Research paperNumerical analysis for a new kind of obstacle problem
Introduction
The obstacle problem is a typical variational inequality of the first kind [9], [15]. It includes the obstacle problem of elastic membrane or plate, elastoplastic torsion of a cylindrical bar, a cavitation problem in hydrodynamic lubrication, the minimal surface problem and so on [7]. Here we only focus on the obstacle problem for an elastic membrane.
The way in which the obstacle problem is constructed has been rigorously described. Assume that the elastic membrane passes through the boundary of a bounded domain ; lies above an obstacle of height with on ; and is subject to the action of a vertical force which is proportional to [15]. Let be the vertical displacement component of the membrane. Under certain assumptions, the following three equivalent forms that the displacement satisfy can be obtained: the energy formthe variational inequality formand the differential equation formHere is the set of admissible displacements. In the past decades, mathematical theories and numerical analysis of this model, including existence, uniqueness, regularity, numerical methods and convergence results, have been established extensively and thoroughly. Details can be found, e.g., in [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [19], [25], [27].
To our best knowledge, the obstacle in most related references is perfectly rigid. In practice, there are no perfectly rigid bodies. Small penetration could occur, due to the body’s and foundation’s asperities [28]. Moreover, contact processes between deformable bodies abound in industry and everyday life, and many important results have been obtained in the modeling and analysis of the corresponding contact problems. In general, the contact problem describes the equilibrium state of a deformable body that (1) is subjected to the action of body forces and surface tractions; and (2) is clamped on part of its surface and in contact with an elastic or elastic-rigid foundation on another part of its surface. We refer to Sofonea and Matei [1], Han and Sofonea [21] for more details. The different contact conditions are where our main interest lies. Different from the contact problem, we consider the contact between the body and the deformable obstacle.
In this paper, we consider elastic-rigid foundation in the obstacle problem of the elastic membrane. More precisely, the elastic membrane is constrained to lie above an obstacle which is made of a rigid body covered by a layer made of soft material. The soft layer is deformable and allows penetration, so it assigns a reactive normal pressure depending on the interpenetration of the elastic membrane and the foundation. Similar to the classical obstacle problem, we give three equivalent descriptions of this kind of obstacle problem, and its well-posedness is discussed. Based on the variational inequality form, we study the numerical solution of the new obstacle problem by the finite element method and the penalty technique. The optimal order error estimate of the linear elements is obtained. We also provide numerical examples to visualize the theoretical analysis.
The rest of the paper is organized as follows. In Section 2, we present three equivalent descriptions for the new kind of obstacle problem and prove the existence and uniqueness of the solution to this problem. We consider numerical approximations in Section 3, including convergence analysis and error estimates for numerical solutions. In Section 4, we give numerical experiments to illustrate the rationality of our theoretical analysis. Finally, a summary is given in the last section.
Section snippets
Description of the new kind of obstacle problem
Let be a bounded domain with a Lipschitz boundary . denotes the Sobolev space with the usual norm, the closure of in and the dual space of . For a normed space we denote by its norm. For a Hilbert space we denote by the inner product of two elements in and simply write instead of when there is no confusion. refers to the Laplace operator, and the gradient operator.
Given non-negative
Numerical analysis
In this section, we consider numerical schemes for variational inequality (11). We first introduce the finite element approximation and derive a priori error estimates. Then, since the penalty method is an effective approach in the numerical solution of problems with constraints, we apply it to the underlying approximation problem and prove its convergence.
Numerical results
In this section, we report simulation results for four numerical examples to support our theoretical analysis. All the simulations are performed on an Intel Dual-core personal computer with MatLab version R2018b. The solution of the penalized discrete problems is based on a kind of Picard iterative. We take and for all the following numerical examples. We adopt the absolute error as the stop criterion in the following tests, where represents the
Conclusion
The goal of this paper is to study a new obstacle problem for the elastic membrane which is constrained to lie above an elastic-rigid obstacle. Under certain assumptions, we deduced three equivalent descriptions of the problem and proved its existence and uniqueness of the solution. Based on the variational inequality form, we derived an optimal order error estimate for the finite element approximate solution. Several numerical experiments illustrated that the simulation results are in good
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. This work is supported by the European Unions Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grand Agreement No. 823731 CONMECH, and by Major Scientifc Research Project of Zhejiang Lab (No. 2019DB0ZX01).
References (28)
- et al.
An algorithm for solving the obstacle problems
Comput Math Appl
(2004) Discretized obstacle problems with penalties on nested grids
Appl Numer Math
(2000)- et al.
A uzawa algorithm with multigrid solver for a bilateral obstacle problem
Appl Math Comput
(2021) - et al.
On friction problems with normal compliance
Nonlinear Anal
(1989) - et al.
Frictional contact problems with normal compliance
Int J Engng Sci
(1988) - et al.
An algorithm for solving the double obstacle problems
Appl Math Comput
(2008) - et al.
Mathematical models in contact mechanics
(2012) - et al.
A penalization method for the elliptic bilateral obstacle problem
IFIP Adv Inform Commun Tech
(2014) - et al.
Equivalence de deux inéquations variationnelles et applications
Arch Ration Mech An
(1971) - et al.
Augmented lagrangian active set methods for obstacle problems
J Optimiz Theory App
(2003)
Inequalities in mechanics and physics
J Appl Mech
Obstacle problems in mathematical physics
Virtual element methods for the obstacle problem
IMA J Numer Anal
Numerical algorithms based on characteristic domain decomposition for obstacle problems
Commun Num Method Eng
Cited by (3)
Optimal control of an elastic–rigid obstacle problem
2024, Communications in Nonlinear Science and Numerical SimulationAnalysis of a new kind of elastic-rigid bilateral obstacle problem
2023, Mathematics and Mechanics of SolidsAnalysis of an elastic–rigid obstacle problem described by a variational–hemivariational inequality
2023, ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik