Nonlocal Heat Conduction Research Articles

Nonlocal thermoelasticity based on nonlocal heat conduction and nonlocal elasticity

Abstract Thermoelastic analysis at micro and nano-scale is becoming important along with the miniaturization of the device and wide application of ultrafast lasers, even the novel laser burst technology, where size effect on heat conduction and elastic deformation increase and classical theory of thermoelastic coupling does not hold any more. In this work, a size-dependent thermoelastic model is established for higher order simple material by adopting both the size effect of heat conduction and elasticity with the aids of extended irreversible thermodynamics and generalized free energy. It is proven that the present model is in essence identical to the coupling of nonlocal heat conductive law (GK model) and nonlocal elastic (stress gradient) model. Also, higher order boundary conditions for stress tensor and heat flux are presented and discussed. For numerical evaluation, a bi-layered structure is considered with interfacial thermal contact resistance and elastic wave impendence incorporated. From numerical results, the effects of size-dependent characteristic lengths and material constants of each layer on the transient responses are discussed, systematically. This study is expected to be helpful for theoretical modeling of thermoelasticity at nano-scale, and may be beneficial to design of nano-sized and multi-layered devices.

A new definition of fractional Laplacian with application to modeling three-dimensional nonlocal heat conduction

This paper proposes a new implicit definition of the fractional Laplacian. Compared with the existing explicit definitions in literature, this novel definition has clear physical significance and is mathematically simple and numerically easy to calculate for multidimensional problems. In stark contrast to a quick increasing and extensive applications of time-fractional derivative to diverse scientific and engineering problems, little has been reported on space-fractional derivative modeling. This is largely because the existing definitions are only feasible for one-dimensional case and become mathematically too complicated and computationally very expensive when applied to higher dimensional cases. In this study, we apply the newly-defined fractional Laplacian for modeling the power law behaviors of three-dimensional nonlocal heat conduction. The singular boundary method (SBM), a recent boundary-only collocation discretization method, is employed to numerically solve the proposed fractional Laplacian heat equation. And the computational costs are observed moderate owing to the proposed new definition of fractional Laplacian and the boundary-only discretization, meshfree, and integration-free natures of the SBM technique. Numerical experiments show the validity of the proposed definition of fractional Laplacian.

Nonlocal two-temperature model: Application to heat transport in metals irradiated by ultrashort laser pulses

Extended two-temperature models with allowance for time (relaxation) and space nonlocal effects have been developed to describe, in particular, heat conduction in metals under ultrashort laser irradiation, which overheats the electron gas in comparison with the lattice leading to energy exchange between them. The time nonlocal effects arise when the electron gas is driven out of local (thermal) equilibrium, whereas the space nonlocal effects begin to be significant when the spatial extent of the temperature gradient is comparable with the mean-free path of the heat carrier. The electron-lattice energy exchange as well as the space–time nonlocal effects in the electron gas leads to a hierarchy of space–time nonlocal heat conduction equations for the electron and lattice temperatures, which are partial differential equation of higher order in comparison with the classical heat conduction equation. The models presented here may be used in combination with molecular dynamic and phase-field approaches.

Determining the thermal diffusivity of sediments using the fractional nonlocal heat conduction equation

Based on the heat conduction equation with fractional-order derivatives and the experimental measurements of temperature distribution in the upper layers of the Earth, the depth dependence of thermal diffusivity is studied at different values of the parameters of nonlocality in time and along the coordinates. It is shown that thermal diffusivity increases with depth, and the values of thermal diffusivity observed in the experiments coincide with the theoretical predictions provided by the solution of the nonlocal heat-conduction equation that allows for the memory effects in fractional-order time derivatives.

Disparate quasiballistic heat conduction regimes from periodic heat sources on a substrate

We report disparate quasiballistic heat conduction trends for periodic nanoscale line heaters deposited on a substrate, depending upon whether measurements are based on the peak temperature of the heaters or the temperature difference between the peak and the valley of two neighboring heaters. The degree of quasiballistic transport is characterized by the effective thermal conductivities of the substrate which are obtained by matching the diffusion solutions to the phonon Boltzmann transport equation results. We find that while the ballistic heat conduction effect based on the peak temperature diminishes as the two heaters become closer, it becomes stronger based on the peak-valley temperature difference. Our results also show that the collective behavior of closely spaced heaters can counteract the nonlocal effects caused by an isolated nanoscale hot spot. These results are relevant to thermal conductivity spectroscopy techniques under development and also have important implications for understanding nonlocal heat conduction in integrated circuits and carbon nanotube array thermal interface materials.

Optimal Distributed Control of Nonlocal Steady Diffusion Problems

A control problem constrained by a nonlocal steady diffusion equation that arises in several applications is studied. The control is the right-hand side forcing function and the objective of control is a standard matching functional. A recently developed nonlocal vector calculus is exploited to define a weak formulation of the state system. When sufficient conditions on certain kernel functions and the volume constraints hold, the existence and uniqueness of the optimal state and control is demonstrated and an optimality system is derived. We demonstrate the convergence, as the nonlocal interactions vanish, of the optimal nonlocal state to the optimal state of a local PDE-constrained control problem. We also define continuous and discontinuous Galerkin finite element discretizations of the optimality system for which we derive a priori error estimates. Numerical examples are provided illustrating these convergence results and also illustrating the differences between optimal controls and states obtained for the nonlocal diffusion equations and for PDEs. In particular, we observe that using nonlocal models result in a better match to nonsmooth target functions and a reduced cost of control. We also provide a one-dimensional numerical example of nonlocal heat conduction in a bar for which we determine the optimal heat source that results in an optimal match to a reference temperature distribution.

Determination of thermal diffusivity and nonlocalization parameters from experimental data

Based on the solution of the nonlocal heat conduction equation in fractional calculus and on experimental data for nonstationary methods of determining temperature distribution, a method of determining the thermal diffusivity and parameters of nonlocalization in time and space has been developed.

OMEGA polar-drive target designs

Low-adiabat polar-drive (PD) [Skupsky et al., Phys. Plasmas 11, 2763 (2004)] implosion designs for the OMEGA [Boehly et al., Opt. Commun. 133, 495 (1997)] laser are described. These designs for cryogenic deuterium–tritium and warm plastic shells use a temporal laser pulse shape with three pickets followed by a main pulse [Goncharov et al., Phys. Rev. Lett. 104, 165001 (2010)]. The designs are at two different on-target laser intensities, with different in-flight aspect ratios (IFARs). These designs permit studies of implosion energetics and target performance closer to ignition-relevant intensities (∼7 × 1014 W/cm2 at the quarter-critical surface, where nonlocal heat conduction and laser–plasma interactions can play an important role) but at lower values of IFAR ∼ 22 or at lower intensity (∼3 × 1014 W/cm2) but at a higher IFAR (IFAR ∼ 32, where shell instability can play an important role). PD geometry requires repointing of laser beams to improve shell symmetry. The higher-intensity designs optimize target performance by repointing beams to a lesser extent, compensating for the reduced equatorial drive by increasing the energies of the repointed beams. They also use custom beam profiles that improve equatorial illumination at the expense of irradiation at higher latitudes. These latter designs will be studied when new phase plates for the OMEGA Laser System, corresponding to the custom beam profiles, are obtained.

The first measurements of soft x-ray flux from ignition scale Hohlraums at the National Ignition Facility using DANTE (invited)

The first 96 and 192 beam vacuum Hohlraum target experiments have been fielded at the National Ignition Facility demonstrating radiation temperatures up to 340 eV and fluxes of 20 TW/sr as viewed by DANTE representing an ∼20 times flux increase over NOVA/Omega scale Hohlraums. The vacuum Hohlraums were irradiated with 2 ns square laser pulses with energies between 150 and 635 kJ. They produced nearly Planckian spectra with about 30±10% more flux than predicted by the preshot radiation hydrodynamic simulations. To validate these results, careful verification of all component calibrations, cable deconvolution, and software analysis routines has been conducted. In addition, a half Hohlraum experiment was conducted using a single 2 ns long axial quad with an irradiance of ∼2×10(15) W/cm(2) for comparison with NIF Early Light experiments completed in 2004. We have also completed a conversion efficiency test using a 128-beam nearly uniformly illuminated gold sphere with intensities kept low (at 1×10(14) W/cm(2) over 5 ns) to avoid sensitivity to modeling uncertainties for nonlocal heat conduction and nonlinear absorption mechanisms, to compare with similar intensity, 3 ns OMEGA sphere results. The 2004 and 2009 NIF half-Hohlraums agreed to 10% in flux, but more importantly, the 2006 OMEGA Au Sphere, the 2009 NIF Au sphere, and the calculated Au conversion efficiency agree to ±5% in flux, which is estimated to be the absolute calibration accuracy of the DANTEs. Hence we conclude that the 30±10% higher than expected radiation fluxes from the 96 and 192 beam vacuum Hohlraums are attributable to differences in physics of the larger Hohlraums.

Mathematical modeling of heat-conduction processes in randomly inhomogeneous bodies using Feynman diagrams

This work is devoted to the mathematical modeling of heat-conduction processes in multiphase bodies with randomly inhomogeneous structure. A nonstationary initial boundary-value problem is formulated on the basis of the Fourier law for a whole body. The technique of Feynman diagrams is applied to the investigation of averaged temperature fields. We obtain the Dyson equation and a nonlocal heatconduction equation for the temperature field averaged over an ensemble of phase configurations in multiphase randomly inhomogeneous bodies.

A stability criterion for a difference scheme in a nonlocal heat conduction problem

A stability criterion for a difference scheme in a nonlocal heat conduction problem

Effects of non-local electron thermal transport on ablative Rayleigh-Taylor instability

A two-dimensional full Fokker-Planck (FP) simulation code coupled with an ideal fluid equation has been constructed. This code was applied for the numerical simulation of the Rayleigh-Taylor (RT) instability on the directly driven ablation front. The simulation results show that an accelerated thin foil is preheated by the non-local electron heat transport, the ablation front density is depleted, and the linear growth rate of the RT instability is suppressed strongly. By investigating the mode structure in the simulation, it is found that the peak of the eigen mode shifts toward the corona region by the non-local heat conduction effects. This is the reason why the growth rate is reduced.

A nonlocal heat conduction problem

We develop a model describing the temperature of a heated wire contained in a vessel. This containing vessel reradiates the heat received, maintaining itself at a steady-state temperature that is not a przori known but whose determination is part of the problem. The resulting Dirichlet problem involves the heat flux through the ends of the wire and the integral of a function of the temperature of the wire; it is thus nonlocal. Under physically motivated hypotheses, we prove that this problem has a unique solution.

The thermal inertia of materials heated with a laser pulse faster than relaxation time

The nonlocal hyperbolic heat conduction equation is used to describe the thermal inertia of thin metal films (TMF) heated with femtosecond laser pulses. It is shown that for TMF the signatures of thermal inertia are (i) the delay of the heating process and (ii) the strong localization of the thermal energy in TMF.

Nonlocal and Nonequilibrium Heat Conduction in the Vicinity of Nanoparticles

Heat transfer around nanometer-scale particles plays an important role in a number of contemporary technologies such as nanofabrication and diagnosis. The prevailing method for modeling thermal phenomena involving nanoparticles is based on the Fourier heat conduction theory. This work questions the applicability of the Fourier heat conduction theory to these cases and answers the question by solving the Boltzmann transport equation. The solution approaches the prediction of the Fourier law when the particle radius is much larger than the heat-carrier mean free path of the host medium. In the opposite limit, however, the heat transfer rate from the particle is significantly smaller, and thus the particle temperature rise is much larger than the prediction of the Fourier conduction theory. The differences are attributed to the nonlocal and nonequilibrium nature of the heat transfer processes around nanoparticles. This work also establishes a criterion to determine the applicability of the Fourier heat conduction theory and constructs a simple approximate expression for calculating the effective thermal conductivity of the host medium around a nanoparticle. Possible experimental evidence is discussed.

Weakly nonlocal heat conduction in rigid solids

A nonlocal heat conduction model, based on nonequilibrium thermodynamics, is proposed. This is achieved by introducing a supplementary state variable, besides the internal energy. Compatibility with the second law of thermodynamics is examined. The analysis is limited to a linear and weakly nonlocal description.

Space-time nonlocal model for heat conduction.

We consider a space-time nonlocal heat conduction model with balance laws in the form of integral equations (so-called strong nonlocality). The model identifies two internal parameters---the time \ensuremath{\tau} and the space h scales of nonlocality. In going from the strong nonlocal model to its approximations of various accuracy in the form of partial differential equations, which correspond to weak nonlocality, we introduce two limiting relations between \ensuremath{\tau} and h as \ensuremath{\tau},h\ensuremath{\rightarrow}0. In the diffusion limit, which preserves the thermal diffusivity a=${\mathit{h}}^{2}$/\ensuremath{\tau}=const as \ensuremath{\tau},h\ensuremath{\rightarrow}0, the strong nonlocal model gives a hierarchy of parabolic equations with an infinite speed of heat waves. In the wave limit, which preserves the ratio v=h/\ensuremath{\tau}=const as \ensuremath{\tau},h\ensuremath{\rightarrow}0, a hierarchy of hyperbolic equations has been obtained. The hyperbolic equations imply a finite speed of heat waves. These results suggest that for diffusion (low-k) and propagative (high-k) regimes distinct models are responsible for the space-time evolution of the temperature and heat flux. The connection with phonon hydrodynamic theory and applications to other problems are discussed.

Stimulated scattering instabilities of electromagnetic waves in collisional plasmas

The effects of nonlocal heat transport on stimulated scattering instabilities of electromagnetic waves are investigated employing the recently derived expression for the electron number density perturbations that are driven by the ponderomotive and thermal forces in high Zi collisional plasmas. The growth rates and the threshold conditions for stimulated Brillouin and stimulated Compton scattering, as well as modulational instabilities, are obtained. It is found that the growth rates of the scattering instabilities of electromagnetic waves are considerably enhanced due to nonlocal electron heat conduction in collisional plasmas.

On nonlocal electron heat conduction

An improvement of the Albritton nonlocal electron heat transport model is proposed for high-Z plasmas. The thermal decay of the temperature perturbation in a uniform plasma as calculated by this model is compared with that obtained by Fokker–Planck simulations. Complete agreement is found up to values kλe≂0.1, where k is the wave number of the perturbation and λe is the thermal electron mean-free path.

Two-temperature discrete model for nonlocal heat conduction

The two-temperature discrete model for heat conduction in heterogeneous media is proposed. It is shown that the discrete model contains as limiting cases both hyperbolic and parabolic heat conduction equations for propagative and diffusive regimes, respectively. To obtain these limiting cases two different laws of continuum limit have been introduced. The evolution of the two-temperature system comprises three stages with distinct time scales: fast relaxation of each subsystem to local equilibrium, energy exchange between the subsystems and classical hydrodynamics