Comparison of the Lagrange Multipliers Function Approximation Methods in Solving Contact Problems by the Independent Contact Boundary Technique

Abstract

The paper considers the contact problem of the elasticity theory in a static spatial two-dimensional formulation without considering friction. For discretization of the elasticity theory equations, the finite element method was introduced using a triangular unstructured grid and linear and quadratic basis functions. To account for the contact boundary conditions, a modified method of Lagrange multipliers with independent contact boundary is proposed. This method implies the ability to construct a contact boundary with the smoothness degree required for the solution precision and to execute approximation of the Lagrange multiplier function independent of the grids inside the contacting bodies. Various types of the Lagrange multiplier function approximations were studied, including piecewise constant, continuous piecewise linear functions and piecewise linear functions with discontinuities at the difference cells boundaries. Examples of test calculations are provided both for problems with rectilinear and curvilinear contact boundaries. In both cases, the use of discontinuous approximations of the Lagrange multiplier function makes it possible to obtain a numerical solution with fewer artificial oscillations and higher rate of convergence at the grid refinement. It is shown that the numerical solution precision could be improved by more detailed discretization of the contact boundary without changing the grids inside the contacting bodies