A solution to the two‐dimensional inverse heat conduction problem

A simple method is developed in this paper to solve two‐dimensional nonlinear steady inverse heat conduction problems. The unknown boundary conditions can be numerically obtained by using the iteration and modification method. The effect of measurement errors of the wall temperature on the algorithm is numerically tested. The results prove that this method has the advantages of fast convergence, high precision, and good stability. The method is successfully applied to estimate the convective heat transfer coefficient in the case of a fluid flowing in an electrically heated helically coiled tube. © 2000 Scripta Technica, Heat Trans Asian Res, 29(2): 113–119, 2000     

Analytical approach with Laplace transform to the inverse problem of one‐dimensional heat conduction transfer: Application to second and third boundary conditions

An analytical method using Laplace transformation has been developed for one‐dimensional heat conduction. This method succeeded in explicitly deriving the analytical solution by which the surface temperature for the first kind of boundary condition can be well predicted. The analytical solutions for the surface temperature and heat flux are applied to the second and third of the boundary conditions. These solutions are also found to estimate the corresponding surface conditions with a high degree of accuracy when the surface conditions smoothly change. On the other hand, when these conditions erratically change such as the first derivative of temperature with time, the accuracy of the estimation becomes slightly less than that for a smooth condition. This trend in the estimation is similar irrespective of any kind of boundary condition. © 2002 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(1): 29–41, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10069     

Analytical method of two‐dimensional inverse heat conduction problem using Laplace transformation: Effect of number of measurement points

An analytical method has been developed for the inverse problem of two‐dimensional heat conduction using the Laplace transform technique. The inverse problem is solved for only two unknown surface conditions and the other surfaces are insulated in a finite rectangular body. In actual temperature measurement, the number of points in a solid is usually limited so that the number of temperature measurements required to approximate the temperature change in the solid becomes too small to obtain an approximate function using a half polynomial power series of time and the Fourier series of the eigenfunction. In order to compensate for this lack of measurement points, the third‐order Spline method is recommended for interpolating unknown temperatures at locations between measurement points. Eight points are recommended as the minimum number of temperature measurement points. The calculated results for a number of representative cases indicate that the surface temperature and the surface heat flux can be predicted well, as revealed by comparison with the given surface condition. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(7): 618–629, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10116     

Application of inverse solution in two‐dimensional heat conduction problem using Laplace transformation

An inverse solution has been explicitly derived for two‐dimensional heat conduction in cylindrical coordinates using the Laplace transformation. The applicability of the inverse solution is checked using the numerical temperatures with a normal random error calculated from the corresponding direct solution. For a gradual temperature change in a solid, the surface heat flux and temperature can be satisfactorily predicted, while for a rapid change in the temperature this method needs the help of a time partition method, in which the entire measurement time is split into several partitions. The solution with the time partitions is found to make an improvement in the prediction of the surface heat flux and temperature. It is found that the solution can be applied to experimental data, leading to good prediction. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(7): 602–617, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10115     

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     

A Novel Variational Formulation of Inverse Problem of Heat Conduction with Free Boundary on an Image

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Improvement of inverse heat conduction solution using Laplace transformation: Method of partial division of time

Any solution for an inverse heat conduction problem makes the estimation of surface temperature and surface heat flux worsen in the case where these values behave like a triangular shape change with time. In order to compensate for this defect, Monde and colleagues, who succeeded in obtaining analytical inverse solutions using the Laplace transform technique, introduce a new idea where these changes over the entire measurement time can be split into several parts depending on the behavior. Therefore, an approximate equation to trace the measured temperature change can be derived, resulting in good estimation of surface temperature and surface heat flux even in the case of the triangular shape change and sharp change. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(7): 630–638, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10117     

一维非稳态导热过程(火用)传递的规律及计算

对平壁在导热第三类边界下的一维非稳态导热过程中的Yong传递进行计算，并将导温系数在定值与变值下的计算结果加以比较，以非稳态条件下Yong传递的时空分布规律进行分析。结果表明，Yong流密度随时间按指数规律递减，而Yong传递数却按指数曲线递增。在密度和比热近似不变的条件下，导温系数对温度场分布、Yong传递系数及Yong流密度有决定性的影响。     

New Type of Basic Functions of FEM in Application to Solution of Inverse Heat Conduction Problem

The work presents the application of heat polynomials for solving an inverse problem. The heat polynomials form the Treffetz Method for non-stationary heat conduction problem. They have been used as base functions in Finite Element Method. Application of heat polynomials permits to reduce the order of numerical integration as compared to the classical Finite Element Method with formulation of the matrix of system of equations.     

An analytical solution for two-dimensional inverse heat conduction problems using Laplace transform

An analytical method has been developed for two-dimensional inverse heat conduction problems by using the Laplace transform technique. The inverse solutions are obtained under two simple boundary conditions in a finite rectangular body, with one and two unknowns, respectively. The method first approximates the temperature changes measured in the body with a half polynomial power series of time and Fourier series of eigenfunction. The expressions for the surface temperature and heat flux are explicitly obtained in a form of power series of time and Fourier series. The verifications for two representative testing cases have shown that the predicted surface temperature distribution is in good agreement with the prescribed surface condition, as well as the surface heat flux.     

The determination temperature-dependent thermal conductivity as inverse steady heat conduction problem

The paper deals with the non-iterative inverse determination of the temperature-dependent thermal conductivity in 2-D steady-state heat conduction problem. The thermal conductivity is modeled as a polynomial function of temperature with the unknown coefficients. The identification of the thermal conductivity is obtained by using the boundary data and additionally from the knowledge of temperature inside the domain. The method of fundamental solutions is used to solve the 2-D heat conduction problem. The golden section search is used to find the optimal place for pseudo-boundary on which are placed the singularities in the frame of method of fundamental solutions.     

Exact variational principle for 3-D unsteady heat conduction with second sound

The exact variational formulation of the extended unsteady heat conduction equation with finite propagationspeed(the 2nd sound speed)of hyperbolic type is derived herein via a systematic and natural way.Moreover,theboundary- and the physically acceptable initial-value conditions are accommodated in the variational principle bya novel method suggested just recently.In this way a perfect justification of the variational theory of transient heatconduction and a rigorous theoretical basis for the finite element analysis of heat conduction are provided.     

The adjoint-solution technique for the calculation of the thermal properties of layers in multilayer walls under transient heat conduction

The thermal properties of the layers of a wall, whether or not exposed to solar radiation, are calculated provided that the boundary conditions and some values of the transient temperature field within the wall are known. The developed procedure is based on the adjoint-solution technique and is applicable both to walls in operation and to the design of walls that are required to meet certain temperature specifications. In the former case, temperature measurements are needed. Theoretical and experimental tests have proved the accuracy of the method. Applications may be found in energy management and thermal storage in buildings, in the improvement of passive systems and in the design of multilayer slabs forming parts of heat-transfer equipment. © 1997 by John Wiley & Sons, Ltd.     

Solution of inverse heat conduction problem using the lattice Boltzmann method

In this paper the D2Q9 lattice Boltzmann method (LBM) was utilized for the solution of a two-dimensional inverse heat conduction (IHCP) problem. The accuracy of the LBM results was validated against those obtained from prevalent numerical methods using a common benchmark problem. The conjugate gradient method was used in order to estimate the heat flux test case. A complete error analysis was performed. As the LBM is attuned to parallel computations, its use is recommended in conjugation with IHCP solution methods.     

Solution to shape identification problem of unsteady heat‐conduction fields

This paper presents a numerical analysis method for shape determination problems of unsteady heat‐conduction fields in which time histories of temperature distributions on prescribed subboundaries or time histories of gradient distributions of temperature in prescribed subdomains have prescribed distributions. The square error integrals between the actual distributions and the prescribed distributions on the prescribed subboundaries or in the prescribed subdomains during the specified period of time are used as objective functionals. Reshaping is accomplished by the traction method that was proposed as a solution to shape optimization problems of domains in which boundary value problems are defined. The shape gradient functions of these shape determination problems are derived theoretically using the Lagrange multiplier method and the formulation of material derivative. The time histories of temperature distributions are evaluated using the finite‐element method for a space integral and the Crank–Nicolson method for a time integral. Numerical analyses of nozzle and coolant flow passage in a wing are demonstrated to confirm the validity of this method. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(3): 212–226, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10086     

Calculation Error of Numerical Solution for a Boundary—Value Inverse Heat Conduction Problem

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Solution of the Stationary 2D Inverse Heat Conduction Problem by Treffetz Method

The paper presents analysis of a solution of Laplace equation with the use of FEM harmonic basic functions. The essence of the problem is aimed at presenting an approximate solution based on possibly large finite element. Introduction of harmonic functions allows to reduce the order of numerical integration as compared to a classical Finite Element Method. Numerical calculations conform good efficiency of the use of basic harmonic functions for resolving direct and inverse problems of stationary heat conduction.Further part of the paper shows the use of basic harmonic functions for solving Poisson's equation and for drawing up a complete system of biharmonic and polyharmonic basic functions     

An experimental and theoretical study of the nonlinear heat conduction in dry porous media

Heat transfer in porous media is important in various engineering fields, including contaminated soil incineration. Most heat transfer models are theoretical in nature. Consequently, this study was undertaken to perform both theoretical and experimental studies of heat transfer in two different sand matrices. A mathematical model based on Fourier's law of heat conduction for a one‐dimensional system with the variable thermal conductivity was developed. The experimental part included heating sand samples placed in a small reactor within an infrared furnace. The transient temperature profiles of the sand layers were monitored by thermocouples. The bulk thermal conductivity was estimated to be linearly proportional to the temperature. The temperature profiles predicted by the model of heat conduction with a variable bulk thermal conductivity was compared by the observed temperatures in Quartz and Sea sands matrices up to 1300 K. Copyright © 1999 John Wiley & Sons, Ltd.