A posteriori error estimation of h-p finite element approximations of frictional contact problems

doi:10.1016/0045-7825(94)90209-7

Computer Methods in Applied Mechanics and Engineering

Volume 113, Issues 1–2, March 1994, Pages 11-45

https://doi.org/10.1016/0045-7825(94)90209-7 Get rights and content

Abstract

Dynamic and static frictional contact problems are described using the normal compliance law on the contact boundary. Dynamic problems are recast into quasistatic problems by time discretization. An a posteriori error estimator is developed for a nonlinear elliptic equation of corresponding static or quasistatic problems. The a posteriori error estimator is applied to a frictionless case and extended to frictional contact problems. Also an adaptive strategy is introduced and h-p finite element meshes are obtained through a procedure based on a priori and a posteriori error estimations. Numerical examples are given to support the theoretical results.

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A simple skew-symmetric Nitsche’s formulation is introduced into the framework of isogeometric analysis (IGA) to deal with various problems in small strain elasticity: essential boundary conditions, symmetry conditions for Kirchhoff plates, patch coupling in statics and in modal analysis as well as Signorini contact conditions. For linear boundary or interface conditions, the skew-symmetric formulation is parameter-free. For contact conditions, it remains stable and accurate for a wide range of the stabilization parameter. Several numerical tests are performed to illustrate its accuracy, stability and convergence performance. We investigate particularly the effects introduced by Nitsche’s coupling, including the convergence performance and condition numbers in statics as well as the extra “outlier” frequencies and corresponding eigenmodes in structural dynamics. We present the Hertz test, the block test, and a 3D self-contact example showing that the skew-symmetric Nitsche’s formulation is a suitable approach to simulate contact problems in IGA.
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To enhance the performance adaptive techniques based on a posteriori error estimators play an important role and are well-established for finite element methods, see [41–44] and the references therein. For abstract variational inequalities, we refer to [45–47], whereas obstacle type problems are considered in [48–51], and early approaches for contact problems can be found in [52–58]. The work is structured as follows: In Section 2, the governing equations and corresponding inequality constraints for frictional contact problems are stated and reformulated as discrete non-smooth equalities.
In this paper, we consider residual and equilibrated error indicators for contact problems with Coulomb friction. The contact problem is handled within the abstract framework of saddle point problems. More precisely, the non-penetration constraint and the friction law is realized as a variationally consistent weak formulation in terms of a localized dual Lagrange multiplier space. Thus from the displacement, we can easily compute in a local post-process the Lagrange multiplier which acts as a Neumann condition on the possible contact zone. Having computed the discrete Lagrange multiplier, we can apply standard error estimators by replacing the unknown Neumann data by its approximation. As it is shown in [1], this results in an error estimator for a one-sided contact problem without friction. Here, we consider more general situations and discuss two additional contact terms which measure the non-conformity of the discrete Lagrange multiplier. Numerical results in two and three dimensions illustrate the flexibility of the approach and show the influence of the material parameters on the adaptive refinement process.
An adaptive NS/ES-FEM approach for 2D contact problems using triangular elements
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Since numerical methods for contact problems yield approximate solutions, it is necessary to control the errors inherited in the method. Based on error estimating and mesh regenerating, the adaptive techniques can provide a desirable numerical solution for contact problems in the content of finite element method (FEM) [1–11]. For FEM model creation, triangular elements are often preferred because of simplicity and strong adaptability for complex boundary.
An adaptive contact analysis approach is presented for 2D solid mechanics problems using only triangular elements and the subdomain parametric variational principle (SPVP). The present approach is implemented for the node-based smoothed FEM (or NS-FEM), the edge-based smoothed FEM (ES-FEM) and the standard FEM models with automatically adaptive refinement scheme. A modified Coulomb frictional contact model and its corresponding discrete equations are introduced. The global discretized system equations are then formulated in an incremental form with the aid of the basic boundary value equations for friction contact and the subdomain parametric variational principle. A simple adaptive refining scheme is presented, and the Voronoi vertices are taken as candidate points to become new nodes because of duality property between the Voronoi diagrams and Delaunay triangulation. The present adaptive approach can properly simulate variable behaviors of a contact interface such as bonding/debonding, contacting/departing, and sticking/slipping. Several examples are presented to numerically validate the proposed approach via the comparison with reference solutions obtained by ABAQUS^®, and to investigate the effects of the various parameters used in the computations on the response of the contact system. The numerical results have demonstrated that the present adaptive contact analysis approach using the ES-FEM has higher accuracy and convergence rate in the strain energy than that using FEM and NS-FEM. However, the latter two methods can provide the lower and upper bound solution for the system strain energy, respectively.
A posteriori error analysis for the normal compliance problem
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In this work, a contact problem between a linear elastic material and a deformable obstacle is numerically analyzed. The contact is modeled using the well-known normal compliance contact condition. The weak formulation leads to a nonlinear variational equation which is approximated by using the finite element method. A priori error estimates are recalled. Then, we define an a posteriori error estimator of residual type to evaluate the accuracy of the finite element approximation of the problem. Upper and lower bounds of the discretization error are proved for this estimator.
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In this paper, we present an adaptive finite element method for steady-state rolling contact in finite deformations along with a residual based a posteriori error estimator for rolling contact problem with Coulomb friction. A general formulation of rolling contact geometry is derived from the point of view of differential geometry. Solvability conditions for the rolling contact problems are discussed. We use Newton's method to solve variational equations derived from a penalty regularization of the finite element approximation of the rolling contact problem. We provide a numerical example to illustrate the method.
A computational methodology for shape optimization of structures in frictionless contact
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How to perform adaptive FEM for frictionless contact problems is still to date largely an open question. A few methods have been developed, see e.g. [17–19]. We have good numerical experiences with the techniques described below.
This paper presents a computational methodology for shape optimization of structures in frictionless contact, which provides a basis for developing user-friendly and efficient shape optimization software. For evaluation it has been implemented as a subsystem of a general finite element software. The overall design and main principles of operation of this software are outlined. The parts connected to shape optimization are described in more detail. The key building blocks are: analytic sensitivity analysis, an adaptive finite element method, an accurate contact solver, and a sequential convex programing optimization algorithm. Results for three model application examples are presented, in which the contact pressure and the effective stress are optimized.

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Computer Methods in Applied Mechanics and Engineering

Abstract

References (22)

Towards a Universal h-p adaptive finite element strategy: Part 1. Constrained approximation and data structure

Comput. Methods Appl. Mech. Engrg.

Toward a universal h-p adaptive finite element strategy: Part 2. A posteriori error estimation

Comput. Methods Appl. Mech. Engrg.

Toward a universal h-p adaptive finite element strategy: Part 3. Design of h-p meshes

Comput. Methods Appl. Mech. Engrg.

Models and computational methods for dynamic friction phenomena

Comput. Methods Appl. Mech. Engrg.

Frictional contact problems with normal compliance

Internat. J. Engrg. Sci.

A priori error estimation of hp-finite element approximations of frictional contact problems with normal compliances

Internat. J. Engrg. Sci.

On friction problems with normal compliance

Nonlinear Anal. Theory Methods Appl.

On some existence and uniqueness results in contact problems with nonlocal friction

Nonlinear Anal. Theory Methods Appl.

A posteriori error estimates for the finite element method

Internat. J. Numer. Methods Engrg.

A posteriori error analysis and adaptive processes in the finite element method: Part I — Error analysis

Internat. J. Numer. Methods Engrg.

Advances in adaptive improvements: a survey of adaptive finite element methods in computational mechanics

Cited by (32)

Skew-symmetric Nitsche's formulation in isogeometric analysis: Dirichlet and symmetry conditions, patch coupling and frictionless contact

Equilibration techniques for solving contact problems with Coulomb friction

An adaptive NS/ES-FEM approach for 2D contact problems using triangular elements

A posteriori error analysis for the normal compliance problem

On the adaptive finite element method of steady-state rolling contact for hyperelasticity in finite deformations

A computational methodology for shape optimization of structures in frictionless contact