改进遗传算法在非线性热传导参数识别中的应用

建立了基于优化算法的估计材料热传导系数和边界条件的热传导反问题求解方法。该方法以观测的温度值与有限元计算模拟的温度值最小二乘极小化原理为基础,然后采用具有全局搜索能力的遗传算法求解。为了加快收敛速度和提高反演识别精度,采用了浮点编码的遗传算法。根据先验信息,建立了高斯变异策略。数值计算结果表明,所建立的数值反演方法可以用来解决未知的热传导系数和边界条件识别问题,并且具有良好的抗观测噪音能力。     

Estimation of temperature distributions and thermal stresses in a functionally graded hollow cylinder simultaneously subjected to inner-and-outer boundary heat fluxes

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to simultaneously estimate the unknown time-dependent inner and outer boundary heat fluxes in a functionally graded hollow circular cylinder from the knowledge of temperature measurements taken within the cylinder. Subsequently, the distributions of temperature and thermal stresses in the cylinder can be determined as well. It is assumed that no prior information is available on the functional forms of the unknown heat fluxes; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements, and the effect of the errors and locations in these measurements upon the precision of the estimated results is also considered. Results show that an excellent estimation on the time-dependent heat fluxes, temperature distributions, and thermal stresses can be obtained for the test case considered in this study.     

Estimation of heat flux and mechanical loads on laminated functionally graded plate

The spatially and temporally varying heat flux and mechanical load on the top (bottom) surface of laminated plates with functionally graded layers are estimated using an inverse algorithm. The temperature and strains at a number of points on the bottom (top) surface of the plate are the only measured input data. The solution of corresponding direct problem is used to simulate the measured temperatures and strains, which are obtained based on a three-dimensional layerwise thermoelastic analysis. The conjugate gradient method as a powerful technique for optimization in conjunction with the discrepancy principle is employed to develop the inverse solution procedure. A semi-analytical approach composed of the layerwise-differential quadrature method and series solution is adopted to discretize the governing differential equations subjected to the related boundary and initial conditions. The influence of measurement errors on the accuracy of the estimated heat flux and mechanical load is investigated. The good accuracy of the results validates the presented inverse approach.     

Parameter estimation in heat conduction using a two-dimensional inverse analysis

This article is concerned with a two-dimensional inverse steady-state heat conduction problem. The aim of this study is to estimate the thermal conductivity, the heat transfer coefficient, and the heat flux in irregular bodies (both separately and simultaneously) using a two-dimensional inverse analysis. The numerical procedure consists of an elliptic grid generation technique to generate a mesh over the irregular body and solve for the heat conduction equation. This article describes a novel sensitivity analysis scheme to compute the sensitivity of the temperatures to variation of the thermal conductivity, the heat transfer coefficient, and the heat flux. This sensitivity analysis scheme allows for the solution of inverse problem without requiring solution of adjoint equation even for a large number of unknown variables. The conjugate gradient method (CGM) is used to minimize the difference between the computed temperature on part of the boundary and the simulated measured temperature distribution. The obtained results reveal that the proposed algorithm is very accurate and efficient.     

Simulation of processes in heat sensors based on solution of inverse problems in heat conduction

The paper deals with nonstationary problems in heat conduction, which arise in connection with the determination of the heat flux density and temperature on the surface of a model in intermittent high-enthalpy wind-tunnel facilities by the results of temperature measurements using intramodel heat sensors. The solution of inverse problem in heat conduction in a one-dimensional formulation with an arbitrary time dependence of the heat flux density is obtained by two methods, namely, by iterations and by integral transformations with finite limits. In the former method, the inverse problem is reduced to a system of two coupled integral and integro-differential equations of the Volterra type relative to the temperature and heat flux density on the external boundary. Calculations demonstrate that the numerical solution asymptotically approaches the exact solution, and the iteration method exhibits smoothing properties and is stable with respect to random errors of measurement. In the integral method, an inverse problem for the class of boundary functions satisfying the Dirichlet conditions and represented by a partial sum of the Fourier series reduces to a set of algebraic equations which has a unique solution. In the absence of measurement errors, the solution of inverse problem is exact. Examples are given of constructing solutions in the presence of random noise; it is demonstrated that, in the case of reasonable restriction of the range of frequencies to be analyzed, the errors in the solution do not exceed the mean-square level of noise.Translated from Teplofizika Vysokikh Temperatur, Vol. 43, No. 1, 2005, pp. 071–085.Original Russian Text Copyright © 2005 by E. P. Stolyarov.     

An inverse optimal control problem in the electrical discharge machining

Electrical discharge machining (EDM) is a thermal material removal process by means of electrical discharge. Because of the stochastic nature of the EDM process, electro-thermal energy conversion in the discharge zone is still not well understood. In this paper, an inverse optimal control problem was used for analysis and optimization of energy conversion processes in order to improve machining efficiency. Modeling and identification of a thermal process were conducted using the inverse heat transfer problem based on the known temperature within a workpiece. In addition to the temperature field, this approach allows the determination of unknown heat flux density distribution on the workpiece surface. By using the heat flux, the inverse optimal control problem based on minimizing a Tikhonov functional allows to obtain the optimal heat source parameters (discharge power and discharge duration) on the discharge energy. In this context, the concept of inverse problem allows reliable determination of the optimal discharge energy to achieve the highest possible productivity with the desired quality. The performance of prediction of the heat affected zone compared to the experimental results showed a good agreement, which confirms the validity of the inverse method compared to the reported models.     

Reconstruction of multiplicative space- and time-dependent sources

This paper presents a numerical regularization approach to the simultaneous determination of multiplicative space- and time-dependent source functions in a nonlinear inverse heat conduction problem with homogeneous Neumann boundary conditions together with specified interior and final time temperature measurements. Under these conditions a unique solution is known to exist. However, the inverse problem is still ill-posed since small errors in the input interior temperature data cause large errors in the output heat source solution. For the numerical discretisation, the boundary element method combined with a regularized nonlinear optimization are utilized. Results obtained from several numerical tests are provided in order to illustrate the efficiency of the adopted computational methodology.     

Identification of a Corroded Boundary and Its Robin Coefficient

An inverse geometric problem for two-dimensional Helmholtz-type equations arising in corrosion detection is considered. This problem involves determining an unknown corroded portion of the boundary of a two-dimensional domain and possibly its surface heat transfer (impedance) Robin coefficient from one or two pairs of boundary Cauchy data (boundary temperature and heat flux), and is solved numerically using the meshless method of fundamental solutions. A nonlinear unconstrained minimisation of the objective function is regularised when noise is added into the input boundary data. The stability of the numerical results is investigated for several test examples, with respect to noise in the input data and various values of the regularisation parameters.     

An inverse problem in simultaneously estimating boundary moisture fluxes in a porous annular cylinder

Summary Based on the conjugate gradient method, this study presents a means of solving the inverse boundary value problem of coupled heat and moisture transport in a porous annular cylinder. While knowing the moisture history at the measuring positions, the unknown time-dependent inner-and-outer boundary moisture fluxes can be simultaneously determined. It is assumed that no prior information is available on the functional form of the unknown moisture fluxes. The accuracy of this inverse heat and moisture transport problem is examined by using the simulated exact and inexact moisture measurements in the numerical experiments. Results show that excellent estimation on the time-dependent boundary moisture fluxes can be obtained with any arbitrary initial guesses. Moreover, the methodology presented in this paper can also be used to calculate the cutting forces in nanomachining by atomic force microscopy (AFM), and to determine the heat sources in an X-ray lithographic process.     

Singular Superposition/Boundary Element Method for Reconstruction of Multi-dimensional Heat Flux Distributions with Application to Film Cooling Holes

A hybrid singularity superposition/boundary element-based inverse problem method for the reconstruction of multi-dimensional heat flux distributions is developed. Cauchy conditions are imposed at exposed surfaces that are readily reached for measurements while convective boundary conditions are unknown at surfaces that are not amenable to measurements such as the walls of the cooling holes. The purpose of the inverse analysis is to determine the heat flux distribution along cooling hole surfaces. This is accomplished in an iterative process by distributing a set of singularities (sinks) inside the physical boundaries of the cooling hole (usually along cooling hole centerline) with a given initial strength distribution. A forward steady-state heat conduction problem is solved using the boundary element method (BEM), and an objective function is defined to measure the difference between the heat flux measured at the exposed surfaces and the heat flux predicted by the BEM under the current strength distribution of the singularities. A Genetic Algorithm (GA) iteratively alters the strength distribution of the singularities until the measuring surfaces heat fluxes are matched, thus satisfying Cauchy conditions. The distribution of the heat flux at the walls of the cooling hole is determined in a post-processing stage after the inverse problem is solved. The advantage of this technique is to eliminate the need of meshing the surfaces of the cooling holes, which requires a large amount of effort to achieve a high quality mesh. Moreover, the use of singularity distributions significantly reduces the number of parameters sought in the inverse problem, which constitutes a tremendous advantage in solving the inverse problem, particularly in the application of film cooling holes.     

Optimal distribution of a heat source within a thermopenetrator

Heat conduction within a heater of an arbitrary shape is investigated. A mathematical model is presented as a mixed boundary-value problem for the Poisson equation converted into a Fredholm boundary integral equation of the first kind which is solved numerically. A closed-form solution for the particular case of a rectangular heater is also found. Provided that the temperature and heat flux on the heater's boundary are given, the problem is treated as an inverse problem where the heat source distribution within the heater is the unknown function. The existence of the unique solution of this inverse problem is proved. Finally, the problem is solved numerically for a one-dimensional heat source.     

An inverse heat transfer problem for optimization of the thermal process in machining

It is evident that machining process causes development of large quantities of thermal energy within a relatively narrow area of the cutting zone. The generated thermal energy and the problems of its evacuation from the cutting zone account for high temperatures in machining. These increased temperatures exert a pronounced negative effect on the tool and workpiece. This paper takes a different approach towards identification of the thermal process in machining, using inverse heat transfer problem. Inverse heat transfer method allows the closest possible experimental and analytical approximation of thermal state for a machining process. Based on a temperature measured at any point within a workpiece, inverse method allows determination of a complete temperature field in the cutting zone as well as the heat flux distribution on the tool/workpiece interface. By knowing the heat flux function, one defines criterium and method of optimization, the inverse heat transfer problem transforms into extreme case. Now, the task of optimization is to determine most favourable ratio between heat flux parameters in order to preserve exploitation properties of the tool and workpiece.     

A method of determining nonstationary heat flux and heat conduction using parametric identification

The formulation and solution of the combined (boundary and coefficient) inverse heat-conduction problem of the simultaneous determination of the thermal characteristics of a material and the heat flux density entering the heat receiver are considered. The solution is based on a parameterization of this problem and the use of an extended digital Kalman filter. The results of a simulation of the simultaneous reestablishment of the heat flux and the heat conduction in a single experiment to measure the temperature of a surface are presented.     

Inverse approach for estimating boundary properties in a transient fin problem

A solution methodology is proposed for an inverse estimation of boundary conditions from the knowledge of transient temperature data. A forward model based on prevalent time-dependent heat conduction fin equation is solved using a fully implicit finite volume method. First, the inverse model is formulated and accomplished for time-invariant heat flux at the fin base, and later extended to transient heat flux, base temperature and average heat transfer coefficient. Secondly, the Nusselt number is then replaced with Rayleigh number in the forward model to realistically estimate the base temperature, which varies with respect to time, based on in-house transient fin heat transfer experiments. This scenario further corroborates the validation of the proposed inverse approach. The experimental set-up consists of a mild steel \(250 \times 150 \times 6\, \hbox {mm}^3\) fin mounted centrally on an aluminium base \(250 \times 150 \times 8\, \hbox {mm}^3\) plate. The base is attached to an electrical heater and insulated with glass-wool to prevent heat loss to surroundings. Five calibrated K-type thermocouples are used to measure temperature along the fin. The functional form of the unknown parameters is not known beforehand; sensitivity studies are performed to determine suitability of the estimation and location of sensors for the inverse approach. Conjugate gradient method with adjoint equation is chosen as the inverse technique and the study is performed as a numerical optimization; subsequently, the estimates show satisfactory results.     

An inverse heat conduction problem with heat flux measurements

Using heat flux measurements as additional information to solve inverse heat conduction problems was and is still rarely employed. Lot of disadvantages linked to heat flux measurement specificities (local disturbance, intrusive measurement, lack of knowledge and proficiency, etc.) make people prefer temperature measurements which are well documented and very widespread. Solving inverse heat conduction problems with heat flux measurements is quite different than the one which uses temperatures and need to be investigated deeply. In this work, this problem is approached through the solution of a bioengineering problem consisting in the development of a non‐invasive blood perfusion probe. The effort here is focused on the development of a methodology for the estimation of time‐dependent blood perfusion from heat flux measurements. The physical probe incorporates a thin heat flux sensor, which is placed in contact with the tissue region where the perfusion is to be measured. The sensor records the heat flux due to an imposed thermal event, which is achieved by air flow. A one‐dimensional mathematical model is used to simulate the thermal event occurring at the contact region holding between the probe and the tissue. A combined parameter and function estimation procedure is developed to estimate simultaneously time‐dependent blood perfusion and thermal contact conductance between the probe and the tissue. The robustness of the method was demonstrated through several test cases using simulated data. The presented examples include various functional changes in the time evolution of blood perfusion. Results from this study have shown the feasibility of solving inverse problems with heat flux measurements and the two unknowns are estimated with no a priori information about their functional forms. Copyright © 2006 John Wiley & Sons, Ltd.     

Proper Generalized Decomposition model reduction in the Bayesian framework for solving inverse heat transfer problems

In this paper, the proper generalized decomposition (PGD) is used for model reduction in the solution of an inverse heat conduction problem within the Bayesian framework. Two PGD reduced order models are proposed and the approximation Error model (AEM) is applied to account for the errors between the complete and the reduced models. For the first PGD model, the direct problem solution is computed considering a separate representation of each coordinate of the problem during the process of solving the inverse problem. On the other hand, the second PGD model is based on a generalized solution integrating the unknown parameter as one of the coordinates of the decomposition. For the second PGD model, the reduced solution of the direct problem is computed before the inverse problem within the parameter space provided by the prior information about the parameters, which is required to be proper. These two reduced models are evaluated in terms of accuracy and reduction of the computational time on a transient three-dimensional two region inverse heat transfer problem. In fact, both reduced models result on substantial reduction of the computational time required for the solution of the inverse problem, and provide accurate estimates for the unknown parameter due to the application of the approximation error model approach.     

Analytical solution of spray cooling characteristics on a hot surface using the Laplace transform

This study applies the Laplace transform method in conjunction with the sequential-in-time concept and experimental temperature data to determine the unknown surface temperature and surface heat flux on a hot surface of the test material. The advantage of the present inverse scheme is that the unknown surface temperature and surface heat flux on a hot surface are determined without any iterative process and the initial guess. More importantly, their functional forms do not need to be assumed in advance. However, the whole time domain may be divided into several sub-time intervals. Later, the appropriate polynomial function in each sub-time interval is selected to fit the experimental temperature data at each measurement location. The results show that the present results are not very sensitive to the measurement location. In order to validate the accuracy of the present method, two experimental examples and an analytical example are illustrated. The comparison between the present results, exact results and other existing estimated results is made.     

Reconstruction of the Surface Temperature of Arctic Glaciers from the Data of Temperature Measurements in Wells

Consideration is given to the problem of reconstruction of the surface temperature of a glacier from the data of measuring the temperature in a well. Mathematically this problem is an inverse problem for a heatconduction equation and refers to a number of incorrectly formulated (illposed) problems. For the reconstruction of the surface temperature, the Tikhonov regularization method has been used. A model that takes into account the vertical advection of the annual layers has been adopted as a mathematical model that describes the propagation of heat in a glacier. The boundary temperatures have been reconstructed from the results of temperature measurements in wells obtained for certain glaciers of the Arctic. The effect of the initial temperature distribution, the accumulation rate, and the magnitude of the geothermal heat flux on the reconstructed boundary temperature has been investigated.     

A comparative study of Domain Embedding Methods for regularized solutions of inverse Stefan problems

In this paper, various Domain Embedding Methods (DEMs) for an inverse Stefan problem are presented and compared. These DEMs extend the moving boundary domain to a larger, but simple and fixed domain. The original unknown interface position is then replaced by a new unknown, which can be a boundary temperature or heat flux, or an internal heat source. In this way, the non-linear identification problem is transformed into a linear one in the enlarged domain. Using different physical quantities as the new unknown leads to different DEMs. They are analysed from various points of view (accuracy, efficiency, etc.) through two test problems, by a comparison with a common Front-Tracking Method (FTM). The first test has a smooth temperature field and the second one has some singularities. The advantage of the DEMs in solving the inverse problem and in computing the corresponding direct mapping is shown. In the direct problem, high-order accurate schemes could be obtained more easily with the DEMs than with the FTM. In the inverse problem, an iterative regularization and a Tikhonov regularization have been employed. For the FTM, the iterative regularization is not efficient—the solution oscillates when the data are noisy. As for the Tikhonov regularization, it requests special care to choose an adequate penalty term. In contrast, both the regularizations give good results with all the considered DEMs, except for the second test problem at the beginning (t=0⁺) when the value of the heat flux and the heat source tends to ∞. Slightly different regularization effects have been obtained when using different DEMs. Finally, an automatic choice of the optimal regularization parameter is also discussed, using data with different noise levels. We propose the use of the curve of the residual norm against the regularization parameter. © 1997 John Wiley & Sons, Ltd.     

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.Paper presented at the Seventeenth European Conference on Thermophysical Properties, September 5–8, 2005, Bratislava, Slovak Republic.M. L?ffler: Deceased