Mathematical Modeling of the Contact Interaction of Two Elastic Bodies Using the Mortar Method

Abstract

The article discusses the development of an algorithm 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 both 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.The results of the constructed algorithm are demonstrated on three test contact problems. They analyze the stress-strain state of differently loaded contacting two-dimensional plates. The examples considered show that continuity of the displacements of displacements and stresses is preserved near the contact line. The versatility of the developed algorithm leaves the possibility of further analysis of the effectiveness of the mortar method using different types of finite elements and form functions.