Linear Inverse Heat Conduction Problem Research Articles

Using a linear inverse heat conduction model to estimate the boundary heat flux with a material undergoing phase transformation

Modeling heat conduction during steel-cooling processes is challenging because solid–solid phase transformations can take place, introducing highly non-linear phenomena to the heat equation. Using a linear model for the inverse problem can be helpful for estimating the dissipated heat flux, especially because complex phase transformations models that use empirical parameters are not always available. However, the performance and applicability of such a simplified inverse model for a strongly non-linear process is still unclear. This study presents several numerical simulations to evaluate the accuracy and limitations of solving a linear inverse heat conduction problem (i.e. constant thermophysical properties and no internal heat source) to estimate the boundary heat flux during cooling of a material undergoing phase transformations. Preliminary simulations with stable materials but with temperature-dependent thermophysical properties showed that the linear model performs well to estimate the boundary heat flux, except when the material has a highly temperature-dependent specific heat, like pure iron near its Curie temperature. Then, we performed several simulations for 42CrMo4 steel, which undergoes phase transformations and, hence, has phase- and temperature-dependent thermophysical properties and latent heat of phase transformations as an internal heat source. The latent heat has a very small effect on the heat flux estimation for fast cooling conditions; however, it can lead to an interpretation bias in medium and slow cooling conditions due to temporary underestimates of the heat flux, reaching errors up to 100%. Even when the estimated heat fluxes seem accurate (average errors smaller than 10%), further estimates of the temperature evolution or phase transformations kinetics are inaccurate (range of uncertainties as large as 200 °C and 20%, respectively) because of the phase-dependent thermophysical properties. Hence, using a linear inverse heat conduction problem for a material undergoing phase transformations is acceptable only for fast cooling conditions ( h > 1500 Wm −2 K −1 ) and exclusively to estimate the boundary heat flux without further simulations. • Numerical simulations of a coupled heat conduction-phase transformation model. • Simulated temperature data were used in a linear inverse model to estimate the heat flux. • The inverse model performs well for AISI 304 and Ni201, which do not undergo phase transformations. • For the 42CrMo4 steel, which undergoes phase transformation, it performs well only for h > 1500 Wm − 2 K − 1 . • For lower h, the latent heat plays an important role, so the simplified model is no longer acceptable.

Automatic selection of regularization parameter in inverse heat conduction problems

This paper presents a fast, robust and automatic approach for selecting the regularization parameter in Tikhonov regularization. The methodology is based on the Generalized Cross-Validation (GCV) method, where the minimum of the GCV curve is sought via a numerical optimization algorithm. Typical convergence issues near the global minimum are addressed by changing the variables in the optimization problem in an appropriate fashion. Evidences for the effectiveness of the proposed approach are presented by performing three numerical experiments, consisting of linear and nonlinear inverse heat conduction problems. The obtained estimates are in good agreement with the reference values, showing that the obtained solutions in both problems were appropriately regularized. In this methodology, inverse problem needs not be solved with multiple regularization parameter candidates. Instead, the inverse problem is solved only once, using the optimal parameter. This feature is even more pressing in nonlinear problems, for the optimal regularization parameter may change during the iterative solution of the least squares problem, thus being adjusted at each iteration. Therefore, manual selection of the regularization parameter is not necessary, thus yielding an automatic selection methodology. Finally, as for the “robust” statement, it was made based on the numerical experiments shown on the paper, where the optimal regularization parameters for each problem vary in orders of magnitude.

Implementation of total variation regularization-based approaches in the solution of linear inverse heat conduction problems concerning the estimation of surface heat fluxes

In this work, we propose the extension of total variation regularization strategies in principle formulated to be implemented in image processing and signal analysis research fields, to the solution of one-dimensional linear inverse heat conduction problems concerning the estimation of surface heat fluxes. Three solution procedures are considered in the current study, and these include the lagged diffusivity fixed point iteration, the iteratively reweighted least-squares, and the split Bregman iteration methods. The performance of such procedures is tested using four cases, and the regularization parameter is selected via the L-curve criterion. The results are compared to the reconstruction obtained via a classical Tikhonov-like strategy. The main outcome is that the staircase effect dominated all the reconstructions. Despite it deteriorated the quality of the solutions in some specific cases, it did not prevent the appropriate fulfillment of the reconstruction task. The results achieved in this communication have been useful to demonstrate the suitability and reliability of extending total variation approaches to the solution of linear inverse heat conduction problems concerning the estimation of surface heat fluxes, as an appropriate and novel alternative to standard procedures.

Transient inverse heat conduction problem of quenching a hollow cylinder by one row of water jets

In this study, a two-dimensional linear transition inverse heat conduction problem (IHCP) was solved using the Generalized Minimal Residual Method (GMRES) in quenching process by water jets. The inverse solution method was validated by set of artificial data and solution sensitivity analysis was done on data noise level, regularization parameter, cell size, etc. An experimental study has been carried out on quenching a rotary hollow cylinder by one row of subcooled water jets. The inverse solution approach enabled prediction of surface temperature and heat flux distribution of test specimen in the quenching experiments by using measured internal specimen temperature. Three different boiling curves were defined in the quenching process of a rotary cylinder. Result obtained by the inverse solution showed clear footprint of rotation in surface temperature and heat flux on each revolution of cylinder and temperature variation damping from quenching surface toward interior of specimen.

Determination of Internal Heat Power during the Induction Heating Based on Solution of Linear Inverse Heat Conduction Problem

In this paper an inverse heat conduction problem for estimating the constant value of power of internal heat source during induction heating of paramagnetic material is developed. The experimentally measured temperature data are used as the input for the inverse heat transfer model. The problem is formulated as a problem of optimal control over an object with distributed parameters, the internal heat power is considered as the control action. The problem is reduced to a problem of mathematical programming; the special optimisation method based on the alternance properties of the sought optimum solutions is applied to estimate the internal heat power.

Comparative application of CGM and Wiener filtering techniques for the estimation of heat flux distribution

The conjugate gradient method (CGM), formulated with the adjoint problem, is adopted in this study as the functional minimization tool for the solution of the 2-D steady-state linear inverse heat conduction problem with the aim of estimating the heat flux density distribution on a given surface using the temperature distribution as input data. The robustness of the solution strategy, based on an iterative regularization scheme, is compared with another consolidated estimation methodology adopted in the literature to handle the same problem: the Wiener filtering technique. The comparison is performed by adopting a noisy test signal aimed at simulating infrared temperature maps. Using a comparative approach, the problem's particular difficulties related to the high number of unknowns and to the deficiency of the estimation techniques at the domain's geometrical boundaries are discussed.

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.

Splitting-based conjugate gradient method for a multi-dimensional linear inverse heat conduction problem

In this paper we consider a multi-dimensional inverse heat conduction problem with time-dependent coefficients in a box, which is well-known to be severely ill-posed, by a variational method. The gradient of the functional to be minimized is obtained by the aid of an adjoint problem, and the conjugate gradient method with a stopping rule is then applied to this ill-posed optimization problem. To enhance the stability and the accuracy of the numerical solution to the problem, we apply this scheme to the discretized inverse problem rather than to the continuous one. The difficulties with large dimensions of discretized problems are overcome by a splitting method which only requires the solution of easy-to-solve one-dimensional problems. The numerical results provided by our method are very good and the techniques seem to be very promising.

Criteria of optimality for sensors' location based on adjoint transformation of observation data interpolation error

AbstractCriteria of optimality for sensors' location are addressed using an interpolation error transformed by especial adjoint problems. The considered criteria correspond to the analysis error in certain Hessian‐based metrics and to the error of some forecast aspect. Both criteria are obtained using adjoint problems that provide computation without the direct use of the Hessian. For a linear inverse heat conduction problem, these criteria are compared and demonstrated promising results when compared with a criterion based on the norm of the interpolation error of observation data. Approaches to sensor set modification using either redistribution of sensors' or refinement of the sensors grid (insertion of additional sensors) are also compared. Copyright © 2009 John Wiley & Sons, Ltd.

Inverse reconstruction of boundary condition coefficients in one-dimensional transient heat conduction

This paper describes the restoration of boundary conditions in one-dimensional transient inverse heat conduction problems (IHCP). In the formulation, convective boundary conditions are represented by linear relations between the temperature and the heat flux, together with an initial condition as a function of space. The temperature inside the solution domain together with the space-dependent ambient temperature or heat transfer coefficient are found from additional boundary temperature or average boundary temperature measurements. The determination of the spacewise dependent ambient temperature is a linear IHCP, whilst the determination of the spacewise dependent heat transfer coefficient is a nonlinear IHCP. For both problems uniqueness theorems are available. Numerical results obtained using the boundary element method are presented and discussed.

Numerical solution to a multi-dimensional linear inverse heat conduction problem by a splitting-based conjugate gradient method

In this paper we consider a multi-dimensional inverse heat conduction problem with time-dependent coefficients in a box, which is well-known to be severely ill-posed, by a variational method. The gradient of the functional to be minimized is obtained by aids of an adjoint problem and the conjugate gradient method with a stopping rule is then applied to this ill-posed optimization problem. To enhance the stability and the accuracy of the numerical solution to the problem we apply this scheme to the discretized inverse problem rather than to the continuous one. The difficulties with large dimensions of discretized problems are overcome by a splitting method which only requires the solution of easy-to-solve one-dimensional problems. The numerical results provided by our method are very good and the techniques seem to be very promising.

An Application of DHW Algorithm for the Solution of Inverse Heat Conduction Problem

A damped heat wave (DHW) algorithm is applied for the temperature distribution calculation in a solution of a linear inverse heat conduction problem (IHCP). A nonlinear least squares algorithm is used for calculation of the unknown boundary heat flux history in a one-dimensional medium. The solution is based on the assumption that the temperature measurements are available, at least, at one point of the medium over the whole time domain. Sample calculations, for a comparison between exact heat sources and estimated ones, are made to confirm the validity of the proposed method. The close agreement between the exact and estimated values calculated for both exact and noisy data shows the potential of the proposed method for finding a relatively accurate heat source distribution in a one-dimensional homogeneous finite medium. The proposed method of solving inverse heat conduction problems is very simple and easy to implement.

Solving inverse heat conduction problem with discrete wavelet transform

We consider as a test case the linear one-dimensional inverse heat conduction problem. Usual regularization methods are equivalent to filtering out high-frequency components, in a uniform way on the whole stretch of data. This sets a limit to the trade-off that can be obtained between good resolution and efficient noise reduction. To overcome this limit, we propose the use of wavelet decomposition, which enables us to perform time–frequency analysis of the data content. A simple thresholding algorithm is used to eliminate noise components. The method is compared with the usual singular-value decomposition. Preliminary results on simple test cases show that wavelet decomposition permits us to enhance the results of the inversion procedure, by giving smoother estimates of slow-varying parts of the solution while maintaining a sharp estimation of fast transients.

DESIGN, FORMULATION, AND SOLUTION OF MULTIDIMENSIONAL INVERSE HEAT CONDUCTION PROBLEMS

ABSTRACT The solution of linear inverse heat conduction problems (IHCP) for the reconstruction of time-dependent, spatially unknown heat fluxes on the boundaries of two- and three-dimensional geometries from several temperature measurements is presented. The solution method is based on the interpretation of the IHCP in the frequency domain. We emphasize the extension of the method to large-scale problems. In particular, we consider the parameterization of the unknown heat fluxes with respect to the number of sensors and their positions and the reduction and stable inversion of the large linear state-space models obtained from semidiscretization of the heat conduction equation. The influence of each of the steps and the choice of their tuning parameters on the quality of the estimated heat flux are discussed. Two- and three-dimensional examples show the good performance of the method with regard to time and space resolution in the presence of noisy measurements.

A numerical solution of the linear multidimensional unsteady inverse heat conduction problem with the boundary element method and the singular value decomposition

In this paper, we present a new method for solving the general linear multidimensional unsteady inverse heat conduction problem. The direct numerical method is based on the Boundary Element Method formulation. Taking into account future time steps, the ill-conditioned linear system is solved using a procedure based on the Singular Value Decomposition technique which handles both spatial and temporal instabilities. The regularization method is essentially a spectral truncation method with a single hyperparameter. The optimal value of this hyperparameter can be chosen a priori from the knowledge of the data uncertainty. In the second part of this paper, an experiment is described which illustrates an application of the method on a two-dimensional problem. The physical problem consists in identifying the heat flux on a plate exposed to a moving front of hot fluid from temperature measurements collected on the opposite side of the plate. Numerical results obtained are discussed in comparison to direct heat flux measurements. The purpose of this experiment was to cross-check two types of heat flux sensors in real unsteady two-dimensional situations. It is also shown that the method can be applied to real 3D problems.

Heat Source Estimation with the Conjugate Gradient Method in Inverse Linear Diffusive Problems

In this work, we present the solution of a class of linear inverse heat conduction problems for the estimation of unknown heat source terms, with no prior information of the functional forms of timewise and spatial dependence of the source strength, using the conjugate gradient method with an adjoint problem. After describing the mathematical formulation of a general direct problem and the procedure for the solution of the inverse problem, we show applications to three transient heat transfer problems: a one-dimensional cylindrical problem; a two-dimensional cylindrical problem; and a one-dimensional problem with two plates.

SOLUTION OF THE INVERSE HEAT CONDUCTION PROBLEM WITH A TIME-VARIABLE NUMBER OF FUTURE TEMPERATURES

The one-dimensional linear inverse heat conduction problem (IHCP) is considered here. In this problem, the accuracy in the determination of the surface heat flux is not only affected by noise occurring in the remote measurement but is also influenced by the bias introduced by the method itself. A modified junction specification algorithm is proposed that automatically balances the sensitivity to measurement error and the bias, for each time step. This algorithm uses a time-variable number of future temperatures. The new method is compared with the classic Beck's function specification method, i.e., with a constant number of future temperatures. Using a clear definition of the different errors, the comparison shows that, in many cases, the modified method significantly improves the IHCP resolution.

Application d'une méthode de perturbation au problème inverse de conduction de la chaleur

A perturbation method coupled with a procedure based on the concept of regularization is presented for the solution of the linear inverse heat conduction problem. The results of a numerical example show the accuracy and efficiency of this approach.

Recent contributions to linear inverse heat conduction problems

Recent contributions to linear inverse heat conduction problems

Calculation error of numerical solution for a boundary-value inverse heat conduction problem

A one-dimensional linear inverse heat conduction problem is studied in this paper. This ill-posed problem is replaced by the perturbed problem with a non-localized boundary condition. After the derivation of its closed-form analytical solution, the calculation error can be determined by the comparison between the numerical and exact solutions.