Investigation on the Use of a Spacetime Formalism for Modeling and Numerical Simulations of Heat Conduction Phenomena

Roula Al Nahas; Alexandre Charles; Benoît Panicaud; Emmanuelle Rouhaud; Israa Choucair; Kanssoune Saliya; Richard Kerner

doi:10.1515/jnet-2019-0074

Article

Investigation on the Use of a Spacetime Formalism for Modeling and Numerical Simulations of Heat Conduction Phenomena

Roula Al Nahas , Alexandre Charles , Benoît Panicaud , Emmanuelle Rouhaud , Israa Choucair , Kanssoune Saliya and Richard Kerner

Published/Copyright: March 5, 2020

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From the journal Journal of Non-Equilibrium Thermodynamics Volume 45 Issue 3

Abstract

The question of frame-indifference of the thermomechanical models has to be addressed to deal correctly with the behavior of matter undergoing finite transformations. In this work, we propose to test a spacetime formalism to investigate the benefits of the covariance principle for application to covariant modeling and numerical simulations for finite transformations. Several models especially for heat conduction are proposed following this framework and next compared to existing models. This article also investigates numerical simulations using the heat equation with two different thermal dissipative models for heat conduction, without thermomechanical couplings. The numerical comparison between the spacetime thermal models derived in this work and the corresponding Newtonian thermal models, which adds the time as a discretized variable, is also performed through an example to investigate their advantages and drawbacks.

Keywords: spacetime modeling; constitutive models; covariance principle; projectors; spacetime thermodynamics; hyperbolic heat conduction; finite propagation; weak integral form

Funding statement: This work was supported by the European Regional Development Funds (FEDER) and the region Grand Est of France and Safran Tech company.

Appendix A

A.1 Tensor densities

In the spacetime formalism, the covariance principle is intrinsically verified with the use of four-tensors densities and four-operators that are, by construction, indifferent to changes of observer (i. e., covariant). We specifically consider a scalar density S and the components of a second-rank tensor density T.

Throughout the article, quantities that vary, such as the base vectors, are called covariant and those that vary, such as the dual base vectors, are called contravariant [67]. The upper indices denote contravariant quantities, whereas lower indices denote covariant quantities. Moreover, the contravariant components of a second-rank tensor density T can be shifted into its covariant components by the use of the metric tensor (for example, Tμν=gμαgνβTαβ).

Through coordinate transformations from xμ to xμ˜, the scalar density S and the components of a second-rank tensor density T verify the following relations:

(46a)S˜=∂xα∂x˜βWS

(46b)T˜μν=∂xα∂x˜βW∂x˜μ∂xλ∂x˜ν∂xκTλκ

(46c)T˜μν=∂xα∂x˜βW∂xλ∂x˜μ∂xκ∂x˜νTλκ,

where ∂x˜μ∂xν is the determinant of the Jacobian matrix ∂x˜μ∂xν. In the coordinate transformation above, the weight of tensor density W has been introduced [67]. Typically, the weight of Cauchy’s stress tensor is equal to one (W=1), whereas the weight of a deformation or strain tensor or temperature is equal to zero (W=0) [67], [68].

A.2 Covariant derivative and rate of deformation

The covariant derivative of, respectively, a scalar density S denoted ∇λS and a second-rank tensor density T, denoted ∇λTμν, is given by

(47a)∇λS=∂S∂xλ−WΓκλκS

(47b)∇λTμν=∂Tμν∂xλ+ΓκλμTκν+ΓκλνTμκ−WΓκλκTμν,

where W is the weight of the tensor density, and Γβγα are the coefficients of the metric connection identified with Christoffel’s symbols, given by

(48)Γκλμ=12gμα∂gακ∂xλ+∂gαλ∂xκ−∂gκλ∂xα=Γλκμ.

Note that in this case ∇λgμν=0, and that for every point of the spacetime domain, for an inertial observer, all Christoffel’s symbols vanish.

It is important to stress that, as a consequence of the absence of gravitation, the Riemann curvature tensor of this spacetime domain is equal to zero, although the Christoffel’s symbols may not. In other words, the considered spacetime is (pseudo-) Euclidean, thus flat, whether the observer is inertial (⇒Γκλμ=0), or not (⇒Γκλμ≠0) [8].

A covariant transport corresponding to the projection of the covariant derivative on the proper time uλ∇λ(.) is also defined. In an inertial coordinate system zμ, in which the Christoffel’s symbols vanish, the covariant transport may be rewritten as

(49)uλ∇λ(.)=uλ∂∂zλ(.)=u4∂∂z4(.)+ui∂∂zi(.)=d(.)ds.

We further introduce the four-tensor rate of deformation d, a spacetime generalization of the symmetric part of the velocity gradient:

(50)daνμ=12(∇νuμ+∇μuν).

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Received: 2019-09-26

Revised: 2020-01-07

Accepted: 2020-02-24

Published Online: 2020-03-05

Published in Print: 2020-07-26

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https://doi.org/10.1515/jnet-2019-0074

Keywords for this article

spacetime modeling; constitutive models; covariance principle; projectors; spacetime thermodynamics; hyperbolic heat conduction; finite propagation; weak integral form