranes,this dissertation first summarizes some basic properties.Then we propose an Uzawa method for the problem.The method introduces the Lagrangian function to transform the original problem into a saddle point problem,which is solved by the Uzawa method.This method is simple in an iterative scheme and easy to obtain the convergence analysis,but the convergence parameter must be small enough and is difficult to estimate.Then,a semi-smooth Newton method for the unilateral contact problem between two membranes is developed.Using a regularization problem of the problem,we apply a method combination of the Newton method with generalized differential to the problem,and the convergence of the method is proved.The semi-smooth Newton method converges to the solution only for sufficiently large parameter and the proof of convergence is tedious.In order to improve the performance of the method,we developed the lagrangian method based on semi-smooth Newton method.In particular,this method converges for all the positive parameters.Differential operators are used to approximate the differential operators,and the numerical results are analyzed and compared for the proposed methods.