Analysis and solution of the ill-posed inverse heat conduction problem

The inverse conduction problem arises when experimental measurements are taken in the interior of a body, and it is desired to calculate temperature and heat flux values on the surface. The problem is shown to be ill-posed, as the solution exhibits unstable dependence on the given data functions. A special solution procedure is developed for the one-dimensional case which replaces the heat conduction equation with an approximating hyperbolic equation. If viewed from a new perspective, where the roles of the spatial and time variables are interchanged, then an initial value problem for the damped wave equation is obtained. Since the formulation is well-posed, both analytic and numerical solution procedures are readily available. Sample calculations confirm that this approach produces consistent, reliable results for both linear and nonlinear problems.     

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.     

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.     

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.     

Numerically solving twofold ill-posed inverse problems of heat equation by the adjoint Trefftz method

The inverse problem endowing with multiple unknown functions gradually becomes an important topic in the field of numerical heat transfer, and one fundamental problem is how to use limited minimal data to solve the inverse problem. With this in mind, in the present article we search the solution of a general inverse heat conduction problem when two boundary data on the space-time boundary are missing and recover two unknown temperature functions with the help of a few extra measurements of temperature data polluted by random noise. This twofold ill-posed inverse heat conduction problem is more difficult than the backward heat conduction problem and the sideways heat conduction problem, both with one unknown function to be recovered. Based on a stable adjoint Trefftz method, we develop a global boundary integral equation method, which together with the compatibility conditions and some measured data can be used to retrieve two unknown temperature functions. Several numerical examples demonstrate that the present method is effective and stable, even for those of strongly ill-posed ones under quite large noises.     

Variational formulation of hyperbolic heat conduction problems applying Laplace transform technique

In this paper, a non-Fourier heat conduction problem is analyzed by employing newly developed theory. Application of conventional numerical schemes leads to strong oscillations of the results around discontinuities in solution domain. To overcome this difficulty the variational formulation of the Laplace-transformed hyperbolic heat conduction equation is developed. The results were used for evaluation of parameters used in approximate transformed temperature profiles. To validate the approach the results were compared with the exact analytical solution solved at special case and with an approach previously reported in the literature. Both showed a close agreement with the proposed approach.     

Reconstruction of a space and time dependent heat source from finite measurement data

This paper deals with an inverse problem of determining a heat source function in heat conduction equations when the solution is known in a discrete point set. Being different from other ordinary inverse source problems which are often dependent on only one variable, the unknown coefficient in this paper not only depends on the space variable x, but also depends on the time t. On the basis of the optimal control framework, the inverse problem is transformed into an optimization problem. The existence and necessary condition of the minimizer for the cost functional are established. The convergence of the minimizer as the mesh parameters tend to zero is also proved. The conjugate gradient method is applied to the inverse problem and some typical numerical experiments are performed in the paper. The numerical results show that the proposed method is stable and the unknown heat source is recovered very well.     

REVERSE TIME TLM MODELING OF THERMAL PROBLEMS DESCRIBED BY THE HYPERBOLIC HEAT CONDUCTION EQUATION

A method for reverse-time transmission line matrix (TLM) modeling of thermal diffusion problems described by the hyperbolic heat conduction equation is presented. The method involves pulse scattering governed by the inverse of the usual scattering matrix. Consideration is given to the accuracy of the reverse time method by comparison with the forward scatter. The numerical behavior of the reverse-time algorithm is investigated, and related to the amplification factor of the inverse scattering matrix.     

An inverse problem in estimating the base heat flux of an annular fin based on the hyperbolic model of heat conduction

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to solve the inverse hyperbolic heat conduction problem in estimating the unknown time-dependent base heat flux of an annular fin from the knowledge of temperature measurements taken within the fin. The inverse solutions will be justified based on the numerical experiments in which two specific cases to determine the unknown base heat flux are examined. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The influence of measurement errors upon the precision of the estimated results is also investigated. Results show that an excellent estimation on the time-dependent base heat flux can be obtained for the test cases considered in this study.     

Application of the method of fundamental solutions and radial basis functions for inverse heat source problem in case of steady-state

The paper deals with the inverse determination of heat sources in steady 2-D heat conduction problem. The problem is described by Poisson equation in which the function of the right hand side is unknown. The identification of the strength of a heat source is given by using the boundary condition and a known value of temperature in chosen points placed inside the domain. For the solution of the inverse problem of identification of the heat source the method of fundamental solution with radial basis functions is proposed. The accurate results have been obtained for five test problems where the analytical solutions were available.     

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.     

Virtual boundary element method in conjunction with conjugate gradient algorithm for three-dimensional inverse heat conduction problems

In this article, the virtual boundary element method (VBEM) in conjunction with conjugate gradient algorithm (CGA) is employed to treat three-dimensional inverse problems of steady-state heat conduction. On the one hand, the VBEM may face numerical instability if a virtual boundary is improperly selected. The numerical accuracy is very sensitive to the choice of the virtual boundary. The condition number of the system matrix is high for the larger distance between the physical boundary and the fictitious boundary. On the contrary, it is difficult to remove the source singularity. On the other hand, the VBEM will encounter ill-conditioned problem when this method is used to analyze inverse problems. This study combines the VBEM and the CGA to model three-dimensional heat conduction inverse problem. The introduction of the CGA effectively overcomes the above shortcomings, and makes the location of the virtual boundary more free. Furthermore, the CGA, as a regularization method, successfully solves the ill-conditioned equation of three-dimensional heat conduction inverse problem. Numerical examples demonstrate the validity and accuracy of the proposed method.     

Solution of inverse heat conduction problems using maximum entropy method

A solution scheme based on the maximum entropy method (MEM) for the solution of one-dimensional inverse heat conduction problem is proposed. The present work introduces MEM in order to build a robust formulation of the inverse problem. MEM finds the solution which maximizes the entropy functional under the given temperature measurements. In order to seek the most likely inverse solution, the present method converts the inverse problem to a non-linear constrained optimization problem. The constraint of the problem is the statistical consistency between the measured temperature and the estimated temperature. Successive quadratic programming (SQP) facilitates the maximum entropy estimation. The characteristic feature of the method is discussed with the sample numerical results. The presented results show considerable enhancement in the resolution of the inverse problem and bias reduction in comparison with the conventional methods.     

Boundary estimation of hyperbolic bio-heat conduction

A sequential method is proposed for estimating the boundary condition in hyperbolic bio-heat conduction. The estimated solution is deduced from a numerical approach combined with the concept of future time. The problem with inverse bio-heat conduction is the slow heat-wave propagation speed, resulting in no temperature measurements obtained. Three cases are presented to demonstrate the features and the validity of the proposed method. Comparison between the exact value and the estimated result is made to confirm the validity and accuracy of the proposed method.     

An inverse problem to estimate relaxation parameter and order of fractionality in fractional single-phase-lag heat equation

In this paper, an inverse analysis is performed for simultaneous estimation of relaxation time and order of fractionality in fractional single-phase-lag heat equation. This fractional heat conduction equation is applied on two physical problems. In inverse procedure, solutions of a previously validated linear dual-phase-lag model on the physical problems under study have been used as the measured temperatures. The inverse fractional single-phase-lag heat conduction problem is solved using the nonlinear parameter estimation technique based on the Levenberg–Marquardt method. The results of the present study show that the Levenberg–Marquardt method can be successfully applied on the inverse fractional heat transfer problem. The solution procedures employed in the present study for direct and inverse problems have greatly increased the reliability and success of parameter estimation problem. In the present study, for the first time, relaxation time and fractionality of a non-homogeneous medium (i.e. processed meat) have been determined. Also, the results of this study show that the fractional single-phase-lag model can predict the same temperature distribution as the linear dual-phase-lag model for the problem under study. This latter result enables us to consider further generalization of the dual-phase-lag model to fractional dual-phase-lag models.     

INVERSE FORCED CONVECTION PROBLEM OF SIMULTANEOUS ESTIMATION OF TWO BOUNDARY HEAT FLUXES IN IRREGULARLY SHAPED CHANNELS

This article deals with the use of the conjugate gradient method of function estimation for the simultaneous identification of two unknown boundary heat fluxes in channels with laminar flows. The irregularly shaped channel in the physical domain is transformed into a parallel plate channel in the computational domain by using an elliptic scheme of numerical grid generation. The direct problem, as well as the auxiliary problems and the gradient equations, required for the solution of the inverse problem with the conjugate gradient method are formulated in terms of generalized boundary-fitted coordinates. Therefore, the solution approach presented here can be readily applied to forced convection boundary inverse problems in channels of any shape. Direct and auxiliary problems are solved with finite volumes. The numerical solution for the direct problem is validated by comparing the results obtained here with benchmark solutions for smoothly expanding channels. Simulated temperature measurements containing random errors are used in the inverse analysis for strict cases involving functional forms with discontinuities and sharp corners for the unknown functions. The estimation of three different types of inverse problems are addressed in the paper: (i) time-dependent heat fluxes; (ii) spatially dependent heat fluxes; and (iii) time and spatially dependent heat fluxes.     

Implementation of an analytical two-dimensional inverse heat conduction technique to practical problems

Two improvements to practical implementation of a solution to the two-dimensional inverse heat conduction problem are presented. The first concept is useful for experimental data with strong or irregular fluctuations in time. The second procedure improves the spatial resolution for problems where the source of the surface heat flux distribution is moving along the surface. The method is tested against analytical solutions and data from quench cooling experiments. Both procedures are found to enhance the quality of the inverse solution results.     

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.     

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.     

A new algorithm for direct and backward problems of heat conduction equation

Direct heat conduction problem (DHCP) and backward heat conduction problem (BHCP) are numerically solved by employing a new idea of fictitious time integration method (FTIM). The DHCP needs to consider the stability of numerical integration in the sense that the solution may be divergent for a specific time stepsize and specific spatial stepsize. The BHCP is renowned as strongly ill-posed because the solution does not continuously depend on the given data. In this paper, we transform the original parabolic equation into another parabolic type evolution equation by introducing a fictitious time variable, and adding a fictitious viscous damping coefficient to enhance the stability of numerical integration of the discretized equations by employing a group preserving scheme. When 10 numerical examples are amenable, we find that the FTIM is applicable to both the DHCP and BHCP. Even under seriously noisy initial or final data, the FTIM is also robust against disturbance. More interestingly, when we use the FTIM, we do not need to use different techniques to treat DHCP and BHCP as that usually employed in the conventional numerical methods. It means that the FTIM can unifiedly approach both the DHCP and BHCP, and the gap between direct problems and inverse problems can be smeared out.