Finite Difference-Peridynamic Differential Operator for Solving Transient Heat Conduction Problems

Transient heat conduction problems widely exist in engineering.In previous work on the peridynamic differential operator(PDDO)method for solving such problems,both time and spatial derivatives were discretized using the PDDO method,resulting in increased complexity and programming difficulty.In this work,the forward difference formula,the backward difference formula,and the centered difference formula are used to discretize the time derivative,while the PDDO method is used to discretize the spatial derivative.Three new schemes for solving transient heat conduction equations have been developed,namely,the forward-in-time and PDDO in space(FT-PDDO)scheme,the backward-in-time and PDDO in space(BT-PDDO)scheme,and the central-in-time and PDDO in space(CT-PDDO)scheme.The stability and convergence of these schemes are analyzed using the Fourier method and Taylor’s theorem.Results show that the FT-PDDO scheme is conditionally stable,whereas the BT-PDDO and CT-PDDO schemes are unconditionally stable.The stability conditions for the FT-PDDO scheme are less stringent than those of the explicit finite element method and explicit finite difference method.The convergence rate in space for these three methods is two.These constructed schemes are applied to solve one-dimensional and two-dimensional transient heat conduction problems.The accuracy and validity of the schemes are verified by comparison with analytical solutions.

Galerkin-based quasi-smooth manifold element(QSME)method for anisotropic heat conduction problems in composites with complex geometry

The accurate and efficient analysis of anisotropic heat conduction problems in complex composites is crucial for structural design and performance evaluation. Traditional numerical methods, such as the finite element method(FEM), often face a trade-off between calculation accuracy and efficiency. In this paper, we propose a quasi-smooth manifold element(QSME) method to address this challenge, and provide the accurate and efficient analysis of two-dimensional(2D) anisotropic heat conduction problems in composites with complex geometry. The QSME approach achieves high calculation precision by a high-order local approximation that ensures the first-order derivative continuity.The results demonstrate that the QSME method is robust and stable, offering both high accuracy and efficiency in the heat conduction analysis. With the same degrees of freedom(DOFs), the QSME method can achieve at least an order of magnitude higher calculation accuracy than the traditional FEM. Additionally, under the same level of calculation error, the QSME method requires 10 times fewer DOFs than the traditional FEM. The versatility of the proposed QSME method extends beyond anisotropic heat conduction problems in complex composites. The proposed QSME method can also be applied to other problems, including fluid flows, mechanical analyses, and other multi-field coupled problems, providing accurate and efficient numerical simulations.

Variational Approach to 2D and 3D Heat Conduction Modeling

The paper proposes an approximate solution to the classical (parabolic) multidimensional 2D and 3D heat conduction equation for a 5 × 5 cm aluminium plate and a 5 × 5 × 5 cm aluminum cube. An approximate solution of the generalized (hyperbolic) 2D and 3D equation for the considered plate and cube is also proposed. Approximate solutions were obtained by applying calculus of variations and Euler-Lagrange equations. In order to verify the correctness of the proposed approximate solutions, they were compared with the exact solutions of parabolic and hyperbolic equations. The paper also presents the research on the influence of time parameters τ as well as the relaxation times τ ∗ to the variation of the profile of the temperature field for the considered aluminum plate and cube.

Variational Approach to Heat Conduction Modeling

It is known that Fourier’s heat equation, which is parabolic, implies an infinite velocity propagation, or, in other words, that the mechanism of heat conduction is established instantaneously under all conditions. This is unacceptable on physical grounds in spite of the fact that Fourier’s law agrees well with experiment. However, discrepancies are likely to occur when extremely short distances or extremely short time intervals are considered, as they must in some modern problems of aero-thermodynamics. Cattaneo and independently Vernotte proved that such process can be described by Heaviside’s telegraph equation. This paper shows that this fact can be derived using calculus of variations, by application of the Euler-Lagrange equation. So, we proved that the equation of heat conduction with finite velocity propagation of the thermal disturbance can be obtained as a solution to one variational problem.

Sterilization Effect of Cooking Process for Guilin Rice Noodles Based on Heat Conduction Model

Guilin rice noodles, a unique cuisine from Guilin, Guangxi, is renowned both domestically and internationally as one of the top ten “Guilin Classics”. Utilizing a heat conduction model, this study explores the effectiveness of the cooking process in sterilizing Guilin rice noodles before consumption. The model assumes that a large pot is filled with boiling water which is maintained at a constant high temperature heat resource through continuous gentle heating. And the room temperature is set as the initial temperature for the preheating process and the final temperature for the cooling process. The objective is to assess whether the cooking process achieves satisfactory sterilization results. The temperature distribution function of rice noodle with time is analytically obtained using the separation of variables method in the three-dimensional cylindrical coordinate system. Meanwhile, the thermal diffusion coefficient of Guilin rice noodles is obtained in terms of Riedel’ theory. By analyzing the elimination characteristics of Pseudomonas cocovenenans subsp. farinofermentans, this study obtains the optimal time required for effective sterilization at the core of Guilin rice noodles. The results show that the potential Pseudomonas cocovenenans subsp. farinofermentans will be completely eliminated through continuously preheating more than 31 seconds during the cooking process before consumption. This study provides a valuable reference of food safety standards in the cooking process of Guilin rice noodles, particularly in ensuring the complete inactivation of potentially harmful strains such as Pseudomonas cocovenenans subsp. farinofermentans.

Analytical solution for the time-fractional heat conduction equation in spherical coordinate system by the method of variable separation

In this paper,using the fractional Fourier law,we obtain the fractional heat conduction equation with a time-fractional derivative in the spherical coordinate system.The method of variable separation is used to solve the timefractional heat conduction equation.The Caputo fractional derivative of the order 0 〈 α≤ 1 is used.The solution is presented in terms of the Mittag-Leffler functions.Numerical results are illustrated graphically for various values of fractional derivative.

Increment-Dimensional Scaled Boundary Finite Element Method for Solving Transient Heat Conduction Problem

An increment-dimensional scaled boundary finite element method (ID-SBFEM) is developed to solve the transient temperature field.To improve the accuracy of SBFEM,the effect of high frequency factor on dynamic stiffness is considered,and the first-order continued fraction technique is used.After the derivation,the SBFE equations are obtained,and the dimensions of thermal conduction,the thermal capacity matrix and the vector of the right side term in the equations are doubled.An example is presented to illustrate the feasibility and good accuracy of the proposed method.

A phase-field study on the oxidation behavior of Ni considering heat conduction

Phase-field modeling approach has been used to study the oxidation behavior of pure Ni when considering heat conduction. In this calculation, the dependence of the coefficient of the Cahn–Hilliard equation Lc on the temperature T was considered. To this end, high-temperature oxidation experiments and phase-field modeling for pure Ni were performed in air under atmospheric pressure at 600,700, and 800？C. The oxidation rate was measured by thermogravimetry and Lc at these temperatures was determined via interactive algorithm. With the Lc-T relationship constructed, oxidation behavior of Ni when considering heat conduction was investigated. The influence of temperature boundaries on the oxidation degree, oxide film thickness, and specific weight gain were discussed. The phase-field modeling approach proposed in this study will give some highlights of the oxidation resistance analysis and cooling measures design of thermal protection materials.

Meshless analysis of three-dimensional steady-state heat conduction problems

Steady-state heat conduction problems arisen in connection with various physical and engineering problems where the functions satisfy a given partial differential equation and particular boundary conditions, have attracted much attention and research recently. These problems are independent of time and involve only space coordinates, as in Poisson＇s equation or the Laplace equation with Dirichlet, Neuman, or mixed conditions. When the problems are too complex, it is difficult to find an analytical solution, the only choice left is an approximate numerical solution. This paper deals with the numerical solution of three-dimensional steady-state heat conduction problems using the meshless reproducing kernel particle method （RKPM）. A variational method is used to obtain the discrete equations. The essential boundary conditions are enforced by the penalty method. The effectiveness of RKPM for three-dimensional steady-state heat conduction problems is investigated by two numerical examples.

Time fractional dual-phase-lag heat conduction equation

We build a fractional dual-phase-lag model and the corresponding bioheat transfer equation, which we use to interpret the experiment results for processed meat that have been explained by applying the hyperbolic conduction. Analytical solutions expressed by H-functions are obtained by using the Laplace and Fourier transforms method. The inverse fractional dual-phase-lag heat conduction problem for the simultaneous estimation of two relaxation times and orders of fractionality is solved by applying the nonlinear least-square method. The estimated model parameters are given. Finally, the measured and the calculated temperatures versus time are compared and discussed. Some numerical examples are also given and discussed.

Theoretical Analysis and Experimental Evidence of Non-Fourier Heat Conduction Behavior

This paper consists of two parts. (1) For a hollow sphere with sudden temperature changes on its inner and outer surfaces, the hyperbolic heat conduction equation is employed to describe this extreme thermal case and an analytical expression of its temperature distribution is obtained. According to the expression, the non-Fourier heat conduction behavior that will appear in the hollow sphere is studied and some qualitative conditions that will result in distinct non-Fourier behavior in the medium is ultimately attained. (2) A novel experiment to observe non-Fourier heat conduction behavior in porous material (mainly ordinary duplicating paper) heated by a microsecond laser pulse is presented. The conditions for observing distinct non-Fourier heat conduction behavior in the experimental sample agree well with the theoretical results qualitatively.

Two exact solutions of the DPL non-Fourier heat conduction equation with special conditions

This paper presents two exact explicit solutions for the three dimensional dual-phase lag （DLP） heat conduction equation, during the derivation of which the method of trial and error and the authors＇ previous experiences are utilized. To the authors＇ knowledge, most solutions of 2D or 3D DPL models available in the literature are obtained by numerical methods, and there are few exact solutions up to now. The exact solutions in this paper can be used as benchmarks to validate numerical solutions and to develop numerical schemes, grid generation methods and so forth. In addition, they are of theoretical significance since they correspond to physically possible situations. The main goal of this paper is to obtain some possible exact explicit solutions of the dual-phase lag heat conduction equation as the benchmark solutions for computational heat transfer, rather than specific solutions for some given initial and boundary conditions. Therefore, the initial and boundary conditions are indeterminate before derivation and can be deduced from the solutions afterwards. Actually, all solutions given in this paper can be easily proven by substituting them into the governing equation.

A CLASS OF TWO-LEVEL EXPLICIT DIFFERENCE SCHEMES FOR SOLVING THREE DIMENSIONAL HEAT CONDUCTION EQUATION

A class of two-level explicit difference schemes are presented for solving three-dimensional heat conduction equation. When the order of truncation error is 0(Deltat + (Deltax)(2)), the stability condition is mesh ratio r = Deltat/(Deltax)(2) = Deltat/(Deltay)(2) = Deltat/(Deltaz)(2) less than or equal to 1/2, which is better than that of all the other explicit difference schemes. And when the order of truncation error is 0((Deltat)(2) + (Deltax)(4)), the stability condition is r less than or equal to 1/6, which contains the known results.

Theoretical solution of transient heat conduction problem in one-dimensional double-layer composite medium

To make heat conduction equation embody the essence of physical phenomenon under study, dimensionless factors were introduced and the transient heat conduction equation and its boundary conditions were transformed to dimensionless forms. Then, a theoretical solution model of transient heat conduction problem in one-dimensional double-layer composite medium was built utilizing the natural eigenfunction expansion method. In order to verify the validity of the model, the results of the above theoretical solution were compared with those of finite element method. The results by the two methods are in a good agreement. The maximum errors by the two methods appear when τ（τ is nondimensional time） equals 0.1 near the boundaries of ζ =1 （ζ is nondimensional space coordinate） and ζ =4. As τ increases, the error decreases gradually, and when τ =5 the results of both solutions have almost no change with the variation of coordinate 4.

A Novel Spacetime Collocation Meshless Method for Solving Two- Dimensional Backward Heat Conduction Problems

In this article,a meshless method using the spacetime collocation for solving the two-dimensional backward heat conduction problem(BHCP)is proposed.The spacetime collocation meshless method(SCMM)is to derive the general solutions as the basis functions for the two-dimensional transient heat equation using the separation of variables.Numerical solutions of the heat conduction problem are expressed as a series using the addition theorem.Because the basis functions are the general solutions of the governing equation,the boundary points may be collocated on the spacetime boundary of the domain.The proposed method is verified by conducting several heat conduction problems.We also carry out numerical applications to compare the SCMM with other meshless methods.The results show that the SCMM is accurate and efficient.Furthermore,it is found that the recovered boundary data on inaccessible boundary can be obtained with high accuracy even though the over specified data are provided only at a 1/6 portion of the spacetime boundary.

Model of Fractional Heat Conduction in a Thermoelastic Thin Slim Strip under Thermal Shock and Temperature-Dependent Thermal Conductivity

The present paper paper,we estimate the theory of thermoelasticity a thin slim strip under the variable thermal conductivity in the fractional-order form is solved.Thermal stress theory considering the equation of heat conduction based on the time-fractional derivative of Caputo of order
is applied to obtain a solution.We assumed that the strip surface is to be free from traction and impacted by a thermal shock.The transform of Laplace(LT)and numerical inversion techniques of Laplace were considered for solving the governing basic equations.The inverse of the LT was applied in a numerical manner considering the Fourier expansion technique.The numerical results for the physical variables were calculated numerically and displayed via graphs.The parameter of fractional order effect and variation of thermal conductivity on the displacement,stress,and temperature were investigated and compared with the results of previous studies.The results indicated the strong effect of the external parameters,especially the timefractional derivative parameter on a thermoelastic thin slim strip phenomenon.

Subdivision Surface-Based Isogeometric Boundary Element Method for Steady Heat Conduction Problems with Variable Coefficient

The present work couples isogeometric analysis(IGA)and boundary element methods(BEM)for three dimensional steady heat conduction problems with variable coefficients.The Computer-Aided Design(CAD)geometries are built by subdivision surfaces,and meantime the basis functions of subdivision surfaces are employed to discretize the boundary integral equations for heat conduction analysis.Moreover,the radial integration method is adopted to transform the additional domain integrals caused by variable coefficients to the boundary integrals.Several numerical examples are provided to demonstrate the correctness and advantages of the proposed algorithm in the integration of CAD and numerical analysis.

Quasi-Reversibility Regularization Method for Solving a Backward Heat Conduction Problem

Non-standard backward heat conduction problem is ill-posed in the sense that the solution(if it exists) does not depend continuously on the data. In this paper, we propose a regularization strategy-quasi-reversibility method to analysis the stability of the problem. Meanwhile, we investigate the roles of regularization parameter in this method. Numerical result show that our algorithm is effective and stable.

Temperature waves in chemical reaction-diffusion-heat conduction systems with two ends respectively subject to Dirichlet and no-flux conditions

Taking the Lindemann model as a sample system in which there exist chemical reactions, diffusion and heat conduction, we found the theoretical framework of linear stability analysis for a unidimensional nonhomogeneous two-variable system with one end subject to Dirichlet conditions, while the other end no-flux conditions. Furthermore, the conditions for the emergence of temperature waves are found out by the linear stability analysis and verified by a diagram for successive steps of evolution of spatial profile of temperature during a period that is plotted by numerical simulations on a computer. Without doubt, these results are in favor of the heat balance in chemical reactor designs.

Inverse Heat Conduction and Data-Type Issues for Advanced Diagnostic Analysis

It is well-known that the local heating/cooling rate implicitly drives the quality and stability of the projection associated with an inverse heat conduction analysis. However, contemporary studies involving inverse heat conduction for heat treatment processes and defense related applications normally use either temperature or strain data. An illustration is presented indicating the merit of using a device capable of either measuring or deducing the heating/cooling rate. Armed with this knowledge certainly suggests that the direct measurement or deduction of the heating/cooling rate can provide a means for developing an alternative and potentially more accurate projection than based on temperature data. A brief theoretical description is offered illustrating that a first-order thermocouple compensation model used in conjunction with an inverse analysis produces a representative prediction of the heating/cooling rate. The presented analysis also offers additional insight into the choice of optimal time-series truncation used by quasi-analytic methods. Using the compensation model, the heating/cooling rate of the desired surface is obtained using the "best" interpretation of the collected thermocouple data. A mathematical formalism is developed and some representative results are presented. Finally, a new mathematical discovery that allows for the time-local temperature errors to be estimated with a high degree of accuracy is briefly elucidated.