Comparison of different error estimators for contact problems (original) (raw)
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An Error Estimator For Adaptive Frictionless Contact Finite Element Analysis
1998
The derivation of an a posteriori error estimator for frictionless contact problems under the hypotheses of linear elastic behavior and infinitesimal deformation is presented. The approximated solution of this problem is obtained by using the Finite Element Method. A Penalization or Augmented-Lagrangian technique is used to deal with the unilateral boundary condition over the contact boundary. An a posteriori error estimator suitable for adaptive mesh refinement in this problem is proposed, together with its mathematical justification. Up to the present time, this mathematical proof is restricted to the Penalization approach. Several numerical results are reported in order to corroborate the applicability of this estimator and to compare it with other a posteriori error estimators.
Residual a posteriori error estimators for contact problems in elasticity
ESAIM: Mathematical Modelling and Numerical Analysis, 2007
This paper is concerned with the unilateral contact problem in linear elasticity. We define two a posteriori error estimators of residual type to evaluate the accuracy of the mixed finite element approximation of the contact problem. Upper and lower bounds of the discretization error are proved for both estimators and several computations are performed to illustrate the theoretical results.
Residual-based a posteriori error estimation for contact problems approximated by Nitsche’s method
Ima Journal of Numerical Analysis, 2017
We introduce a residual-based a posteriori error estimator for contact problems in two and three dimensional linear elasticity, discretized with linear and quadratic finite elements and Nitsche's method. Efficiency and reliability of the estimator are proved under a saturation assumption. Numerical experiments illustrate the theoretical properties and the good performance of the estimator.
An adaptive nite element approach for frictionless contact problems
SUMMARY The derivation of an a posteriori error estimator for frictionless contact problems under the hypotheses of linear elastic behaviour and innitesimal deformation is presented. The approximated solution of this problem is obtained by using the nite element method. A penalization or augmented-Lagrangian technique is used to deal with the unilateral boundary condition over the contact boundary. An a posteriori error estimator suitable for adaptive mesh renement in this problem is proposed, together with its mathematical justication. Up to the present time, this mathematical proof is restricted to the penalization approach. Several numerical results are reported in order to corroborate the applicability of this estimator and to compare it with other a posteriori error estimators. Copyright ? 2001 John Wiley & Sons, Ltd.
A Posteriori Error Estimator and Error Control for Contact Problems
2009
Abstract. In this paper, we consider two error estimators for one-body contact problems. The first error estimator is defined in terms of H(div)-conforming stress approximations and equilibrated fluxes while the second is a standard edge-based residual error estimator without any modification with respect to the contact. We show reliability and efficiency for both estimators. Moreover, the error is bounded by the first estimator with a constant one plus a higher order data oscillation term plus a term arising from the contact that is shown numerically to be of higher order. The second estimator is used in a control-based AFEM refinement strategy, and the decay of the error in the energy is shown. Several numerical tests demonstrate the performance of both estimators. 1.
Goal-oriented error estimation and mesh adaptivity in three-dimentional elasticity problems
In finite element simulation of engineering applications, accuracy is of great importance considering that generally no analytical solution is available. Conventional error estimation methods aim to estimate the error in energy norms or the global L 2 -norm. These values can be used to estimate the accuracy of the model or to guide how to adapt the model to achieve more accuracy. However, in engineering applications specific quantities are required to be accurate. The novel error estimation approach which is called Dual-Weighted Residual (DWR) error estimation, approximates the error with respect to the quantity of interest which can be mean stress or displacement in a subspace or the solution ('s gradient) on a specific point, etc. DWR error estimation is a dual-based scheme which requires an adjoint (dual) problem. The dual problem is described by defining the quantity of interest in a functional form. Then by solving the primal and dual problems, errors in terms of the specified quantities are calculated. In this paper the DWR error estimation besides the conventional residual-based error estimation and a recovery-based error estimation are applied in a three-dimensional elasticity problem. Local estimated errors are exploited in order to accomplish the mesh adaptivity procedure. The goaloriented mesh adaptivity control the local errors in terms of the prescribed quantities. Both refinement and coarsening processes are applied to raise the efficiency. The convergence rates are plotted to illustrate the superiorities of the goal-oriented adaptivity over the traditional ones.
A Posteriori Error Control of Finite Element Approximations for Coulomb's Frictional Contact
SIAM Journal on Scientific Computing, 2001
This paper is concerned with the frictional unilateral contact problem governed by Coulomb's law. We define an a posteriori error estimator based on the concept of error in the constitutive relation to quantify the accuracy of a finite element approximation of the problem. We propose and study different mixed finite element approaches and discuss their properties in order to compute the estimator. The information given by the error estimates is then coupled with a mesh adaptivity technique which provides the user with the desired quality and minimizes the computation costs. The numerical implementation of the error estimator as well as optimized computations are performed.
Computational Mechanics, 2007
An r-h adaptive scheme has been proposed and formulated for analysis of bimaterial interface problems using adaptive finite element method. It involves a combination of the configurational force based r-adaption with weighted laplacian smoothing and mesh enrichment by h-refinement. The Configurational driving force is evaluated by considering the weak form of the material force balance for bimaterial inerface problems. These forces assembled at nodes act as an indicator for r-adaption. A weighted laplacian smoothing is performed for smoothing the mesh. The h-adaptive strategy is based on a modifed weighted energy norm of error evaluated using supercovergent estimators. The proposed method applies specific non sliding interface strain compatibility requirements across inter material boundaries consistent with physical principles to obtain modified error estimators. The best sequence of combining r-and h-adaption has been evolved from numerical study. The study confirms that the proposed combined r-h adaption is more efficient than a purely h-adaptive approach and more flexible than a purely r-adaptive approach with better convergence characteristics and helps in obtaining optimal finite element meshes for a specified accuracy.
Finite Elements in Analysis and Design, 1997
This paper presents a study of two types of a posteriori error estimator based on equilibrium defaults in finite element solutions of planar linear elastic problems. The (~-type estimator, developed at LTAS-Infographie, uses explicit relations between the equilibrium defaults and the energy of the error. This type is represented here with a new physical interpretation, and with the reference tractions on the sides of elements redefined so as to satisfy equilibrium. The d-type estimator uses a reference solution for stress d, which has no equilibrium defaults, i.e. d is a statically admissible stress field. The reference solutions considered in this paper are based on the formulation first proposed by Ladeveze. This present study considers several alternative formulations of equilibrating tractions suitable for both types of estimator, and it is restricted to errors in models composed of 4-noded bilinear Lagrange elements. Numerical results are presented for several examples with smooth or singular stress fields. It is observed that significant improvements to the effectiveness of the G-estimator can be obtained, particularly for coarser meshes. With judicious choice of equilibrating tractions it is concluded that this estimator can be more effective than the 6"-estimator.
International Journal of Engineering Science, 1993
An analysis of a class of contact problems with frie1ion characterized by interFaces obeying a normal compliance law is given. A priori error estimations arc developed for general cases and for the special case in which only steady sliding occurs. These estimates are applicable to It-, po, and Itp-finite element approximations and give estimated rates of convergence for generalltp-methods. A regularization of the Friction law transforms variational inequalities corresponding to the general problem of contact with friction into nonlinear equations which can he solved by standard methods. A priori error estimations arc also developed for these regularized problems. Numerical examples arc given to support the theoretical results.