On existence and numerical approximation in phase-lag thermoelasticity with two temperatures

Marco Campo; Maria I. M. Copetti; José R. Fernández; Ramón Quintanilla

doi:10.3934/dcdsb.2021130

Article Contents

2022, Volume 27, Issue 4: 2221-2245. Doi: 10.3934/dcdsb.2021130

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On existence and numerical approximation in phase-lag thermoelasticity with two temperatures

1.
Departamento de Matemáticas, ETS de Ingenieros de Caminos, Canales y Puertos, Universidade da Coruña, Campus de Elviña, 15071 A Coruña, Spain
2.
Laboratório de Análise Numérica e Astrofísica, Departamento de Matemática, Universidade Federal de Santa Maria, 97105-900, Santa Maria, RS, Brazil
3.
Departamento de Matemática Aplicada I, Universidade de Vigo, Escola de Enxeñería de Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain
4.
Departament de Matemàtiques, Universitat Politècnica de Catalunya, C. Colom 11, 08222 Terrassa, Barcelona, Spain

^* Corresponding author: José R. Fernández
^* Corresponding author: José R. Fernández

Received: August 2020

Revised: March 2021

Early access: April 2021

Published: April 2022

The work of M. Campo and J.R. Fernández has been partially supported by Ministerio de Ciencia, Innovación y Universidades under the research project PGC2018-096696-B-I00 (FEDER, UE). The work of M.I.M. Copetti has been partially supported by the Brazilian institution CNPq (grant 304709/2017-4). The work of R. Quintanilla has been supported by Ministerio de Economía y Competitividad under the research project "Análisis Matemático de Problemas de la Termomecánica" (MTM2016-74934-P), (AEI/FEDER, UE), and Ministerio de Ciencia, Innovación y Universidades under the research project "Análisis matemático aplicado a la termomecánica" (PID2019-105118GB-I00). The authors want to thank to the anonymous referees their useful comments which have allowed us to improve the paper

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Abstract

In this work we study from both variational and numerical points of view a thermoelastic problem which appears in the dual-phase-lag theory with two temperatures. An existence and uniqueness result is proved in the general case of different Taylor approximations for the heat flux and the inductive temperature. Then, in order to provide the numerical analysis, we restrict ourselves to the case of second-order approximations of the heat flux and first-order approximations for the inductive temperature. First, variational formulation of the corresponding problem is derived and an energy decay property is proved. Then, a fully discrete scheme is introduced by using the finite element method for the approximation of the spatial variable and the implicit Euler scheme for the discretization of the time derivatives. A discrete stability

Keywords:

Mathematics Subject Classification: Primary: 35Q74, 74H10, 80A20, 65M15, 65M60; Secondary: 74F05.

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Figure 1. Example 1: Asymptotic behavior of the numerical scheme

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Figure 2. Example 1: Energy evolution in absolute and semilogarithmic scales

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Figure 3. Example 2: Evolution in time of the temperature and inductive temperature at point $ {\boldsymbol{x}} = (4, 0.5) $ for different values of parameter $ m $

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Figure 4. Example 2: Evolution in time of the temperature and inductive temperature at point $ {\boldsymbol{x}} = (1, 0.5) $ for different values of parameter $ m $

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Figure 5. Example 2: Evolution in time of the horizontal and vertical displacements at point $ {\boldsymbol{x}} = (1, 0.5) $ for different values of parameter $ m $

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Table 1. Example 1: Numerical errors ($ \times 100 $) for some $ nd $ and $ k $

$ n_{el} \downarrow k \to $	0.02	0.01	0.005	0.001	0.0001
8	0.0986766	0.0985441	0.0993091	0.1004371	0.1018615
16	0.0368700	0.0313313	0.0303459	0.0308803	0.0315329
32	0.0246722	0.0145489	0.0105503	0.0090745	0.0093450
64	0.0244400	0.0124476	0.0066725	0.0030114	0.0028566
128	0.0245822	0.0124424	0.0063666	0.0016409	0.0009504

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