A stabilized Lagrange multiplier method for the enriched finite-element approximation of Tresca contact problems of cracked elastic bodies

Abstract

In this paper we propose a local projection stabilized Lagrange multiplier method in order to approximate the two-dimensional linear elastostatics unilateral contact problem with Tresca friction in the framework of the eXtended Finite Element Method X-FEM. This last method allows to perform finite-element computations on cracked domains by using meshes of the non-cracked domain. The advantage of the used stabilization technique is to affect only the equation on multipliers and thus to be equation independent. We study the existence, uniqueness and a priori error estimate of three hybrid discrete formulations.

Boundary augmented Lagrangian method for contact problems in linear elasticity

“…We can use properties of projection and Steklov-Poincaré operator (also known as the Dirichlet-to-Neumann mapping) to analyse the convergence of the method. Usually the augmented Lagrangian method (ALM) needs to solve a nonlinear problem in every iteration step, but the semismooth Newton method can be applied for the solution [11,12,14]. Numerical results show that our method is accurate and efficient.…”

“…The second approach to solve the problem is to use the Lagrange multiplier. Then the contact problem is transformed into a sequence of linear elasticity problems [11][12][13][14][15][16][17]. The development of new, fast, and reliable methods for the numerical simulation of contact problems is still an area of frequent research [18][19][20][21][22][23][24][25][26].…”

Boundary Element and Augmented Lagrangian Methods for Contact Problem with Coulomb Friction

“…An option is to discrete the problem by the finite element method (FEM) [2][3][4][5] or the boundary element method (BEM) [6][7][8][9][10] and obtain a convex optimization problem in the finite dimensional space. Another option is to use the Lagrange multiplier which replaces the nonlinear problem with a sequence of linear problems in function spaces, and this idea has been introduced in [11][12][13][14][15][16][17][18]. Recently, some new methods for the numerical simulation of the friction contact problem have also been developed, and we mention the penalty method and Nitsche's method [19][20][21][22][23][24][25][26][27].…”

“…When compared with other methods, there is no inequality constraint, and a boundary integral equation is introduced, which is useful from both theoretical and numerical points of view. At each iteration step, our method only needs updating boundary values and solving a boundary variational problem, and we can apply the semismooth Newton method for the solution [11,12,15]. Using properties of projection and the Steklov-Poincaré operator (also known as the Dirichlet-to-Neumann mapping), we obtain the convergence in function spaces.…”