A symmetrizable extension of polyconvex thermoelasticity and applications to zero-viscosity limits and weak-strong uniqueness

Christoforou, Cleopatra; Galanopoulou, Myrto; Tzavaras, Athanasios E.

doi:10.1080/03605302.2018.1456551

Cited by 10 publications

(17 citation statements)

References 30 publications

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“…A nice summary of the "fluid mechanics" applications of this method is in the monograph by Málek et al [23]. Recently, Tzavaras and coauthors adapted these ideas to problems in elastodynamics, thermoelasticity, and other related problems [5], [7], [21]. Another application to the Cucker-Smale equation is due to Mucha and Peszek [25].…”

Section: Navier-stokes-fourier Systemmentioning

confidence: 99%

Stability of Strong Solutions to the Navier--Stokes--Fourier System

Březina

Feireisl²,

Novotný

2020

SIAM J. Math. Anal.

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We identify a large class of objectsdissipative measure-valued (DMV) solutions to the Navier-Stokes-Fourier system -in which the strong solutions are stable. More precisely, a DMV solution coincides with the strong solution emanating from the same initial data as long as the latter exists. The DMV solutions are represented by parameterized families of measures satisfying certain compatibility conditions. They can be seen as an analogue to the dissipative measure-valued solutions introduced earlier in the context of the (inviscid) Euler system.

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Section: Navier-stokes-fourier Systemmentioning

confidence: 99%

Stability of Strong Solutions to the Navier--Stokes--Fourier System

Březina

Feireisl²,

Novotný

2020

SIAM J. Math. Anal.

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“…Above condition (8) is strongly connected with the notion of material stability [22] and ensures the stability of the system of governing equations (4) and (6) [13,14]. A sufficient condition for (8) to hold is obtained when the extended representation W (F , H, J, θ) is convex with respect to the set V = {F , H, J} (namely, polyconvex with respect to the mechanics [1,3,4]) and concave with respect to θ, namely…”

Section: The Helmholtz Free Energymentioning

confidence: 99%

“…An alternative but equivalent definition of the directional derivatives of Ψ (C, θ) to those in (13) Finally, inserting (16) into (15) a and comparison with (13) enables to obtain an equivalent expression for S and η to those in equations (14) as…”

Section: The Helmholtz Free Energymentioning

confidence: 99%

A new energy–momentum time integration scheme for non-linear thermo-mechanics

Ortigosa

Gil

Martínez-Frutos

et al. 2020

Computer Methods in Applied Mechanics and Engineering

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The aim of this paper is the design a new one-step implicit and thermodynamically consistent Energy-Momentum (EM) preserving time integration scheme for the simulation of thermo-elastic processes undergoing large deformations and temperature fields. Following [12], we consider wellposed constitutive models for the entire range of deformations and temperature. In that regard, the consideration of polyconvexity inspired constitutive models and a new tensor cross product algebra are shown to be crucial in order to derive the so-called discrete derivatives, fundamental for the construction of the algorithmic derived variables, namely the second Piola-Kirchoff stress tensor and the entropy (or the absolute temperature). The proposed scheme inherits the advantages of the EM scheme recently published by Franke et al. [17] (i.e. consistency, stability, conservation) whilst resulting in dramatically far simpler algorithmic expressions, thus circumventing a bottleneck and paving the way for the incorporation of further physics into the model. A series of numerical examples will be presented in order to demonstrate the robustness and applicability of the new EM scheme. Although the examples presented will make use of a temperature-based version of the EM scheme (using the Helmholtz free energy as the thermodynamical potential and the temperature as the thermodynamical state variable), we also include in an Appendix an entropy-based analogue EM scheme (using the internal energy as the thermodynamical potential and the entropy as the thermodynamical state variable).

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“…However, the recently introduced class of dissipative measure-valued solutions is particularly suitable since the weak-strong uniqueness holds and the dissipative measure-valued solution coincides with the classical solution as far as the latter exists [11,12,28,34]. Similar concept has been adopted by Tzavaras et al [21], [14], in the context of elastodynamics, thermoelasticity, and other related problems.…”

Section: Introductionmentioning

confidence: 99%

Convergence of Finite Volume Schemes for the Euler Equations via Dissipative Measure-Valued Solutions

2019

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The Cauchy problem for the complete Euler system is in general ill posed in the class of admissible (entropy producing) weak solutions. This suggests there might be sequences of approximate solutions that develop fine scale oscillations. Accordingly, the concept of measure-valued solution that capture possible oscillations is more suitable for analysis. We study the convergence of a class of entropy stable finite volume schemes for the barotropic and complete compressible Euler equations in the multidimensional case. We establish suitable stability and consistency estimates and show that the Young measure generated by numerical solutions represents a dissipative measure-valued solution of the Euler system. Here dissipative means that a suitable form of the Second law of thermodynamics is incorporated in the definition of the measure-valued solutions. In particular, using the recently established weak-strong uniqueness principle, we show that the numerical solutions converge pointwise to the regular solution of the limit systems at least on the lifespan of the latter.

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A symmetrizable extension of polyconvex thermoelasticity and applications to zero-viscosity limits and weak-strong uniqueness

Cited by 10 publications

References 30 publications

Stability of Strong Solutions to the Navier--Stokes--Fourier System

Stability of Strong Solutions to the Navier--Stokes--Fourier System

A new energy–momentum time integration scheme for non-linear thermo-mechanics

Convergence of Finite Volume Schemes for the Euler Equations via Dissipative Measure-Valued Solutions

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