A hybrid Green’s function method for the hyperbolic heat conduction problems

The present study is devoted to propose a hybrid Green’s function method to investigate the hyperbolic heat conduction problems. The difficulty of the numerical solutions of hyperbolic heat conduction problems is the numerical oscillation in the vicinity of sharp discontinuities. In the present study, we have developed a hybrid method combined the Laplace transform, Green’s function and ε-algorithm acceleration method for solving time dependent hyperbolic heat conduction equation. From one- to three-dimensional problems, six different examples have been analyzed by the present method. It is found from these examples that the present method is in agreement with the Tsai-tse Kao’s solutions [Tsai-tse Kao, Non-Fourier heat conduction in thin surface layers, J. Heat Transfer 99 (1977) 343–345] and does not exhibit numerical oscillations at the wave front. The propagation of the two- and three-dimensional thermal wave becomes so complicated because it occur jump discontinuities, reflections and interactions in these numerical results of the problem and it is difficult to find the analytical solutions or the result of other study to compare with the solutions of the present method.     

Numerical solution of hyperbolic heat conduction problems in the cylindrical coordinate system by the hybrid Green’s function method

In the present study, we have analyzed the hyperbolic heat conduction problems in the cylindrical coordinate system using a hybrid Green’s function method. The major difficulty encountered in the numerical solutions of hyperbolic heat conduction problems is the suppression of the numerical oscillations in vicinity of sharp discontinuities (Chen and Lin (1993) [11]). The proposed method combines the Laplace transform for the time domain, Green’s function for the space domain and ε-algorithm acceleration method for fast convergence of the series solution. Six different examples included the one-, two- and three-dimensional problems have been analyzed by the present method. It is found from these examples that the present method does not exhibit numerical oscillations at the wave front and the propagation of the two- and three-dimensional thermal wave becomes so complicated because it occur jumping discontinuities, reflections and interactions in these numerical results of the hyperbolic heat conduction problem.     

Numerical solution for the hyperbolic heat conduction problems in the radial–spherical coordinate system using a hybrid Green's function method

The hyperbolic heat conduction problems in the radial–spherical coordinate system are investigated by the hybrid Green's function method. The present method combines the Laplace transform for the time domain, Green's function for the space domain and ?-algorithm acceleration method for fast convergence of the series solution. Three different examples problems have been analyzed by the present method. It is found that the present method does not exhibit numerical oscillations at the wave front and the numerical solutions are stable.     

SPECIALLY TAILORED TRANS FINITE-ELEMENT FORMULATIONS FOR HYPERBOLIC HEAT CONDUCTION INVOLVING NON-FOURIER EFFECTS

The phenomenon of hyperbolic heat conduction in contrast to the classical (parabolic) form of Fourier heat conduction involves thermal energy transport that propagates only at finite speeds as opposed to an infinite speed of thermal energy transport. To accommodate the finite speed of thermal wave propagation, a more precise form of heat flux law is involved, thereby modifying the heat flux originally postulated in the classical theory of heat conduction. As a consequence, for hyperbolic heat conduction problems, the thermal energy propagates with very sharp discontinuities at the wave front. The primary purpose of the present paper is to provide accurate solutions to a class of one-dimensional hyperbolic heat conduction problems involving non-Fourier effects that can precisely help understand the true response and furthermore can be used effectively for representative benchmark tests and for validating alternate schemes. As a consequence, the present paper purposely describes modeling/analysis formulations via specially tailored hybrid computations for accurately modeling the sharp discontinuities of the propagating thermal wave front. Comparative numerical test models are presented for various hyperbolic heat conduction models involving non-Fourier effects to demonstrate the present formulations.     

LATTICE BOLTZMANN SCHEME FOR HYPERBOLIC HEAT CONDUCTION EQUATION

An extended lattice Boltzmann (LB) equation, the lattice Boltzmann equation with a source term, is developed for the system of equations governing the hyperbolic heat conduction equation. Mathematical consistence between the proposed extended LB equation and the governing equations are accomplished by the Chapman-Enskog expansion. Four illustrative examples, with both finite and semi-infinite computational domains and subjected to linear and nonlinear boundary conditions, are simulated. All numerical predications agree very well with the existing solutions in the literature. It is also demonstrated that the present scheme is stable and free of numerical oscillations especially around the wave front, where sharp change in temperature occurs.     

Integral equation solution for hyperbolic heat conduction with surface radiation

Temperature distributions for the hyperbolic heat conduction in a semi-infinite medium with surface radiation are found from the solutions of a nonlinear Volterra equation for the surface temperature. The integral equation is obtained by the Laplace transform. This method has the advantage that the temperature distributions do not involve numerical oscillations around the thermal wave front.     

AN EXPLICIT TVD SCHEME FOR HYPERBOLIC HEAT CONDUCTION IN COMPLEX GEOMETRY

Two-dimensional hyperbolic heat conduction problems of complex geometry are investigated numerically. A second-order total variation diminishing (TVD) scheme is introduced and its application to the hyperbolic heat conduction is developed in detail using the knowledge of characteristics. In current work primitive variables, rather than characteristic variables, are used as the dependent variables. The governing equations of two-dimensional heat conduction are transformed from the physical coordinates to the computational coordinates, so that the hyperbolic heat conduction problems of irregular geometry can be solved numerically by the present TVD scheme. Three examples with different geometry are used to verify the accuracy of the present numerical scheme. Results show the explicit TVD scheme can predict the thermal wave without oscillation.     

Difference Scheme for Hyperbolic Heat Conduction Equation with Pulsed Heating Boundary

horotctionIt is well known that the basic law of heatconduction is the Fourier law. It has the formq = --k' vT, and one-dimensional heat conduchondifferenhal equation is. The aboveequations are derived from the hypothesis that thevelocity to establish the thermal balance is infinitelygreat. In the modem heat conduchon theory heattransacts in materials in a licited velocity. The factorsto affect the velocity are the thermal propelles of thematerials. In order tO describe this Problem, the sch…     

Multiple scattering of thermal waves from double subsurface cylinders in semi‐infinite slab

In this paper, combined with the non‐Fourier equation of heat conduction and expansion method of wave functions, the multiple scattering of thermal waves from a subsurface cylinder in a semi‐infinite body is investigated. A general solution of scattered fields based on hyperbolic equations of heat conduction is presented for the first time. The effects of physical and geometric parameters on the temperature are analyzed. The thermal waves are excited at the front surface of opaque material by modulated optical beams. The circular cylinder is taken as a cavity with thermal insulation conditions. © 2007 Wiley Periodicals, Inc. Heat Trans Asian Res, 36(7): 398– 407, 2007; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20174     

CHARACTERISTICS-BASED,HIGH-ORDER ACCURATE AND NONOSCILLATORY NUMERICAL METHOD FOR HYPERBOLIC HEAT CONDUCTION

For extremely short durations oral very low temperatures (near absolute zero), the classical Fourier heat conduction equation fails and has to be replaced by a hyperbolic equation to account for finite thermal wave propagation. During the last few years, there has been a growing interest in numerical simulation of the hyperbolic heal conduction problem. The schemes used in previous studies were either the classical upwind and central difference or the MacCormack's predictor-corrector schemes. As a result, spurious oscillations or excessive diffusion appeared near the discontinuities. This paper presents a method that makes use of the characteristics to suppress these oscillations or diffusions. The equations governing the hyperbolic heat conduction problem are first transformed into characteristic equations, and the first-order upwind, second-order upwind (Beam-Warming)and second-order central (Lax-Wendroff)schemes are applied based on the direction of the characteristic velocity. It is shown that simple implementation of these schemes cannot lead to satisfactory results. Subsequently, the high-order TVD (Total Variation Diminishing) schemes are introduced to solve the problem. These TVD schemes are oscillation free and can give high-order accuracy without introducing wiggles. The principle of the proposed scheme is demonstrated by proper switching between Lax-Wendroff and Beam-Warming schemes. Basically, the higher-order scheme is used in the calculation domain, except at discontinuities and local extrema where it is switched to the first-order scheme. Several examples are used to demonstrate the success of the numerical method.     

SOLUTION OF TWO-DIMENSIONAL HYPERBOLIC HEAT CONDUCTION BY HIGH-RESOLUTION NUMERICAL METHODS

Studies of hyperbolic heat conduction have so far been limited mostly to one-dimensional frameworks. For two-dimensional problems, the reflection and interaction of oblique thermal waves and complicated geometries present a challenge. This paper describes a numerical solution of two-dimensional hyperbolic heat conduction by high-resolution schemes. First, the governing equations are transformed from Cartesian coordinates into generalized curvilinear coordinates. Then the dependent variables are cast in a characteristic form that decouples the original system equation into scalar equations. Two-dimensional high-resolution numerical schemes, suck as total variational diminishing ( TVD) are built up by forming symmetrical products of one-dimensional difference operators on each individual wave. Three examples are used to demonstrate the unique feature of complicated interaction of two-dimensional thermal waves.     

Transient Thermal Stresses in an Orthotropic Cylinder under the Hyperbolic Heat Conduction Model

An important consideration in design involving high temperature variation is the determination of the thermal stresses developed. The numerical solution for thermoelastic transient response of orthotropic cylinder subjected to a constant temperature at the surface is presented. The thermoelastic equations with one relaxation time developed by Lord and Shulman with uncoupled thermoelasticity assumption are used in the present work. The hyperbolic heat conduction model is used for the prediction of the temperature history. Thermally induced displacement and stresses are determined. A numerical method based on implicit finite difference scheme is used to calculate the temperature, displacement, and stress distributions within the cylinder. Numerical examples for orthotropic, transverse isotropic, and isotropic cylinders were carried out for the stresses. Furthermore, the results of the numerical solution and the exact solution at the steady state condition are compared.     

Analysis of boundary inverse heat conduction problems using space marching with Savitzky-Gollay digital filter

This paper presents a method by which boundary inverse heat conduction problems can be analyzed. A space marching algorithm is used for formulating and solving parabolic and hyperbolic inverse heat conduction problems. The solution of numerical examples shows that a combination of the digital filter with the hyperbolic approximation of inverse heat conduction problem increases the stability of the results without loss of resolution. The validity of numerical solution for the inverse problem is examined by comparing the obtained results with the direct solution of the problem.     

An inverse radiation boundary design problem for an enclosure filled with an emitting,absorbing, and scattering media

This work is an inverse radiative design problem in which the objective is to determine the spatial distribution of heat source strengths which produces a desired temperature and heat flux distribution on the design surface. The furnace whose walls are diffuse-grey is assumed to be filled with an absorbing, emitting, and scattering medium. The function to be minimized is the sum of squares of the differences between the desired and calculated radiative heat fluxes at the design surface. Radiative heat flux calculations are accomplished by means of the Modified Discrete Transfer Method MDTM using the correction factors suggested by Coelho and Carvalho [P.J. Coelho, M.G. Carvalho, Conservative formulation of the discrete transfer method, ASME J. Heat Transfer, 119 (1997) 118–128.] and Cumber [P.S. Cumber, Improvements to the discrete transfer method of calculating radiative heat transfer, Int. J. Heat Mass Transfer, 38 (12) (1995) 2251–2258.]. For inverse design calculations the Conjugate Gradient Method CGM is employed, in which the sensitivity coefficients are defined and used as needed by the algorithm. Our investigation shows that the presented algorithm is able to estimate heater strengths accurately.     

Application of CESE method to simulate non-Fourier heat conduction in finite medium with pulse surface heating

This study employs the space–time conservation element and solution element (CESE) method to simulate the temperature and heat flux distributions in a finite medium subject to various non-Fourier heat conduction models. The simulations consider three specific cases, namely a single phase lag (SPL) thermal wave model with a pulsed temperature condition, a SPL model with a surface heat flux input, and a dual phase lag (DPL) thermal wave model with an initial deposition of thermal energy. In every case, the thermal waves are simulated with respect to time as the thermal wave propagates through the medium with a constant velocity. In general, the simulation results are found to be in good agreement with the exact analytical solutions. Furthermore, it is shown that the CESE method yields low numerical dissipation and dispersion errors and accurately models the propagation of the wave form even in its discontinuous portions. Significantly, compared to traditional numerical schemes, the CESE method provides the ability to model the behavior of the SPL thermal wave following its reflection from the boundary surface. Further, a numerical analysis is performed to establish the CESE time step and mesh size parameters required to ensure stable solutions of the SPL and DPL thermal wave models, respectively.     

RAPID TRANSIENT HEAT CONDUCTION IN MULTILAYER MATERIALS WITH PULSED HEATING BOUNDARY

ABSTRACT

Rapid transient heat conduction in multilayer materials under pulsed heating is solved numerically based on a hyperbolic heat conduction equation and taking into consideration the non-Fourier heat conduction effects. An implicit difference scheme is presented and a stability analysis conducted, which shows that the implicit scheme for the hyperbolic equation is stable. The code is validated by comparing the numerical results with an existing exact solution, and the physically unrealistic conditions placed on the time and space increments are identified. Using the validated model, the numerical solution of thermal wave propagation in multilayer materials is presented. By analyzing the results, the necessary conditions for observing non-Fourier phenomena in the laboratory can be inferred. The results are also compared with the numerical results from the parabolic heat conduction equation. The difference between them is clearly apparent, and this comparison provides new insight for the management of thermal issues in high-energy equipment. The results also illustrate the time scale required for metal films to establish equilibrium in energy transport, which makes it possible to determine a priori the time response and the measurement accuracy of metal film, thermal-resistant thermometers.     

Inverse heat conduction problems by using particular solutions

Based on the method of fundamental solutions, we develop in this paper a new computational method to solve two‐dimensional transient heat conduction inverse problems. The main idea is to use particular solutions as radial basis functions (PSRBF) for approximation of the solutions to the inverse heat conduction problems. The heat conduction equations are first analyzed in the Laplace transformed domain and the Durbin inversion method is then used to determine the solutions in the time domain. Least‐square and singular value decomposition (SVD) techniques are adopted to solve the ill‐conditioned linear system of algebraic equations obtained from the proposed PSRBF method. To demonstrate the effectiveness and simplicity of this approach, several numerical examples are given with satisfactory accuracy and stability. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com ). DOI 10.1002/htj.20335     

Real-time solution of heat conduction in a finite slab for inverse analysis

Laplace transform is used to solve the problem of heat conduction over a finite slab. The transfer functions relating the temperature and heat flux on the front and back surfaces of the finite slab are developed. Although there are many competing methods for constructing the inverse Laplace transform, we use polynomial approximation of the transfer function. Therefore, transient solutions for given boundary conditions are easily obtained using SIMULINK. This process is much simpler than other numerical solution methods for the heat equation. Most importantly, our method of solution allows us to obtain, in real-time, the front surface temperature and heat flux based on the thermodynamic measurements on the back surface. We also demonstrate the feasibility of reconstructing the front surface temperature when sensor noise is incorporated to the back surface measurements.     

Numerical Simulation of Non-Fourier Temperature Behavior of Functionally Graded Materials in a Slab with Surface Radiation

Functionally graded materials (FGMs) are used in many applications that presumably produce the wave nature of thermal energy transport. This study investigates the hyperbolic and parabolic heat conduction problem for a solid slab made of FGM numerically. A constant heat flux is considered at both sides of the slab, and boundaries dissipate heat by radiation into an ambient. An exponential space-dependent function of volume fraction is considered. MacCormack's explicit predictor-corrector scheme is used to solve the nonlinear equation in order to handle discontinuities at the wave front quite satisfactorily with small oscillations. Results are compared to the results obtained with the assumption of constant and linear spatial variation of volume fraction function. Further effects of different nondimensional numbers on the temperature distribution is sought. Numerical results are validated by the analytical solution of a special case that shows excellent agreement.     

Application of a new finite integral transform method to the wave model of conduction

A new finite integral transform method [Int. J. Heat Mass Transfer 44 (2001) 3307] is applied to the wave model of conduction. It is compared with a standard method of solution of the hyperbolic conduction equation. The temperature fields coincide. The chosen test problem and its results bring to the foreground some of the difficulties of standard technique applications. These difficulties are by-passed when using the new method.The Cattaneo Vernotte model is then tested through a comparison of its results with transient molecular dynamics simulations taken from Volz [Transferts de chaleur aux temps ultra-courts par la technique de la dynamique moléculaire, Thèse, Univ. de Poitiers, 1996]. When the used parameters of the continuous model are near their equilibrium values, the agreement remains weakly qualitative. An adaptation of these parameter values, notably the diffusion time scale, can give a quantitative coincidence; but never is the agreement obtained for both studied variables (internal energy and flux density). These observations are discussed. Various causes liable to justify the realized adaptations of parameters are considered. None of them gives a right explanation. Concerning the impossibility of making both variables coincide, the source of conflicts is as much in the constitutive law as in the energy conservation law. The key seems to be in thermodynamics.