Numerical analysis of a type III thermo-porous-elastic problem with microtemperatures
- PDF / 1,650,361 Bytes
- 27 Pages / 439.37 x 666.142 pts Page_size
- 32 Downloads / 172 Views
Numerical analysis of a type III thermo-porous-elastic problem with microtemperatures Noelia Bazarra1 · José R. Fernández1
· Ramón Quintanilla2
Received: 6 October 2019 / Revised: 18 May 2020 / Accepted: 5 July 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this work, we consider, from the numerical point of view, a poro-thermoelastic problem. The thermal law is the so-called of type III and the microtemperatures are also included into the model. The variational formulation of the problem is written as a linear system of coupled first-order variational equations. Then, fully discrete approximations are introduced by using the classical finite-element method and the implicit Euler scheme. A discrete stability property and an a priori error estimates result are proved, from which the linear convergence of the algorithm is derived under suitable additional regularity conditions. Finally, some one- and two-dimensional numerical simulations are presented to show the accuracy of the approximation and the behavior of the solution. Keywords Type III thermoelasticity with voids · Microtemperatures · Numerical approximation · Error estimates · Numerical solutions Mathematics Subject Classification 65M60 · 65M12 · 74F05 · 74B05
1 Introduction The most useful model to describe the heat conduction is based on the Fourier law that proposes a linear relation between the heat flux vector and the gradient of temperature. If we combine this equation with the usual energy equation, we obtain the existence of thermal
Communicated by Cassio Oishi.
B
José R. Fernández [email protected] Noelia Bazarra [email protected] Ramón Quintanilla [email protected]
1
Departamento de Matemática Aplicada I, Universidade de Vigo, Escola de Enxeñería de Telecomunicación Campus as Lagoas Marcosende s/n, 36310 Vigo, Spain
2
Departament de Matemátiques, Universitat Politècnica de Catalunya, C. Colom 11, 08222 Terrassa, Barcelona, Spain 0123456789().: V,-vol
123
242
Page 2 of 27
N. Bazarra et al.
waves propagating with an unbounded speed. That is, a thermal perturbation at one point is instantaneously felt at any other point of the space for every distance. It is clear that this effect contradicts the causality principle. For this reason, a big deal trying to overcome this paradox has been developed in the last and current centuries. It seems that the first works in this aspect correspond to Cattaneo and Maxwell (Cattaneo 1958). They proposed the introduction of a relaxation time in the Fourier law. Recently, in the 1990s decade, Green and Naghdi proposed several alternative models (Green and Naghdi 1992, 1993). In fact, they proposed these new theories in the context of the thermoelasticity and the main difference concerning the classical theory corresponds to the thermal effects. The most general is the so-called type III and it contains the Fourier model as a limit case. It is also worth recalling the type II which is also called without energy dissipation. It also corresponds t
Data Loading...
