Implicit numerical schemes for generalized heat conduction equations

Rieth, Ágnes; Kovács, Róbert; Fülöp, Tamás

doi:10.1016/j.ijheatmasstransfer.2018.06.067

Cited by 35 publications

(24 citation statements)

References 45 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where is mass density (constant in the small-strain approximation), (1) tells how the spatial derivative of stress σ determines the time derivative of the velocity field v (volumetric force density being omitted 1 Hence, there is no need to distinguish between Lagrangian and Eulerian variables, and between material manifold vectors/covectors/tensors/. .…”

Section: Properties Of the Continuum Modelmentioning

confidence: 99%

See 1 more Smart Citation

Thermodynamical Extension of a Symplectic Numerical Scheme with Half Space and Time Shifts Demonstrated on Rheological Waves in Solids

Fülöp

Kovács

Szücs

et al. 2020

Entropy

View full text Add to dashboard Cite

On the example of the Poynting-Thomson-Zener rheological model for solids, which exhibits both dissipation and wave propagation -with nonlinear dispersion relation -, we introduce and investigate a finite difference numerical scheme. Our goal is to demonstrate its properties and to ease the computations in later applications for continuum thermodynamical problems. The key element is the positioning of the discretized quantities with shifts by half space and time steps with respect to each other. The arrangement is chosen according to the spacetime properties of the quantities and of the equations governing them. Numerical stability, dissipative error and dispersive error are analysed in detail. With the best settings found, the scheme is capable of making precise and fast predictions. Finally, the proposed scheme is compared to a commercial finite element software, COMSOL, which demonstrates essential differences even on the simplest -elastic -level of modelling.

show abstract

Section: Properties Of the Continuum Modelmentioning

confidence: 99%

“…Methods that were born with reversibility in mind may apparently fail for a nonreversible problem. For example, a finite element software is able to provide, at the expense of large run time, quantitatively and even qualitatively wrong outcome while a simple finite difference scheme solves the same problem fast and precisely [1].…”

Section: Introductionmentioning

confidence: 99%

Thermodynamical Extension of a Symplectic Numerical Scheme with Half Space and Time Shifts Demonstrated on Rheological Waves in Solids

Fülöp

Kovács

Szücs

et al. 2020

Entropy

View full text Add to dashboard Cite

show abstract

“…The non-dimensional model (19)- (25) is numerically solved using a semi-implicit finite difference method on a staggered grid [57]. The transformations ξ = r/R(t) and η = (r − R(t))/(1 − R(t)) are used to map the evolving domains 0 < r < R(t) and R(t) < r < 1 to stationary domains 0 < η, ξ < 1, respectively.…”

Section: Methodsmentioning

confidence: 99%

Modelling ultra-fast nanoparticle melting with the Maxwell–Cattaneo equation

Hennessy

Calvo-Schwarzwälder

Myers

2019

Applied Mathematical Modelling

View full text Add to dashboard Cite

The role of thermal relaxation in nanoparticle melting is studied using a mathematical model based on the Maxwell-Cattaneo equation for heat conduction. The model is formulated in terms of a two-phase Stefan problem. We consider the cases of the temperature profile being continuous or having a jump across the solid-liquid interface. The jump conditions are derived from the sharp-interface limit of a phase-field model that accounts for variations in the thermal properties between the solid and liquid. The Stefan problem is solved using asymptotic and numerical methods. The analysis reveals that the Fourier-based solution can be recovered from the classical limit of zero relaxation time when either boundary condition is used.However, only the jump condition avoids the onset of unphysical "supersonic" melting, where the speed of the melt front exceeds the finite speed of heat propagation. These results conclusively demonstrate that the jump condition, not the continuity condition, is the most suitable for use in models of phase change based on the Maxwell-Cattaneo equation. Numerical investigations show that thermal relaxation can increase the time required to melt a nanoparticle by more than a factor of ten. Thus, thermal relaxation is an important process to include in models of nanoparticle melting and is expected to be relevant in other rapid phase-change processes.can be compared against experimental data and used to assess new theories of nanoscale heat transport and phase change. From a practical perspective, nanoparticles play fundamental roles in novel drug delivery systems [3], materials with modified properties [4,5], and for improving the efficiency of solar collectors [6].Many current and future applications of nanoparticles require quantitative knowledge of how they respond to their thermal environment and their behaviour during melting.The thermal response of a nanoparticle differs from that of a macroscopic body for two main reasons.Firstly, the large ratio of surface to bulk atoms leads to many key thermophysical properties, such as melt temperature [7-9], latent heat [10][11][12], and surface energy [13] becoming dependent on the size of the nanoparticle. Secondly, the mechanism of thermal transport changes between the macroscale and the nanoscale. At the macroscale, heat is transported by a diffusive process that is driven by frequent collisions between thermal energy carriers known as phonons. Diffusive transport of heat across macroscopic length scales is well described by Fourier's law. On nanometer length scales, thermal energy is transported by a ballistic process driven by infrequent collisions between phonons. The finite time between phonon collisions results in a wave-like propagation of heat with finite speed [14,15]. Since Fourier's law leads to an infinite speed of heat propagation, it is not suitable for describing ballistic energy transport.Recent theoretical studies of nanoparticle melting have investigated the role of size-dependent material properties [16][17][18][19][20][21][22] u...

show abstract

“…In time, Theqrefore the difference equations are only an explicit forward differencing scheme is used. Figure 1: Concept of the spatial discretization [39].…”

Section: Difference Equationsmentioning

confidence: 99%

“…It is successfully applied in the evaluation of room temperature experiments as well [37,38]. Moreover, due to the nonlocal term, its numerical solution required a particular spatial discretization to embed the boundary conditions appropriately [39].…”

Section: Introductionmentioning

confidence: 99%

Numerical treatment of nonlinear Fourier and Maxwell-Cattaneo-Vernotte heat transport equations

Kovács

Rogolino

2020

International Journal of Heat and Mass Transfer

Self Cite

View full text Add to dashboard Cite

The second law of thermodynamics is a useful and universal tool to derive the generalizations of the Fourier's law. In many cases, only linear relations are considered between the thermodynamic fluxes and forces, i.e., the conduction coefficients are independent of the temperature. In the present paper, we investigate a particular nonlinearity in which the thermal conductivity depends on the temperature linearly. Also, that assumption is extended to the relaxation time, which appears in the hyperbolic generalization of Fourier's law, namely the Maxwell-Cattaneo-Vernotte (MCV) equation. Although such nonlinearity in the Fourier heat equation is well-known in the literature, its 1 arXiv:1910.09175v2 [cond-mat.stat-mech] 24 Dec 2019 extension onto the MCV equation is rarely applied. Since these nonlinearities have significance from an experimental point of view, an efficient way is needed to solve the system of partial differential equations. In the following, we present a numerical method that is first developed for linear generalized heat equations. The related stability conditions are also discussed.

show abstract

Implicit numerical schemes for generalized heat conduction equations

Cited by 35 publications

References 45 publications

Thermodynamical Extension of a Symplectic Numerical Scheme with Half Space and Time Shifts Demonstrated on Rheological Waves in Solids

Thermodynamical Extension of a Symplectic Numerical Scheme with Half Space and Time Shifts Demonstrated on Rheological Waves in Solids

Modelling ultra-fast nanoparticle melting with the Maxwell–Cattaneo equation

Numerical treatment of nonlinear Fourier and Maxwell-Cattaneo-Vernotte heat transport equations

Contact Info

Product

Resources

About