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.     

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.     

Two dimensional steady-state inverse heat conduction problems

In this work we estimate the surface temperature in two dimensional steady-state in a rectangular region by two different methods, the singular value decomposition (SVD) with boundary element method (BEM) and the least-squares approach with integral transform method (ITM). The BEM method is efficient for solving inverse heat conduction problems (IHCP) because only the boundary of the region needs to be discretized. Furthermore, both temperature and heat flux at the unknown boundary are estimated at the same time. The least-squares technique involves solving the equations constructed from the measured temperature and the exact solution. The measured data are simulated by adding random errors to the exact solution of the direct problem. The effects of random errors on the accuracy of the predictions are examined. The sensitivity coefficients are also presented to illustrate the effect of sensor location on the estimated surface conditions. Numerical experiments are given to demonstrate the accuracy of the present approaches.     

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     

An inversion approach for the inverse heat conduction problems

The inverse heat conduction problems (IHCP) analysis method provides an efficient approach for estimating the thermophysical properties of materials, the boundary conditions, or the initial conditions. Successful applications of the IHCP method depend mainly on the efficiency of the inversion algorithms. In this paper, a generalized objective functional, which has been developed using a generalized stabilizing functional and a combinational estimation that integrates the advantages of the least trimmed squares (LTS) estimation and the M-estimation, is proposed. The objective functional unifies the regularized M-estimation, the regularized least squares (LS) estimation, the regularized LTS estimation, the regularized combinational estimation of the LTS estimation and the M-estimation, and the regularized combinational estimation of the LS estimation and the M-estimation into a concise formula. The filled function method, which is coupled with the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm, is developed for searching a possible global optimal solution. Numerical simulations are implemented to evaluate the feasibility and effectiveness of the proposed algorithm. Favorable numerical performances and satisfactory results are observed, which indicates that the proposed algorithm is successful in solving the IHCP.     

A technique for uncertainty analysis for inverse heat conduction problems

A technique is presented for the uncertainty analysis of the linear Inverse Heat Conduction Problem (IHCP) of estimating heat flux from interior temperature measurements. The selected IHCP algorithm is described. The uncertainty in thermal properties and temperature measurements is considered. A propagation of variance equation is used for the uncertainty analysis. An example calculation is presented. Parameter importance factors are defined and computed for the example problem; the volumetric heat capacity is the dominant parameter and an explanation is offered. Thoughts are presented on extending the analysis to include the non-linear problem of temperature dependent properties.     

Fourier analysis of conjugate gradient method applied to inverse heat conduction problems

The convergence and regularization mechanism of the conjugate gradient algorithm applied to inverse heat conduction problems are studied within the context of a Fourier analysis, for a square enclosure subjected to an unknown time-varying heat flux on one side, and to known boundary conditions on the remaining sides. Analytic solutions are derived for the Fourier components of the unknown flux over a given time interval. The convergence rate of the algorithm is thereby shown to depend essentially on the time frequency of the data. Numerical solutions are also presented to describe in details the convergence process and solution regularization power of the conjugate gradient method, when the unknown heat flux contains many frequency components and the measurement data are noisy. It is found that an unknown time-dependent heat flux may be satisfactorily recovered using a single sensor even when the temperature field becomes two-dimensional, and that the sensor should be placed in a symmetric manner for better results.     

Intelligent fuzzy weighted input estimation method applied to inverse heat conduction problems

The innovative intelligent fuzzy weighted input estimation method which efficiently and robustly estimates the unknown time-varying heat flux in real-time is presented in this paper. The algorithm includes the Kalman Filter (KF) and the recursive least square estimator (RLSE), which is weighted by the fuzzy weighting factor proposed based on the fuzzy logic inference system. To directly synthesize the Kalman filter with the estimator, this work presents an efficient robust forgetting zone, which is capable of providing a reasonable compromise between the tracking capability and the flexibility against noises. The capability of this inverse method are demonstrated in one- and two-dimensional time-varying estimation cases and the proposed algorithm is compared by alternating between the constant and adaptive weighting factors. The results show that this method has the properties of faster convergence in the initial response, better target tracking capability and more effective noise reduction.     

A fitting algorithm for solving inverse problems of heat conduction

The paper presents an algorithm for solving inverse problems of heat transfer. The method is based on iterative solving of direct and adjoint model equations with the aim to minimize a fitting functional. An optimal choice of the step length along the descent direction is proposed. The algorithm has been used for the treatment of a steady-state problem of heat transfer in a region with holes. The temperature and the heat flux density were known on the outer boundary of the region, whereas these values on the boundaries of the holes are to be determined. The idea of the algorithm consist in solving of Neumann problems where the heat flux on the outer boundary is prescribed, whereas the heat flux on the inner boundary is guessed. The guess is being improved iteratively to minimize the mean quadratic deviation of the solution on the outer boundary from the given distribution.The results obtained show that the algorithm provides smooth, non-oscillating, and stable solutions to inverse problems of heat transfer, that is, it avoids disadvantages inherent in other computational methods for such problems. The ill-conditioning of inverse problems in the Hadamard sense is exhibited in that a very quick convergence of the fitting functional to its minimum does not imply a comparable rate of convergence of the recovered temperature on the inner boundary to the true distribution.The considered method can easily be extended to nonlinear problems.Numerical calculation has been carried out with the FE program Felics developed at the Chair of Mathematical Modelling of the Technical University of Munich.     

On the iterative regularization of inverse heat conduction problems by conjugate gradient method

The convergence and regularization properties of the conjugate gradient algorithm applied to the inverse heat conduction problem are considered for a time-dependent boundary heat flux. An analysis based on both numerical and analytical results clearly shows that the convergence process of the algorithm is strongly frequency-dependent and provides in this way a very efficient regularization mechanism against the destabilizing effect of random errors in the input data.     

Numerical solution of hyperbolic heat conduction in thin surface layers

The purpose of the present paper is to propose a new hybrid method investigating the effect of the surface curvature of a solid body on hyperbolic heat conduction. The difficulty encountered in the numerical solutions of hyperbolic heat conduction problems is the numerical oscillation in vicinity of sharp discontinuities. In the present study, we have developed a new hybrid method combined the Laplace transform, the weighting function scheme [Shong-leih Lee, Weighting function scheme and its application on multidimensional conservation equations, Int. J. Heat Mass Transfer 32 (1989) 2065–2073], and the hyperbolic shape function for solving time dependent hyperbolic heat conduction equation with a conservation term. Four different examples have been analyzed by the present method. It is found from these examples that the present method is in good agreement in the analytical solutions [Tsai-tse Kao, Non-Fourier heat conduction in thin surface layers, J. Heat Transfer 99 (May) (1977) 343–345] and does not exhibit numerical oscillations at the wave front and the surface temperature is modified by the surface curvature during the short period when the non-Fourier effect is significant. The curvature will increase or decrease the temperature of the wave front, depending on whether the surface is concave or convex.     

Estimation of two-sided boundary conditions for two-dimensional inverse heat conduction problems

A hybrid numerical algorithm of the Laplace transform technique and finite-difference method with a sequential-in-time concept and the least-squares scheme is proposed to predict the unknown surface temperature of two-sided boundary conditions for two-dimensional inverse heat conduction problems. In the present study, the functional form of the estimated surface temperatures is unknown a priori. The whole time domain is divided into several analysis sub-time intervals and then the unknown surface temperatures in each analysis interval are estimated. To enhance the accuracy and efficiency of the present method, a good comparison between the present estimations and previous results is demonstrated. The results show that good estimations on the surface temperature can be obtained from the transient temperature recordings only at a few selected locations even for the case with measurement errors. It is worth mentioning that the unknown surface temperature can be accurately estimated even though the thermocouples are located far from the estimated surface. Owing to the application of the Laplace transform technique, the unknown surface temperature distribution can be estimated from a specific time.     

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.     

A general method for the solution of inverse heat conduction problems with partially unknown system geometries

An auxiliary problem is introduced in the solution of inverse heat conduction problems with geometries not fully specified. Resolving the position of the unknown boundary subject to a Dirichlet condition leads to the solution of a nonlinear algebraic equation. Imposing Neumann or Robin conditions at the unknown boundary requires the solution of a first-order nonlinear, ordinary differential equation. The method yields accurate results for exact data, while measurement errors render the Neumann problem insoluble. The Dirichlet and Robin problems are still solvable, and for these problems, the errors in the investigated boundaries increase with their depth, a nature of the problem being investigated.     

Shape identification for inverse geometry heat conduction problems by FEM without iteration

The boundary geometry shape is identified by the finite element method (FEM) without iteration and mesh reconstruction for two-dimensional (2-D) and three-dimensional (3-D) inverse heat conduction problems. First, the direct heat conduction problem with the exact domain is solved by the FEM and the temperatures of measurement points are obtained. Then, by introducing a virtual boundary, a virtual domain is formed. By minimizing the difference between the temperatures of measurement points in the exact domain and those in the virtual domain, the temperatures of the points on the virtual boundary are calculated based on the least square error method and the Tikhonov regularization. Finally, the objective geometry shape can be estimated by the method of searching the isothermal curve or isothermal surface for 2-D or 3-D problems, respectively. In the process, no iterative calculation is needed. The proposed method has a tremendous advantage in reducing the computational time for the inverse geometry problems. Numerical examples are presented to test the validity of the proposed approach. Meanwhile, the influences of measurement noise, virtual boundary, measurement point number, and measurement point position on the boundary geometry prediction are also investigated in the examples. The solutions show that the method is accurate and efficient to identify the unknown boundary geometry configurations for 2-D and 3-D heat conduction problems.     

A robust and fast algorithm for three-dimensional transient inverse heat conduction problems

Despite numerous studies of inverse heat conduction problems (IHCP) over the last several decades, their solutions still suffer from the mathematical difficulties and the bottleneck of currently available numerical methods for large-scale problems. In this paper, we present a robust and efficient algorithm for the solution of a specific type of three-dimensional (3D) IHCP commonly involved in various engineering applications. The solution method incorporates the Tikhonov regularization for tackling the severe ill-posedness and the conjugate gradient (CG) method for solving the resulting minimization problems. A model function approach is used to significantly reduce the effort needed to find the optimal Tikhonov regularization parameter. The proposed solution method requires no a priori knowledge of the measurement noise and is much more computationally efficient than the traditional Tikhonov regularization-based inversion approaches. Thus, it can be used for the efficient solution of large-scale practical problems. Two simulation case studies of practical significance are presented to validate and assess the performance of the proposed method. Finally, the solution method is successfully applied to the reconstruction of instantaneous heat fluxes from experimentally measured temperature data.     

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.     

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.     

Particle Swarm Optimization-based algorithms for solving inverse heat conduction problems of estimating surface heat flux

An inverse analysis of estimating a time-dependent surface heat flux for a three-dimensional heat conduction problem is presented. A global optimization method known as Particle Swarm Optimization (PSO) is employed to estimate the unknown heat flux at the inner surface of a crystal tube from the knowledge of temperature measurements obtained at the external surface. Three modifications of the PSO-based algorithm, PSO with constriction factor, PSO with time-varying acceleration of the cognitive and social coefficients, and PSO with mutation are carried out to implement the optimization process of the inverse analysis. The results show that the PSO with mutation algorithm is significantly better than other PSO-based algorithms because it can overcome the drawback of trapping in the local optimum points and obtain better inverse solutions. The effects of measurement errors, number of dimensionalities, and number of generations on the inverse solutions are also investigated.     

Local RBF-DQ method for two-dimensional transient heat conduction problems

The meshless local radial basis function-based differential quadrature (RBF-DQ) method is applied on two-dimensional heat conduction for different irregular geometries. This method is the combination of differential quadrature approximation of derivatives and function approximation of radial basis function. Four different geometries with regular and irregular boundaries are considered, and numerical results are compared with those gained by finite element (FE) solution achieved by COMSOL commercial code. Outcomes prove that current technique is in very good agreement with FEM and this fact that RBF-DQ method is an accurate and flexible method in solution of heat conduction problems.