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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.
MSC:
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
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