Gradient method for inverse heat conduction problem in nanoscale

An inverse heat conduction problem for nanoscale structures was studied. The conduction phenomenon is modelled using the Boltzmann transport equation. Phonon‐mediated heat conduction in one dimension is considered. One boundary, where temperature observation takes place, is subject to a known boundary condition and the other boundary is exposed to an unknown temperature. The gradient method is employed to solve the described inverse problem. The sensitivity, adjoint and gradient equations are derived. Sample results are presented and discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

A Gaussian RBFs method with regularization for the numerical solution of inverse heat conduction problems

In this paper, a recursion numerical technique is considered to solve the inverse heat conduction problems, with an unknown time-dependent heat source and the Neumann boundary conditions. The numerical solutions of the heat diffusion equations are constructed using the Gaussian radial basis functions. The details of algorithms in the one-dimensional and two-dimensional cases, involving the global or partial initial conditions, are proposed, respectively. The Tikhonov regularization method, with the generalized cross-validation criterion, is used to obtain more stable numerical results, since the linear systems are badly ill-conditioned. Moreover, we propose some results of the condition number estimates to a class of positive define matrices constructed by the Gaussian radial basis functions. Some numerical experiments are given to show that the presented schemes are favourably accurate and effective.     

Global space–time multiquadric method for inverse heat conduction problem

In this paper, a radial basis collocation method (RBCM) based on the global space–time multiquadric (MQ) is proposed to solve the inverse heat conduction problem (IHCP). The global MQ is simply constructed by incorporating time dimension into the MQ function as a new variable in radial coordinate. The method approximates the IHCP as an over‐determined linear system with the use of two sets of collocation points: one is satisfied with the governing equation and another is for the given conditions. The least‐square technique is introduced to find the solution of the over‐determined linear system. The present work investigates two types of the ill‐posed heat conduction problems: the IHCP to recover the surface temperature and heat flux history on a source point from the measurement data at interior locations, and the backward heat conduction problem (BHCP) to retrieve the initial temperature distribution from the known temperature distribution at a given time. Numerical results of four benchmark examples show that the proposed method can provide accurate and stable numerical solutions for one‐dimensional and two‐dimensional IHCP problems. The sensitivity of the method with respect to the measured data, location of measurement, time step, shape parameter and scaling factor is also investigated. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Simultaneous reconstruction of the time-dependent Robin coefficient and heat flux in heat conduction problems

This paper aims to solve an inverse heat conduction problem in two-dimensional space under transient regime, which consists of the estimation of multiple time-dependent heat sources placed at the boundaries. Robin boundary condition (third type boundary condition) is considered at the working domain boundary. The simultaneous identification problem is formulated as a constrained minimization problem using the output least squares method with Tikhonov regularization. The properties of the continuous and discrete optimization problem are studied. Differentiability results and the adjoint problems are established. The numerical estimation is investigated using a modified conjugate gradient method. Furthermore, to verify the performance of the proposed algorithm, obtained results are compared with results obtained from the well-known finite-element software COMSOL Multiphysics under the same conditions. The numerical results show that the proposed algorithm is accurate, robust and capable of simultaneously representing the time effects on reconstructing the time-dependent Robin coefficient and heat flux.     

An inverse problem in estimating the relaxation time for nanoscale phonon radiative transfer problem

An inverse nanoscale phonon radiative transfer problem is solved in this study by using conjugate gradient method (CGM) to estimate the unknown frequency‐ and temperature‐dependent relaxation time, based on the simulated phonon intensity measurements. The CGM in dealing with the present integro‐differential governing equations is not as straightforward as for the normal differential equations; special treatments are needed to overcome the difficulties. Results obtained in this inverse analysis will be justified based on the numerical experiments where two different unknown distributions of relaxation time are to be estimated. Finally, it is shown that the reliable frequency and temperature‐dependent relaxation time can be obtained with CGM. Copyright © 2007 John Wiley & Sons, Ltd.     

Direct error in constitutive equation formulation for inverse heat conduction problem

We present a mixed numerical formulation that handles discontinuities well for scalar hyperbolic partial differential equations. The formulation is based on a least‐square error in the constitutive equation. It is motivated by scalar inverse diffusion problems with interior data and applies to convection of a passive scalar in a discontinuous compressible flow field. We motivate the need for a mixed formulation by discretizing using an irreducible finite element method and discuss some of the limitations of that approach. We then develop and prove that the mixed formulation is well posed and verify that it works for problems with continuous and discontinuous thermal conductivity distributions.     

Multivariate numerical derivative by solving an inverse heat source problem

A method for approximating multivariate numerical derivatives is presented from multidimensional noise data in this paper. Starting from solving a direct heat conduction problem using the multidimensional noise data as an initial condition, we conclude estimations of the partial derivatives by solving an inverse heat source problem with an over-specified condition, which is the difference of the solution to the direct problem and the given noise data. Then, solvability and conditional stability of the proposed method are discussed for multivariate numerical derivatives, and a regularized optimization is adopted for overcoming instability of the inverse heat source problem. For achieving partial derivatives successfully and saving amount of computation, we reduce the multidimensional problem to a one-dimensional case, and give a corresponding algorithm with a posterior strategy for choosing regularization parameters. Finally, numerical examples show that the proposed method is feasible and stable to noise data.     

橡胶注射机模具加热系统温度场模拟仿真

 热板是橡胶注射机不可缺少的模具加热元件,电热管为热板的热源,并按一定的热功率排布在热板内,使模具受热均匀．在热板的适当位置放置热电偶,用来自动控制热板温度,要求热板表面温差保持在+2~-2 ℃范围内．但一直以来,单凭人工排布形式,往往导致热板加热不均匀和温度控制精度低,直接影响热板使用性能,也是目前生产注射机企业碰到的最大难题．因此,采用有限元法,对热板进行瞬态温度场模拟仿真,得到了合理的热管排布方式,并在实际生产中证明切实可行．     

模拟退火算法在线热源反问题数值求解中的应用

提出采用模拟退火算法(simulated annealing,SA)来数值求解线热源反问题.探讨了如何设计算法使之适合反问题求解,并给出了算法求解的伪代码;通过线源正问题的模拟数据,使用设计的SA算法进行反问题求解,以此来验证算法求解的准确性和可靠性,并对一组实测数据进行了计算.结果表明,该算法不但可以实现两个参数同时、快速反演,而且具有求解精度高,对初始条件依赖少,编制容易等优点.     

Effects of wall thickness and material property on inverse heat conduction analysis of a hollow cylindrical tube

In this study, we examined the effects of a hollow cylindrical tube’s thickness and material properties on estimated time delay and waveform distortion in a one-dimensional inverse heat transfer analysis model using the thermal resistance method and an input estimation algorithm. Results indicated a persistent time delay for various heat flux amounts applied to different tube thicknesses. As the tube thickness increased, the numerically determined temperature data also experienced a time delay, which affected the inverse heat transfer response curve. Results also indicated that the transient heat flux waveform estimated for different material properties showed higher levels of distortion for materials having relatively low thermal conductivity. These materials also exhibited greater time delays. To address these issues, we applied a Fourier number (a dimensionless number representing the tube’s thickness and material properties) and proposed an equation to calculate sharpness, which can subsequently be used to predict probable time delays and heat flux waveform distortion. In conclusion, a correction is required when a low Fourier number is used in inverse heat transfer analysis.     

Regularization of the inverse heat conduction problem by the discrete Fourier transform

Two methods of solving the inverse heat conduction problem with employment of the discrete Fournier transform are presented in this article. The first one operates similarly to the SVD algorithm and consists in reducing the number of components of the discrete Fournier transform which are taken into account to determine the solution to the inverse problem. The second method is related to the regularization of the solution to the inverse problem in the discrete Fournier transform domain. Those methods were illustrated by numerical examples. In the first example, an influence of the boundary conditions disturbance by a random error on the solution to the inverse problem (its stability) was examined. In the second example, the temperature distribution on the inner boundary of the multiply connected domain was determined. Results of calculations made in both ways brought very good outcomes and confirm the usefulness of applying the discrete Fournier transform to solving inverse problems.     

An inverse solution for reconstruction of the heat transfer coefficient from the knowledge of two temperature values in a solid substrate

Reconstruction of the heat transfer coefficient from the knowledge of temperature distribution is an inverse problem. The main focus of this study was to develop an inverse model that could be used to determine the heat transfer coefficient associated with a fluid in contact with a solid surface from the knowledge of two measured temperature values (T₁ and T_M) in the solid substrate. The temperature distribution for the inverse problem was numerically generated, for a situation with a known heat transfer coefficient, using an implicit finite-differencing scheme. The solution domain was first discretized in to finite number of small regions and nodes. Conservation of energy was then applied to each of the control volume about the nodal regions. This approach resulted in a set of linear equations that was solved simultaneously. Two nodal temperatures in the substrate, from the direct solution, were then used in the inverse problem to reconstruct the heat transfer coefficient. To solve the inverse problem, the solution domain was divided into two distinct regions (Region I and Region II). Region I contained the solution domain between the two known temperatures (T₁ and T_M), and Region II included the region between T_M and the surface with the convective boundary condition. Again, a finite-differencing scheme was employed to generate a set of linear equations in each region. First, the set of linear equations in Region I was solved simultaneously and the results were then used to reconstruct the nodal temperatures in Region II. The convective surface temperature was then utilized to determine the heat transfer coefficient. A series of numerical experiments were conducted to test the validity of the inverse model. Comparison of the inverse solutions with the direct solutions confirms that the heat transfer coefficient can be reconstructed, with good accuracy, from the knowledge of two temperature points in the solid substrate.     

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     

The method of fundamental solutions for inverse source problems associated with the steady‐state heat conduction

This paper presents the use of the method of fundamental solutions (MFS) for recovering the heat source in steady‐state heat conduction problems from boundary temperature and heat flux measurements. It is well known that boundary data alone do not determine uniquely a general heat source and hence some a priori knowledge is assumed in order to guarantee the uniqueness of the solution. In the present study, the heat source is assumed to satisfy a second‐order partial differential equation on a physical basis, thereby transforming the problem into a fourth‐order partial differential equation, which can be conveniently solved using the MFS. Since the matrix arising from the MFS discretization is severely ill‐conditioned, a regularized solution is obtained by employing the truncated singular value decomposition, whilst the optimal regularization parameter is determined by the L‐curve criterion. Numerical results are presented for several two‐dimensional problems with both exact and noisy data. The sensitivity analysis with respect to two solution parameters, i.e. the number of source points and the distance between the fictitious and physical boundaries, and one problem parameter, i.e. the measure of the accessible part of the boundary, is also performed. The stability of the scheme with respect to the amount of noise added into the data is analysed. The numerical results obtained show that the proposed numerical algorithm is accurate, convergent, stable and computationally efficient for solving inverse source problems in steady‐state heat conduction. Copyright © 2006 John Wiley & Sons, Ltd.     

Fuzzy interval perturbation method for uncertain heat conduction problem with interval and fuzzy parameters

This paper proposes a fuzzy interval perturbation method (FIPM) and a modified fuzzy interval perturbation method (MFIPM) for the hybrid uncertain temperature field prediction involving both interval and fuzzy parameters in material properties and boundary conditions. Interval variables are used to quantify the non‐probabilistic uncertainty with limited information, whereas fuzzy variables are used to represent the uncertainty associated with the expert opinions. The level‐cut method is introduced to decompose the fuzzy parameters into interval variables. FIPM approximates the interval matrix inverse by the first‐order Neumann series, while MFIPM improves the accuracy by considering higher‐order terms of the Neumann series. The membership functions of the interval temperature field are eventually derived using the fuzzy decomposition theorem. Three numerical examples are provided to demonstrate the feasibility and effectiveness of the proposed methods for solving heat conduction problems with hybrid uncertain parameters, pure interval parameters, and pure fuzzy parameters, respectively. Copyright © 2015 John Wiley & Sons, Ltd.     

The discrepancy in the prediction of surface temperatures by inverse heat conduction models for different quenching processes from very high initial surface temperature

In the current work, an attempt has been made to study the effect of different parameters on the accuracy of the prediction at a very high initial surface temperature by developing two different heat conduction models. The result depicts that MSSE (minimum sum squared error) in the prediction decreases with increasing number of sensors used in the prediction. The accuracy of the prediction enhances with decreasing plate thickness and distance between the thermocouple and quenched surface. Up to a cooling rate of 60?K/s, the selection of model dimension (1-D or 2-D) does not affect, but beyond the previously mentioned cooling rate, 2-D model induces less error than 1-D. Moreover, the inclusion of thermo-physical properties in the model reduces the error in the MSSE. By using Box–Behnken methodology, the optimum conditions (d/D?=?0.81, n/Y?=?0.5 and Y*/Y?=?0.65) for the least MSSE have also been determined.     

Numerical method for the solution of non‐linear two‐dimensional inverse heat conduction problem using unstructured meshes

A new method of solving multidimensional heat conduction problems is formulated. The developed space marching method allows to determine quickly and exactly unsteady temperature distributions in the construction elements of irregular geometry. The method which is based on temperature measurements at the outer surface, is especially appropriate for determining transient temperature distribution in thick‐wall pressure components. Two examples are included to demonstrate the capabilities of the new approach. Copyright © 2000 John Wiley & Sons, Ltd.