Analytical and numerical analyses of a viscous strain gradient problem involving type Ⅱ thermoelasticity

Noelia Bazarra; José R. Fernández; Jaime E. Muñoz-Rivera; Elena Ochoa; Ramón Quintanilla; Noelia Bazarra; José R. Fernández; Jaime E. Muñoz-Rivera; Elena Ochoa; Ramón Quintanilla

doi:10.3934/math.2024825

AIMS Mathematics

2024, Volume 9, Issue 7: 16998-17024. doi: 10.3934/math.2024825

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Research article

Analytical and numerical analyses of a viscous strain gradient problem involving type Ⅱ thermoelasticity

1.
Departamento de Matemática Aplicada I, Universidade de Vigo, Escola de Enxeñería de Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain
2.
Departamento de Matemática, Universidad del Bío-Bío, Avenida Collao 1202, Casilla 5-Concepción, Chile
3.
National Laboratory for Scientific Computation, Petrópolis, Rio de Janeiro, Brazil
4.
Departamento de Matemáticas, Facultad de Ciencias Exactas, Universidad Andres Bello, Sede Concepción, Autopista Concepción-Talcahuano 7100, Talcahuano, Chile
5.
Departament de Matemàtiques, Escuela Superior de Ingenierías Industrial, Aeroespacial y Audiovisual de Terrassa, Universitat Politécnica de Catalunya, Colom 11, 08222 Terrassa, Barcelona, Spain

Received: 26 March 2024 Revised: 26 April 2024 Accepted: 28 April 2024 Published: 15 May 2024
MSC : 35B40, 65M60, 74F05, 74H55, 74K10

In this paper, a thermoelastic problem involving a viscous strain gradient beam is considered from the analytical and numerical points of view. The so-called type Ⅱ thermal law is used to model the heat conduction and two possible dissipation mechanisms are introduced in the mechanical part, which is considered for the first time within strain gradient theory. An existence and uniqueness result is proved by using semigroup arguments, and the exponential energy decay is obtained. The lack of differentiability for the semigroup of contractions is also shown. Then, fully discrete approximations are introduced by using the finite element method and the backward time scheme, for which a discrete stability property and a priori error estimates are proved. The linear convergence is derived under suitable additional regularity conditions. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximations and the behavior of the discrete energy decay.
- strain gradient,
- viscoelasticity,
- type Ⅱ thermoelasticity,
- existence and uniqueness,
- finite elements,
- a priori error analysis
Citation: Noelia Bazarra, José R. Fernández, Jaime E. Muñoz-Rivera, Elena Ochoa, Ramón Quintanilla. Analytical and numerical analyses of a viscous strain gradient problem involving type Ⅱ thermoelasticity[J]. AIMS Mathematics, 2024, 9(7): 16998-17024. doi: 10.3934/math.2024825

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Abstract

In this paper, a thermoelastic problem involving a viscous strain gradient beam is considered from the analytical and numerical points of view. The so-called type Ⅱ thermal law is used to model the heat conduction and two possible dissipation mechanisms are introduced in the mechanical part, which is considered for the first time within strain gradient theory. An existence and uniqueness result is proved by using semigroup arguments, and the exponential energy decay is obtained. The lack of differentiability for the semigroup of contractions is also shown. Then, fully discrete approximations are introduced by using the finite element method and the backward time scheme, for which a discrete stability property and a priori error estimates are proved. The linear convergence is derived under suitable additional regularity conditions. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximations and the behavior of the discrete energy decay.

References

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