We discuss an algorithm for the solution of variational inequalities associated with simply supported plates in contact with a rigid obstacle. Our approach has a fixed domain character, uses just linear equations and approximates both the solution and the corresponding coincidence set. Numerical examples are also provided.
DuvautGLionsJL.Inequalities in mechanics and physics. Berlin: Springer-Verlag, 1976.
2.
KikuchiNOdenJT.Contact problems in elasticity: A study of variational inequalities and finite element methods (SIAM Studies in Applied Mathematics, vol. 8). Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1988.
3.
SofoneaMMateiA.Variational inequalities with applications: A study of antiplane frictional contact problems (Advances in Mechanics and Mathematics, vol. 18). Berlin: Springer, 2009.
4.
ElliottCMOckendonJR.Weak and variational methods for moving boundary problems (Research Notes in Mathematics, vol. 59). London: Pitman, 1982.
5.
RodriguesJF.Obstacle problems in mathematical physics. Amsterdam: North-Holland Publishing Co., 1987.
6.
KinderlehrerDStampacchiaG.An introduction to variational inequalities and their applications (Classics in Applied Mathematics, vol. 31). Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2000 (reprint).
7.
GlowinskiRLionsJLTrémolièresR.Numerical analysis of variational inequalities (Studies in Mathematics and its Applications, vol. 8). Amsterdam: North-Holland Publishing Co., 1981.
8.
FortinMGlowinskiR.Augmented Lagrangian methods: Applications to the numerical solution of boundary value problems (Studies in Mathematics and its Applications, vol. 15). Amsterdam: North-Holland Publishing Co., 1983.
9.
SofoneaMMigorskiSOchalA.Nonlinear inclusions and hemivariational inequalities: Models and analysis of contact problems. New York, NY: Springer, 2012.
10.
BarbuV.Optimal control of variational inequalities (Research Notes in Mathematics, vol. 100). Boston, MA: Pitman, 1984.
11.
TibaD.Optimal control of nonsmooth distributed parameter systems. Berlin: Springer, 1990.
12.
NeittaanmakiPTibaD.Optimal control of nonlinear parabolic systems: Theory, algorithms, and applications (Monographs and Textbooks in Pure and Applied Mathematics, vol. 179). New York, NY: Marcel Dekker Inc., 1994.
13.
NeittaanmakiPSprekelsJTibaD.Optimization of elliptic systems: Theory and applications (Springer Monographs in Mathematics). New York, NY: Springer, 2006.
NeittaanmakiPPennanenATibaD.Fixed domain approaches in shape optimization problems with Dirichlet boundary conditions. Inv Prob2009; 25: 1–18.
16.
HalanayAMureaCMTibaD.Existence and approximation for a steady fluid-structure interaction problem using fictitious domain approach with penalization. Ann Acad Rom Sci Ser Math Appl2013; 5: 120–147.
17.
HalanayAMureaCMTibaD.Existence of a steady flow of Stokes fluid past a linear elastic structure using fictitious domain. J Math Fluid Mech2016; 18: 397–413.
18.
MureaCMTibaD.A direct algorithm in some free boundary problems. J Numer Math2016; 24: 253–271.
19.
GrisvardP.Elliptic problems in nonsmooth domains. London: Pitman, 1985.
20.
TibaD.A duality approximation of some nonlinear PDE’s. Ann Acad Rom Sci Ser Math Appl2016; 8: 68–77.
21.
BarbuV.Nonlinear semigroups and differential equations in Banach spaces. Leyden, Netherlands: Noordhoff International Publishing, 1976.
22.
EvansLC.Partial differential equations. Providence, RI: American Mathematical Society, 2010.
23.
DautrayRLionsJL.Mathematical analysis and numerical methods for science and technology (Integral Equations and Numerical Methods, vol. 4). Berlin: Springer-Verlag, 1990.
24.
BoffiDBrezziFFortinM.Mixed finite element methods and applications (Springer Series in Computational Mathematics, vol. 44). Heidelberg, Germany: Springer, 2013.
25.
BrennerSCGuSGudiT. A partition of unity method for the obstacle problem of simply supported Kirchhoff plates. In: GriebelMSchweitzerMA (eds) Meshfree methods for partial differential equations VII (Lecture Notes in Computational Science and Engineering, vol. 100). New York, NY: Springer, 2015, 23–41.
26.
HechtF.New development in FreeFem++. J Numer Math2012; 20: 251–265.
27.
BrennerSCSungLZhangH. A quadratic C0 interior penalty method for the displacement obstacle problem of clamped Kirchhoff plate. SIAM J Numer Anal2012; 50: 3329–3350.
28.
ThéodorR.Initiation à l’analyse numérique. Paris: Masson, 1982.
