Sébastien Boyaval
email: sebastien.boyaval@enpc.fr

Mark Dostalík
email: dostalik@karlin.mff.cuni.cz

Non-isothermal viscoelastic flows with conservation laws and relaxation

Keywords: viscoelasticity, Maxwell fluid, balance laws, mathematical entropy, short-time well-posedness

We propose a system of conservation laws with relaxation source terms (i.e. balance laws) for non-isothermal viscoelastic flows of Maxwell fluids.

The system is an extension of the polyconvex elastodynamics of hyperelastic bodies using additional structure variables. It is obtained by writing the Helmholtz free energy as the sum of a volumetric energy density (function of the determinant of the deformation gradient det F and the temperature θ like the standard perfect-gas law or Noble-Abel stiffened-gas law) plus a polyconvex strain energy density function of F, θ and of symmetric positive-definite structure tensors that relax at a characteristic time scale. One feature of our model is that it unifies various ideal materials ranging from hyperelastic solids to perfect fluids, encompassing fluids with memory like Maxwell fluids.

We establish a strictly convex mathematical entropy to show that the system is symmetric-hyperbolic. Another feature of the proposed model is therefore the short-time existence and uniqueness of smooth solutions, which define genuinely causal viscoelastic flows with waves propagating at finite speed.

In heat-conductors, we complement the system by a Maxwell-Cattaneo equation for an energy-flux variable. The system is still symmetric-hyperbolic, and smooth evolutions with finite-speed waves remain well-defined.

Introduction

Continuum mechanics has proved a useful theory for a number of real materials, when constitutive assumptions allow one to predict motions unequivocally see e.g. [START_REF] Marsden | Mathematical foundations of elasticity[END_REF][START_REF] Boyaval | Viscoelastic flows of Maxwell fluids with conservation laws[END_REF]. However, for many materials, it remains a challenge to propose constitutive assumptions that both yield unequivocal predictions and realistic behaviours. It is the case of polymeric materials which exhibit viscoelastic stress relaxation see e.g. [START_REF] Mackay | On the derivation of macroscopic models for compressible viscoelastic fluids using the generalized bracket framework[END_REF][START_REF] Bollada | On the mathematical modelling of a compressible viscoelastic fluid[END_REF].

Recently, one of us proposed a symmetric-hyperbolic system of conservation laws with relaxation source terms for the isentropic or isothermal compressible viscoelastic flows of Maxwell fluid [START_REF] Boyaval | Viscoelastic flows of Maxwell fluids with conservation laws[END_REF]. The system basically extends the polyconvex elastodynamics conservation laws of hyperelastic materials by introducing new structure variables and using Maxwell definition of viscosity. It contains a compressible version of the Upper-Convected Maxwell model, which is known to describe polymer flows see e.g. [START_REF] Gupta | Modeling of nonisothermal polymer processes[END_REF][START_REF] Mackay | On the derivation of macroscopic models for compressible viscoelastic fluids using the generalized bracket framework[END_REF]. A salient feature of the proposed model is the existence and uniqueness (on short-times) of smooth solutions to the initial-value problem, with waves propagating at finite-speed. However, for practical applications, many polymer flows are non-isothermal.

Here, we first propose an extension of the recent model of [START_REF] Boyaval | Viscoelastic flows of Maxwell fluids with conservation laws[END_REF] to nonisothermal flows. This is achieved by letting the two terms in the Helmholtz free energy of [START_REF] Boyaval | Viscoelastic flows of Maxwell fluids with conservation laws[END_REF] depend on the temperature θ. On the one hand, the volumetric term function of the determinant of the deformation gradient det F can be chosen as the standard perfect-gas law (or Noble-Abel stiffened-gas law, see variations of our model in Section 4). On the other hand, the polyconvex Hookean energy density term that is a function of F and of positive-definite structure tensors has a molecular interpretation (see e.g. [9,[START_REF] Boyaval | Viscoelastic flows of Maxwell fluids with conservation laws[END_REF]) which suggests how it depends on θ. The dependence on θ is also suggested by phenomenological models for rubbery materials [7]. Heat-conduction is allowed at finite propagation-speed using Maxwell-Cattaneo law.

We propose a symmetric-hyperbolic system of conservation laws with relaxation. Therefore, the system unequivocally models smooth non-isothermal viscoelastic flows (on short times) using finite-speed waves.

Second, we consider important variations of our model. For instance, we add a term function of the inverse deformation gradient to the polyconvex Hookean elastic energy density, as in models for polymer melts like the celebrated K-BKZ fluids. It allows for an equivalent Lagrangian description of our Eulerian model, and in turn the consistency of our smooth solutions to the governing system of conservation laws with relaxation [START_REF] Wagner | Symmetric-hyperbolic equations of motion for a hyperelastic material[END_REF]. We also describe how to further modify the polyconvex strain energy term and account for finiteextensibility e.g. in applications to polymer flows, using what is called Gent law for rubbery materials or FENE-P law for diluted polymer suspensions.

The paper is organized as follows. In Section 2 we first recall the continuum thermomechanics setting and the Maxwell-Cattaneo heat-conduction model. Then, we proceed with the derivation of constitutive relations in a thermodynamically consistent way, looking for closure relations that guarantee the nonnegativity of the entropy production. (We presume that the identification of the energy storage mechanisms and entropy storage mechanisms provides a complete characterisation of the fluid, deferring to the Section 3 a mathematical proof that thermomechanical evolutions are unequivocally well-defined.)

In Section 3 auxiliary tensorial quantities are introduced, which can be interpreted as structure variables (a material metric) and which enable us to find an additional balance law for a strictly convex scalar quantity. As an application of the theory of first-order systems of conservation laws with source terms, that additional law yields symmetrizability of our system of balance laws, and short-time well-posedness of the associated Cauchy problem.

In Section 4 we discuss generalizations of our compressible heat-conducting model for Maxwell fluids to (i) models with an equivalent Lagrangian description, in the spirit of other fluids of the K-BKZ class for polymer melts, to (ii) models with finite-extensibility effects, like in FENE-P fluids or Gent elastomers, and to (iii) models with a volumetric energy term more suitable for liquids than the perfect-gas law, using Noble-Abel stiffened-gas law namely.

2 Compressible Maxwell model, with heat-conduction

Balance laws of continuum thermomechanics

Let us briefly recall the concepts of thermomechanics of continuous media while introducing the notation. We refer to e.g. [START_REF] Marsden | Mathematical foundations of elasticity[END_REF][START_REF] Šilhavý | The mechanics and thermodynamics of continuous media[END_REF] for a proper introduction to the theory. To begin with, we assume that the balance of total energy holds,

ρ dE dt = div T ⊤ v -j e + ρf • v, (2.1)
where ρ denotes the density of the fluid, E denotes its specific total energy, T denotes the Cauchy stress tensor, v denotes the spatial velocity field, j e denotes the energy flux, and f denotes the specific body force. Note that we assume an Eulerian description in Euclidean space with divergence operator div and the material derivative

d dt def = ∂ ∂t + v • ∇. (2.2)
Assuming that the specific total energy E is given by

E def = e + 1 2 |v| 2 , (2.3)
where e denotes the specific internal energy of the material, and that the balance of total energy (2.1) is Galilean invariant, one can derive

dρ dt = -ρ div v, (2.4a)
ρ dv dt = div T + ρf , (2.4b)
i.e. the balance of mass and the balance of linear momentum, respectively. An equation for the specific internal energy can be deduced from (2.4b) and (2.1)

ρ de dt + div j e = T : L, (2.5)
where L denotes the velocity gradient, i.e. L def = ∇v. Here, A : B def = tr(A ⊤ B) denotes the Frobenius inner product as usual. Lastly, we assume the evolution equation for the specific entropy η,

ρ dη dt + div j η = ξ, (2.6)
where j η denotes the entropy flux, and ξ denotes the entropy production of the given fluid. Note that to guarantee the second law of thermodynamics, the entropy production must be a nonnegative quantity. We also require a well-defined, positive thermodynamic temperature θ def = ∂ η e > 0. Our goal here is to predict "viscoelastic" fluid motions as the solutions to Cauchy initial-value problems that use the equations above, plus closure relations between the physical quantities introduced. By viscoelastic motions, we mean motions where the stress tensor is governed by an evolution with features similar to the seminal viscoelastic model of Maxwell [START_REF] Maxwell | IV. On the dynamical theory of gases[END_REF], see also [START_REF] Boyaval | Viscoelastic flows of Maxwell fluids with conservation laws[END_REF].

Thermodynamic of compressible Maxwell fluids, with heat conduction

Specific Helmholtz free energy

For compressible Maxwell fluids we propose a fundamental thermodynamic relation (or complete equation of state) given by a specific Helmholtz free energy ψ that is the sum of three terms as follows

ψ(ρ, θ, C, j e ) def = ψ s (ρ, θ) + α 2 (K(θ) tr C -k B θ log det C) + τ 0 2κ |j e | 2 .
(2.7)

We have denoted k B the Boltzmann constant, α > 0 a degree of elasticity per unit mass, τ 0 > 0 a relaxation time and κ > 0 a thermal conductivity.

For heat-conducting fluids, we have added a dependence of ψ on internal degrees of freedom through an energy flux vector j e . The particular case without heat conduction coincides with the formal limit j e = κ∇θ, κ → 0.

In any case, the fundamental variables of the specific Helmholtz free energy other than the energy flux vector j e are the temperature θ, and the density ρ together with an additional symmetric positive-definite tensorial quantity C to be linked further with the deformations of the body (a measure of strain beyond the purely volumetric term ρ).

The volumetric contribution ψ s to the specific Helmholtz free energy ψ is for the moment left unspecified. However, the choice of ψ s matters in wellposedness. We use particular (analytical) formulae in Sections 3.2 and 4.

The second term in the right-hand side of (2.7) characterizes Hookeanelastic fluids with stiffness K(θ) > 0 like Maxwell fluids. For polymer suspensions, it can be derived from a molecular theory when C is the conformation tensor of diluted Hookean-elastic dumbbells with spring factor K(θ), see e.g. [START_REF] Boyaval | Viscoelastic flows of Maxwell fluids with conservation laws[END_REF] for a reference. The tensor then expresses the degree of orientation and elongation of diluted polymer chains, and following the seminal work on nonisothermal viscoelastic fluid models [START_REF] Marrucci | The free energy constitutive equation for polymer solutions from the dumbbell model[END_REF] we can assume here

K(θ) def = K 0 + K 1 θ, (2.8)
with constant parameters K 0 , K 1 > 0. Moreover, C has the meaning of a strain measure similar to the left Cauchy-Green deformation tensor FF ⊤ ,

F def = ∂χ t ∂a • χ -1 t
being the gradient with respect to material coordinates a of a deformation

χ t : a ∈ R 3 → R 3 ∋
x associated with the continuous motion of velocity v:

dF dt = LF. (2.9)
A natural time rate for C is then given by the upper convected derivative

∇ C def = dC dt -LC -CL ⊤ , (2.10)
an objective time derivative which ensures material frame-indifference of our constitutive assumptions. Lastly, for entangled polymer melts and other rubber-like materials, one rather interprets C as a measure of strain similar to the inverse right Cauchy-Green tensor F -⊤ F -1 , by a network kinetic theory which still suggest K(θ) to be an affine function of the temperature in a first approach [START_REF] Marrucci | Testing of a constitutive equation for entangled networks by elongational and shear data of polymer melts[END_REF]. We shall inspect generalizations of our model using various strain measures C -like in the so-called K-BKZ fluid models [START_REF] Bernstein | A study of stress relaxation with finite strain[END_REF]-in our last Section 4. Note that for nonconducting polymer melts, the Helmholtz free-energy formula with affine dependence on temperature is similar to phenomenological models proposed for rubbery materials [7]. Moreover, using the inverse right Cauchy-Green measure of strain, with a time rate different than (2.10), shall turn out mathematically useful for a consistent interpretation of our model, see Section 4.1.

Refined physical phenomenas like the finite-extensibility of polymers can be introduced in (2.7) using e.g. the so-called Gent or FENE-P law instead of the Hookean law for the elastic energy term. We also refer to Section 4.

First, let us specify the simpler non-isothermal compressible Maxwell model, possibly heat-conducting, and its mathematical properties for predicting time evolutions.

In any case, a non-linear dependence of K on the temperature as suggested in e.g. [START_REF] Gupta | Modeling of nonisothermal polymer processes[END_REF] appears to be a non-trivial modification of our model; its analysis is not easily generalized from the one to be developed here.

Specific internal energy

We can derive a fundamental thermodynamic relation of compressible Maxwell fluids in terms of the internal energy e = ψ + θη. Using (2.7) we find that

η(ρ, θ, C, j e ) def = - ∂ψ ∂θ = - ∂ψ s ∂θ (ρ, θ) - α 2 (K 1 tr C -k B log det C). (2.11)
We assume ∂ 2 ψs ∂θ 2 < 0 so the relation (2.11) can be inverted for the temperature

θ(ρ, η, C, j e ) = ∂e s ∂η ρ, η + α 2 (K 1 tr C -k B log det C) , (2.12)
where e s denotes the volumetric internal energy in variables (ρ, η) such that

η s (ρ, θ) def = - ∂ψ s ∂θ (ρ, θ) e s (ρ, η s (ρ, θ)) def = ψ s (ρ, θ) -θ ∂ψ s ∂θ (ρ, θ) . (2.13)
A full thermodynamic description of compressible Maxwell fluids, equivalent to the fundamental thermodynamic relation in terms of the Helmholtz free energy (2.7), is given by the full internal energy of the fluid, i.e.

e(ρ, η, C, j e ) = e s ρ, η

+ α 2 (K 1 tr C -k B log det C) + α 2 K 0 tr C + τ 0 2κ |j e | 2 .
(2.14) On postulating the specific internal energy e, we assume ∂ 2 es ∂η 2 > 0 to formulate an explicit formula for the specific Helmholtz free energy ψ in turn.

Constitutive closure relations

Our objective now is to derive a compressible heat-conducting Maxwell model by closing the generic balance laws of Section 2.1 consistently with the thermodynamics postulated in Section 2.2. The derivation outlined below follows [START_REF] Rajagopal | A thermodynamic frame work for rate type fluid models[END_REF] see also [START_REF] Málek | On a variant of the Maxwell and Oldroyd-B models within the context of a thermodynamic basis[END_REF], [START_REF] Hron | On thermodynamics of viscoelastic rate type fluids with temperature dependent material coefficients[END_REF], or [START_REF] Málek | Derivation of equations for continuum mechanics and thermodynamics of fluids[END_REF]. By specifying a fundamental thermodynamic relation describing the material one can determine possible constitutive relations for the Cauchy stress tensor T, the evolution equation for the conformation tensor C, and the energy and entropy fluxes j e , j η , in a thermodynamically consistent way. Indeed, a suitable choice of the aforementioned relations yields an explicit formula for the entropy production ξ which is nonnegative, thus assuring the validity of the second law of thermodynamics. Of course, such a model derivation is only formal and it remains to show that motions can be well-defined, typically as solutions to well-posed Cauchy problems: this will be the subject of Section 3. To proceed, let us formulate the evolution equation for the specific entropy from the postulated fundamental thermodynamic relation (2.14).

Evolution equation for entropy

Given e = e(ρ, η, C, j e ) supposedly well-defined by (2.14)

∂η ρ, η + α 2 (K 1 tr C -k B log det C) , (2.17a)
p(ρ, η, C, j e ) def = ρ 2 ∂e ∂ρ (ρ, η, C, j e ) = ρ 2 ∂e s ∂ρ ρ, η + α 2 (K 1 tr C -k B log det C) . (2.17b)
Exploiting (2.14) and the formula (2.10), equation (2.16) can then be rewritten as

T : L -div j e = ρθ dη dt -p div v + αρ 2 K(θ)I -k B θC -1 : ∇ C +LC + CL ⊤ + τ 0 ρ κ j e • dj e dt . (2.18)
To cast (2.18) into the general evolution equation for the specific entropy (2.6), we divide (2.18) by the temperature and we arrive at

ρ dη dt + div j e θ = 1 θ (T -αρK(θ)C) : D + (p + αρk B θ) div v - αρ 2 K(θ)I -k B θC -1 : ∇ C -j e • ∇θ θ + τ 0 ρ κ dj e dt , (2.19)
where we used the fact that the Cauchy stress tensor T and the conformation tensor C are symmetric. Finally, we split the Cauchy stress tensor T as

T = mI + T δ (2.20)
where m def = 1 3 tr T is the mean normal stress and T δ def = T -1 3 (tr T)I the traceless part. The manipulation allows us to separate the volume-changing and volume-preserving deformations, see e.g. [START_REF] Málek | Derivation of equations for continuum mechanics and thermodynamics of fluids[END_REF]. Rewriting (2.19) as

ρ dη dt + div j e θ = 1 θ (T δ -αρK(θ)C δ ) : D δ + m + p - 1 3 αρK(θ) tr C + αρk B θ div v - αρ 2 K(θ)I -k B θC -1 : ∇ C -j e • ∇θ θ + τ 0 ρ κ dj e dt (2.21)
we have finally identified four "independent" entropy-producing mechanisms on the right-hand side of (2.21). It allows one to straightforwardly propose constitutive relations in a thermodynamically consistent way, so as to assure the nonnegativity of the individual contributions to the entropy production.

Entropy production and viscoelastic constitutive relations

A possible closure for the constitutive relations that yields the desired heatconducting Maxwell model reads as follows

m = -p + 1 3 αρK(θ) tr C -αρk B θ, (2.22a)
T δ = αρK(θ)C δ , (2.22b)
∇ C = - 4K(θ) ζ C + 4k B θ ζ I, (2.22c)
τ 0 ρ κ dj e dt = - ∇θ θ - j e κθ , (2.22d
)

j η = j e θ , (2.22e)
where the parameter ζ > 0 can be interpreted as the hydrodynamic drag coefficient in molecular theories for polymer suspensions, recall Section 2.2.1. Such a friction coefficient ζ a priori depends on the temperature θ (so ζ/k B θ actually scales like a relaxation time, characteristic of a polymer fluid) and possibly also on the conformation C. That dependence has important quantitative consequences for the numerical values of predicted motions, see e.g. [START_REF] Gupta | Modeling of nonisothermal polymer processes[END_REF]. However, the dependence of ζ on θ is only algebraic a priori (i.e. it is not through its derivatives), so it has no influence on our (short-time) wellposedness result provided it is smooth. Furthermore, we shall come back to the dependence of ζ on C later in Section 4, to introduce finite-extensibility effects using the so-called Gent or FENE-P law. Extensions in that direction beyond the Gent or FENE-P law are certainly desirable physically but are left here as issues that could possibly be addressed in the future, see our conclusion in Section 5.

Entropy production ξ, i.e. the right-hand side of (2.21), now reads

ξ(ρ, θ, C, j e ) = |j e | 2 κθ 2 + 2αρ ζθ K(θ)I -k B θC -1 : (K(θ)C -k B θI) (2.23) = |j e | 2 κθ 2 + 2αρ ζθ K(θ)C 1 2 -k B θC -1 2 2 ≥ 0, (2.24)
and the second law of thermodynamics is satisfied. Here, |A| def = √ A : A denotes the Frobenius norm as usual.

Further, combining (2.20), (2.22a), and (2.22b) we can write

T = -pI + S, (2.25a)
S def = αρ(K(θ)C -k B θI), (2.25b)
where S denotes the so-called extra stress tensor for the compressible nonisothermal Maxwell fluids, see e.g. [9]. A straightforward computation shows that S satisfies a multi-dimensional version of Maxwell seminal model [START_REF] Maxwell | IV. On the dynamical theory of gases[END_REF], so our model is indeed of that viscoelastic type.

Remark 1 Fourier's law of heat conduction

j e = -κ∇θ (2.26)
is widely used for heat-conducting continuous media (assumed homogeneous and thermally isotropic here for the sake of simplicity). However, Fourier's law formally yields propagation of thermal signals with infinite speed, see e.g. [START_REF] Joseph | Heat waves[END_REF], which in turn violates the principle of causality. To address the causality issue, the following Galilean-invariant Maxwell-Cattaneo law [START_REF] Cattaneo | Sulla Conduzione Del Calore[END_REF]

τ 0 ρ dj e dt = - κ∇θ θ - j e θ (2.27)
is one standard possible alternative, which reduces to the standard Fourier's law when τ 0 → 0 + . A reason why we opt for the heat conduction law in the form of (2.27) is the quadratic dependence of the corresponding specific internal energy on the heat flux j e , see Section 2.2.2. It allows a simple proof of the strict convexity in Section 3.2. It is also noteworthy that our model covers heat-insulators (non-conducting materials) in the formal limits τ 0 → 0 + (so (2.26) holds), κ → 0.

Having derived the constitutive closure relations (2.22) we can now formulate a formally-closed system of equations for the unknowns (ρ, η, j e , v, C) that govern the thermomechanical evolution of the material:

dρ dt = -ρ div v, (2.28a)
ρ dη dt + div j e θ(ρ, η, C, j e ) = ξ(ρ, η, C, j e ), (2.28b)
ρ dj e dt = - 1 τ 0 θ(ρ, η, C, j e )
(κ∇θ(ρ, η, C, j e ) + j e ) (2.28c)

ρ dv dt = div(T(ρ, η, C)) + ρf , (2.28d)
∇ C = - 4K(θ(ρ, η, C, j e )) ζ C + 4k B θ(ρ, η, C, j e ) ζ I, (2.28e)
where the Cauchy stress tensor T, the temperature θ, the pressure p and the entropy production ξ are assumed to be smooth functions of the main unknown variables given by (2.25), (2.17) and (2.23). We also recall that ζ can be a smooth function of θ here at the present stage, and we refer to Section 4 for a possible (physically-meaningful) additional dependence of ζ on C. The governing equations (2.28) shall define a mechanical model of compressible heat-conducting Maxwell fluids provided unequivocal (i.e. univocal) motions can be defined. To that aim, we consider here the solutions to wellposed Cauchy problems for (2.28) complemented by initial conditions.

Well-posedness

Let us consider the short-time existence of unique continuous-in-time solutions to Cauchy problems for (2.28) like in Th. 1 below as a minimal well-posedness requirement. To use Th. 1, the balance laws (2.28a)-(2.28d) must be reformulated as a first-order system of conservation laws (3.7) with smooth flux and source terms. But the evolution equation for the conformation tensor (2.28e) cannot be cast into such a conservative form. This makes the construction of univocal solutions to the associated multidimensional Cauchy problems very difficult. Following [START_REF] Boyaval | Viscoelastic flows of Maxwell fluids with conservation laws[END_REF], we thus introduce a new tensor variable, to embed (2.28) in a system of conservation laws with smooth flux and relaxation source terms.

Governing equations as conservation laws with relaxation source terms

Let us define the tensorial quantity A by the formula

A def = F -1 CF -⊤ . (3.1)
The tensor A is assumed symmetric positive-definite, consistently with the definition of C. The definition of the tensorial quantity A is motivated by the K-BKZ theory, see the seminal works [START_REF] Kaye | Non-Newtonian flow in incompressible fluids[END_REF][START_REF] Bernstein | A study of stress relaxation with finite strain[END_REF] where A is used implicitly to derive an integro-differential isothermal viscoelastic fluid model similar to elastodynamics. More specifically, it is motivated by the polyconvex elastodynamics, which actually allows one to define univocal motions of hyperelastic materials after embedding the governing equations into a symmetric-hyperbolic system of conservation laws, see e.g. [START_REF] Wagner | Symmetric-hyperbolic equations of motion for a hyperelastic material[END_REF]. The tensor A can be interpreted as a material metric in the context of viscoelasticity, see [START_REF] Boyaval | Viscoelastic flows of Maxwell fluids with conservation laws[END_REF] for a brief description of its physical meaning. In any case, the tensor A satisfies

dA dt = - 4K(θ) ζ A + 4k B θ ζ F -1 F -⊤ (3.2)
upon using the differential identity

d(F -1 ) dt = -F -1 dF dt F -
∂ρ ∂t + div(ρv) = 0, (3.3a) ∂(ρη) ∂t + div ρηv + j e θ = ξ, (3.3b)
∂(ρj e ) ∂t + div ρj e ⊗ v + κ τ 0 e -θ I = - j e τ 0 θ (3.3c) ∂(ρv) ∂t + div(ρv ⊗ v -T) = ρf , (3.3d)
∂(ρF) ∂t + div ρ F ⊗ v -v ⊗ F ⊤ = 0, (3.3e)
∂(ρA) ∂t + div(ρA ⊗ v) = - 4ρK(θ) ζ A + 4ρk B θ ζ F -1 F -⊤ , (3.3f)
where the entropy production ξ, the temperature θ, and the Cauchy stress tensor T are given as functions of the unknowns (ρ, ρη, ρj e , ρv, ρF, ρA) by the relations (2.23), (2.17a), and (2.25), respectively. Here, the symbol ⊗ denotes the outer product (of tensors) and the divergence operator is always taken with respect to the last coordinate. This means that we have, in coordinates,

[div(ρF ⊗ v)] i α = ∂ ρ[F] i α v k ∂x k , (3.4)
and analogously in the remaining cases. The conservation law (3.3e) is classically derived from (2.9) using (2.4a), the so-called Piola identity div

F ⊤ det F = 0, (3.5)
assuming that we are dealing with a homogeneous material, so that

ρ R def = ρ det F, (3.6)
i.e. the density of the material in the reference configuration, is constant. Now, we would like to use the standard results for (3.3) which we rewrite in the form

∂u ∂t + d j=1 ∂f j (u) ∂x j = c(u) (3.7)
with smooth flux f j : R n → R n and relaxation source terms c : R n → R n see e.g. [START_REF] Serre | Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves[END_REF][START_REF] Dafermos | Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften[END_REF][START_REF] Benzoni-Gavage | Multi-dimensional hyperbolic partial differential equations: First-order systems and applications[END_REF]. Precisely, we consider smooth fields u : R d × [0, +∞) → R n that are equivalently solutions of the quasilinear system

∂u ∂t + d j=1 A j (u) ∂u ∂x j = c(u), (3.8)
where the n × n matrices A j are given by

A j (u) def = ∂f j (u) ∂u .
(3.9) Theorem 1 (Short-time well-posedness) Let U be an open subset of R n . We assume that A j and c are C ∞ functions of u ∈ U and that (3.8) is Friedrichs-symmetrizable in U i.e. there exists a C ∞ mapping S : U → R n×n such that S(u) is symmetric positive-definite, and the matrices S(u)A j (u) are symmetric for all u ∈ U. Let u 0 ∈ U and

u 0 ∈ H s (R d ; R n ) with s > 1 + d 2 such that u 0 + u 0 is compactly supported in U.
Then, there exists T > 0 and a unique classical solution

u ∈ C 1 (R d × [0, T ]; U) to (3.7) with the initial data u(0) = u 0 + u 0 . Furthermore, u -u 0 ∈ C([0, T ]; H s ) ∩ C 1 ([0, T ]; H s-1 ).
Proof See Theorem 10.1 and Theorem 13.1 in [START_REF] Benzoni-Gavage | Multi-dimensional hyperbolic partial differential equations: First-order systems and applications[END_REF].

To apply Theorem 1 to (3.8), one can show that the quasilinear system is Friedrichs-symmetrizable using a mathematical entropy s:

Theorem 2 (Godunov-Mock) Assume that U is convex and that the system of balance laws (3.7) implies the existence of an additional balance law

∂s(u) ∂t + d j=1 ∂F j (u) ∂x j = b(u), (3.10)
where s : U → R, F j : U → R, j ∈ {1, . . . , d}, and b : U → R are smooth functions and s is strictly convex. Then, the associated quasilinear system (3.8) is Friedrichs-symmetrizable.

Proof See [START_REF] Friedrichs | Systems of conservation equations with a convex extension[END_REF], or [START_REF] Godunov | An interesting class of quasi-linear systems[END_REF], [START_REF] Godunov | Elements of continuum mechanics[END_REF], and [START_REF] Godunov | Lois de conservation et integrales d'energie des equations hyperboliques[END_REF].

The quantity ρE is a natural candidate for the mathematical entropy of (3.3) because (2.1) has the conservation form

∂(ρE) ∂t + div ρEv -T ⊤ v + j e = ρf • v (3.11)
with, by virtue of (2.3) and (2.14), a specific total energy that reads

E = 1 2 |v| 2 + e s ρ, η + α 2 K 1 tr FAF ⊤ -k B log det FAF ⊤ + τ 0 2κ |j e | 2 + α 2 K 0 tr FAF ⊤ . (3.12)
But ρE is not strictly convex in u. Precisely, ρE is strictly convex with respect to (ρ, ρη, ρj e , ρv, ρF, ρA) if and only if the function E is strictly convex with respect to (ρ -1 , η, j e , v, F, A), see [START_REF] Wagner | Symmetric-hyperbolic equations of motion for a hyperelastic material[END_REF] or [START_REF] Bouchut | Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources[END_REF]. However, it can be easily shown that the last term in (3.12) is not strictly convex with respect to (F, A). The situation is similar to the elastodynamics of hyperelastic materials see e.g. [START_REF] Wagner | Symmetric-hyperbolic equations of motion for a hyperelastic material[END_REF]: to establish well-posedness, we are led to similarly use polyconvexity and rewrite our system with additional independent quantities.

Governing equations as a symmetrizable system

Let us define a symmetric positive-definite tensor Y via the formula

Y def = A -2 .
(3.13)

As long as Y and A remain symmetric positive-definite, Y is equivalently defined by the evolution equation (3.2) for A and (3.13) as follows

dY dt = 8K(θ) ζ Y - 4k B θ ζ Y F -1 F -⊤ Y -1 2 + Y -1 2 F -1 F -⊤ Y def = f. (3.14)
Moreover, recalling the additional conservation law (3.12), we also introduce the scalar quantity Y def = det Y -1 as an independent variable, with evolution equation

dY dt = 4 ζ Y tr k B θ F -1 F -⊤ Y -1 2 + Y -1 2 F -1 F -⊤ Y -2K(θ)I def = h (3.15)
compatible with the evolution equations for A or Y when Y > 0.

The thermomechanical evolution of compressible Maxell fluids is governed by a system for the new set of unknowns (ρ, ρη, ρj e , ρv, ρF, ρY, ρY ):

∂ρ ∂t + div(ρv) = 0, (3.16a) ∂(ρη) ∂t + div ρηv + j e θ = ξ, (3.16b)
∂(ρj e ) ∂t + div ρj e ⊗ v + κ log θ τ 0 I = - j e τ 0 θ (3.16c) ∂(ρv) ∂t + div(ρv ⊗ v -T) = ρf , (3.16d)
∂(ρF) ∂t + div ρ F ⊗ v -v ⊗ F ⊤ = 0, (3.16e)
∂(ρY) ∂t + div(ρY ⊗ v) = ρf, (3.16f)
∂(ρY ) ∂t + div(ρY v) = ρh. (3.16g)
the entropy production ξ, the temperature θ, the Cauchy stress tensor T, and the quantities f, h are given by equations (2.23), (2.17a), (2.25), (3.14), (3.15) as functions of the unknowns (ρ, ρη, ρj e , ρv, ρF, ρY, ρY ).

It remains to identify a mathematical entropy for (3.16). Denoting e ref a constant reference value of the internal energy, we propose ρ E where

E def = 1 2 |v| 2 + e s (ρ, η, F, Y, Y ) + τ 0 2κ |j e | 2 + α 2 K 0 tr FY -1 2 F ⊤ + e ref 2 |Y| 2 , (3.17
e s (ρ, η, F, Y, Y ) def = e s ρ, η + α 2 K 1 tr FY -1 2 F ⊤ -k B log ρ R ρ 2 Y 1 2 . (3.18)
Note that by virtue of (3.12) and (3.13) it holds

E = E + e ref 2 |Y| 2 (3.19)
and thus, using the balance of the total energy (3.11) we obtain

∂(ρ E) ∂t + div ρ Ev -T ⊤ v + j e = ρf • v + ρe ref Y : f, (3.20)
where the right-hand side of (3.20) is an algebraic function of the unknowns (ρ, ρη, ρj e , ρv, ρF, ρY, ρY ).

Theorem 3 Assume that the volumetric energy functional e s = e s (ρ, η) is chosen so that the function e s defined by (3.18) is convex with respect to (ρ -1 , η, F, Y, Y ) and strictly convex with respect to (ρ -1 , η, Y ).

Then, the scalar quantity ρ E, where E is given by the formula (3.17), is strictly convex with respect to (ρ, ρη, ρj e , ρv, ρF, ρY, ρY ).

Proof We already know that the function ρ E is strictly convex with respect to (ρ, ρη, ρj e , ρv, ρF, ρY, ρY ) if and only if the function E is strictly convex with respect to (ρ -1 , η, j e , v, F, Z, Y, Y ), see [START_REF] Wagner | Symmetric-hyperbolic equations of motion for a hyperelastic material[END_REF] or [START_REF] Bouchut | Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources[END_REF]. Next, note that the function tr(FY -1 2 F ⊤ ) is convex with respect to (F, Y), see [START_REF] Lieb | Convex trace functions and the wigner-yanase-dyson conjecture[END_REF], and strictly convex with respect to F. As a consequence, the sum

α 2 K 0 tr FY -1 2 F ⊤ + e ref 2 |Y| 2 (3.21)
is strictly convex with respect to (F, Y). Exploiting the assumption about the function e s we then obtain that the sum

e s (ρ, η, F, Y, Y ) + α 2 K 0 tr FY -1 2 F ⊤ + e ref 2 |Y| 2 (3.22)
is strictly convex with respect to (ρ -1 , η, F, Y, Y ). Further, the terms 1 2 |v| 2 and τ0 2κ |j e | 2 are strictly convex with respect to v and j e , respectively. Trivially, these terms as well as the expression (3.22) are also convex functions with respect to (ρ -1 , η, j e , v, F, Y, Y ). Consequently, E is strictly convex with respect to (ρ -1 , η, j e , v, F, Y, Y ).

We are now ready to formulate our first result.

Theorem 4 Let the volumetric term of the energy be described by the polytropic gas equation of state

e s (ρ, η) = c V,s θ ref ρ ρ ref γ-1 exp η c V,s , (3.23)
where c V,s denotes a specific heat capacity at constant volume, γ > 1 is a constant referred to as the adiabatic exponent, and θ ref , ρ ref denote a constant reference temperature and a constant reference density, respectively, which can be chosen at will as they serve for normalization purposes only. Then, the scalar quantity ρ E, where E is given by the formula (3.17), is a mathematical entropy of the system of balance laws (3.16).

To prove Theorem 4, it is useful to recall an elementary fact from convex optimization see e.g. [27, Theorem 2.3.10].

Lemma 1 Let X ⊂ R n , n ∈ N be a convex set and let g : X → R be convex (resp. strictly convex) and f : R → R be non-decreasing (resp. increasing) and convex. Then, f • g is convex (resp. strictly convex).

Proof (of Theorem 4) We already know that the quantity ρ E satisfies an additional balance law, see (3.20). By Theorem 3, to prove strict convexity of ρ E, it suffices to show that the function e s defined by (3.18) satisfies the assumptions of the theorem. The fundamental relation (3.23) yields an explicit formula for e s ,

e s (ρ, η, F, Y, Y ) = c V,s θ ref ρ ρ ref γ-1 exp η c V,s + αK 1 2c V,ref tr FY -1 2 F ⊤ ρ R ρ - αk B c V,s Y - αk B 4c V,s . (3.24)
First, we show that e s is strictly convex with respect to (ρ -1 , η, Y ). Let us define an auxiliary function

f (x, y, z) def = e qy x p z r , (3.25)
where (x, y, z) ∈ D f def = {(x, y, z) ∈ (0, +∞) × R × (0, +∞)} and p, q, r > 0. To prove that the function e s is strictly convex with respect to (ρ -1 , η, Y ) for arbitrary values of the material parameters α, c V,s , γ, θ ref , ρ ref it suffices to prove that the function f is strictly convex with respect to (x, y, z) for arbitrary values of the parameters p, q, r. Let H i denote the i-th leading principal minor of the Hessian matrix H corresponding to the function f . An elementary calculation yields

H 1 = p(p + 1)e qy x p+2 z r , H 2 = q 2 pe 2qy x 2p+2 z 2r , H 3 = q 2 pre 3qy x 3p+2 z 3r+2 , (3.26)
and we immediately see that, by Sylvester's criterion, the Hessian matrix H is positive definite on D f . Consequently, the function f is strictly convex on D f . Second, we prove that e s is convex with respect to (ρ -1 , η, F, Y, Y ). Note that the formula (3.24) can be conveniently written as

e s (ρ, η, F, Y, Y ) = C exp -p log ρ -1 + qη -r log(Y ) + s tr FY -1 2 F ⊤ , (3.27)
where C, p, q, r, s > 0 are positive constants whose explicit formulae can be found easily. Since the scalar function exp is non-decreasing and convex, let us only concentrate on the inner function

g(ρ, η, F, Y, Y ) def = -p log ρ -1 + qη -r log(Y ) + s tr FY -1 2 F ⊤ . (3.28)
As we already know since [START_REF] Lieb | Convex trace functions and the wigner-yanase-dyson conjecture[END_REF], the function tr(FY -1 2 F ⊤ ) is convex with respect to (F, Y), recall the proof of Th. 3, so g is clearly convex with respect to (ρ -1 , η, F, Y, Y ). Employing Lemma 1 we then conclude that e s is convex with respect to the unknowns (ρ -1 , η, F, Y, Y ).

Finally, having proved Theorem 4, it is now possible to formulate a shorttime well-posedness result of the Cauchy problem associated with the system (3.16) for u = (ρ, ρη, ρj e , ρv, ρF, ρY, ρY ), when the volumetric specific internal energy is given by the polytropic gas equation of state (3.23). Recall Theorems 1&2, and the conservation form (3.7) of (3.16) using smooth flux f j (u), j = 1 . . . d and source c(u) well-defined for all u ∈ V in the convex open set of R 24 6 sym,> × (0, +∞) ∩ {θ(u) > 0} (3.29) where R 6 sym,> denotes the convex subset of R 6 whose elements can be rearranged by convention to form 3 × 3 symmetric positive definite matrices.

V def = (0, +∞) × R × R 3 × R 3 × R 9 × R
Theorem 5 Let U be an open subset of the convex set V given by (3.29). Consider the system of balance laws (3.16) governing the motion of compressible heat-conducting Maxwell viscoelastic fluid with a volumetric specific internal energy given by the polytropic gas equation of state (3.23). Assume that u 0 ∈ U and u 0 ∈ H s (R 3 ; R 30 R 24 ) with s > 5 2 such that u 0 + u 0 is compactly supported in U. Then, there exists T > 0 and a unique classical solution u ∈ C 1 (R 3 × [0, T ]; U) of the Cauchy problem associated with (3.16) and the initial data u(0

) = u 0 + u 0 . Furthermore, u -u 0 ∈ C([0, T ]; H s ) ∩ C 1 ([0, T ]; H s-1 ).
At this point, we should stress that the components of the solution u = (ρ, ρη, ρj e , ρv, ρF, ρY, ρY ) act as independent quantities. However, we are interested by the case where initial conditions for the Cauchy problem

u 0 + u 0 = (ρ 0 , ρ 0 η 0 , ρ 0 j e 0 , ρ 0 v 0 , ρ 0 F 0 , ρ 0 Y 0 , ρ 0 Y 0 ) satisfy Y 0 = 1/ det Y 0 , and ρ R = ρ 0 det F 0 with F 0 = ∂χt|t=0 ∂a for some initial deformation χ t | t=0 : R 3 → R 3 such that v 0 • χ t | t=0 = ∂ t χ t | t=0
, consistently with physical interpretation. Moreover, we would like to preserve the interpretation for t > 0, that is Y (x, t) = 1/ det Y(x, t), and ρ R = ρ(x, t) det F(x, t) with F = ∂χt ∂a for some deformation χ t : R 3 → R 3 such that v•χ t = ∂ t χ t holds almost everywhere, consistently with physical interpretation. Now, whereas Y (x, t) = 1/ det Y(x, t) is obviously preserved for t > 0 by the unique smooth solutions to (3.14), (3.15), it is neither the case of F = ∂χt ∂a , v t • χ t = ∂ t χ t (i.e. the existence of a Lagrangian description equivalent almost everywhere to our Eulerian description) nor of ρ R = ρ(x, t) det F(x, t) by consequence. Indeed, our system of conservation laws has not preserved (2.9), see e.g. [START_REF] Wagner | Conservation laws, coordinate transformations, and differential forms[END_REF]. This motivates a variation of our model in Section 4 which allows to preserve equivalence between Eulerian and Lagrangian descriptions with a conservation law, moreover with physical grounds in applications to polymer melts.

Lastly, to conclude this section, we would like to stress that our wellposedness result holds for short times where θ > 0 is preserved, with θ interpreted as the temperature. Note that it preserves K(θ) > 0 in turn, with K interpreted as the stiffness. Using η = -∂ψ ∂θ , the evolution equation for the specific entropy (2.28b) rewrites where we recall p(ρ, θ, C, j e ) def = ρ 2 ∂ψ ∂ρ (ρ, θ, C, j e ) = ρ 2 ∂ψs ∂ρ (ρ, θ) using (2.17b) and the standard thermodynamic relations, and where we have introduced the specific heat capacity at constant volume of the solvent c V,s

ρ - ∂ 2 ψ ∂θ 2 dθ dt - ∂ 2 ψ ∂θ∂ρ dρ dt - ∂ 2 ψ
c V,s (ρ, θ) def = -θ ∂ 2 ψ s ∂θ 2 (ρ, θ) = -θ ∂ 2 ψ ∂θ 2 (ρ, θ, C, j e ). (3.32)
Finally, using the definition of the specific Helmholtz free energy (2.7), the formula (2.23) for the entropy production, the evolution equations for the energy flux (2.28c) and for the conformation tensor (2.28e), we arrive at

ρc V,s dθ dt + div j e = -θ ∂p ∂θ (ρ, θ, C, j e ) + αρk B div v + j e • ∇ log θ + |j e | 2 κ + 2αρ ζ K(θ)I : (K(θ)C -k B θI). (3.33)
Clearly, the evolution equation (3.33) preserves θ > 0 for the smooth short time evolutions in Theorem 5. Note in passing that due to the first two terms on the right-hand side, (3.33) cannot be formulated as a conservation law.

Some variations of the Maxwell model

We consider here three variations of the Maxwell model in Theorem 5. First, we consider conservation laws that extend the elastodynamics to viscoelastic flows using a different measure of strain than C used in the previous section for compressible Maxwell fluids. This is motivated mathematically by [START_REF] Wagner | Conservation laws, coordinate transformations, and differential forms[END_REF][START_REF] Wagner | Symmetric-hyperbolic equations of motion for a hyperelastic material[END_REF] where it is shown that only such a variation allows a correct interpretation of the (small-time) smooth solutions. This is also motivated physically by the success of such strain measures in some applications to polymer fluids; melts in particular. That is why we formulate the first variation by adding an energy term to the Helmholtz free-energy which is a function of that new strain measure, like in the useful class of K-BKZ fluids which inspires our formulation. To obtain a well-posedness result similar to Theorem 5 with that variation, it suffices to establish a mathematical entropy, which is done in Section 4.1. Noteworthily, in addition to a well-posedness result similar to Theorem 5, that variation can be shown fully consistent with its physical interpretation as time evolves (smooth solutions preserve their initial interpretation).

Second, we consider an independent variation which uses a different elastic energy term motivated by physics: real strain is bounded above. We show that our well-posedness result still holds with a standard elastic energy term that accounts for polymers finite-extensibility.

Last, we consider in Section 4.3 another independent variation regarding the volumetric energy term, since the polytropic gas equation of state is not very well-suited for polymeric liquids. A small-time well-posedness result similar to Theorem 5 can be established provided a mathematical entropy is established, and ρ ∈ (0, b) is ensured on choosing an initial condition ρ 0 ∈ (0, b).

Many other variations are certainly possible. Note our three ones here have mathematical and/or physical motivation. Different interesting variations to be considered in the future are mentionned in our conclusive Section 5.

Adding another measure of strain like in K-BKZ fluids

Let us postulate the specific Helmholtz free energy in the form

ψ(ρ, θ, C 1 , C 2 , j e ) = ψ s (ρ, θ) + α 2 K (1) (θ) tr C 1 -k B θ log det C 1 + α 2 K (2) (θ) tr C 2 -k B θ log det C 2 + τ 0 2κ |j e | 2 , (4.1)
where C 1 , C 2 are two symmetric positive-definite tensors measuring strain and K (1) , K (2) are two corresponding stiffness, affine functions of the temperature

K (1) def = K (1) 0 + K (1) 1 θ, (4.2a)
K (2) def = K (2) 0 + K (2) 1 θ, (4.2b)
where K

(1) 0 , K

0 , K

> 0 are constant parameters. Following Section 2.2, a fundamental thermodynamic relation in terms of the internal energy e(ρ, η, C 1 , C 2 , j e ) = e s ρ, η + α 2 K

(1)

1 tr C 1 + K (2) 1 tr C 2 -k B log det(C 1 C 2 ) + α 2 K (1) 0 tr C 1 + α 2 K (2) 0 tr C 2 + τ 0 2κ |j e | 2 (4.3)
is equivalent upon assuming differentiability of ψ s , e s . Further, we assume that C 1 is a measure of strain similar to C in the previous sections (i.e. similar to the left Cauchy-Green deformation tensor FF ⊤ ) while we assume C 2 /ρ 2 similar to the inverse right Cauchy-Green deformation tensor F -⊤ F -1 . Precisely, we postulate time rates are given by the following objective time derivatives

∇ C 1 def = dC 1 dt -LC 1 -C 1 L ⊤ , (4.4a)
∆ C 2 def = dC 2 dt + C 2 L + L ⊤ C 2 -2(tr L)C 2 . ( 4
The mass balance (2.4a) and the equation (2.5) for internal energy then yield

ρ dη dt + div j e θ = 1 θ T : D + p div v - ∂e ∂C 1 : dC 1 dt - ∂e ∂C 2 : dC 2 dt -j e • ∇θ θ + τ 0 ρ κ dj e dt , (4.6)
where we introduced the temperature θ and the pressure p defined by the standard thermodynamic relations (2.17). Exploiting the explicit relation for the internal energy (4.3) and (4.4), equation (4.6) rewrites

ρ dη dt + div j e θ = 1 θ T -αρ K (1) (θ)C 1 -K (2) (θ)C 2 : D + p -αρ K (2) (θ) tr C 2 -3k B θ div v - j e • ∇θ θ - αρ 2 K (1) (θ)I -k B θC -1 1 : ∇ C 1 - αρ 2 K (2) (θ)I -k B θC -1 2 : ∆ C 2 . (4.7)
This finally leads to the following set of constitutive relations

T = -pI + S, (4.8a)
S = αρ K (1) (θ)C 1 -K (2) (θ)C 2 -K (2) (θ) tr C 2 -3k B θ I , (4.8b)
∇ C 1 = - 4K (1) (θ) ζ C 1 + 4k B θ ζ I, (4.8c)
∆ C 2 = - 4K (2) (θ) ζ C 2 + 4k B θ ζ I, (4.8d)
which guarantee that the right-hand side of (4.7), i.e. the entropy production, is non-negative. From (4.8) one also recognizes the viscoelastic character of the Hookean material considered here, insofar as S follows Maxwell law in the infinitesimal strain limit [START_REF] Boyaval | Viscoelastic flows of Maxwell fluids with conservation laws[END_REF].

Next, with a view to applying Theorem 4, we proceed as in Section 3.2 and introduce symmetric positive-definite tensors A 1 , A 2 via the formulae

A 1 def = F -1 C 1 F -⊤ , (4.9a)
A 2 def = (Cof F) -1 C 2 (Cof F) -⊤ . (4.9b)
Using (4.8c), (4.8d) for C 1 , C 2 , along with the kinematic relations

dF dt = LF, (4.10a) d(Cof F) dt = (tr L)I -L ⊤ Cof F, (4.10b)
one can easily identify evolution equations

dA 1 dt = - 4K (1) (θ) ζ A 1 + 4k B θ ζ F -1 F -⊤ , (4.11a
)

dA 2 dt = - 4K (2) (θ) ζ A 2 + 4k B θ ζ (Cof F) -1 (Cof F) -⊤ , (4.11b)
so we can formulate a system of conservation laws with relaxation source terms in the unknowns (ρ, ρη, ρj e , ρv, ρF, ρ Cof F ⊤ , ρA 1 , ρA 2 ) which governs our new compressible Maxwell fluids of a more general type, with two strain measures. We recall that ρ Cof F ⊤ ≡ F -1 is governed by the conservation law (see [START_REF] Wagner | Symmetric-hyperbolic equations of motion for a hyperelastic material[END_REF])

∂(F -1 ) ∂t + ∇ ⊗ F -1 v = 0, (4.12)
where we have used the notation

∇ ⊗ F -1 v α i = ∂ ∂x i [F -1 ] α j v j .
To establish (short-time) well-posedness, it remains to identify a symmetric-hyperbolic system of conservation laws, typically endowed with a mathematical entropy.

Introducing symmetric positive-definite tensors

Y 1 def = A -2 1 , Y 2 def = A -2 2 and scalars Y 1 def = det Y -1 1 , Y 2 def = det Y -1
2 as in Section 3.2, we obtain a system of conservation laws with relaxation source terms for the set of unknowns (ρ, ρη, ρj e , ρv, ρF, ρ Cof F ⊤ , ρY 1 , ρY 2 , ρY 1 , ρY 2 ).

(4.13)

That system is endowed with a mathematical entropy

E def = 1 2 |v| 2 + e s (ρ, η, F, Cof F ⊤ , Y 1 , Y 2 , Y 1 , Y 2 ) + τ 0 2κ |j e | 2 + α 2 K (1) 0 tr Y -1 2 1 F ⊤ F + α 2 K (2) 0 tr Y -1 2 2 Cof F ⊤ Cof F + 1 2 e ref |Y 1 | 2 + |Y 2 | 2 , (4.14)
where -exploiting (3.6) and det(Cof F) = (det F) 2 -e s is defined as

e s (ρ, η, F, Cof F ⊤ , Y 1 , Y 2 , Y 1 , Y 2 ) def = e s ρ, η + α 2 K (1) 1 tr Y -1 2 1 F ⊤ F + K (2) 1 tr Y -1 2 2 Cof F ⊤ Cof F -k B log ρ R ρ 6 Y 1 2 1 Y 1 2 2 (4.15) if e s is convex with respect to (ρ, η, F, Cof F ⊤ , Y 1 , Y 2 , Y 1 , Y 2
) and strictly convex with respect to (ρ -1 , η, Y 1 , Y 2 ). The procedure for verifying that e s satisfies the convexity properties for specific instances of the solvent contribution is completely analogous to the one in Section 3.2 and we refrain from giving a detailed derivation. Let us only prove that the polytropic gas equation of state yields the quantity e s being strictly convex with respect to (ρ -1 , η, Y 1 , Y 2 ).

From the fundamental thermodynamic relation for the polytropic gas in terms of the internal energy, see (3.23), we obtain

e s (ρ, η, F, Cof F ⊤ , Y 1 , Y 2 , Y 1 , Y 2 ) = c V,s θ ref ρ ρ ref γ-1 × exp η c V,s + αK (1) 1 2c V,ref tr FY -1 2 1 F ⊤ + αK (2) 1 2c V,ref tr (Cof F)Y -1 2 2 (Cof F) ⊤ × ρ R ρ 6 Y 1 2 1 Y 1 2 2 - αk B 2c V,s . (4.16)
We are interested in proving the strict convexity of e s with respect to the unknowns (ρ -1 , η, Y 1 , Y 2 ). For this purpose, let us define an auxiliary function

f (x, y, z 1 , z 2 ) def = e qy x p z r 1 z r 2 , (4.17)
where (x, y, z 1 , z 2 ) ∈ D f def = {(x, y, z 1 , z 2 ) ∈ (0, +∞) × R × (0, +∞) × (0, +∞)} and p, q, r > 0. To prove that the function e s is strictly convex with respect to (ρ -1 , η, Y 1 , Y 2 ) for arbitrary values of the material (or reference) parameters α, c V,s , γ, θ ref , ρ ref it suffices to prove that the function f is strictly convex with respect to (x, y, z 1 , z 2 ) for arbitrary values of the parameters p, q, r. Let H i denote the i-th leading principal minor of the Hessian matrix H corresponding to the function f . An elementary calculation yields

H 1 = p(p + 1)e qy x p+2 z r 1 z r 2 , H 2 = q 2 pe 2qy x 2p+2 z 2r 1 z 2r 2 , (4.18)
H 3 = q 2 pre 3qy x 3p+2 z 3r+2 1 z 3r 2 , H 4 = q 2 pr 2 e 4qy x 4p+2 z 4r+2 1 z 4r+2 2 , (4.19)
and we see that, by Sylvester's criterion, the Hessian matrix H is positive definite on D f . Consequently, the function f is strictly convex on D f . We can finally use Theorem 1 to establish well-posedness of the system of conservation laws for (4.13), which is symmetric-hyperbolic by application of Theorem 2, as in Theorem 4 for our previous (simpler) Maxwell fluid. Moreover, in addition to Theorem 4, we can establish the full consistency for all existence times t > 0 of the smooth solutions with their physical interpretation set initially. Indeed, equation (4.12) obviously allows smooth solutions to preserve the Piola's identity rot F -1 = 0, (4.20) thus the interpretation of F -1 as a gradient ∇χ -1 t , see [START_REF] Wagner | Conservation laws, coordinate transformations, and differential forms[END_REF]. In turn, it allows the preservation of all other desired interpretations that were missing after our Theorem 4 in the previous Section.

A last comment is in order here, before we shift to another variation of our initial Maxwell model. We have chosen here a Helmholtz free energy (4.1) that is one particular sum of three energy terms, functions of three particular strain measures:

ψ = ψ s (ρ, θ) + α 2 K (1) (θ) tr Y -1 2 1 F ⊤ F + K (2) (θ) tr Y -1 2 2 Cof F ⊤ Cof F -k B log ρ R ρ 6 Y 1 2 1 Y 1 2 2 + τ 0 2κ |j e | 2 .
We have showed that this choice provides a fully consistent system of conservation laws that naturally extends standard elastodynamics to fading memory materials with a mathematical entropy-thereby ensuring (short time) well-posedness. Certainly, this is only one choice among many possible other Helmholtz free energies using different functions of different strain measures. We show below two physical variations of the Hookean energy term and of the volumetric energy term, respectively, keeping the same strain measures. But one could also use different strain measures, for instance assuming C 2 (and not C 2 /ρ 2 ) similar to the inverse right Cauchy-Green deformation tensor F -⊤ F -1 (sometimes termed Finger deformation), then with another time rate than (4.4) (the objective lower-convected derivative, in effect). And we are not aware of a clear rationale that would lead one to choose a particular formula function of various strain measures.

Introducting finite-extensibility effects

We propose a variation of the Hookean elastic energy term in the compressible Maxwell models above that takes into account the finite-extensibility of polymers. For the sake of simplicity, we consider only one "distortional" strain measure (beyond the purely volumetric variable ρ), chosen similar to the left Cauchy-Green deformation tensor as in Section 3. We consider the fundamental thermodynamic relation in terms of the specific Helmholtz free energy with an elastic term of FENE-P type [START_REF] Peterlin | Hydrodynamics of macromolecules in a velocity field with longitudinal gradient[END_REF]

ψ(ρ, θ, C, j e ) = ψ s (ρ, θ) + τ 0 2κ |j e | 2 + α 2 -K(θ)b 2 log 1 - tr C b 2 -k B θ log det C , (4.21)
where the additional constant parameter b denotes the maximum allowable extension of the polymer chains. We again assume the stiffness K affine in the temperature, see (2.8). But contrary to [9], we opt for the simple setting in which the material parameter b does not depend on the temperature. The new (distortional) elastic energy term, in comparison to the specific Helmholtz free energy for the Maxwell model (2.7), yields a different constitutive relations for the Cauchy stress tensor T, and a different evolution equation for the conformation tensor C (a different source term, in effect) as well as a different formula for the entropy production ξ. However, these changes can be derived easily and they are not essential to the well-posedness of the model.

Regarding well-posedness, the only substantial difference appears in the formulation of the mathematical entropy, see (3.17) and (3.18). There, the terms tr

FY -1 2 F ⊤ (4.22)
are replaced by

-b 2 log   1 - tr FY -1 2 F ⊤ b 2   . (4.23)
Since the rest of the formula for E stays the same, it remains to verify that the convexity properties of the term (4.22) are preserved for (4.23) as well. Specifically, (4.23) must be convex with respect to (F, Y) and strictly convex with respect to F. However, this is a simple consequence of Lemma 1 since the scalar function -log(1-x/b 2 ) is increasing and convex and the aforementioned convexity properties hold for the term tr(FY -1 2 F ⊤ ). We thus conclude that an analogous short-time well-posedness result as for the Maxwell model, see Section 3, can be obtained for the FENE-P model.

Noble-Abel stiffened-gas equation of state

We propose a variation of the volumetric term in our Maxwell model, the Noble-Abel stiffened-gas equation of state [START_REF] Métayer | The Noble-Abel stiffened-gas equation of state[END_REF], that is capable of describing gaseous as well as liquid phases by a suitable choice of parameters.

The fundamental thermodynamic relation for the Noble-Abel stiffened gas in terms of the specific internal energy (a complete equation of state) reads

e s (ρ, η) = c V,s θ ref ρ ρ ref 1 1 -bρ γ-1 e η c V,s + 1 ρ -b p ∞ + q, (4.24)
where c V,s denotes the specific heat capacity at constant volume of the solvent, γ > 1 is a constant referred to as the adiabatic exponent, and b, q, and p ∞ denote constant material parameters specific for the given fluid. The symbols θ ref and ρ ref denote constant reference temperature and constant reference density, respectively. An analogous (short-time) well-posedness result as Theorem 4 in Section 3.2 can be achieved. It suffices to prove: Theorem 6 If the volumetric energy term is the Noble-Abel stiffened-gas equation of state (4.24), then the scalar quantity ρ E, where E is given by (3.17), is a mathematical entropy of the system of balance laws (3.16).

Proof Similarly as in the proof of Theorem 4 it suffices to show that the function e s defined by (3.18) satisfies the assumptions of Theorem 3, i.e. that e s is strictly convex with respect to (ρ -1 , η, Y ) and convex with respect to (ρ -1 , η, F, Y, Y ). The fundamental relation (4.24) yields an explicit formula for the quantity e s , e s (ρ, η,

F, Y, Y ) = c V,s θ ref ρ ρ ref 1 1 -bρ γ-1 exp η c V,s + αK 1 2c V,ref tr FY -1 2 F ⊤ × ρ R ρ Y - αk B c V,s + 1 ρ -b p ∞ + q. (4.25)
Note that the last two terms on the right-hand side of (4.25) are convex with respect to (ρ -1 , η, F, Y, Y ). Hence, it suffices to prove (strict) convexity of the remaining term. First, we show that e s is strictly convex with respect to (ρ -1 , η, Y ). Let us define an auxiliary function f (x, y, z) def = e qy (x -b) p x r z r , (

where (x, y, z) ∈ D f def = {(x, y, z) ∈ (b, +∞) × R × (0, +∞)} and p, q, r > 0. To prove that the first term on the right-hand side of (4.25) is strictly convex with respect to (ρ -1 , η, Y ) for arbitrary values of the material parameters α, c V,s , γ, b, θ ref , ρ ref it suffices to prove that the function f is strictly convex with respect to (x, y, z) for arbitrary values of the parameters p, q, r. Let H i denote the i-th leading principal minor of the Hessian matrix H corresponding to the function f . An elementary yet tedious calculation yields

H 1 =
e qy (x -b) p+2 x r+2 z r (p + r)(1 + p + r)x 2 -2br(1 + p + r) + b 2 r(1 + r) , (4.27a)

H 2 =
q 2 e 2qy (x -b) 2p+2 x 2r+2 z 2r (p + r)x 2 -2brx + b 2 r , (4.27b)

H 3 = q 2 re 3qy (x -b) 3p+2 x 3r+2 z 3r+2 (p + r)x 2 -2brx + b 2 r . (4.27c)
By a direct computation one can verify that the discriminants of the quadratic functions appearing in (4.27a) and (4.27b) read -4b 2 pr(1 + p + r) and -4b 2 pr, respectively. Hence, by Sylvester's criterion, the Hessian matrix H is positive definite on D f . Consequently, the function f is strictly convex on D f . Second, we prove that e s is convex with respect to (ρ -1 , η, F, Y, Y ). Note that the first term on the right-hand side of (4.25) can be conveniently written C exp -p log ρ -1 -b -r log ρ -1 + qη -r log(Y ) + s tr FY -1 2 F ⊤ , (4.28) where C, p, q, r, s > 0 are positive constants whose explicit formulae can be found easily. Similarly as in the proof of Theorem 4, by showing that the argument of the exponential function in (4.28) is convex and by exploiting Lemma 1 we finally conclude that e s is convex with respect to (ρ -1 , η, F, Y, Y ).

Conclusion

We have pursued here our extension of the elastodynamics system of conservation laws for compressible viscoelastic fluids with fading memory using relaxing structural tensors [START_REF] Boyaval | Viscoelastic flows of Maxwell fluids with conservation laws[END_REF] to non-isothermal flows.

We have also considered generalized (Hookean-elastic) Maxwell fluids using different measures of strain. This is physically justified (for application to polymer melts and other rubbery materials beyond polymer solutions e.g.) and mathematically interesting insofar as it ensures smooth solutions that are fully consistent with their interpretation (through a Lagrangian description equivalent to our Eulerian model).

Our model asymptotically contains numerous known models in the isothermal incompressible limits. It offers a sound mathematical framework to answer physical issues raised by usual models e.g. [START_REF] Mackay | On the derivation of macroscopic models for compressible viscoelastic fluids using the generalized bracket framework[END_REF][START_REF] Bollada | On the mathematical modelling of a compressible viscoelastic fluid[END_REF], insofar as univocal (smooth) solutions can be defined (on short times).

One question to be addressed in the future is the class of Helmholtz freeenergy formulae, functions of various strain measures, that can be considered. For many real materials, adding terms (volumetric or Hookean-elastic, possibly with finite-extensibility effects, but functions of only one strain measure among various possible, plus of independent structural parameters with different time scales) might suffice to cover useful applications.

A second-related-question is how to improve the description of thermal effects, especially on large temperature ranges throughout "phase transitions". On large temperature ranges, one has to cover various time scales, so a Helmholtz free-energy sum of many terms, using many structural tensors with many different relaxation time scales, is certainly one option to be considered. To correctly match thermal effects then, letting those relaxation times ζ depend on θ might suffice, e.g. to model glass-forming polymers. Precise mathematical formulations seem desirable for sharp numerical predictions.

  ) and where e s is defined-exploiting the relation (3.6)-as

  .4b) Next, we can derive constitutive relations following Section 2.3. We first take the material derivative of the relation e = e(ρ, η, C 1 , C 2 , j e )

	ρ	de dt	= ρ	∂e ∂ρ	dρ dt	+	∂e ∂η	dη dt	+	∂e ∂C 1	:	dC 1 dt	+	∂e ∂C 2	:	dC 2 dt	+	∂e ∂j e	•	dj e dt	.

Acknowledgements Sébastien Boyaval has been supported by ANR JCJC SEDIFLO Project-ANR-15-CE01-0013. Mark Dostalík has been supported by the Czech Science Foundation, Grant Number 20-11027X. Additional funding was provided by institutional grants Charles University Grant Agency, Grant Number 1652119, and by Charles University Research Programme No. UNCE/SCI/023.


Estevan Lara

Allan Constante

Juliano Benfica

Fabian Vargas

Alexandre Boyer

Sonia Ben Dhia

Andreas Gleinser

Gunter Winkler

Bernd Deutschmann

Preliminary Study on the Impact of Place and Route Strategy on FPGA Electromagnetic Emission

Keywords: Commercial Field-Programmable Gate Array (FPGA), Reliability, Logic place and route process, Electromagnetic emission (EME), GTEM Cell Test Method, Surface Scan Test Method

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

As semiconductor technologies have been shrinking, the speed of circuits, integration density, and the number of I/O interfaces have been significantly increased. As consequence, electromagnetic emission (EME) became a paramount issue in the integrated circuit (IC) design. In this scenario, engineers must guarantee the ability of electronic components to operate safely in an increasingly hostile electromagnetic environment. Therefore, dealing to better understand the relationship between ICs and EME, several studies have been found in the literature dealing to define measurement procedures and standards, laboratory setup and countermeasures to understand and minimize EME from ICs. King Lee Chua at al. [START_REF] Chua | Characterizations of FPGA chip emissions based on GTEM cell measurements[END_REF] presented an alternative approach to measure electromagnetic emissions by locating the entire FPGA test board inside the GTEM cell instead of mounting it at the wall of the cell. In [START_REF] Van Toan Nguyen | Electromagnetic emanation exploration in FPGA-based digital design[END_REF], authors evaluate and analyze the impact of I/O switching activities on the EME of a Xilinx Spartan-3 FPGA. In [START_REF] Ramdani | The electromagnetic compatibility of integrated circuits-past, present, and future[END_REF], Mohamed Ramdani at al. provide a non-exhaustive review of the research work conducted in the field of electromagnetic compatibility (EMC) at the IC level over the past 40 years with focus on measurement methods and modeling approaches. In [START_REF] Mocha | Electromagnetic emission of microprocessors systems implemented in FPGAs" Dec. 2009[END_REF], Jan Mocha et al. presented an approach that compared the conducted and radiated EME generated by two processor systems implemented in a FPGA. The same program was used in both processor systems. However, one of the systems was build using only internal resources of the FPGA, while the other used external SRAM as the program memory. In [START_REF] Szczęsny | Electromagnetic emission of digital circuits implemented in Xilinx FPGAs XC 4025E and XCV 800[END_REF], authors presented and discussed the results of testing EME generated by circuits implemented in Xilinx FPGAs XC 4025E and XCV 800 types. Both radiated and conducted emissions of the circuits were measured as a function of clock frequency. At the end of this work, authors presented possible practical recommendations to minimize EME level. In [START_REF] Benfica | Customized Platform for TID and EMI IC Combined Measurements: Case-Study and Experimental Results[END_REF] and [START_REF] Benfica | Analysis of SRAM-Based FPGA SEU Sensitivity to Combined EMI and TID-Imprinted Effects[END_REF], authors developed a customized platform tailored for combined tests of total ionizing dose (TID) radiation and electromagnetic interference (EMI). This platform is dedicated to Xilinx Spartan and Virtex FPGAs. In the sequence, they validated the platform by analyzing the single-event upset (SEU) sensitivity of a SRAM-based FPGA to the combined effects of EMI and TID-imprinted values.

Nevertheless, from the best of our knowledge it is worth noting that none of the previous works found in the literature was dedicated to study the effect of logic place and route (P&R) on the EME level of FPGA. Dealing to minimize this gap, this paper analyzes the impact of the P&R process of logic inside the FPGA on the emission level of such component. Though, a softcore version of the processor Hellfire RISC [START_REF] Aguiar | Hellfire: A design framework for critical embedded systems' applications[END_REF]9] was placed and routed in three different ways in a commercial FPGA (Xilinx/Spartan 3E, part number XC3S500E-4PQ208). In order to measure EME of such embedded system, three experiments were performed. The first experiment measured the far-field emission from voltage raw measurements at GTEM cell terminal [START_REF]Integrated circuits -Measurement of electromagnetic emissions, 150 kHz to 1GHz -Part 2: Measurement of radiated emissions -TEM cell and wideband TEM cell method" TC 47/SC 47A[END_REF]. The second experiment was based on measuring the near magnetic field using a fixed H-field probe within a small anechoic chamber as shielded housing to minimize ambient noise. Then, the last experiment was based on the so-called Surface Scan Method [START_REF]Integrated circuits -Measurement of electromagnetic emissions -Part 3: Measurement of radiated emissions -Surface scan method[END_REF]. This measurement technique enables the visualization of the near magnetic field directly at the surface of an IC package or microchip. In this case an H-field probe is automatically The remainder of this work is organized as follows: Section II describes the performed experiments, proceedings and laboratory set-up. Section III presents the case-study based on the Hellfire RISC softcore processor, application code and P&R strategies for the FPGA. Then, Section IV discusses the obtained results and suggests further directions of the research. Finally, Section V draws the final conclusions of the work.

II. PERFORMED EXPERIMENTS

In order to observe the effect of the place and route (P&R) algorithm on the FPGA emission level, three experiments were performed as described in the sequence.

A. GTEM Cell Test Method

The first experiment, based on the GTEM Cell Test Method [START_REF]Integrated circuits -Measurement of electromagnetic emissions, 150 kHz to 1GHz -Part 2: Measurement of radiated emissions -TEM cell and wideband TEM cell method" TC 47/SC 47A[END_REF][START_REF] De Smedt | TEM cell measurements of an active EMC test chip[END_REF][START_REF] King | Radiated emissions estimation of an integrated circuit based on measurements in GTEM cell[END_REF], was performed at the LAAS-CNRS and is depicted in Fig. 1. The FPGA board was placed within the GTEM cell in a fixed position. Its activity was controlled by a Test Host Computer (THC) placed outside the GTEM cell. The voltage induced at the GTEM cell terminal by the FPGA radiation was measured with a spectrum analyzer. If the FPGA board is assumed to be electrically small and equivalent to an elementary electric dipole, the far-field emission from the FPGA board can be estimated at any distance (in this experiment, at 1m) [START_REF] Wilson | On correlating TEM cell and OATS emission measurements[END_REF]. In spite of these assumptions, the results can be compared with typical radiated emission requirements at electronic equipment level. The GTEM cell primarily estimates the far-field emission of the FPGA including the emission generated by the PCB on which the IC is mounted on. In contrast to that, the two near field measurement techniques (Sections B and C described hereafter) have been chosen to have a good comparison between far and near field of the generated emission by measuring only the silicon die contributions. For the experiment described in Section B, a big near field probe was located in a fixed center position on top of the IC package to measure the magnetic field contribution of the whole die. Unlike to this, in the last experiment (Section C) the focus was set to localize the near magnetic field distribution over the die surface by using a very small, moving near field probe, which offered a very high local resolution. These methods constitute effective methodologies to characterize integrated circuits in terms of EME.

B. Anechoic Chamber with Fixed Near-Field Probe

The second experiment, based on a near-H field probe located within a small anechoic chamber which was used as shielded housing to minimize ambient noise, was performed at the TU Graz University (see Fig. 2). The H-field probe was placed approximately 2mm above the surface of the ICs package. For all measurements the probe was at the same position. The output voltage of the near-field probe was measured using a spectrum analyzer.

C. Surface Scan Method with a Moving Near-Field Probe

The third experiment, based on the Surface Scan Method [START_REF]Integrated circuits -Measurement of electromagnetic emissions -Part 3: Measurement of radiated emissions -Surface scan method[END_REF][START_REF] Bauer | Time Domain FEM Computational Approach for Calibration of Surface Scan Method[END_REF][START_REF] Shi | Using near-field scanning to predict radiated fields[END_REF], was also performed at the TU Graz University (Fig. 3). This method constitutes an effective methodology to characterize printed circuit boards and integrated circuits in terms of electromagnetic emission (and immunity as well). It is a useful technique to locate areas of critical radiation or susceptibility, which could influence the performance of devices nearby or the device under test itself. Fig. 3 depicts a photo of the test set-up. The equipment seen in this figure consists of a micromanipulation (wafer probing) that is equipped with stepper motors to control the probe movement in XYZ directions (in our experiment, Z was fixed to a constant value of 2mm). Note that while the aforementioned experiment measured emission based on a fixed probe position with respect to the FPGA package, this experiment scanned 4,000 points above the surface of the package with a moving near-field probe where the chip is located, hence adding a much higher precision and sensitivity to the emission measurement.

III. CASE-STUDY: HW, SW AND PLACE & ROUTE STRATEGIES

The case-study was based on a Spartan 3E FPGA (part number XC3S500E-4PQ208), from Xilinx, which was mounted on a dedicated board specially designed to support EMC test [START_REF] Benfica | Customized Platform for TID and EMI IC Combined Measurements: Case-Study and Experimental Results[END_REF]. The board design and software development were carried out by the Catholic University -PUCRS. In order to analyze the effect of the P&R algorithm on the FPGA emission level, a softcore processor: Hellfire RISC running a Bubble-Sort program over an operating system (HF-OS) [START_REF] Aguiar | Hellfire: A design framework for critical embedded systems' applications[END_REF]9] was placed and routed in three different configurations in the Spartan 3E FPGA. In the first configuration, the processor was placed and routed automatically by the ISE-EDK Design Tool of Xilinx without designer assistance (let us say this is the "Automated" P&R version). In the second and third configurations, the processor was completely and intentionally sat in the periphery region ("Center" version) and in the lefthand side ("Right" version) of the configurable logic block (CLB) array of the FPGA. With this purpose, the designer prevented the ISE-EDK Tool from instantiating the processor core in the center of the CLB array (Center version) and in the right side of the CLB array (Right version), respectively. Fig. 4 depicts these three P&R strategies. See also Fig. 5, which depicts the embedded system basic block diagram. In order to prevent board-level signals from jeopardizing FPGA EME measurements, the whole system (i.e., processor and memory containing operating system, program and data) was mapped inside the FPGA. The only signals flowing on and outside board during EM emission measurement were those associated to the serial communication connecting the FPGA to the user test host computer (THC). This serial port was used to monitor processor operation in real time during the experiments. In order to prevent communication from being disrupted by electromagnetic interference during the experiments, the serial communication was implemented by means of a two-module optic fiber link (Tx/Rx). Figs. 4 and5 (five blue dots) depict the pin positions configured around the FPGA CLB: "Read" and "Write" (for the serial communication), "LED" (to monitor FPGA operation), "Clock" and "Reset". The pin positions were fixed for all three P&R configurations. Additionally, a JTAG connection was used to configure the FPGA (Fig. 5, red dot). Once the configuration was complete, the cable was removed before starting EME measurements.

IV. OBTAINED RESULTS Fig. 6 summarizes the EME measurements. Fig. 6a depicts results for the LAAS-CNRS experiment, whereas Figs. 6b and 6c present results for the two TU Graz experiments. Table I summarizes the ambient noise values measured for the three experiments. The ambient noise was measured by disconnecting the board under test from power supply lines and then, the Spectrum Analyzer captured the remaining noise around the FPGA. By observing Fig. 6 and Table I, the following conclusions can be taken: a) In Fig. 6a, the emission level of the "Automated" place and route version is 21.8% higher than the average of the other two configurations ("Centered" and "Right" versions). Note that the GTEM Cell Test Method measures EM emission from the whole system, i.e., not only from the FPGA but also from the board (including tracks and other components such the serial/optical converter logic sitting around the FPGA). b) In Fig. 6b, the emission level of the three P&R strategies is similar: 1.649 x10 -2 volts. This experiment was not able to identify different emission levels among the threes strategies, as did the other two experiments of Fig. 6a and6c. This could be explained by the fact that the measurements in the small anechoic chamber were performed under 44.56% higher ambient noise than the other near-field technique (4.766x10 -3 volts against 3.297x10 -3 volts, Table I). This high noise level may have jeopardized the results in such a way that EME measured from the FPGA was observed to be in the same order of magnitude for the three P&R strategies. c) In Fig. 6c, the EM emission level of the "Automated" place and route version is 8.4% higher than the other two configurations ("Centered" and "Right" versions).

V. CONCLUSIONS

This work presented a preliminary study about the impact of the place and route (P&R) strategy on the electromagnetic (EM) emission level of a commercial FPGA. The considered device was a Xilinx Spartan 3E, part number XC3S500E-4PQ208, which was configured with the Hellfire softcore RISC processor available in the github public domain. The FPGA was mapped by three different strategies: "Automated", in which the processor was placed and routed automatically by the ISE-EDK Design Tool of Xilinx. In the second and third versions, the processor was completely and intentionally placed in the left-hand side ("Right" strategy), and in the periphery region ("Center" strategy) of the configurable logic block (CLB) array of the FPGA. The processor executed a Bubble-Sort program running over an operating system (OS).

The obtained results suggest that the P&R strategy can affect the FPGA emission level. For the developed experiments, the influence was in the order of 21.8% (GTEM Cell Test Method) and 8.4% (Surface Scan Method). The anechoic chamber with fixed near-field probe experiment presented high ambient noise at laboratory, which prevented us from getting measurements with sensitiveness high enough to discriminate the emission levels generated by the three P&R strategies.

Fig. 1 .

 1 Fig. 1. GTEM Cell Test Method experiment set-up at LAAS-CNRS: (a) general overview; (b) detail of the FPGA under test.

Fig. 2 .

 2 Fig. 2. Anechoic chamber with near-field probe experiment set-up at TU Graz: (a) general overview; (b) and (c) details of the FPGA under test.

Fig. 3 .

 3 Fig. 3. Surface Scan Method experiment set-up at TU Graz: (a) general overview; (b) detail of the FPGA under test.

Fig. 4 .

 4 Fig. 4. Three P&R strategies to instantiate the processor core inside the CLB array of the FPGA: (a) "Automated", (b) "Center", (c) "Right".

Fig. 5 .

 5 Fig. 5. Overview of the embedded system: FPGA and board.

  ) JTAG Port (used for FPGA configuration, cable removed during testing) from THC (download of FPGA configuration bitstream) Optical link Tx/Rx (to THC for online monitoring) Serial Port (used for monitoring processor operation during EME measurement)

Fig. 6 .

 6 Fig. 6. EME measurements for the three FPGA P&R strategies (C: "Center", A: "Automated", R: "Right"), performed during the three experiments: (a) GTEM Cell Test Method; (b) Anechoic Chamber with Fixed Near-Field Probe; (c) Surface Scan Method (Moving Near-Field Probe).

TABLE I .

 I AMBIENT NOISE MEASURED FOR THE THREE EXPERIMENTS

ACKNOWLEDGEMENT

This work has been supported in part by CNPq (National Science Foundation, Brazil) under contract n. 300124/2019-0 (PQ), and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001.