Applications of Mathematics, Vol. 68, No. 4, pp. 405-424, 2023
Solution of 3D contact shape optimization problems with Coulomb friction based on TFETI
Alexandros Markopoulos, Petr Beremlijski, Oldřich Vlach, Marie Sadowská
Received June 2, 2022. Published online December 6, 2022.
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.
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Affiliations: Alexandros Markopoulos, Safran Aircraft Engines, 10, allée du Brévent, 91019 Evry, France; Petr Beremlijski (corresponding author), Department of Applied Mathematics, VSB-Technical University of Ostrava, 17. listopadu 2172/15, 708 00 Ostrava-Poruba, Czech Republic, e-mail: petr.beremlijski@vsb.cz; Oldřich Vlach, Department of Applied Mathematics, VSB-Technical University of Ostrava, 17. listopadu 2172/15, 708 00 Ostrava-Poruba, Czech Republic and IT4Innovations National Supercomputing Center, VSB-Technical University of Ostrava, Studentská 6231/1B, 708 00 Ostrava-Poruba, Czech Republic, e-mail: oldrich.vlach2@vsb.cz; Marie Sadowská, Department of Applied Mathematics, VSB-Technical University of Ostrava, 17. listopadu 2172/15, 708 00 Ostrava-Poruba, Czech Republic, e-mail: marie.sadowska@vsb.cz
