Numerical analysis of a thermoelastic problem with dual-phase-lag heat conduction

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Abstract

In this paper we study, from the numerical point of view, a thermoelastic problem with dual-phase-lag heat conduction. The variational formulation is written as a coupled system of hyperbolic linear variational equations. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. A discrete stability result is proved and a priori error estimates are obtained, from which the linear convergence of the algorithm is deduced under suitable additional regularity conditions. Finally, some two-dimensional numerical simulations are presented to demonstrate the accuracy of the approximation and the behaviour of the solution.

Introduction

The heat conduction theory based on the Fourier law is compatible with the fact that the thermal perturbations at some point will be felt instantly anywhere. This is a drawback of the model and for this reason several authors have tried to overcome this difficulty proposing alternative theories. Most celebrated one is the Cattaneo–Maxwell law [5]. There exist two thermoelastic theories based in this law. Green and Lindsay [13] proposed the first one and the second one is due to Lord and Shulman [20]. Both cases introduce thermoelastic theories described by means of hyperbolic equations. In the decade of the 90's Green and Naghdi [14], [15], [16] suggested three alternative theories which were based on an entropy balance which represents an alternative approach to the usual one based on the entropy inequality. The proposition was based on the axioms of thermomechanics and it was developed from a rational point of view. The main difference between these three theories came from the choice of the independent variables.

In 1995, Tzou [27], [28] proposed a modification of the Fourier law where the delay parameters were present. That is, the constitutive equation takes the formq(x,t+τq)=κθ(x,t+τθ), where θ is the temperature, q is the heat flux vector and τθ and τq represent the phase-lag of the temperature and the heat-flux. One thinks that the time delay τθ is caused by microstructural interactions such as phonon scattering or phonon–electron interactions, meanwhile τq can be seen as the relaxation time due to the fast transient effects of thermal inertia. However, if we adjoin our equation with the heat equationθt+divq=0 the problem becomes ill-posed in the sense of Hadamard (see [9] for details). At the same time, as it has been pointed out in [10], this model is not in agreement with the second law of thermodynamics. The solutions have a very explosive behaviour and we may conclude that the problem cannot be a good candidate to describe the heat conduction nor from the mathematical point of view neither the thermomechanical one. Nevertheless, many people have been attracted by the theories obtained when we substitute the proposed constitutive equation by the Taylor approximations with respect to the delay parameters. Many papers have been published dealing with mathematical and numerical issues as existence, uniqueness, energy decay, spatial behaviour, numerical resolution and so on (see, for instance, [2], [4], [7], [10], [11], [12], [18], [19], [21], [22], [24], [26], [30], [31], [32]).

In this paper, we are going to consider the thermoelastic theory associated to the constitutive equationq+τqqt+τq22qtt=κθκτθθt. It is worth noting that this model is in agreement with the second law of thermodynamics under suitable conditions for the delay parameters (see [10], [11]). As it has been pointed out in [32] the involvement of high-order terms in the delay parameters are the natural consequence of the handling of systems in which multiple energy carriers are involved (see also [29, p. 376]).

In this work we revisit the thermoelastic model based on the previous constitutive equation which can be obtained following the arguments of Chandrasekharaiah [6]. The system of equations was studied in [24], where the existence of a unique solution was proved as well as the spatial behaviour of the solutions. The exponential stability for the one-dimensional case was also obtained. We here continue the research of this problem, providing the numerical analysis of the variational problem, including a discrete stability property and a priori error estimates, and performing two-dimensional numerical simulations which demonstrate the accuracy of the approximation and the behaviour of the solution.

The paper is structured as follows. The mechanical and variational models are presented in Section 2 following [24], and an existence and uniqueness result is recalled. Then, in Section 3 a fully discrete approximation is introduced, based on the finite element method to approximate the spatial domain and the backward Euler scheme to discretize the time derivatives. A discrete stability property and a priori error estimates are proved, from which, under suitable additional regularity conditions, the linear convergence of the algorithm is deduced. Finally, some two-dimensional numerical simulations are presented in Section 4, and some conclusions are shown in Section 5.

Section snippets

The mechanical and variational problems: existence and uniqueness

In this section, we present a brief description of the model and we obtain its mechanical and variational formulations (details can be found in [24]). We also recall an existence and uniqueness result.

Let ΩRd, d=1,2,3, be the domain and denote by [0,T], T>0, the time interval of interest. The boundary of the body Γ=Ω is assumed to be Lipschitz, with outward unit normal vector ν=(νi)i=1d. Moreover, let xΩ and t[0,T] be the spatial and time variables, respectively. In order to simplify the

Fully discrete approximations: an a priori error analysis

In this section, we now consider a fully discrete approximation of Problem VP. This is done in two steps. First, we assume that the domain Ω is polyhedral and we denote by Th a regular triangulation in the sense of [8]. Thus, we construct the finite dimensional spaces VhV and EhE given byVh={zh[C(Ω)]d;z|Trh[P1(Tr)]dTrTh,zh=0onΓ},Eh={rhC(Ω);r|TrhP1(Tr)TrTh,rh=0onΓ}, where P1(Tr) represents the space of polynomials of degree less or equal to one in the element Tr, i.e. the finite

Numerical results

In this final section, we describe the numerical scheme implemented in the well-known finite element code FreeFem++ for solving Problem VPhk, and we show some numerical examples to demonstrate the accuracy of the approximations and the behaviour of the solution.

Conclusions

In this paper we analyzed, from the numerical point of view, a dynamic problem involving a thermoelastic body. The dual-phase-lag heat conduction theory was used to model the thermal effects. The variational formulation was written as a hyperbolic system of coupled linear variational equations in terms of the thermal acceleration and the velocity field. Then, we introduced a fully discrete scheme using the finite element method to approximate the spatial variable and the implicit Euler scheme

Acknowledgements

The work of N. Bazarra, M. Campo and J.R. Fernández has been supported by the Ministerio de Economía y Competitividad of the Spanish Governement under the research project MTM2015-66640-P (with the participation of FEDER).

The work of R. Quintanilla has been supported by the Ministerio de Economía y Competitividad of the Spanish Governement under the research project “Análisis Matemático de Problemas de la Termomecánica” (MTM2016-74934-P), (AEI/FEDER, UE).

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