On finite element uniqueness studies for Coulombs frictional contact model

Patrick Hild

International Journal of Applied Mathematics and Computer Science (2002)

  • Volume: 12, Issue: 1, page 41-50
  • ISSN: 1641-876X

Abstract

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We are interested in the finite element approximation of Coulomb's frictional unilateral contact problem in linear elasticity. Using a mixed finite element method and an appropriate regularization, it becomes possible to prove existence and uniqueness when the friction coefficient is less than Cε^{2}|log(h)|^{-1}, where h and ε denote the discretization and regularization parameters, respectively. This bound converging very slowly towards 0 when h decreases (in comparison with the already known results of the non-regularized case) suggests a minor dependence of the mesh size on the uniqueness conditions, at least for practical engineering computations. Then we study the solutions of a simple finite element example in the non-regularized case. It can be shown that one, multiple or an infinity of solutions may occur and that, for a given loading, the number of solutions may eventually decrease when the friction coefficient increases.

How to cite

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Hild, Patrick. "On finite element uniqueness studies for Coulombs frictional contact model." International Journal of Applied Mathematics and Computer Science 12.1 (2002): 41-50. <http://eudml.org/doc/207567>.

@article{Hild2002,
abstract = {We are interested in the finite element approximation of Coulomb's frictional unilateral contact problem in linear elasticity. Using a mixed finite element method and an appropriate regularization, it becomes possible to prove existence and uniqueness when the friction coefficient is less than Cε^\{2\}|log(h)|^\{-1\}, where h and ε denote the discretization and regularization parameters, respectively. This bound converging very slowly towards 0 when h decreases (in comparison with the already known results of the non-regularized case) suggests a minor dependence of the mesh size on the uniqueness conditions, at least for practical engineering computations. Then we study the solutions of a simple finite element example in the non-regularized case. It can be shown that one, multiple or an infinity of solutions may occur and that, for a given loading, the number of solutions may eventually decrease when the friction coefficient increases.},
author = {Hild, Patrick},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {Coulomb's friction law; non-uniqueness example; mesh-size dependent uniqueness conditions; finite elements; non-uniqueness; linear elasticity; mixed finite element method; regularization},
language = {eng},
number = {1},
pages = {41-50},
title = {On finite element uniqueness studies for Coulombs frictional contact model},
url = {http://eudml.org/doc/207567},
volume = {12},
year = {2002},
}

TY - JOUR
AU - Hild, Patrick
TI - On finite element uniqueness studies for Coulombs frictional contact model
JO - International Journal of Applied Mathematics and Computer Science
PY - 2002
VL - 12
IS - 1
SP - 41
EP - 50
AB - We are interested in the finite element approximation of Coulomb's frictional unilateral contact problem in linear elasticity. Using a mixed finite element method and an appropriate regularization, it becomes possible to prove existence and uniqueness when the friction coefficient is less than Cε^{2}|log(h)|^{-1}, where h and ε denote the discretization and regularization parameters, respectively. This bound converging very slowly towards 0 when h decreases (in comparison with the already known results of the non-regularized case) suggests a minor dependence of the mesh size on the uniqueness conditions, at least for practical engineering computations. Then we study the solutions of a simple finite element example in the non-regularized case. It can be shown that one, multiple or an infinity of solutions may occur and that, for a given loading, the number of solutions may eventually decrease when the friction coefficient increases.
LA - eng
KW - Coulomb's friction law; non-uniqueness example; mesh-size dependent uniqueness conditions; finite elements; non-uniqueness; linear elasticity; mixed finite element method; regularization
UR - http://eudml.org/doc/207567
ER -

References

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