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This paper further develops a localized nonsingular method of fundamental solutions (LNMFS) for effectively solving heat conduction problems in functionally graded materials (FGMs). In this numerical framework, the LNMFS approach is proposed as a solution for the inhomogeneous boundary value problems, where a pseudo-spectral Chebyshev collocation scheme (CCS) is employed for the approximation of the corresponding particular solutions. In the proposed LNMFS, the artificial boundary that is present in the existing method of fundamental solutions (MFS) is eliminated. Accordingly, the singularities are replaced by the normalized area integrals of the fundamental solution over small circular disks, which encompass the source nodes that intersect with the collocation nodes. The proposed method inherits the high accuracy of the nonsingular method of fundamental solutions (NMFS). Additionally, in contrast to the NMFS, the LNMFS not only possesses the capability to achieve solutions with a notable level of accuracy in nonharmonic boundary condition but also overcomes the constraint of exclusively simulating heat conduction issues in FGMs that exhibit specific spatial variations. The feasibility and accuracy of the recently developed method are demonstrated by means of comparing it with analytical solutions for several numerical examples.

The process of heat conduction (HC) has recently found application in the information filtering [Zhang et al., Phys. Rev. Lett.99, 154301 (2007)], which is of high diversity but low accuracy. The classical HC model predicts users' potential interested objects based on their interesting objects regardless to the negative opinions. In terms of the users' rating scores, we present an improved user-based HC (UHC) information model by taking into account users' positive and negative opinions. Firstly, the objects rated by users are divided into positive and negative categories, then the predicted interesting and dislike object lists are generated by the UHC model. Finally, the recommendation lists are constructed by filtering out the dislike objects from the interesting lists. By implementing the new model based on nine similarity measures, the experimental results for MovieLens and Netflix datasets show that the new model considering negative opinions could greatly enhance the accuracy, measured by the average ranking score, from 0.049 to 0.036 for Netflix and from 0.1025 to 0.0570 for Movielens dataset, reduced by 26.53% and 44.39%, respectively. Since users prefer to give positive ratings rather than negative ones, the negative opinions contain much more information than the positive ones, the negative opinions, therefore, are very important for understanding users' online collective behaviors and improving the performance of HC model.

We study, by means of numerical simulations, the model of dissipative particle dynamics with energy conservation for the simple case of thermal conduction. It is shown that the model displays correct equilibrium fluctuations and reproduces Fourier law. The connection between "mesoscopic coarse-graining" and "resolution" is clarified.

The field equations within the framework of Lyra's geometry with a time-dependent displacement vector field for a Bianchi type-V space–time filled with a perfect fluid and heat flow are presented. Two different classes of physically viable solutions are obtained by using a special law of variation for the generalized mean Hubble's parameter which correspond to singular and nonsingular models with constant deceleration parameter. These models are found to be consistent with the observations on the present day universe. Some thermodynamical relations are studied. The physical and kinematical behaviors of the models are also discussed.

Tilted Bianchi type-V barotropic fluid cosmological models with heat conduction and variable bulk viscosity is investigated. To get the deterministic model of the universe, we have assumed two conditions: (i) A = (BC)ⁿ and (ii) ζθ = const where A, B, C are metric potentials, n is a constant, ζ the coefficient of bulk viscosity, θ the expansion in the model. The behavior of the model in presence and absence of bulk viscosity and singularities in the model are discussed. Some thermodynamic relations and physical aspects of the model related to the observation on the present day universe are also discussed.

We theoretically investigate the lattice thermal conductivity of a hollow Si nanowire under the relaxation time approximation. The results show that the thermal conductivity in such structure is decreased markedly below the bulk value due to phonon confinement and boundary scattering. The thermal conductivities under different scattering mechanisms are given, and it is found that the boundary scattering is dominant resistive process for the decrease of the thermal conductivity.

A model of a thermal transistor to control heat flow is reported. Like its electronic counterpart, the thermal transistor is a three-terminal device with the important feature that the heat current through two terminals can be switched or modulated by the temperature of the third terminal. The thermal transistor model is possible because of the negative differential thermal resistance.

The structure and the dynamics of the gas-liquid phase interface of the three-dimensional Lennard-Jones (12-6) particle system are studied using nonequilibrium molecular dynamics simulation. Heat flux maintains the system into a gas-liquid coexisting state with a steady interface. In the steady state, the interface shows an asymmetric structure and this is well described by a free energy density model with an asymmetric double-well form. When the system approaches to the steady state, a gas of the temperature profile appears between each phase and the gap value is relaxed to that in the steady state following for large t. It is observed that heat resistance exists in gas-liquid interface in this scale.

We investigate the heat conduction in a modified Lorentz gas with freely rotating disks periodically placed along one-dimensional channel. The heat conductivity is dependent on the moment of inertia η of the disks, with a power-law decay when η > 1. By plotting the Poincaré surface of the section, we observe a contraction of phase space over the range of η > 1, which is sensitive to the initial condition. We find that the power-law decay of the heat conductivity is relevant to the mixing phase space. As a possible application, we model the heterostructure by connecting the segments of different η, and predict the analytical results of the temperature profiles and the heat conductivity, which are in good agreement with the numerical ones.

By means of the Ford–Kac–Mazur formalism, the heat current and local kinetic energy of a dielectric chain with Fermi–Pasta–Ulam–β nonlinear interactions are derived out in the stationary equilibrium and analyzed numerically. We find that the anharmonicity reduces the heat current for repulsive interactions and enhances the heat current for attractions. The magnitudes of heat current are relevant strong at low frequencies, while the local kinetic energy is bigger in the high-frequency regime. The local kinetic energy displays a periodic oscillation due to the interference of the thermal waves. In particular, the transmission matrix exhibits oscillations due to scattering of phonons for the imperfect contacts. In general, the thermal transport is qualitatively altered by the differences of temperature between reservoirs, nonlinear interactions and the size of chain as well as the contacts.

Heat conduction in one-dimensional lattice with double-well interaction potential is studied numerically in different temperature regions. In the low temperature case, different structures such as order, period-2, and disorder structure phases, lead to different anomalous heat conduction. In a shallow intermediate temperature region, the heat conductivity is finite in a large system size. When temperature increases high enough, the heat conduction is anomalous, as well as FPU-β model.

We present the theory of spacetime elasticity and demonstrate that it involves traditional thermoelasticity. Assuming linear-elastic constitutive behavior and using spacetime transversely-isotropic elastic constants, we derive all principal thermodynamic relations of classical thermoelasticity. We introduce the spacetime principle of virtual work, and use it to derive the equations of motion for both reversible and dissipative thermoelastic dynamics. We show that spacetime elasticity directly implies the Fourier and the Maxwell–Cattaneo laws of heat conduction. However, spacetime elasticity is richer than classical thermoelasticity, and it advocates its own equations of motion for coupled thermoelasticity, complemented by the spectrum of boundary and interface conditions. We argue that the presented framework of spacetime elasticity should prove adequate for describing the thermoelastic phenomena occurring at low temperatures, for interpreting the results of molecular simulations of heat conduction in solids, and also for the optimal heat and stress management in the microelectronic components and the thermoelectric devices.

We investigate, using the exact Langevin equation, the properties of the heat conduction of an individual 1D harmonic chain with harmonic on-site potential. Each particle interacts with its own independent thermal reservoir. The results indicate that there is no rectification effect in this model even with mass gradient in the case of weak interparticle interaction. It may suggest that a system with harmonic on-site potential and asymmetric structure does not guarantee an asymmetric heat transport.

We study the transport of heat in a three-dimensional, harmonic crystal of slab geometry whose boundaries and the intermediate surfaces are connected to stochastic, white noise heat baths at different temperatures. Heat baths at the intermediate surfaces are required to fix the initial state of the slab in respect of its surroundings. We allow the flow of energy fluxes between the intermediate surfaces and the attached baths and impose conditions that relate the widths of Gaussian noises of the intermediate baths. The radiated heat obeys Newton's law of cooling when intermediate baths collectively constitute the environment surrounding the slab. We show that Fourier's law holds in the continuum limit. We obtain an exponentially falling temperature profile from high to low temperature end of the slab and this very nature of the profile was already confirmed by Ingen-Hausz's experiment. Temperature profile of similar nature is also obtained in the one-dimensional version of this model.

We have investigated the thermal conductivity of defective fullerene (C${}_{60})$ by using the nonequilibrium molecular dynamics (MD) method. It is found that the thermal conductivity of C${}_{60}$ with one defect is lower than the thermal conductivity of perfect C${}_{60}$. However, double defects in C${}_{60}$ have either positive or negative influence on the thermal conductivity, which depends on the positions of the defects. The phonon spectra of perfect and defective C${}_{60}$ are also provided to give corresponding supports. Our results can be extended to long C${}_{60}$ chains, which is helpful for the thermal management of C${}_{60}$.

We study transport of heat in an ordered, anharmonic crystal in the form of slab geometry in three dimensions. Apart from attaching baths of Langevin type to two extreme surfaces, we also attach baths of same type to the intermediate surfaces of the slab. Since the crystal is uninsulated, it exchanges energy with the intermediate heat baths. We find that both Fourier’s law of heat conduction and the Newton’s law of cooling hold to leading order in anharmonic coupling. The leading behavior of the temperature profile is exponentially falling from high to low temperature surface of the slab. As the anharmonicity increases, profiles fall more below the harmonic one in the log plot. In the thermodynamic limit thermal conductivity remains independent of the environment temperature and its leading order anharmonic contribution is linearly proportional to the temperature change between the two extreme surfaces of the slab. A fast crossover from one-dimensional (1D) to three-dimensional (3D) behavior of the thermal conductivity is observed in the system.

A CMOS phase-shift oscillator that uses a phase shift in the conduction of heat is proposed. The oscillator consists of an inverting amplifier and a feedback thermal filter that are integrated on a silicon chip. The thermal filter consists of a polysilicon heater and a MOSFET thermosensor separated from each other by a S_iO₂ heat-conducting layer.¹ An input signal for the filter travels from the heater to the thermosensor through the S_iO₂ layer in the form of heat. The filter accepts the output of the inverting amplifier, produces a phase shift due to the heat conduction, and returns a 180°-shifted feedback signal to the amplifier. The oscillator produces oscillation at a specific frequency determined by the dimensions of the filter and the thermal conductivity and specific heat of Si and S_iO₂.

To minimize the thermal resistance of the fractal treelike heat conduction network and develop an optimization principle applicable for the network with an arbitrary cross-sectional shape, this paper first establishes a theoretical model regarding the total thermal resistance of the symmetric treelike network with arbitrary cross-sectional shapes and then studies the effects of the geometric and structural parameters of the network on its total thermal resistance. The numerical simulations are also performed to analyze the influences of the geometric and structural parameters of symmetric treelike networks with circular, rectangular and triangular cross-sectional shapes on the total thermal resistance. Both the theoretical model and the numerical simulation show that the total thermal resistance of the network with an arbitrary cross-sectional shape first decreases and then increases with increasing cross-sectional area ratio but always increases with increasing length ratio of branches at two successive branching levels when the total branch volume is constant. When the cross-sectional area ratio is equal to the reciprocal of the branching number, the treelike network has the minimum total thermal resistance. This scaling law is applicable for the treelike network with an arbitrary cross-sectional shape to achieve the minimum total thermal resistance.

In this paper, we investigate the decay properties of the Bresse–Cattaneo system in the whole space. We show that the coupling of the Bresse system with the heat conduction of the Cattaneo theory leads to a loss of regularity of the solution and we prove that the decay rate of the solution is very slow. In fact, we show that the $L^2$-norm of the solution decays with the rate of ${(1+t)}^{-1/12}$. The behavior of solutions depends on a certain number $\alpha$ (which is the same stability number for the Timoshenko–Cattaneo system [Damping by heat conduction in the Timoshenko system: Fourier and Cattaneo are the same, J. Differential Equations255(4) (2013) 611–632; The stability number of the Timoshenko system with second sound, J. Differential Equations253(9) (2012) 2715–2733]) which is a function of the parameters of the system. In addition, we show that we obtain the same decay rate as the one of the solution for the Bresse–Fourier model [The Bresse system in thermoelasticity, to appear in Math. Methods Appl. Sci.].

The h-p version of the finite element method (FEM) is considered to determine the transient temperature distribution in functionally graded materials (FGM). The h-p version may be regarded as the marriage of conventional h-version and p-version. The graded Fourier p-element is used to set up the two-dimensional heat conduction equations. The temperature is formulated in terms of linear shape functions used generally in FEM plus a variable number of trigonometric shape functions representing the internal degrees of freedom (DOF). In the graded Fourier p-element, the function of the thermal conductivity is computed exactly within the conductance matrix and thus overcomes the computational errors caused by the space discretization introduced by the FEM. Explicit and easily programmed trigonometric enriched capacitance, conductance matrices and heat load vectors are derived for plates and cylinders by using symbolic computation. The convergence properties of the h-p version proposed and the results of the numbers of test problems are in good agreement with the analytical solutions. Also, the effect of the non-homogeneity of the FGM on the temperature distribution is considered.