zbMATH — the first resource for mathematics

Mathematical analysis and numerical simulation of a nonlinear thermoelastic system. (English) Zbl 1411.74026
Summary: In this paper, we give a theoretical and numerical analysis of a model for small vertical vibrations of an elastic membrane coupled with a heat equation and subject to linear mixed boundary conditions. We establish the existence, uniqueness, and a uniform decay rate for global solutions to this nonlinear non-local thermoelastic coupled system with linear boundary conditions. Furthermore, we introduced a numerical method based on finite element discretization in a spatial variable and finite difference scheme in time which results in a nonlinear system to be solved by Newton’s method. Numerical experiments for one-dimensional and two-dimensional cases are presented in order to estimate the rate of convergence of the numerical solution that confirm our analysis and show the efficiency of the method.
74H45 Vibrations in dynamical problems in solid mechanics
74K15 Membranes
74S05 Finite element methods applied to problems in solid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
74F05 Thermal effects in solid mechanics
Full Text: DOI
[1] Chipot, M.; Lovat, B., On the asymptotic behaviour of some nonlocal problems, Positivity, 3, 1, 65-81, (1999) · Zbl 0921.35071
[2] Clark, H.; Jutuca, L. S. G.; Miranda, M., Global existence, uniqueness and exponential stability for a nonlinear thermoelastic system, Electron. J. Differ. Equ, 1998, 4, 1-20, (1998)
[3] Apolaya, R.; Clark, H.; Feitosa, A., On a nonlinear coupled system with internal damping, Electron. J. Differ. Equ, 2000, 1-17, (2000) · Zbl 0962.35002
[4] Clark, H.; Clark, M.; Louredo, A.; Oliveira, A., A nonlinear thermoelastic system with nonlinear boundary conditions, J. Evol. Equ, 15, 4, 895-911, (2015) · Zbl 1337.35145
[5] Clark, H.; Guardia, R., On a nonlinear thermoelastic system with nonlocal coefficients, J. Math. Anal. Appl, 433, 1, 338-354, (2016) · Zbl 1327.35071
[6] Clark, H., Global existence, uniqueness and exponential stability for a nonlinear thermoelastic system, Appl. Anal, 66, 1-2, 39-56, (1997)
[7] Medeiros, L.; Miranda, M., On a boundary value problem for wave equations: existence, uniqueness-asymptotic behavior, Rev. Mat. Apl, 17, 47-73, (1996) · Zbl 0859.35070
[8] Dafermos, C., On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal, 29, 4, 241-271, (1968) · Zbl 0183.37701
[9] Slemrod, M., Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity, Arch. Rational Mech. Anal, 76, 2, 97-133, (1981) · Zbl 0481.73009
[10] Chou, S.; Wang, C., Estimates of error in finite element approximate solutions to problems in linear thermoelasticity. Part 1. Computationally coupled numerical schemes, Arch. Rational Mech. Anal, 77, 3, 263-299, (1981) · Zbl 0494.73071
[11] Chou, S.; Wang, C., Estimates of error in finite element approximate solutions to problems in linear thermoelsticity. Part ii. Computationally uncoupled numerical schemes, Arch. Rational Mech. Anal, 85, 1, 27-40, (1984) · Zbl 0555.73075
[12] Rincon, M.; Santos, B.; Límaco, J., Numerical method, existence and uniqueness for thermoelasticity system with moving boundary, Mat. Apl. Comput, 24, 3, 439-460, (2005) · Zbl 1213.35016
[13] Zhelezovskii, S., Error estimate for a symmetric scheme of the projection difference method for an abstract hyperbolic-parabolic system of the type systems of thermoelasticity equations, Diff. Equ, 48, 7, 950-964, (2012) · Zbl 1332.65071
[14] Brezis, H., Analyse Fonctionnelle, théorie et Applications, (1999), Paris: Dunod, Paris
[15] Lions, J., Quelques méthodes de résolution des problèmes aux limites non-linéaires, (1960), Paris: Dunod, Paris
[16] Aubin, J., Un théorème de compacité, C.R.A. Sci. Paris, 256, 5042-5044, (1963) · Zbl 0195.13002
[17] Haraux, A.; Zuazua, E., Decay estimates for some semi-linear damped hyperbolic problems, Arch. Rational Mech. Anal, 100, 2, 191-206, (1988) · Zbl 0654.35070
[18] Komornik, V.; Zuazua, E., A direct method for boundary stabilization of the wave equation, J. Math. Pure Appl, 69, 1, 33-54, (1990) · Zbl 0636.93064
[19] Rincon, M.; Liu, I.-S, Introdução ao método de elementos finitos, (2011), Rio De Janeiro: IM/UFRJ, Rio De Janeiro
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.