A modified perturbed Lagrangian formulation for contact problems

Abstract

The aim of this work is to propose a formulation to solve both small and large deformation contact problems using the finite element method. We consider both standard finite elements and the so-called immersed boundary elements. The method is derived from a stabilized Nitsche formulation. After introduction of a suitable Lagrange multiplier discretization the method can be simplified to obtain a modified perturbed Lagrangian formulation. The stabilizing term is iteratively computed using a smooth stress field. The method is simple to implement and the numerical results show that it is robust. The optimal convergence rate of the finite element solution can be achieved for linear elements.

On the effect of the contact surface definition in the Cartesian grid finite element method

“…The Lagrange multipliers in the second Equation in (25) can be condensed element-wise [7] when considering the numerical integration, obtaining the following result:…”

“…In the first alternative we will consider a linear approximation of . The second alternative, first introduced in [7], includes the CAD definition of (i) c in the reference configuration, combined with the FE approximation of the displacements. Finally we present an alternative in which the CAD surface is deformed such that it fits the FE solution.…”

“…This study is focused on the solution of frictionless 3D contact problems using the cgFEM, so we recall the stabilized Lagrange functional presented in [7]. The solution of the contact problem is the displacement field u and the Lagrange multipliers field λ N that optimizes the following stabilized Lagrangian:…”

“…In order to solve the contact problem with cgFEM we use a stabilized Lagrangian formulation first presented in [7]. The method has similarities with Nitsche-based formulations proposed in [8][9][10][11] with a relevant difference in the stabilizing stress field.…”

Large deformation frictional contact analysis with immersed boundary method

“…The novelties of the present work are the use of a smooth stress field to iteratively evaluate the stabilizing term and the inclusion of the NURBS surface in the contact kinematics. The work presented in this paper represents an extension of a previous work [41], in which a stabilized formulation for solving frictionless contact problems was introduced and applied to body-fitted Finite Element meshes.…”