Solution of the contact problem using mortar-method and Schwarz alternating method on mismatched grids

The article discusses the implementation of the algorithms based on mortar-method and Schwarz alternating method for solving contact problems of elasticity theory. Solving such problems is often associated with necessity of using mismatched grids. Their joining can be carried out both with the help of iterative procedures that form the so-called Schwarz alternating methods, and with the help of the Lagrange multipliers method or the penalty method. The algorithm constructed in the article uses the mortar method for matching the finite elements on the contact line. All these methods of joining the grids make it possible to ensure continuity of displacements and stresses near the contact line. However, one of the main advantages of the mortar method is the possibility of independent choice of different types of finite elements and form functions on both boundaries of two bodies on the contact line, and when integrating along it. The application of this method in conjunction with the classical formulation of the finite element method based on the minimization of the Lagrange functional leads to a system of linear algebraic equations with a saddle point. The article discusses in detail its numerical solution based on the modified symmetric successive upper relaxation method.
One of the main advantages of the Schwarz method is the ability to reduce the solution of the general contact problem of several bodies to a sequence of solutions of standard problems of mechanics for each body separately. But the competitiveness of this method compared to the methods of penalty functions and Lagrange multipliers is largely determined by the convergence rate of the considered iterative process.
On the example of a test problem with various combinations of grid steps, some regularities are revealed. The influence of the master and slave bodies choice on the distribution of displacements and stresses on the contact line is investigated. In the case of matched grids, there are no oscillations in the graphs of the distribution of displacements and stresses, regardless of the choice of active and passive bodies. In the case of mismatched grids, the choice of a master body with a finer mesh leads to a significant decrease in oscillations of both displacements and stresses. When using the Schwarz method, fluctuations in the graphs of distributions of displacements and stresses are absent.