带源参数的二维热传导反问题的无网格方法

利用无网格有限点法求解带源参数的二维热传导反问题，推导了相应的离散方程. 与 其它基于网格的方法相比，有限点法采用移动最小二乘法构造形函数，只需要节点信息，不 需要划分网格，用配点法离散控制方程，可以直接施加边界条件，不需要在区域内部求积分. 用有限点法求解二维热传导反问题具有数值实现简单、计算量小、可以任意布置节点等优点. 最后通过算例验证了该方法的有效性.     

Retrieving the heat transfer coefficient for jet impingement from transient temperature measurements

Algorithm of retrieving the heat transfer coefficient (HTC) from transient temperature measurements is presented. The unknown distributions of two types of boundary conditions: the temperature and heat flux are parameterized using a small number of user defined functions. The solutions of the direct heat conduction problems with known boundary temperature and flux are expressed as a superposition of auxiliary temperature fields multiplied by unknown parameters. Inverse problem is formulated as a least squares fit of calculated and measured temperatures and is cast in a form of a sum of two objective functions. The first results originates from an inverse problem for retrieving the boundary temperature the second comes from the inverse problem for reproducing the boundary heat flux. The final form of the objective function is obtained by enforcing constant in time value of the heat transfer coefficient. This approach leads to substantial regularization of the results, when compared with the standard technique, where HTC is calculated from separately reconstructed temperature and heat flux on the boundary. The validation of the numerical procedure is carried out by reconstructing a known distribution of the HTC using simulated measurements laden by stochastic error. The proposed approach is also used to reconstruct the distribution of the HTC in a physical experiment of heating a cylindrical sample using an impinging jet.     

New analytic solutions to 2D transient heat conduction problems with/without heat sources in the symplectic space

The two-dimensional (2D) transient heat conduction problems with/without heat sources in a rectangular domain under different combinations of temperature and heat flux boundary conditions are studied by a novel symplectic superposition method (SSM). The solution process is within the Hamiltonian system framework such that the mathematical procedures in the symplectic space can be implemented, which provides an exceptional direct rigorous derivation without any assumptions or predetermination of the solution forms compared with the conventional inverse/semi-inverse methods. The distinctive advantage of the SSM offers an access to new analytic heat conduction solutions. The results obtained by the SSM agree well with those obtained from the finite element method (FEM), which confirms the accuracy of the SSM.     

Analytical solution to the unsteady one-dimensional conduction problem with two time-varying boundary conditions: Duhamel’s theorem and separation of variables

An analytical resolution of the time-dependent one-dimensional heat conduction problem with time-dependent boundary conditions using the method of separation of variables and Duhamel’s theorem is presented. The two boundary conditions used are a time-dependent heat flux at one end and a varying temperature at the other end of the one-dimensional domain. It is put forth because the author found that the prescribed resolution method using separation of variables and Duhamel’s theorem presented in heat conduction textbooks is not directly applicable to problems with more than one time-dependent boundary condition. The analytical method presented in this paper makes use of one of the property of the heat conduction equation: the apparent linearity of the solutions. For that reason, in order to solve a problem with two time-dependent boundary conditions, the author first separates the initial problem into two independent but complementary problems, each with only one time-dependent boundary condition. Doing that, both simpler problems can be solved independently using a prescribed method that is known to work and the final solution can be obtained by joining the two independent solutions from the simpler separated problems. Every step of the resolution method is presented in this paper, along with a numerical validation of the final solution of three test case problems.     

二阶非定常多宗量热传导反问题的正则解

引入Bregman距离函数及其加权函数作为正则项，应用Tikhonov正则 化方法，对二阶非定常多宗量热传导反问题进行求解. 利用测量信息和计算信息构造最小二 乘函数，将多宗量反演识别问题转化为一个优化问题. 空间上采用8节点等参元进行离散， 时域上采用时域精细算法进行离散，建立了二阶非定常多宗量热传导问题的有限元正/反演数 值模型. 该模型不仅考虑了非均质和参数分布的影响，而且也便于正反演问题的敏度分析， 可对导热系数和边界条件等宗量进行有效的单一和组合识别. 给出了相关的数值验证，对信 息测量误差以及不同正则项的计算效率作了探讨. 数值结果表明，该方法能够对二阶非定常 多宗量热传导反问题进行有效的求解，并具有较高的计算精度.     

Periodic heat conduction with relaxation time in cylindrical geometry

Steady-periodic heat conduction with relaxation time in an infinitely long hollow cylinder is considered. Four boundary value problems, with boundary conditions of the first and of the second kind, are solved analytically. The solution for a solid cylinder with a sinusoidally varying surface temperature is obtained as a special case of a solution found for the hollow cylinder. The effects of the relaxation time on the steady-periodic temperature field are analysed, in details, for a solid cylinder with a sinusoidally varying surface temperature and for a hollow cylinder with a sinusoidally varying heat flux at the inner surface and with a constant temperature at the outer surface. The results show that thermal resonances may occur and suggest that accurate measurements of the relaxation time could be obtained by means of experiments on steady-periodic heat conduction in cylindrical geometry. Received on 15 April 1997     

PRECISE INTEGRAL ALGORITHM BASED SOLUTION FOR TRANSIENT INVERSE HEAT CONDUCTION PROBLEMS WITH MULTI-VARIABLES

IntroductionIHCPs (InverseHeatConductionProblems)arecloselyassociatedwithmanyengineeringaspects,andwelldocumentedintheliteratures,coveringtheidentificationsofthermalparameters[1,2 ],boundaryshapes[3],boundaryconditions[4 ]andsource_relatedterms[5 ,6 ]etc .Howeveritseemsthatonlylittleworkisdirectlyconcernedwithmulti_variablesidentificationsbyauthors’knowledge.Tsengetal.presentedanapproachtodeterminingtwokindsofvariables[7],butonlygavefewnumericalexamplestodeterminethemsimultaneously .Thesol…     

Heat flux sensor with minimal impact on boundary conditions

A technique for determining the heat transfer on the far surface of a wall based on measuring the heat flux and temperature on the near wall is presented. Although heat transfer measurements have previously been used to augment temperature measurements in inverse heat conduction methods, the sensors used alter the heat flow through the surface, disturbing the very quantity that is desired to be measured. The ideal sensor would not alter the boundary condition that would exist were the sensor not present. The innovation of this technique in that it has minimal impact on the wall boundary condition. Since the sensor is placed on the surface of the wall, no alteration of the wall is needed. The theoretical basis for the experimental technique as well as experimental results showing the heat flux sensor performance is presented.     

一种求解瞬态热传导方程的无条件稳定方法

瞬态热传导问题普遍存在于航空航天、土木和冶金等领域中, 对这类问题精确、高效的数值求解方法一直备受关注. 为此, 本文提出了一种无条件稳定的单步时间积分方法. 在所提出的方法中, 拉格朗日插值函数被用来近似真实的温度场及其一次导数场, 之后, 加权残量法被用来建立二者之间的关系. 通过对算法参数的巧妙设计, 本文提出的方法具有二阶精度和L型数值耗散, 其中, L型耗散使得本文方法能够快速过滤掉虚假的高频振荡. 多数现有时间积分方法仅对线性系统具有无条件稳定性, 对非线性系统则是条件稳定的. 为此, 本文改进了Hughes针对一阶非线性热传导问题提出的时间积分方法稳定性评估理论, 并将改进的理论用于方法的参数设计中. 理论分析的结果表明本文方法对线性和非线性热传导系统均是无条件稳定的. 即使对于著名的Crank-Nicolson方法失稳的非线性热传导问题, 本文方法仍能给出稳定且精确的预测. 数值测试结果显示, 所提出的方法相较于当前流行的方法具有明显的精度、耗散和稳定性优势.      

共轭梯度法求解非线性多宗量稳态传热反问题

应用共轭梯度法求解非线性多宗量稳态热传导反问题。采用八节点的等参单元在空间上进行离散,建立了便于敏度分析的非线性正演和反演的有限元模型,可直接求导进行敏度分析。给出了相关的数值验证,对测量误差及测点数目的影响作了初步探讨,结果表明,采用的算法能够对非线性稳态热传导中导热系数和边界条件联合反问题进行有效的求解,并具有较高精度。     

A boundary identification method for an inverse heat conduction problem with an application in ironmaking

A boundary identification problem in inverse heat conduction is studied, based on data from internal measurement of temperature and heat flux. Formulated as a sideways heat conduction equation, a spatial continuation technique is applied to extend the solution to a known boundary condition at the desired boundary position. Recording the positions traversed in the continuation for each time instant yields the boundary position trajectory and hence the solution of the identification problem. A prospective application of the method can be found in the ironmaking blast furnace, where it is desired to monitor the thickness of the accreted refractory wall based on measurement of its internal state. Simulations featuring noisy measurement data demonstrate the feasibility of the identification method for blast furnace wall thickness estimation.     

二阶非定常热传导反问题的多宗量辨识

引入Bregman距离构造同伦函数,建立了二阶非定常多宗量热传导反问题的一种求解模式,可对导热系数和边界条件等宗量进行识别。时域上采用精细算法,建立了便于敏度分析的有限元正/反演模型,对各宗量进行有效的组合识别。对信息测量误差和初值选取作了初步探讨,数值验证取得了满意的结果。     

Controlled random search technique for estimation of convective heat transfer coefficient

This paper is concerned with a method for solving inverse heat conduction problem. The method is based on the controlled random search (CRS) technique in conjunction with modified Newton–Raphson method. The random search procedure does not need the computation of derivative of the function to be evaluated. Therefore, it is independent of the calculation of the sensitivity coefficient for nonlinear parameter estimation. The algorithm does not depend on the future-temperature information and can predict convective heat transfer coefficient with random errors in the input temperature data. The technique is first validated against an analytical solution of heat conduction equation for a typical rocket nozzle. Comparison with an earlier analysis of inverse heat conduction problem of a similar experiment shows that the present method provides solutions, which are fully consistent with the earlier results. Once validated, the technique is used to investigate another estimation of heat transfer coefficient for an experiment of short duration, high heating rate, and employing indepth temperature measurement. The CRS procedure, in conjunction with modified Newton–Raphson method, is quite useful in estimating the value of the convective heat-transfer coefficient from the measured transient temperature data on the outer surface or imbedded thermocouple inside the rocket nozzle. Some practical examples are illustrated, which demonstrate the stability and accuracy of the method to predict the surface heat flux.     

Comparison of interfacial heat transfer coefficient estimated by two different techniques during solidification of cylindrical aluminum alloy casting

The interfacial heat transfer coefficient (IHTC) is necessary for accurate simulation of the casting process. In this study, a cylindrical geometry is selected for the determination of the IHTC between aluminum alloy casting and the surrounding sand mold. The mold surface heat flux and temperature are estimated by two inverse heat conduction techniques, namely Beck’s algorithm and control volume technique. The instantaneous cast and mold temperatures are measured experimentally and these values are used in the theoretical investigations. In the control volume technique, partial differential heat conduction equation is reduced to ordinary differential equations in time, which are then solved sequentially. In Beck’s method, solution algorithm is developed under the function specification method to solve the inverse heat conduction equations. The IHTC was determined from the surface heat flux and the mold surface temperature by both the techniques and the results are compared.     

多宗量瞬态热传导反演识别

引入Bregman距离加权函数,建立了多宗量瞬态热传导反演的一种求解模式.时域上采用精细算法,分别建立了便于敏度分析的有限元正/反演模型,应用同伦算法进行反问题求解,对导热系数和边界条件等宗量进行有效的组合识别.对信息误差和计算效率作了探讨,并给出了相应的数值验证.     

Axisymmetric non-fourier temperatures in cylindrically bounded domains

Heat conduction solutions are presented for the case where the material obeys a non-Fourier conduction law. In contrast to the Fourier law which predicts an infinite speed of heat propagation, the non-Fourier theory implies that the speed of thermal signals are finite. Axisymmetric problems for regions interior and exterior to a circular cylinder are investigated by using methods of Laplace transformation and asymptotic analysis. Comparisons of the temperature profiles are made with Fourier theory for the case of step function temperature boundary conditions.     

A semi-numerical method for solving inverse heat conduction problems

A semi-numerical method is presented for solving the inverse heat conduction problems in homogeneous and composite bodies. The presented solution does not require both the initial temperature distribution in the body and the whole temperature-time history at the temperature sensor locations. Sample calculations confirm that this approach produces stable and accurate results for both exact and noisy data. The extension of the method presented to two or three dimensions is straightforward.     

Inverse approach and sensitivity analysis for identification of ingot-mould thermal resistance in continuous casting of metals

The problem of appropriate location of the sensors for identification of ingot – mould thermal resistance during continuous casting of metals is the subject of the paper. Analysed problem belongs to the group of inverse problems. The present work shows also the method of identification of unknown thermal resistance using the temperature measurements at the number of sensors located in the wall of the mould. The influence of the location of the sensors on the results of identification is analysed. The best location of the sensors results from the sensitivity analysis for the steady-state inverse heat conduction problem. Validation of the proposed inverse method is realized by comparison of the results taken from solution of inverse and direct problems. Several numerical examples are presented and analysed.     

Heat wave phenomena in solids subjected to time dependent surface heat flux

Hyperbolic heat conduction in a plane slab, infinitely long solid cylinder and solid sphere with a time dependent boundary heat flux is analytically studied. The solution is based on the separation of variables method and Duhamel’s principle. The temperature distribution, the propagation and reflection of the temperature wave and the effect of geometry on the shape of the wave front are studied for the case of a rectangular pulsed boundary heat flux. Comparisons with the solution obtained for Fourier heat conduction are performed by considering the limit of a vanishing thermal relaxation time.     

Cavity detection in a heat conductor using linear sampling method

In this paper, solution of inverse problems in heat conduction transient fields is investigated. For this purpose, a new time-domain version of linear sampling method (TDLSM) is developed for cavity detection in a heat conductor. The linear sampling method (LSM) is an effective approach to image the geometrical features of unknown targets. Although this method has been used in the context of inverse scattering problems such as solid, acoustics, and electromagnetism, there is no specific attempt to apply this method to identification of cavities in heat conductors. This study emphasizes the implementation of the LSM in the time-domain fields using finite element method. A set of numerical simulations on two-dimensional transient heat conduction problems is presented to highlight many effective features of the proposed TDLSM fast qualitative identification method.