non-Fourier Heat Conduction Problems Research Articles

A hybrid shape functions based temporal finite element method to solve transient heat conduction problems

A novel temporal finite element method is presented to solve Fourier and non-Fourier heat conduction problems. Two kinds of temporal finite element models are developed using Gurtin variational principle and weighted residual technique, respectively. A kind of hybrid shape function with polynomial and trigonometric basis is stressed to give more flexible and appropriate description of temporal behavior of temperature. A recursive algorithm is developed by which the temperature solution at a specific time can be obtained only via a matrix power product with the initial condition, and a criterion of stability analysis is derived, which is can numerically be conducted when the constitution of shape function and time step size are prescribed. The proposed approach is available for Fourier and non-Fourier heat transfer problems, and can conveniently be combined with well-developed numerical algorithms for the boundary value problem, such as FE, SBFEM, etc. Various numerical examples, including those with the nonlinear thermal conductivity, radiative boundary condition, etc. are provided to illustrate the efficiency of proposed approaches, and impacts of the temporal FE model, constitution of shape functions, and step size, etc. are taken into account. Satisfactory results are achieved in comparison with the analytical and the ABAQUS based numerical solutions.

A Reduced-Order FEM Based on POD for Solving Non-Fourier Heat Conduction Problems under Laser Heating

A Reduced-Order FEM Based on POD for Solving Non-Fourier Heat Conduction Problems under Laser Heating

Fractional Modeling of Fin on non-Fourier Heat Conduction via Modern Fractional Differential Operators

The enhancement of heat transfer for electronic kits and automobiles has become highly dependent on the finned heat exchangers; this is because fin provides high heat transfer rate and superior performance with a significant temperature reduction. In this manuscript, a fractional modeling of non-Fourier heat conduction problem of a fin is proposed within the periodic temperature boundary condition. The mathematical modeling is performed via classical theory of heat conduction that is directly proportional to temperature gradient through which hyperbolic heat conduction equation for a fin is generated. The hyperbolic heat conduction equation for a fin is fractionalized via modern approaches of fractional differentiations, namely Atangana–Baleanu and Caputo–Fabrizio differential operators. In order to have analyticity of hyperbolic heat conduction equation for a fin, we invoked the mathematical techniques of Laplace transform. The exact solutions of temperature distribution have been obtained in terms of Fox-H and Mittag–Leffler functions with the product of convolution. The solutions of temperature distribution have been classified into integer verses non-integer theories by making fractional parameters $$ \alpha = \beta = 1 $$ and $$ \alpha \ne \beta \ne 1 $$ , respectively. Our results suggest that due to variability of different rheological parameters on temperature distribution, the cooling process is faster via fractional models in comparison to non-fractional model. Additionally, it is also observed that thermal wave propagates at a specific time results the reciprocal trend in temperature distribution.

Analysis of Non-Fourier Heat Conduction Problem with Suddenly Applied Surface Heat Flux

In literature, the well-acknowledged initial condition is generally used when solving the non-Fourier problems. However, in references, some cases with a suddenly applied heat flux boundary under such an initial condition violate the first law of thermodynamics. This unreasonable situation is first demonstrated in the cases of constant heat flux using the Cattaneo–Vernotte model of a one-dimensional finite medium with the other side being isothermal or insulated. Then it is also demonstrated in the cases of arbitrary heat flux by reduction to absurdity. To adjust the unreasonable situation, an innovative definition of initial condition is proposed to obtain accurate solutions. Besides, the modified cases of a constant heat flux boundary are also discussed in this work with different relaxation time. In those cases, this study, for the first time, discovers that the heat wave can be inversely reflected under the isothermal boundary. Then when is extremely big, the heat flux input may result in a cooling response.

Nonlinear analysis of non-Fourier heat conduction problems by a locally stabilized explicit approach

In this paper, a stabilized, locally defined, explicit approach is considered to analyse nonlinear, non-Fourier heat conduction problems. In this sense, a modified central difference method is applied, which performs adapting itself along the solution process, considering the properties and results of the model, as well as the relations between the adopted temporal and spatial discretizations. The proposed technique is highly accurate, versatile and efficient. In fact, the new approach stands as a non-iterative, single-solve framework based on reduced systems of equations; thus, it demands very low computational efforts. In addition, the procedure is very simple to implement and entirely automatized, requiring no decision or expertise from the user. Numerical results are presented at the end of the manuscript, illustrating the performance and effectiveness of the novel methodology.

Lattice Boltzmann simulation of non-Fourier heat conduction with phase change

The lattice Boltzmann method (LBM) combined with the enthalpy method is a very effective method to solve the solid–liquid phase transition problem. However, when the heat flux is very high or the time of the process is in the same order of magnitude as the relaxation time, it is necessary to consider the non-Fourier effect in heat conduction. At this time, whether the LBM-BGK format based on Bhatnagar-Gross-Krook (BGK) approximation is still valid remains to be discussed. In this paper, the hyperbolic lattice Boltzmann method (HLBM) is combined with the enthalpy method to solve the non-Fourier solid–liquid phase change problem. By solving the non-Fourier heat conduction problem and the Fourier solid–liquid phase change problem, the numerical solution is compared with the analytical solution to verify the accuracy of the algorithm. The effect of different relaxation times on the solid–liquid phase transition is analyzed. In addition, the effect of changes in thermal diffusivity due to state changes on the non-Fourier solid–liquid phase transition is discussed.

Analysis and Modelling of Non-Fourier Heat Behavior Using the Wavelet Finite Element Method.

Non-Fourier heat behavior is an important issue for film material. The phenomenon is usually observed in some laser induced thermal responses. In this paper, the non-Fourier heat conduction problems with temperature and thermal flux relaxations are investigated based on the wavelet finite element method and solved by the central difference scheme for one- and two-dimensional media. The Cattaneo–Vernotte model and the Dual-Phase-Lagging model are used for finite element formulation, and a new wavelet finite element solving formulation is proposed to address the memory requirement problem. Compared with the current methodologies for the Cattaneo–Vernotte model and the Dual-Phase-Lagging model, the present model is a direct one which describe the thermal behavior by one equation about temperature. Compared with the wavelet method proposed by Xiang et al., the developed method can be used for arbitrary shapes. In order to address the efficient computation problems for the Dual-Phase-Lagging model, a novel iteration updating methodology is also proposed. The proposed iteration algorithms on time avoids the use the global stiffness matrix, which allows the efficient calculation for title issue. Numerical calculations have been conducted in the manner of comparisons with the classical finite element method and spectral finite element method. The comparisons from accuracy, efficiency, flexibility, and applicability validate the developed method to be an effective and alternative tool for material thermal analysis.

Non-Fourier fractional heat conduction in two bonded dissimilar materials with a penny-shaped interface crack

This paper studies a generalized non-Fourier fractional heat conduction problem of a bi-material with a penny-shaped interface crack under sudden heat flux shock. The Hankel transform and Laplace transform are employed to solve an initial-boundary value problem associated with a time-fractional partial differential equation. Closed-form temperature field and the dynamic intensity factors of heat flux and temperature gradient near the front of a penny-shaped crack are obtained in the Laplace transform domain. Numerical results in the time domain are obtained by using a numerical inversion of the Laplace transform. Two cases of a cracked homogeneous platinum and a bi-material composed of platinum and quartz glass with an interface crack are analyzed to show the effects of fractional order, crack size and material properties on the temperature change and the intensity factors of heat flux and temperature gradient. Obtained results based on the non-Fourier fractional order heat conduction are compared with those using the hyperbolic heat conduction graphically. It is found that transient temperature gradient and heat flux exhibit the inverse-square-root singularity near the crack front, and their intensity factors depend on the bi-material property. Moreover, the temperature in magnitude on the crack face of platinum is much smaller than that on the crack face of quartz glass. The heat flux is continuous across the interface, while the temperature gradient is discontinuous across the interface. Transient response of temperature, its gradient, and heat flux for the case of fractional order is noticeable for the earlystage of application of sudden heat flux.

Identification of boundary conditions for non-Fourier heat conduction problems by differential transformation DRBEM and improved cuckoo search algorithm

The differential transformation method is combined with the dual reciprocity boundary element method to solve the non-Fourier heat conduction problems in functionally gradient materials. The cuckoo search algorithm is improved by the Broyden–Fletcher–Goldfarb–Shanno algorithm to identify the boundary conditions for the heat conduction problems. The polynomial function related to coordinate and time is proposed to approximate the unknown boundary conditions. Numerical examples discuss the influences of measurement point numbers and measurement errors on inverse solutions. Numerical results demonstrate the effectiveness and accuracy of the proposed method.

Generalized Haar wavelet operational matrix method for solving hyperbolic heat conduction in thin surface layers

It is remarkably known that one of the difficulties encountered in a numerical method for hyperbolic heat conduction equation is the numerical oscillation within the vicinity of jump discontinuities at the wave front. In this paper, a new method is proposed for solving non-Fourier heat conduction problem. It is a combination of finite difference and pseudospectral methods in which the time discretization is performed prior to spatial discretization. In this sense, a partial differential equation is reduced to an ordinary differential equation and solved implicitly with Haar wavelet basis. For the pseudospectral method, Haar wavelet expansion has been using considering its advantage of the absence of the Gibbs phenomenon at the jump continuities. We also derived generalized Haar operational matrix that extend usual domain (0,1] to (0,X]. The proposed method has been applied to one physical problem, namely thin surface layers. It is found that the proposed numerical results could suppress and eliminate the numerical oscillation in the vicinity jump and in good agreement with the analytic solution. In addition, our method is stable, convergent and easily coded. Numerical results demonstrate good performance of the method in term of accuracy and competitiveness compare to other numerical methods.

Nonlinear analysis of a non-Fourier heat conduction problem in a living tissue heated by laser source

The effect of laser, as a heat source, on a one-dimensional finite living tissue was studied in this paper. The dual phase lagging (DPL) non-Fourier heat conduction model was used for thermal analysis. The thermal conductivity was assumed temperature-dependent, resulting in a nonlinear equation. The obtained equations were solved using the approximate-analytical Adomian decomposition method (ADM). It was concluded that the nonlinear analysis was important in non-Fourier heat conduction problems. Moreover, a good agreement between the present nonlinear model and experimental result was obtained.

Orthogonal Eigenfunction Expansion Method for One-Dimensional Dual-Phase Lag Heat Conduction Problem With Time-Dependent Boundary Conditions

The separation of variables (SOV) can be used for all Fourier, single-phase lag (SPL), and dual-phase lag (DPL) heat conduction problems with time-independent source and/or boundary conditions (BCs). The Laplace transform (LT) can be used for problems with time-dependent BCs and sources but requires large computational time for inverse LT. In this work, the orthogonal eigenfunction expansion (OEEM) has been proposed as an alternate method for non-Fourier (SPL and DPL) heat conduction problem. However, the OEEM is applicable only for cases where BCs are homogeneous. Therefore, BCs of the original problem are homogenized by subtracting an auxiliary function from the temperature to get a modified problem in terms of a modified temperature. It is shown that the auxiliary function has to satisfy a set of conditions. However, these conditions do not lead to a unique auxiliary function. Therefore, an additional condition, which simplifies the modified problem, is proposed to evaluate the auxiliary function. The methodology is verified with SOV for time-independent BCs. The implementation of the methodology is demonstrated with illustrative example, which shows that this approach leads to an accurate solution with reasonable number of terms in the expansion.

Nonlinear Analysis of a One-Dimensional Non-Fourier Heat Conduction Problem

In this paper, the problem of Non-Fourier heat conduction in a one-dimensional finite body was studied. Non-Fourier heat conduction model of thermal wave was used for thermal analysis of the problem. Temperature-dependent heat conductivity was assumed and a nonlinear equation was obtained. To solve the equations, semi-analytical methods of Reduced Differential Transform Method (RDTM) and Homotopy Perturbation Method (HPM) were applied. It was concluded that nonlinear analysis of Non-Fourier heat conduction problems is highly important and higher accuracy of RDTM with respect to HPM was specified.

A new solution for nonlinear Dual Phase Lagging heat conduction problem

The application of new methods to solution of non-Fourier heat transfer problems has always been one of the interesting topics among thermal science researchers. In this paper, the effect of laser, as a heat source, on a thin film was studied. The Dual Phase Lagging (DPL) non-Fourier heat conduction model was used for thermal analysis. The thermal conductivity was assumed temperature-dependent which resulted in a nonlinear equation. The obtained equations were solved using the approximate-analytical Adomian Decomposition Method (ADM). It was concluded that the nonlinear analysis is important in non-Fourier heat conduction problems. Significant differences were observed between the Fourier and non-Fourier solutions which stress the importance of non-Fourier solutions in the similar problems.

Nonlinear Solution to a Non-Fourier Heat Conduction Problem in a Slab Heated by Laser Source

Abstract The effect of laser, as a heat source, on a one-dimensional finite body was studied in this paper. The Cattaneo-Vernotte non-Fourier heat conduction model was used for thermal analysis. The thermal conductivity was assumed temperature-dependent which resulted in a non-linear equation. The obtained equations were solved using the approximate-analytical Adomian Decomposition Method (ADM). It was concluded that the non-linear analysis is important in non-Fourier heat conduction problems. Significant differences were observed between the Fourier and non-Fourier solutions which stresses the importance of non-Fourier solutions in the similar problems.

Tickhonov based well-condition asymptotic waveform evaluation for dual-phase-lag heat conduction

The Tickhonov based well condition asymptotic waveform evaluation (TWCAWE) is presented here to study the non-Fourier heat conduction problems with various boundary conditions. In this paper, a novel TWCAWE method is proposed to overwhelm ill-conditioning of the asymptotic waveform evaluation (AWE) technique for thermal analysis and also presented for time-reliant problems. The TWCAWE method is capable to evade the instability of AWE and also efficaciously approximates the initial high frequency and delay similar as well-established numerical method, such as Runge-Kutta (R-K). Furthermore, TWCAWE method is found 1.2 times faster than the AWE and also 4 times faster than the traditional R-K method.

Precise Time-Domain Expanding BEM for Solving Non-Fourier Heat Conduction Problems

The precise time-domain expanding boundary-element method (BEM) is presented for solving non-Fourier heat conduction problems. The recursive boundary integral equation is obtained via the precise time-domain expanding method and solved by the BEM, where the radial integral method is used to transform the domain integral into the boundary integral. Also, a self-adaptive judging criterion is used in the solving process. The transformation matrices of domain integrals need to be computed only once, except those related to the heat source. Finally, numerical results show that the present method can obtain stable and accurate results with different time steps.

Radial integration BEM for solving non-Fourier heat conduction problems

The radial integral BEM (RIBEM) with a step-by-step integration method is presented for solving non-Fourier heat conduction problems in this paper. First, the system of second-order ordinary differential equations is obtained by using the RIBEM to discretize the space domain. Then, the Newmark method and the central difference method are adopted to solve the system of ordinary differential equations with respect to time. Finally, several numerical examples with laser heat sources are performed to demonstrate the performance of the present method. The results show that the present approach can obtain accurate and stable numerical results.

Non-Fourier Heat Conduction Problems and the Use of Exponential Basis Functions

A solution method using exponential basis functions (EBFs) is proposed for transient one-/two-dimensional non-Fourier heat conduction problems having particular application in bio-heat fields. A summation of EBFs satisfying the governing differential equation is considered in time and space. The presented method uses a noniterative algorithm for the solution of direct/inverse problems. It is demonstrated that the use of extra EBFs in the form of enrichment functions significantly improves the results when some jumps are seen in the input data. Four numerical examples, including bio-heat conduction problems, are provided to investigate the accuracy and performance of the method presented.

Trefftz Functions Applied to Direct and Inverse Non-Fourier Heat Conduction Problems

The paper presents a new approximate method of solving non-Fourier heat conduction problems. The approach described here is suitable for solving both direct and inverse problems. The way of generating Trefftz functions for non-Fourier heat conduction equation has been shown. Obtained functions have been used for solving direct and boundary inverse problems (identification of boundary condition). As a rule, inverse problems are ill-posed. Therefore, each method of solving these problems has to be checked according to disturbance of the input data. Presented examples confirm high usability of the presented approach for solving direct and inverse non-Fourier heat conduction problems.