Numerical analysis of a problem in micropolar thermoviscoelasticity

Noelia Bazarra; José R. Fernández; Ramón Quintanilla; Noelia Bazarra; José R. Fernández; Ramón Quintanilla

doi:10.3934/era.2022036

Electronic Research Archive

2022, Volume 30, Issue 2: 683-700. doi: 10.3934/era.2022036

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Numerical analysis of a problem in micropolar thermoviscoelasticity

1.
Departamento de Matemática Aplicada I, Universidade de Vigo, Escola de Enxeñería de Telecomunicación, Campus As Lagoas Marcosende s/n, Vigo 36310, Spain
2.
Departamento de Matemáticas, E.S.E.I.A.A.T.-U.P.C., Colom 11, Terrassa 08222, Barcelona, Spain

Academic Editor: Lei Wang

Received: 14 December 2021 Revised: 28 January 2022 Accepted: 08 February 2022 Published: 18 February 2022

In this work, we study, from the numerical point of view, a dynamic thermoviscoelastic problem involving micropolar materials. The model leads to a linear system composed of parabolic partial differential equations for the displacements, the microrotation and the temperature. Its weak form is written as a linear system made of first-order variational equations, in terms of the velocity field, the microrotation speed and the temperature. Fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved, from which the linear convergence is derived under some additional regularity conditions. Finally, some two-dimensional numerical simulations are presented to demonstrate the accuracy of the approximation and the behavior of the solution.
- Micropolar materials,
- themoviscoelasticity,
- finite elements,
- a priori error analysis,
- numerical simulations
Citation: Noelia Bazarra, José R. Fernández, Ramón Quintanilla. Numerical analysis of a problem in micropolar thermoviscoelasticity[J]. Electronic Research Archive, 2022, 30(2): 683-700. doi: 10.3934/era.2022036

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Abstract

In this work, we study, from the numerical point of view, a dynamic thermoviscoelastic problem involving micropolar materials. The model leads to a linear system composed of parabolic partial differential equations for the displacements, the microrotation and the temperature. Its weak form is written as a linear system made of first-order variational equations, in terms of the velocity field, the microrotation speed and the temperature. Fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved, from which the linear convergence is derived under some additional regularity conditions. Finally, some two-dimensional numerical simulations are presented to demonstrate the accuracy of the approximation and the behavior of the solution.

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