We study the homogenization of the Poisson equation in a periodically perforated domain, of period , with a friction type boundary condition on the holes’ boundary. This non-linear condition allows the solution to be non-zero on the periodic boundary if some conditions are satisfied. Using two-scale convergence results we prove that the solution of the mixed variational formulation converges, as ε goes to 0, to the solution of a two-scale mixed problem. We also prove that this homogenized problem is well-posed. A numerical test is done, using the Finite Element Method and a quadratic programming algorithm, in order to compare the heterogeneous and homogenized solutions.
G.Allaire, Homogenization and two scale convergence, SIAM J. Math. Anal.23(6) (1992), 1482–1518.
2.
G.Allaire and M.Amar, Boundary layer tails in periodic homogenization, ESAIM: COCV4 (1999), 209–243. doi:10.1051/cocv:1999110.
3.
G.Allaire, A.Damlamian and U.Hornung, Two-scale convergence of periodic surfaces and applications, in: Proceedings of the International Conference on Mathematical Modelling of Flow Through Porous Media, May 1995, A.Bourgeatet al., eds, World Scientific Publishing, Singapore, 1996, pp. 15–25.
4.
A.Bensoussan, J.-L.Lions and G.Papanicoulaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.
5.
D.Cioranescu, A.Damlamian and J.Orlik, Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptot. Anal.82(3-4) (2013), 201–232.
6.
C.Conca, F.Murat and C.Timofte, A generalized strange term in Signorini’s type problems, ESAIM: Math. Model. and Numer. Anal.37(5) (2003), 773–805.
7.
G.Duvaut and J.-L.Lions, Inequalities in Mechanics and Physics, Springer, Berlin, 1976.
8.
H.Fujita, A coherent analysis of Stokes flows under boundary conditions of friction type, J. Comput. Appl. Math. (English)149(1) (2002), 57–69.
9.
R.Glowinsky, Lectures on Numerical Methods for Nonlinear Variational Problems, M.G.Vijayasundaram and M.Adimurthi, eds, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Vol. 65, Tata Institute of Fundamental Research, Springer, Bombay, Berlin, New York, 1980.
10.
J.Haslinger, I.Hlaváček and J.Nečas, Numerical methods for unilateral problems in solid mechanics, in: Handbook of Numerical Analysis, Vol. IV, North-Holland, Amsterdam, 1996, pp. 313–485.
11.
J.Haslinger and T.Sassi, Mixed finite element approximation of 3D contact problem with given friction: Error analysis and numerical realisation, M2AN38(3) (2004), 563–578.
12.
F.Hecht, FreeFEM++, 3rd edn, Version 3.18, 2012, available at: http://www.freefem.org/ff++/ftp/freefem++doc.pdf.
13.
R.Kučera, Convergence rate of an optimization algorithm for minimizing quadratic functions with separable convex constraints, SIAM J. Optim.19(2) (2008), 846–862.
14.
J.-L.Lions, Inéquations variationnelles d’évolution, Actes, Congrès intern. Math., Tome 2, Gauthier-Villars, Paris, 1970, pp. 841–851.
15.
G.Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal.20 (1989), 608–623.
16.
G.Yosifian, On some homogenization problems in perforated domains with nonlinear boundary conditions, Appl. Anal.65(3-4) (1997), 257–288.
17.
G.Yosifian, Homogenization of some contact problems for the system of elasticity in perforated domains, Rend. Sem. Mat. Univ. Padova105 (2001), 37–64.
