Solution of 3D contact shape optimization problems with Coulomb friction based on TFETI

Alexandros Markopoulos; Petr Beremlijski; Oldřich Vlach; Marie Sadowská

Applications of Mathematics (2023)

  • Volume: 68, Issue: 4, page 405-424
  • ISSN: 0862-7940

Abstract

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The present paper deals with the numerical solution of 3D shape optimization problems in frictional contact mechanics. Mathematical modelling of the Coulomb friction problem leads to an implicit variational inequality which can be written as a fixed point problem. Furthermore, it is known that the discretized problem is uniquely solvable for small coefficients of friction. Since the considered problem is nonsmooth, we exploit the generalized Mordukhovich’s differential calculus to compute the needed subgradient information.The state problem is solved using successive approximations combined with the Total FETI (TFETI) method. The latter is based on tearing the bodies into “floating” subdomains, discretization by finite elements, and solving the resulting quadratic programming problem by augmented Lagrangians. The presented numerical experiments demonstrate our method’s power and the importance of the proper modelling of 3D frictional contact problems. The state problem solution and the sensitivity analysis process were implemented in parallel.

How to cite

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Markopoulos, Alexandros, et al. "Solution of 3D contact shape optimization problems with Coulomb friction based on TFETI." Applications of Mathematics 68.4 (2023): 405-424. <http://eudml.org/doc/299601>.

@article{Markopoulos2023,
abstract = {The present paper deals with the numerical solution of 3D shape optimization problems in frictional contact mechanics. Mathematical modelling of the Coulomb friction problem leads to an implicit variational inequality which can be written as a fixed point problem. Furthermore, it is known that the discretized problem is uniquely solvable for small coefficients of friction. Since the considered problem is nonsmooth, we exploit the generalized Mordukhovich’s differential calculus to compute the needed subgradient information.The state problem is solved using successive approximations combined with the Total FETI (TFETI) method. The latter is based on tearing the bodies into “floating” subdomains, discretization by finite elements, and solving the resulting quadratic programming problem by augmented Lagrangians. The presented numerical experiments demonstrate our method’s power and the importance of the proper modelling of 3D frictional contact problems. The state problem solution and the sensitivity analysis process were implemented in parallel.},
author = {Markopoulos, Alexandros, Beremlijski, Petr, Vlach, Oldřich, Sadowská, Marie},
journal = {Applications of Mathematics},
keywords = {shape optimization; nonsmooth optimization; contact problem; Coulomb's friction; TFETI method},
language = {eng},
number = {4},
pages = {405-424},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Solution of 3D contact shape optimization problems with Coulomb friction based on TFETI},
url = {http://eudml.org/doc/299601},
volume = {68},
year = {2023},
}

TY - JOUR
AU - Markopoulos, Alexandros
AU - Beremlijski, Petr
AU - Vlach, Oldřich
AU - Sadowská, Marie
TI - Solution of 3D contact shape optimization problems with Coulomb friction based on TFETI
JO - Applications of Mathematics
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 4
SP - 405
EP - 424
AB - The present paper deals with the numerical solution of 3D shape optimization problems in frictional contact mechanics. Mathematical modelling of the Coulomb friction problem leads to an implicit variational inequality which can be written as a fixed point problem. Furthermore, it is known that the discretized problem is uniquely solvable for small coefficients of friction. Since the considered problem is nonsmooth, we exploit the generalized Mordukhovich’s differential calculus to compute the needed subgradient information.The state problem is solved using successive approximations combined with the Total FETI (TFETI) method. The latter is based on tearing the bodies into “floating” subdomains, discretization by finite elements, and solving the resulting quadratic programming problem by augmented Lagrangians. The presented numerical experiments demonstrate our method’s power and the importance of the proper modelling of 3D frictional contact problems. The state problem solution and the sensitivity analysis process were implemented in parallel.
LA - eng
KW - shape optimization; nonsmooth optimization; contact problem; Coulomb's friction; TFETI method
UR - http://eudml.org/doc/299601
ER -

References

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