On inverse boundary heat-conduction problems for recovery of heat fluxes to anisotropic bodies with nonlinear heat-transfer characteristics

A closed method is proposed for recovering heat fluxes to anisotropic bodies under conditions of aero-gasdynamic heating from experimental temperature data at spatial-temporal nodes. The thermal protection of a body is made of anisotropic materials with components of thermal-conductivity tensor, which are dependent of temperature, i.e., are nonlinear. The method is based on approximating a spatial dependence of a heat flux by a linear combination of basis functions with sought coefficients (parameters), which are found by minimization of a quadratic functional of the residual (the discrepancy between experimental and theoretical temperature values) using the implicit method of gradient descent, as well as on constructing and numerically solving problems for the determination of sensitivity coefficients. To increase the degree of correctness of an inverse problem, along with a main functional, the regularizing functionals have been constructed and utilized on the basis of smoothness requirements for spatial functions of heat fluxes to have continuous first and second derivatives, which allowed heat fluxes with the coupled heat transfer to be recovered in the form of arbitrary functions: monotonic, nonmonotonic, having extrema, inflection points, etc. Numerous results of recovering heat fluxes to anisotropic bodies are obtained and discussed, with the regularization parameter being selected for every case.     

Functional Identification of the Nonlinear Thermal-Conductivity Coefficient by Gradient Methods. II. Numerical Modeling

Algorithms for the gradient method of solution of the inverse problem on determination of the nonlinear thermal-conductivity coefficients are given. Results of numerical experiments are discussed. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 78, No. 4, pp. 75–81, July–August, 2005.     

Uncertainties Quantification of Effective Thermal Conductivity for Ceramic Fiber Blanket

In the present paper, a probabilistic propagation model for assessing the uncertainty of the effective thermal conductivity was developed based on a combined conduction and radiation heat transfer model of a ceramic fiber blanket composite. The Monte Carlo technique was used to cope with the uncertainties in the material density, radiative properties, and boundary temperatures observed in experimental tests. The calculated effective thermal-conductivity distribution for the sample was compared with the experimental measurements performed on multiple samples, and the predicted mean values were in good agreement with the measured data. The result validates the thermal predictive model and demonstrates the suitability of the stochastic model containing statistical distributions in the input variables. Statistical information also indicates that the uncertainty effect can be enlarged at high temperatures. Response sensitivity analysis between the random inputs and the effective thermal conductivity demonstrates that the randomness in the hot-side temperature, the cold-side temperature, and extinction coefficient of the sample has a significant influence on the variability of thermal-conductivity properties. The extinction coefficient becomes more and more important with an increase of temperature due to the dominant radiative heat transfer contribution at high temperature. The analysis provides good insight into the scattering control in the experimental measurement and theoretical prediction of the effective thermal conductivity of a ceramic fiber composite.     

An exact analytical solution of the inverse problem for a plasma cylinder expanding in a compressible medium

A strict analytical solution of the wave equation with cylindrical symmetry in a region with mobile boundaries was obtained by the method of inverse problems with allowance for the interaction of nonlinear arguments. The proposed method is universal and applicable to solving both inverse and direct problems for arbitrary values of the initial radius and displacements. The solution describes the near wave field of an expanding plasma piston, including the field formed in the initial moments of a pulsed expansion process. The solution gives exact values of a given pressure and velocity wave profile at a fixed point of the wave zone in the initial moment, as well as particular finite values of the pressure and velocity at a mobile boundary of the expanding plasma piston at the moments of time approaching zero. The solution is obtained with allowance for additional nonlinear conditions.     

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     

Determining Thermal Conductivity and Thermal Diffusivity of Low-Density Gases Using the Transient Short-Hot-Wire Method

The transient short-hot-wire method for measuring thermal conductivity and thermal diffusivity makes use of only one thermal-conductivity cell, and end effects are taken into account by numerical simulation. A search algorithm based on the Gauss–Newton nonlinear least-squares method is proposed to make the method applicable to high-diffusivity (i.e., low-density) gases. The procedure is tested using computer-generated data for hydrogen at atmospheric pressure and published experimental data for low-density argon gas. Convergence is excellent even for cases where the temperature rise versus the logarithm of time is far from linear. The determined values for thermal conductivity from experimental data are in good agreement with published values for argon, while the thermal diffusivity is about 10 % lower than the reference data. For the computer-generated data, the search algorithm can return both thermal conductivity and thermal diffusivity to within 0.02 % of the exact values. A one-dimensional version of the method may be used for analysis of low-density gas data produced by conventional transient hot-wire instruments.     

基于遗传算法的混凝土一维瞬态导热反问题

基于Laplace积分变换法和遗传算法,提出了一种混凝土一维瞬态导热反问题求解的新方法。运用Laplace变换将温度的求解表示为只与空间坐标及浇筑时间有关的函数,使得反问题的求解具有测点布置灵活的特点。运用遗传算法寻求非线性反演问题全局最优解,只需要若干点温度实测值便可实现多个热学参数的同时反演,算例对反演方法的反演精度及数值稳定性给出了满意的证明。     

Estimation of constant thermal conductivity by use of Proper Orthogonal Decomposition

An inverse approach is developed to estimate the unknown heat conductivity and the convective heat transfer coefficient. The method relies on proper orthogonal decomposition (POD) in order to filter out the higher frequency error. The idea is to solve a sequence of direct problems within the body under consideration. The solution of each problem is sampled at a predefined set of points. Each sampled temperature field, known in POD parlance as a snapshot, is obtained for an assumed value of the retrieved parameters. POD analysis, as an efficient mean of detecting correlation between the snapshots, yields a small set of orthogonal vectors (POD basis), constituting an optimal set of approximation functions. The temperature field is then expressed as a linear combination of the POD vectors. In standard applications, the coefficients of this combination are assumed to be constant. In the proposed approach, the coefficients are allowed to be a nonlinear function of the retrieved parameters. The result is a trained POD base, which is then used in inverse analysis, resorting to a condition of minimization of the discrepancy between the measured temperatures and values calculated from the model. Several numerical examples show the robustness and numerical stability of the scheme.     

Measurements of Hydrogen Thermal Conductivity at High Pressure and High Temperature

The thermal conductivity for normal hydrogen gas was measured in the range of temperatures from 323 K to 773 K at pressures up to 99 MPa using the transient short hot-wire method. The single-wire platinum probes had wire lengths of 10 mm to 15 mm with a nominal diameter of 10 μm. The volume-averaged transient temperature rise of the wire was calculated using a two-dimensional numerical solution to the unsteady heat conduction equation. A non-linear least-squares fitting procedure was employed to obtain the values of the thermal conductivity required for agreement between the measured temperature rise and the calculation. The experimental uncertainty in the thermal-conductivity measurements was estimated to be 2.2 % (k = 2). An existing thermal-conductivity equation of state was modified to include the expanded range of conditions covered in the present study. The new correlation is applicable from 78 K to 773 K with pressures to 100 MPa and is in agreement with the majority of the present thermal-conductivity measurements within ±2 %.     

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.     

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.     

A possible method for finding the value of indeterminate near wave field of an expanding cylinder

A strict analytical solution of the wave equation with cylindrical symmetry in a region with mobile boundaries was obtained by the method of inverse problems with an allowance for the interaction of nonlinear arguments. The method is universal and applicable to solving both inverse and direct problems for arbitrary values of the initial radius and displacements. The solution describes a near wave field of an expanding plasma piston, including that formed in the initial moments of a pulsed process.     

Reduced‐order modeling of parameterized PDEs using time–space‐parameter principal component analysis

This paper presents a methodology for constructing low‐order surrogate models of finite element/finite volume discrete solutions of parameterized steady‐state partial differential equations. The construction of proper orthogonal decomposition modes in both physical space and parameter space allows us to represent high‐dimensional discrete solutions using only a few coefficients. An incremental greedy approach is developed for efficiently tackling problems with high‐dimensional parameter spaces. For numerical experiments and validation, several non‐linear steady‐state convection–diffusion–reaction problems are considered: first in one spatial dimension with two parameters, and then in two spatial dimensions with two and five parameters. In the two‐dimensional spatial case with two parameters, it is shown that a 7 × 7 coefficient matrix is sufficient to accurately reproduce the expected solution, while in the five parameters problem, a 13 × 6 coefficient matrix is shown to reproduce the solution with sufficient accuracy. The proposed methodology is expected to find applications to parameter variation studies, uncertainty analysis, inverse problems and optimal design. Copyright © 2009 John Wiley & Sons, Ltd.     

Thermal-Diffusivity Measurement in Low Thermal-Conductivity Solids by a Transient Heating Method

A simple method for thermal-diffusivity measurement in low thermal-conductivity solids is proposed on a basis of the analytical solution of the heat diffusion equation for a solid plate with uniform continuous heating at one of its surfaces. The method involves combined measurements of the temperature evolution at both the front (illuminated) and rear surfaces of the sample in both the thermally thin and thick regimes. The principal advantage of the method is its independence on a knowledge of the convection and radiation heat loss coefficient, and hence, the non-necessity of performing measurements in blackbody similar samples under vacuum conditions. If these conditions are achieved, the thermal conductivity and specific (volume) heat capacity could also be achieved.     

An overdeterrnined two-dimensional transient inverse heat conduction problem

The method of solving the two-dimensional nonlinear inverse heat conduction problems is presented. The time- and space temperature distribution inside a solid and heat transfer coefficient distribution on the boundary of the solid is determined based on the temperature histories measured in several selected inside locations. The problem is overdetermined because the number of measuring points is higher than the number of calculated parameters (components of heat transfer coefficient).     

Inverse estimation of variable thermal parameters in a functionally graded annular fin using dragon fly optimization

This paper presents an inverse study of heat transfer of a conductive, convective and radiative annular fin made of a functionally graded material. Three major parameters such as conductive–convective parameter, conductive–radiative parameter and the parameter describing the variation of thermal conductivity are inversely estimated from a specified temperature field. The forward solution of temperature field is obtained from the closed form solution of nonlinear heat transfer equation using Homotopy perturbation method (HPM). A dragonfly algorithm that simulates the swarming behaviour of dragonflies, as analogous, is employed in finding out the inverse parameters. The temperature values of the forward solution are used as input data for the inverse analysis. The inverse parameters are then estimated iteratively by minimizing the objective function until the guessed temperature field approximately satisfies the preassigned temperature field of the forward solution. The inverse simulation following HPM-based forward solution converges faster than ordinary differential equation-based forward solution. The reconstructed temperature fields obtained from the various combination of inverse parameters give good agreement (～1% error) with the desired temperature field. Thus, the presented inverse model provides an opportunity to the fin designer for selecting the several feasible combinations of thermal parameters suggesting the material design that result in a prescribed temperature field.     

Solution of an Inverse Problem of Heat Conduction of 45 Steel with Martensite Phase Transformation in High Pressure during Gas Quenching

In order to simulate thermal strains,thermal stresses,residual stresses and microstructure of the steel during gas quenching by means of the numerical method,it is necessary to obtain an accurate boundary condition of temperature field.The surface heat transfer coefficient is a key parameter.The explicit finite difference method,nonlinear estimation method and the experimental relation between temperature and time during gas quenching have been used to solve the inverse problem of heat conduction.The relationship between surface temperature and surface heat transfer coefficient of a cylinder has been given.The nonlinear surface heat transfer coefficients include the coupled effects between martensitic phase transformation and temperature.     

基于同伦延拓的冗余机械臂逆运动学优化算法研究

目的 针对偏置冗余机械臂的逆运动学,采用传统数值法存在依赖初始值、奇异位姿收敛性差等问题,提出一种改进数值法。方法 首先将非线性方程组转化为同伦方程组,引入同伦延拓算法能够有效避免依赖初始值的问题,同时能够获取逆运动学解空间。然后考虑奇异位姿,将同伦方程组转化为最小二乘问题,采用Levenberg Marquardt算法对同伦方程组进行路径追踪,以获取逆运动学解空间。最后将关节极限避免问题映射为解空间优化问题,引入二进制改进粒子群优化算法,获得最优逆运动学解。结果 实验结果表明,相较于传统数值法,文中所提数值法针对逆运动学求解具有更高的收敛率、更快的收敛速度,同时二进制改进粒子群算法能够有效避免关节极限问题。结论 采用文中所提数值法求解逆运动学的精度较高,能够满足实时性要求,对于机械臂用于包装作业具有一定的理论意义和工程应用价值。     

Approximate analytic relationship for the efficiency of a thin radiating bar

An approximate relationship between the efficiency of a thin radiating bar and the dimensionless thermal-conductivity parameter is proposed. The temperature distribution over the bar length is obtained for a wide range of values of this parameter.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 17, No. 2, pp. 320–324, August, 1969.     

The resistivity, temperature coefficient of resistivity and thermoelectric power of thin continuous metal films II: A methodology for computer processing of thin film data to extract parameters with associated error estimates

A straightforward computer-based general methodology is presented which will enable parameter values and associated error estimates to be extracted from experimental thin film data points. The methodology operates on exact thin film relationships and overcomes problems in interpreting results, such as having to resort to the use of approximate thin film relationships.

The methodology is presented within the framework of the well-known Fuchs-Sondheimer model for conduction in thin continuous metal films. However, its general nature means that it is equally applicable to other theoretical thin film models. An illustration of the methodology's use is given by applying it to a set of thin film resistive, temperature coefficient of resistivity and thermoelectric power data obtained from measurements on thin continuous copper films.