An Approximation of Solutions for the Problem with Quasistatic Contact in the Case of Dry Friction
In this paper, we discuss the question of finding an optimal control for the solutions of the problem with dry friction quasistatic contact, in the case that the friction law is modeled by a nonlocal version of Coulomb’s law. In order to get the necessary optimality conditions, we use some regulariz...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-04-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/9/8/904 |
| _version_ | 1797537210769080320 |
|---|---|
| author | Nicolae Pop Miorita Ungureanu Adrian I. Pop |
| author_facet | Nicolae Pop Miorita Ungureanu Adrian I. Pop |
| author_sort | Nicolae Pop |
| collection | DOAJ |
| description | In this paper, we discuss the question of finding an optimal control for the solutions of the problem with dry friction quasistatic contact, in the case that the friction law is modeled by a nonlocal version of Coulomb’s law. In order to get the necessary optimality conditions, we use some regularization techniques, and this leads us to a problem of control for an inequality of the variational type. The optimal control problem consists, in our case, of minimizing a sequence of optimal control problems, where the control variable is given by a Neumann-type boundary condition. The state system is represented by a limit of a sequence, whose terms are obtained from the discretization, in time with finite difference and space with the finite element method of a regularized quasistatic contact problem with Coulomb friction. The purpose of this optimal control problem is that the traction force (the control variable) acting on one side of the boundary (the Neumann boundary condition) of the elastic body produces a displacement field (the state system solution) close enough to the imposed displacement field, and the traction force from the boundary remains small enough. |
| first_indexed | 2024-03-10T12:11:51Z |
| format | Article |
| id | doaj.art-c17462dfe0ee4991ad6c13d0435581c7 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T12:11:51Z |
| publishDate | 2021-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-c17462dfe0ee4991ad6c13d0435581c72023-11-21T16:09:07ZengMDPI AGMathematics2227-73902021-04-019890410.3390/math9080904An Approximation of Solutions for the Problem with Quasistatic Contact in the Case of Dry FrictionNicolae Pop0Miorita Ungureanu1Adrian I. Pop2Institute of Solid Mechanics of Romanian Academy, Constantin Mille Str. 15, 010141 Bucharest, RomaniaNorth University Centre at Baia Mare, Faculty of Engineering, Technical University of Cluj Napoca, V. Babes Str. 62, 430083 Baia Mare, RomaniaNorth University Centre at Baia Mare, Faculty of Engineering, Technical University of Cluj Napoca, V. Babes Str. 62, 430083 Baia Mare, RomaniaIn this paper, we discuss the question of finding an optimal control for the solutions of the problem with dry friction quasistatic contact, in the case that the friction law is modeled by a nonlocal version of Coulomb’s law. In order to get the necessary optimality conditions, we use some regularization techniques, and this leads us to a problem of control for an inequality of the variational type. The optimal control problem consists, in our case, of minimizing a sequence of optimal control problems, where the control variable is given by a Neumann-type boundary condition. The state system is represented by a limit of a sequence, whose terms are obtained from the discretization, in time with finite difference and space with the finite element method of a regularized quasistatic contact problem with Coulomb friction. The purpose of this optimal control problem is that the traction force (the control variable) acting on one side of the boundary (the Neumann boundary condition) of the elastic body produces a displacement field (the state system solution) close enough to the imposed displacement field, and the traction force from the boundary remains small enough.https://www.mdpi.com/2227-7390/9/8/904unilateral quasistatic contact problemfinite element methodfinite differenceCoulomb lawboundary controlregularized optimal |
| spellingShingle | Nicolae Pop Miorita Ungureanu Adrian I. Pop An Approximation of Solutions for the Problem with Quasistatic Contact in the Case of Dry Friction Mathematics unilateral quasistatic contact problem finite element method finite difference Coulomb law boundary control regularized optimal |
| title | An Approximation of Solutions for the Problem with Quasistatic Contact in the Case of Dry Friction |
| title_full | An Approximation of Solutions for the Problem with Quasistatic Contact in the Case of Dry Friction |
| title_fullStr | An Approximation of Solutions for the Problem with Quasistatic Contact in the Case of Dry Friction |
| title_full_unstemmed | An Approximation of Solutions for the Problem with Quasistatic Contact in the Case of Dry Friction |
| title_short | An Approximation of Solutions for the Problem with Quasistatic Contact in the Case of Dry Friction |
| title_sort | approximation of solutions for the problem with quasistatic contact in the case of dry friction |
| topic | unilateral quasistatic contact problem finite element method finite difference Coulomb law boundary control regularized optimal |
| url | https://www.mdpi.com/2227-7390/9/8/904 |
| work_keys_str_mv | AT nicolaepop anapproximationofsolutionsfortheproblemwithquasistaticcontactinthecaseofdryfriction AT mioritaungureanu anapproximationofsolutionsfortheproblemwithquasistaticcontactinthecaseofdryfriction AT adrianipop anapproximationofsolutionsfortheproblemwithquasistaticcontactinthecaseofdryfriction AT nicolaepop approximationofsolutionsfortheproblemwithquasistaticcontactinthecaseofdryfriction AT mioritaungureanu approximationofsolutionsfortheproblemwithquasistaticcontactinthecaseofdryfriction AT adrianipop approximationofsolutionsfortheproblemwithquasistaticcontactinthecaseofdryfriction |