Solving the Temperature Distribution Field in Nonlinear Heat Conduction Problems Using the Hopfield Neural Network

This article employs the continuous-time analog Hopfield neural network (CHNN) to compute the temperature distribution in nonlinear heat conduction problems. The relationship between the CHNN synaptic connection weights and the governing equations of the nonlinear heat conduction problems is established and a corresponding network connectivity circuit design scheme proposed. The CHNN algorithm is used to solve the heat equation for conduction problems with a power-law nonlinearity. The results confirm that the proposed CHNN scheme provides an accurate means of solving the transient temperature distributions of nonlinear heat conduction problems on a real-time basis.     

Transient heat flux function estimation utilizing the variable metric method

A new method is presented to solve inverse heat conduction problems (IHCP's). The method belongs to the whole domain type of IHCP and utilizes the variable metric method (VMM) instead of the conjugate gradient method (CGM) in order to minimize the function of sum of square errors. Appropriate formulations for sensitivity coefficients, objective function and its gradient are set up in a manner suitable for computer programming of VMM. Numerically simulated data are utilized to assess the effectiveness of the method in comparison with the conjugate gradient method in estimation of space and time varying heat flux. Results indicate that the presented method is quite faster and more accurate than the conjugate gradient method in estimation of unknowns in the whole domain inverse problems.     

NONLINEAR TRANSIENT HEAT CONDUCTION ANALYSIS WITH PRECISE TIME INTEGRATION METHOD

The precise time integration (PTI) method is introduced and then its extension to the transient heat conduction problem is presented. The symmetry of the matrix exponential of PTI in heat conduction is proved and used in subdomain integration to reduce computational expense a great deal. For nonlinear heat conduction, the predictor-corrector algorithm is employed to solve the nonlinear equations. Numerical examples validate the method.     

Structured Multiblock Grid Solution of Two-Dimensional Transient Inverse Heat Conduction Problems in Cartesian Coordinate System

ABSTRACT

In this study a structured multiblock grid is used to solve two-dimensional transient inverse heat conduction problems. The multiblock method is implemented for geometric decomposition of the physical domain into regions with blocked interfaces. The finite-element method is employed for direct solution of the transient heat conduction equation in a Cartesian coordinate system. Inverse algorithms used in this research are iterative Levenberg-Marquardt and adjoint conjugate gradient techniques for parameter and function estimations. The measured transient temperature data needed in the inverse solution are given by exact or noisy data. Simultaneous estimation of unknown linear/nonlinear time-varying strengths of two heat sources in two joined surfaces with equal and different heights is obtained for the solution of the inverse problems, and the results of the present study for unknown heat source functions are compared to those of exact functions. This study is an attempt to challenge the goal of combining a multiblock technique with inverse analysis methods. In fact, the structured multiblock grid is capable of providing accurate solutions of inverse heat conduction problems (IHCPs) in industrial configurations, including composite structures. In addition, the multiblock IHCP solver is suitable to estimate unknown parameters and functions in these structures.     

Heterogeneous Heat Conduction Problems by an Improved Element-Free Galerkin Method

In this article an improved element-free Galerkin method is proposed to solve heat conduction problems with heterogeneous media. Because the method almost possesses interpolation property, the implementation of essential boundary condition is as simple as that in the finite-element method. In order to validate the proposed method, several heat conduction problems with different degrees of heterogeneity are presented. In these test problems, we focus on the influence of nodal distribution to the proposed method for heat conduction problems with heterogeneous media. It is shown that, for different degrees of heterogeneity, regardless of matter whether the node is located on the interface, accurate solutions can be obtained by the proposed method for heterogeneous heat conduction problems without a source term.     

Applying the Hopfield Neural Network to the Solution of the Temperature Distribution Field in Heat Conduction Problems

This article employs the continuous-time analog Hopfield neural network (CHNN) to compute the temperature distribution in one- and two-dimensional transient heat conduction problems. The relationship between the CHNN synaptic connection weights and the governing equations of the problems is established and a corresponding network connectivity circuit design scheme proposed. The CHNN algorithm is initially applied to the solution of conventional problems and is then used to solve more complicated problems involving time-varying heat flux profiles. The results confirm that the CHNN scheme provides an accurate means of solving the transient temperature distributions of heat conduction problems on a real-time basis.     

Integral transform solutions of diffusion problems with nonlinear equation coefficients

A hybrid numerical-analytical procedure is described for the accurate and reliable solution of nonlinear diffusion problems due to potential dependent equation coefficients. A sufficiently general formulation of a transient multidimensional problem is first considered, and formal solutions provided in terms of the related transformed potentials, obtained from the numerical solution of a denumerable system of coupled nonlinear ordinary differential equations. An application related to heat conduction with temperature dependent thermal conductivity is then more closely studied, and numerical results presented to illustrate convergence behavior for increasing truncation order of the associated infinite O.D.E. system.     

Estimation of the time-dependent heat flux using the temperature distribution at a point by conjugate gradient method

In this paper, the conjugate gradient method coupled with adjoint problem is used in order to solve the inverse heat conduction problem and estimation of the time-dependent heat flux using the temperature distribution at a point. Also, the effects of noisy data and position of measured temperature on final solution are studied. The numerical solution of the governing equations is obtained by employing a finite-difference technique. For solving this problem the general coordinate method is used. We solve the inverse heat conduction problem of estimating the transient heat flux, applied on part of the boundary of an irregular region. The irregular region in the physical domain (r,z) is transformed into a rectangle in the computational domain (ξ,η). The present formulation is general and can be applied to the solution of boundary inverse heat conduction problems over any region that can be mapped into a rectangle. The obtained results for few selected examples show the good accuracy of the presented method. Also the solutions have good stability even if the input data includes noise and that the results are nearly independent of sensor position.     

NUMERICAL CALCULATIONS FOR COMBINED CONDUCTION AND RADIATION TRANSIENT HEAT TRANSFER IN A SEMITRANSPARENT MEDIUM

A numerical computer code was developed for calculating the combined conduction and radiation transient heat transfer in cylindrical, semitransparent materials that have temperature-dependent thermal properties. The radiative component is combined with the equation of conduction heat transfer by adding it as a heat source. The finite element method (FEM) was used for calculating the radiative component and for solving the temperature field in the medium. Very good agreement was observed between results obtained by using our code and those that exist in the literature for several steady-state cases. The advantage of the code is due to the fact that it incorporates temperature-dependent properties; thus it leads to more realistic and accurate results. The code was applied to calculate the cooling path of a large cylindrical sapphire boule while using varying, transient, temperature-dependent, combined heat transfer coefficients.     

Peridynamic simulation of transient heat conduction problems in functionally gradient materials with cracks

In this article, a modified state-based peridynamic (PD) method is proposed to solve transient heat conduction problems in functionally gradient materials (FGMs) with extending insulated cracks. The PD formulation of transient heat conduction has been derived by using the time integration through the forward difference technique. Numerical simulations have been performed to verify the accuracy and effectiveness of the proposed method. The analytical solution and the finite element method results are used for comparison. In this work, the material properties of functionally graded materials are assumed to vary exponentially in z-direction. Our PD results show good agreement with analytical solutions and results from the finite element method. Hence the proposed PD method is suitable to deal with the transient heat conduction problem in FGMs with extending insulated cracks.     

A simple direct estimation of temperature-dependent thermal conductivity with kirchhoff transformation

A direct method is proposed to estimate the temperature-dependent thermal conductivity without internal measurements. In the proposed method, the steady-state nonlinear heat conduction equation is transformed into the Laplace equation via the Kirchhoff transformation. The thermal conductivity is modeled as a linear combination of known functions with unknown coefficients, which are directly determined from the imposed heat flux and measured temperatures at the boundary. Several inverse heat conduction problems are successfully introduced to confirm the validity of the proposed method.     

Total heat transfer coefficients for canned foods during sterilization

An analytical mathematical model for determining the total heat transfer coefficient of a cylindrically shaped canned food subjected to sterilization was developed. There is a need to determine these coefficients in a simple and accurate form for process heat transfer analysis and energy optimization. In the mathematical modelling, a new technique for heat sterilization conduction problem was used by considering the boundary condition of the third kind in transient heat conduction. The temperature data at the centres of the cylindrically shaped cans were obtained in the experimental investigation at the medium temperatures of 115 and 121°C and were used to determine the total heat transfer coefficients of the individual canned products. The total heat transfer coefficient for an individual canned product increased with increasing medium temperature. The results of this study shows that the present analytical model is a simple tool for determining the total heat transfer coefficients for the individual canned products.     

A regularized Fourier sine series solution of a 3D backward heat conduction problem with extremal long time span

Abstract

In the article, we solve an extremal long time span backward heat conduction problem (BHCP) of a 3D nonhomogeneous heat conduction equation with nonhomogeneous boundary conditions in a cuboid. We first derive a time-dependent 3D homogenization function, such that in terms of the new variable by a variable transformation we can find the expansion coefficients in a closed form by using the Fourier sine series method. After a simple regularization technique, a stable analytic series solution of the 3D BHCP is available. We also develop a regularized Fourier sine series solution of temperature in the whole space-time domain. Numerical tests for the BHCPs in a large space-time domain reveal that the present method is very accurate to recover the initial temperature and the whole solution.     

Two-Dimensional Inverse Heat Transfer Analysis of Functionally Graded Materials in Estimating Time-Dependent Surface Heat Flux

Two-dimensional transient inverse heat conduction problem (IHCP) of functionally graded materials (FGMs) is studied herein. A combination of the finite element (FE) and differential quadrature (DQ) methods as a simple, accurate, and efficient numerical method for FGMs transient heat transfer analysis is employed for solving the direct problem. In order to estimate the unknown boundary heat flux in solving the inverse problem, conjugate gradient method (CGM) in conjunction with adjoint problem is used. The results obtained show good accuracy for the estimation of boundary heat fluxes. The effects of measurement errors on the inverse solutions are also discussed.     

A finite element analysis for inverse heat conduction problems

This paper presents an efficient inverse analysis technique based on a sensitivity coefficient algorithm to estimate the unknown boundary conditions of multidimensional steady and transient heat conduction problems. Sensitivity coefficients were used to represent the temperature response of a system under unit loading conditions. The proposed method, coupled with the sensitivity analysis in the finite element formulation, is capable of estimating both the unknown temperature and heat flux on the surface provided that temperature data are given at discrete points in the interior of a solid body. Inverse heat conduction problems are referred to as ill-posed because minor inaccuracy or error in temperature measurements cause a drastic effect on the predicted surface temperature and heat flux. To verify the accuracy and validity of the new method, two-dimensional steady and transient problems are considered. Their surface temperature and heat flux are evaluated. From a comparison with the exact solution, the effects of measurement accuracy, number and location of measuring points, a time step, and regularization terms are discussed. © 1998 Scripta Technica. Heat Trans Jpn Res, 26(6): 345–359, 1997     

Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body

An analytic approach based on the homotopy analysis method is proposed to solve a nonlinear model of combined convective and radiative cooling of a spherical body. An explicit series solution is given, which agrees well with the exact or numerical solutions. Our series solutions indicate that, for the nonlinear model of combined convective and radiative cooling of a spherical body, the temperature on the surface of the body decays more quickly for larger values of the Biot number Bi and/or the radiation–conduction parameter N_rc. Different from traditional analytic techniques based on eigenfunctions and eigenvalues for linear problems, our approach is independent of the concepts of eigenfunctions and eigenvalues, and besides is valid for nonlinear problems in general. This analytic method provides us with a new way to obtain series solutions of unsteady nonlinear heat conduction problems, which are valid for all dimensionless times 0 ⩽ τ < +∞.     

The BEM for point heat source estimation: application to multiple static sources and moving sources

This paper deals with an inverse problem, which consists of the identification of point heat sources in a homogeneous solid in transient heat conduction. The location and strength of the line heat sources are both unknown. For a single source we examine the case of a source which moves in the system during the experiment. The two-dimensional and three-dimensional linear heat conduction problems are considered here. The identification procedure is based on a boundary integral formulation using transient fundamental solutions. The discretized problem is non-linear if the location of the line heat sources is unknown. In order to solve the problem we use an iterative procedure to minimize a quadratic norm. The proposed numerical approach is applied to experimental 2D examples using measurements provided by an infrared scanner for surface temperatures and heat fluxes. A numerical example is presented for the 3D application.     

Computational design of metadevices for heat flux manipulation considering the transient regime

The present work introduces the optimization-based approach for the design of metadevices to manipulate the heat flux in transient regime. It consists of solving a continuous, nonlinear, constrained, large-scale optimization problem where the objective function (to be minimized) is the error in accomplishing a given heat flux manipulation task along a transient heat conduction process. The response of the metadevice is modeled by using the finite element method, and its design is characterized by a set of parameters defining the material at all the finite elements in the device. These parameters are the design variables of the optimization problem, being chosen from an admissible design set in order to guarantee the feasibility of the optimal solution. As an example, this optimization-based approach is applied to the design of a heat flux shielding metadevice. Compared to a metadevice designed under the classical thermodynamics transformation approach and intuition, the current device performs the shielding task with considerably higher success. In order to highlight the versatility of the proposed optimization-based design method, this approach is also applied to the design of metadevices to satisfy multiple different simultaneous tasks, particularly shielding and cloaking.     

Firefly algorithm combined with Newton method to identify boundary conditions for transient heat conduction problems

Firefly algorithm combined with Newton method (FA–NM) is proposed for identifying time-dependent boundary conditions of two-dimensional transient heat conduction problems with a heat source. The dual reciprocity boundary element method (DRBEM) is applied to solve the direct problem. The improved firefly algorithm has not only a good global search ability, but also a good local search ability. FA–NM can acquire accurate results with much less iterations. Furthermore, different measurement points and noises are also considered. A small number of measurement points cannot obtain precise results. With the increase in the measurement points to a certain value, the results are more accurate. When the measurement noises are not very large, FA–NM can get desirable results. With the decrease in measurement noises, the results are in better agreement with the exact solutions. Numerical results also indicate that the more front the time substep is, the more accurate the estimated boundary conditions are.