Nonlinear Inverse Heat Conduction Problem Research Articles

Genetic Algorithm as the Solution of Non-Linear Inverse Heat Conduction Problems: A Novel Sequential Approach

Abstract Direct measurement of surface heat flux could be extremely challenging, if not impossible, in numerous applications. In such cases, the use of temperature measurement data from subsurface locations can facilitate the determination of surface heat flux and temperature through the solution of the inverse heat conduction problem (IHCP). Several different techniques have been developed over the years for solving IHCPs, with different levels of complexity and accuracy. The filter coefficient technique has proved to be a promising approach for solving IHCPs. Inspired by the filter coefficient approach, a novel method is presented in this paper for solving one-dimensional IHCPs in a domain with temperature-dependent material properties. A test case is developed in COMSOL Multiphysics where the front side of a slab is subject to known transient heat flux and the temperature distributions within the domain are calculated through numerical simulation. The IHCP solution in the form of filter coefficients is defined and a genetic algorithm (GA) is used for the calculation of filter matrix. The number of significant filter coefficients required to evaluate surface heat flux at each time-step is determined through trial and error and the optimal number is selected for evaluating the solution. The structure of the filter matrix is assessed and discussed. The resulting filter coefficients are then used to evaluate the surface heat flux for several different test cases and the performance of the proposed approach is assessed in detail. The results showed that the presented approach is robust and can result in finding optimal filter coefficients that can accurately estimate various types of surface heat flux profiles using temperature data from a limited number of time steps and in a near real-time fashion.

A variable future-time-steps method for solving nonlinear unsteady inverse heat conduction problems

In some non-linear unsteady inverse problems, the inverse solution will oscillate violently in the whole time domain due to the sharp change of the sensitivity coefficients. To deal with this problem, a new sequential function specification method with variable future time steps is proposed in this paper. The future time steps are adjusted by the error amplification coefficients which are defined as the reciprocal of the square sum of the sensitivity coefficients. When the error amplification coefficients are small, a small number of future time steps is used to reduce the deterministic error. While in the period with large error amplification coefficient, a large number of future time steps is used to reduce stochastic error. Finally, the total error of estimated heat flux is reduced. Avoid the sharp fluctuation of estimated heat flux in time domain due to the sharp change of sensitivity coefficients. The variable future-time-steps method is applied to the estimation of 1-D non-linear unsteady heat flux without and with ablation through numerical experiments. Numerical experiments show that the proposed method can not only estimate various forms of heat flux, but also its inversion results are significantly better than those of the fixed future time steps method based on the discrepancy principle, and also better than those of the fixed future time step method based on the minimum relative error of heat flux.

The Fourier-based dimensional reduction method for solving a nonlinear inverse heat conduction problem with limited boundary data

The objective of this article is to introduce a novel technique for computing numerical solutions to the nonlinear inverse heat conduction problem. This involves solving nonlinear parabolic equations with Cauchy data provided on one side Γ of the boundary of the computational domain Ω. The key step of our proposed method is the truncation of the Fourier series of the solution to the governing equation. The truncation technique enables us to derive a system of 1D ordinary differential equations. Then, we employ the well-known Runge–Kutta method to solve this system, which aids in addressing the nonlinearity and the lack of data on ∂Ω∖Γ. This new approach is called the Fourier-based dimensional reduction method. By converting the high-dimensional problem into a 1D problem, we achieve exceptional computational speed. Numerical results are provided to support the effectiveness of our approach.

Studying the effects of coatings on the thermal protection of cutting tools during turning via a nonlinear inverse heat conduction problem

Studying the effects of coatings on the thermal protection of cutting tools during turning via a nonlinear inverse heat conduction problem

Estimation of the Total Heat Exchange Factor for the Reheating Furnace Based on the First-Optimize-Then-Discretize Approach and an Improved Hybrid Conjugate Gradient Algorithm

The total heat exchange factor is one of the most important thermal physical parameters in the heat transfer model for a reheating furnace machine. In this paper, a novel general strategy, which is combined with the first-optimize-then-discretize (FOTD) approach and an improved hybrid conjugate gradient (IHCG) algorithm, is proposed to identify the total heat exchange factor by solving a nonlinear inverse heat conduction problem (IHCP). Firstly, a nonlinear IHCP with the Dirichlet-type boundary condition Tm(t)=T(0,t) is built to determine the unknown total heat exchange factor w(t). Secondly, the analysis of the Fréchet gradient of the cost functional is given and the gradient is proved as Lipschitz continuous by the FOTD approach. Thirdly, based on the gradient information by FOTD, a new IHCG algorithm, whose global convergence is proved by us, is proposed for fast solving of the optimization problem. Finally, simulation experiments are given to verify the effectiveness of the proposed strategy. Compared with the first-discretize-then-optimize (FDTO) approach, the FOTD approach can reduce running time and iteration number. Compared with other CG algorithms, the proposed IHCG algorithm has better convergence performance. The experimental data by the thermocouples experiments from a reheating furnace are also given to identify the total heat exchange factor.

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.

Inverse estimation of hot-wall heat flux using nonlinear artificial neural networks

Compared with cold-wall heat flux measurement, hot-wall heat flux indicates more information from dynamic heat transfer between the aerothermodynamic boundary layer and the true model’s surface; thus, hot-wall heat flux measurement has become predominant in the aerothermodynamic measurement field. In harsh aerothermodynamic in-flight or ground tests, the hot-wall heat flux must be determined from time history temperature measurements at one or more interior locations. Therefore, accounting for the temperature dependence of the thermophysical properties, hot-wall heat flux measurement essentially results in the solution of the nonlinear inverse heat conduction problem (IHCP). In this paper, a novel inverse estimation of hot-wall heat flux using nonlinear artificial neural networks (ANN) is presented. First, motivated by the hybrid method proposed by Clayton A. Pullins, David O. Hubble, and Tom E. Diller et al., [2010], a modified hybrid heat flux measurement method using two in-depth thermocouples is proposed, which avoid to directly measure surface temperature of gauge or model under harsh aerodynamic heating environments; accounting for the unknown temperature dependent thermophysical properties, a new heat flux inverse estimation model of a nonlinear ANN is proposed and identified to approximate the modified hybrid measurement method through calibration experiment. This heat flux inverse estimation method does not need to solve a first kind Volterra integral equation and to obtain the information about the thermophysical properties of heat conduction body, and the thermal inertia, locations of thermocouples. This paper presents xenon lamp calibration and arc-heated wind tunnel experiments that validate the new inverse estimation method combined with the fabricated hot-wall heat flux sensor and its probe. These experimental results show that, in general, the dynamical hot-wall heat flux estimated based on the proposed method agree well with the known calibrated values and the stagnation heat fluxes of the classic slug calorimeter.

A sequential conjugate gradient method to estimate heat flux for nonlinear inverse heat conduction problem

In this paper, a sequential conjugate gradient method is presented to reconstruct the undetermined surface heat flux for nonlinear inverse heat conduction problem (IHCP). This modified method combines the merits of sequential function specification method (SFSM) and conjugate gradient method (CGM). The proposed method can effectively eliminate the leading error which is inevitably generated in SFSM due to the existence of temporary assumption of the unknown parameters and reduce the computing time of nonlinear IHCP. The square wave and sine wave heat flux are used to evaluate the proposed method and the influence of the number of future time steps and measurement errors on the predicted results is studied. The numerical experiments verify the effectiveness and timeliness of the proposed nonlinear IHCP algorism in reconstructing the heat flux. The comparisons with the typical SFSM and CGM are conducted to demonstrate the superiority of the proposed method. The results show that proposed method is more accurate compared with that of the SFSM under the condition of large measurement errors and the computing time is much faster than that of CGM.

Sequential particle filter estimation of a time-dependent heat transfer coefficient in a multidimensional nonlinear inverse heat conduction problem

In the applied mathematical modelling of heat transfer systems, the heat transfer coefficient (HTC) is one of the most important parameters. This paper proposes a combination of the Method of Fundamental Solutions (MFS) with particle filter Sequential Importance Resampling (PF-SIR) to estimate the time-dependent HTC in two-dimensional transient inverse heat conduction problems from non-standard boundary integral measurements. These measurements ensure the unique solvability of the boundary coefficient identification problem. Numerical results show high performance on several test cases with both linear and nonlinear Robin boundary conditions, supporting the synergy between the MFS and simulation-based particle filter sequential analysis methods.

Experimental Verification of the Inverse Method of the Heat Transfer Coefficient Calculation

The purpose of this work is to formulate a method which can be used to solve nonlinear inverse heat conduction problems and to calculate the heat transfer coefficient distribution on the unknown boundary. The domain under consideration is divided into control volumes in polar coordinates, and heat balance equations are written. Based on temperature transients measured in selected points on the outer surface, temperature values in other points of the domain are determined. Finally, the heat transfer coefficient distribution on the inner surface with the unknown boundary condition is calculated from the presented heat balance equation. The proposed inverse method was verified experimentally using a collector that is part of a semi-industrial laboratory system. This collector is a horizontal, cylindrical thick-walled tank with flat side walls with outlets that enable oil supply and removal. Each side wall has an additional connector to ensure venting. The calculations made it possible to identify the phenomena occurring inside the collector during the experiment. The transient temperature distribution identified by the proposed inverse method was verified by a comparison of the calculated and the measured temperature transients in points inside the collector wall. Very good agreement is observed between the calculated and the measured temperature transients, which confirms the correctness of the identification. This proposed inverse method of the temperature and the heat transfer coefficient calculation is fast enough to apply in online thermal state monitoring systems. The proposed algorithm presented in this paper can easily be implemented industrially.

Experimental design for estimation of the distribution of the convective heat transfer coefficient for a bubbly impinging jet

In this study, a two-dimensional inverse algorithm is developed to determine the heat transfer coefficient distribution of a two-phase air–water bubbly jet impinging on a steel cylindrical thermal mass. All procedures of thermal mass heating and cooling are simulated by solving two-dimensional, transient heat conduction equations using finite difference method. Afterward, the nonlinear inverse heat conduction problem is implemented to directly predict the local convective heat transfer coefficient of the bubbly jet. The sum of squared differences between calculated and measured temperature data is the objective function. Conjugate gradient method is employed sequentially in every time step to optimize the objective function at four gas Reynolds numbers which represent four different two-phase jets. The inverse scheme is validated using exact temperature data without noise. Local heat transfer coefficients are then estimated by inverse technique at five data acquisition times and four initial temperatures of thermal mass in the presence of noise. Furthermore, the effects of uncertainties due to indefinite lateral boundary conditions, temperature dependency of thermal conductivity, and the non-uniformity of the initial temperature distribution are investigated. A satisfactory agreement between exact and estimated heat transfer coefficients is achieved. However, the results show a greater sensitivity to the highest value of initial temperature, the shortest data acquisition time, and the lowest gas Reynolds number allowing a better estimation of heat transfer distribution for the bubbly jet.

A general approach for solving three-dimensional transient nonlinear inverse heat conduction problems in irregular complex structures

A novel general approach is proposed for solving three-dimensional transient nonlinear inverse heat conduction problems in irregular complex structures. The complex-variable-differentiation method (CVDM) is introduced into the commercial finite element method (FEM) software ABAQUS, for solving three-dimensional inverse heat conduction problems for the first time. First, the complex variable finite elements are set up, and the complex variable FEM is innovatively developed through the user element subroutine (UEL) in ABAQUS. Then, the key parameters in the Levenberg-Marquardt algorithm for solving inverse problems – sensitivity matrix coefficients – are accurately evaluated by utilizing the CVDM and ABAQUS standard solvers with the UEL. Finally, several numerical tests are carried out to verify the performances of the new approach for solving the three-dimensional transient nonlinear inverse heat conduction problem in an irregular complex structure.

Derivation and application of the adjoint method for estimation of both spatially and temporally varying convective heat transfer coefficient

In this study, a numerical algorithm is developed to determine the heat transfer coefficient distribution of mixed convection on a vertical plate with a two dimensional inverse method. Cooling procedures of hot plate include forced convection with buoyancy effect is simulated by solving two dimensional, transient heat conduction equation using finite difference method while the Grashof number is varied through natural convection in combination with forced convection resulting in a variation of the Richardson number from 0.1 to 1. The nonlinear inverse heat conduction problem is then implemented to directly predict the time-space-varying convective heat transfer coefficient of cooling in the mixed convection regime. The sum of squared differences between calculated and measured temperature data at thermocouples’ locations is the objective functional. The adjoint method is employed to optimize the functional using conjugate gradient method via the solutions of the direct, adjoint and sensitivity sub-problems in a whole time-domain optimization process. The inverse scheme is validated using exact temperature data without noise. Local heat transfer coefficients are then estimated by the adjoint method at four cooling strategies for ten minutes of data acquisition in the presence of noise with standard deviations of σ=0.01 °C and σ=0.1 °C. Although results are affected by noisy simulated temperatures, a satisfactory agreement between exact and estimated heat transfer coefficients is achieved. Furthermore, increasing data acquisition time to sixty minutes reveals that the inverse scheme is able to predict 1200 convective heat transfer coefficient components efficiently employing the rapidly convergent adjoint method.

Estimation of thermal conductivity of low thermal conductive solid materials using the jet flush method

A novel and practical transient-type technique named as “Jet Flush” is proposed for measuring temperature-dependent thermal conductivity of solid materials with low thermal conductivity. In order to estimate thermal conductivity of the sample, its surface is exposed to an impinging air jet flow with higher (or lower) temperature than the sample for a short period of time, while an infrared thermometer records the heated (or cooled) surface temperature history. A nonlinear 1D inverse heat conduction problem with constant convective heat transfer coefficient on the boundary condition is then employed for determining the temperature-dependent thermal conductivity using the measured temperatures. As a requirement for solving the inverse problem, the convective coefficient on the surface of the sample is measured in a separate experiment with the same flow condition using a standard sample. Sensitivity and error analyses are performed to discuss the effects of the estimation points (the temperatures in which the thermal conductivity is to be estimated), and the simplification assumption of constant convective heat transfer coefficient (with respect to time) on the solution of the inverse problem and the accuracy of the estimated results. In order to demonstrate the applicability of the proposed method, the thermal conductivities of three polymer-type materials (HDPE, PMMA and PA6) are measured over the temperature range of ~22–62 °C, and the results are found to be in good agreement with literature data. Moreover, the uncertainty analysis performed in this study showed that the accuracy of the proposed method can be better than 5% for an appreciable temperature range.

Determination of the time-dependent reaction coefficient and the heat flux in a nonlinear inverse heat conduction problem

ABSTRACTDiffusion processes with reaction generated by a nonlinear source are commonly encountered in practical applications related to ignition, pyrolysis and polymerization. In such processes, determining the intensity of reaction in time is of crucial importance for control and monitoring purposes. Therefore, this paper is devoted to such an identification problem of determining the time-dependent coefficient of a nonlinear heat source together with the unknown heat flux at an inaccessible boundary of a one-dimensional slab from temperature measurements at two sensor locations in the context of nonlinear transient heat conduction. Local existence and uniqueness results for the inverse coefficient problem are proved when the first three derivatives of the nonlinear source term are Lipschitz continuous functions. Furthermore, the conjugate gradient method (CGM) for separately reconstructing the reaction coefficient and the heat flux is developed. The ill-posedness is overcome by using the discrepancy principle to stop the iteration procedure of CGM when the input data is contaminated with noise. Numerical results show that the inverse solutions are accurate and stable.

A new approach to speed up the conjugate gradient method by applying the complex variable differentiation method

ABSTRACTThis study represents a novel approach to speed up the solution of nonlinear inverse heat conduction problems (IHCPs) by the implementation of the complex variable differentiation method (CVDM). A difficulty appeared in the solution of nonlinear problems is the great amount of computational time. To handle this problem, a new scheme is introduced to improve the conjugate gradient method (CGM). The main contribution in the improvement of conventional CGM is the simultaneous solution of direct and sensitivity problems by applying the complex variable method. The derivation of sensitivity problem by utilizing the analytic derivative is difficult or impossible, since the direct problem is complicated mathematically in many cases. By this approach, the analytic derivation of sensitivity problem is circumvented, while the Jacobian matrix components are obtained accompanied by the solution of the direct problem. Therefore, the developed scheme results in the reduction of mathematical manipulations. Due to the high nonlinearity of moving boundary inverse problems, the ablation problem is considered as the benchmark to examine the accuracy and effectiveness of the proposed scheme. The simulated results illustrate that the developed scheme has the potential to significantly reducing of the computational expenses while maintaining the quality of numerical solution.

FEATURES OF THE APPLICATION OF THE IQLAB PROGRAM FOR SOLVING THE INVERSE HEAT CONDUCTION PROBLEM FOR CHROMIUM-NICKEL CYLINDRICAL THERMOSONDES

The paper presents an experimental-computational study of the results of using the IQLab program to solve inverse heat conduction problem and restore the surface temperature of cylindrical thermosondes from heat-resistant chromium-nickel alloys while cooling them in liquid media. The purpose of this paper is to verify the correct operation of the IQLab program when restoring the surface temperature of thermosondes with 1-3 thermocouples. The IQLab program is also designed to solve one-dimensional nonlinear direct lines and inverse heat conduction problems with constant initial and boundary conditions specified as a function of time in a tabular form with a constant and variable time step. A finite-difference method is used to solve the heat equation. Experiments were carried out on samples D = 10-50 mm in liquids with different cooling capacities such as aqueous solutions of NaCl and Yukon-E polymer, rapeseed oil and I-20A mineral oil. For the calculation we used the readings of thermocouples installed at internal points of cylindrical thermosondes. The advantages of solving inverse heat conduction problems with the IQLab program include the possibility of restoring the surface temperature for cylindrical samples with a diameter of 10 mm to 50 mm with practical accuracy according to the indications of a single thermocouple located in the geometrical center of the thermosonde, which simplifies the manufacture of the probe. For larger dimensions with a diameter D ≥ 50 mm, it is necessary to install control intermediate thermocouples and perform additional tests. The solution of inverse heat conduction problems and restoration of the surface temperature of the sample makes it possible to calculate other important characteristics of the cooling process: the heat flux density and the heat transfer coefficient.

An extended approach of a Kalman smoothing technique applied to a transient nonlinear two-dimensional inverse heat conduction problem

An extended Kalman smoother (EKS) introducing the concept of future measurement information is developed to handle nonlinear inverse heat conduction problems. In the present study, the formulation of the EKS algorithm is generalized in the case of a two-dimensional problem in order to reconstruct a non-homogeneous heating condition on the front surface of a cylindrical sample. This leads to apply the new method in a multiple sensor case for the simultaneous reconstruction of several parameters at each time. The analyzed inverse problem is nonlinear due to the variability of the thermophysical properties with temperature and to the presence of radiation boundary conditions. Inverse estimation is successfully performed where temporal and spatial variations of the front surface heat flux are recovered based on non intrusive transient temperature measurements made on the back surface. Numerical experiments show that the use of an optimal number of future data greatly improves the solution of the EKS compared to the extended Kalman filter (EKF) estimates, where a noticeable reduction of time lag and sensitivity to measurement errors is observed. Inversion results show that the optimal number of future data, chosen on the basis of a minimum measure of the heat flux bias, depends on the modeling error, the measurement time step, the distance between the sensors and the front heated surface and the measurement noise level. The proposed algorithm is robust in recovering different heat flux profiles and provides for all the examined patterns symmetric and stable solutions that superposes well with the exact functions. The EKS formulation based on an augmented state vector allows to efficiently recover the front surface temperature simultaneously with the heat flux.

Solution of an inverse transient heat conduction problem in a part of a complex domain

The purpose of this work is to formulate two simple methods which can be used to solve nonlinear inverse heat conduction problems in a part of a complex-shaped component in the on-line mode. The proposed methods can be useful if temperature measurements are carried out only in a selected part of an outsize element or if the whole element cannot be analysed due to the inverse heat conduction problem being ill-conditioned. It is assumed that the conductive heat transfer occurs through the surfaces separating the domain from the rest of the component. It is shown that the simplifying assumption of thermal insulation on these surfaces, which is often presented in literature, can cause significant errors. Compared to works published previously, if two additional unknown boundary conditions are introduced on the separating surfaces, the inverse problem conditioning deteriorates substantially. Despite this, stable solutions are achieved for “noisy measured data”.The presented methods can be used to optimize the power unit start-up and shutdown operation. They may also enable a reduction in the heat loss arising during the process and extend the power unit life. The methods presented herein can be applied in monitoring systems working both in conventional and nuclear power plants.

An iterative finite-element algorithm for solving two-dimensional nonlinear inverse heat conduction problems

It is often useful to determine temperature and heat flux in multidimensional solid domains of arbitrary shape with inaccessible boundaries. In this study, an effective algorithm for solving boundary inverse heat conduction problems (IHCPs) is implemented: transient temperatures on inaccessible boundaries are estimated from redundant simulated measurements on accessible boundaries. A nonlinear heat equation is considered, where some of the material properties are dependent on temperature. The IHCP is reformulated as an optimization problem. The resulting functional is iteratively minimized using a conjugate gradient method together with an adjoint (dual) problem approach. The associated partial differential equations are solved using the finite-element package FEniCS. Tikhonov regularization is introduced to mitigate the ill-posedness of the IHCP. The accuracy of the implemented algorithm is assessed by comparing the solutions to the IHCP with the correct temperature values, on the inaccessible boundaries. The robustness of our method is tested by adding Gaussian noise to the initial conditions and redundant boundary data in the inverse problem formulation. A mesh independence study is performed.