Newton-type methods and their modifications for inverse heat conduction problems

This paper studies to numerical solutions of an inverse heat conduction problem.The effect of algorithms based on the Newton-Tikhonov method and the Newton-implicit iterative method is investigated,and then several modifications are presented.Numerical examples show the modified algorithms always work and can greatly reduce the computational costs.

Resolving double-sided inverse heat conduction problem using calibration integral equation method

In this paper, a novel calibration integral equation is derived for resolving double-sided, two-probe inverse heat conduction problem of surface heat flux estimation. In contrast to the conventional inverse heat conduction techniques, this calibration approach does not require explicit input of the probe locations, thermophysical properties of the host material and temperature sensor parameters related to thermal contact resistance, sensor capacitance and conductive lead losses. All those parameters and properties are inherently contained in the calibration framework in terms of Volterra integral equation of the first kind. The Laplace transform technique is applied and the frequency domain manipulations of the heat equation are performed for deriving the calibration integral equation. Due to the ill-posed nature, regularization is required for the inverse heat conduction problem, a future-time method or singular value decomposition (SVD) can be used for stabilizing the ill-posed Volterra integral equation of the first kind.

Measurement of boiling heat transfer coefficient in liquid nitrogen bath by inverse heat conduction method

Inverse heat conduction method (IHCM) is one of the most effective approaches to obtaining the boiling heat transfer coefficient from measured results. This paper focuses on its application in cryogenic boiling heat transfer. Experiments were conducted on the heat transfer of a stainless steel block in a liquid nitrogen bath, with the assumption of a 1D conduction condition to realize fast acquisition of the temperature of the test points inside the block. With the inverse-heat conduction theory and the explicit finite difference model, a solving program was developed to calculate the heat flux and the boiling heat transfer coefficient of a stainless steel block in liquid nitrogen bath based on the temperature acquisition data. Considering the oscillating data and some unsmooth transition points in the inverse-heat-conduction calculation result of the heat-transfer coefficient, a two-step data-fitting procedure was proposed to obtain the expression for the boiling heat transfer coefficients. The coefficient was then verified for accuracy by a comparison between the simulation results using this expression and the verifying experimental results of a stainless steel block. The maximum error with a revised segment fitting is around 6%, which verifies the feasibility of using IHCM to measure the boiling heat transfer coefficient in liquid nitrogen bath.

A dimension-reduced neural network-assisted approximate Bayesian computation for inverse heat conduction problems

Due to the flexibility and feasibility of addressing ill-posed problems,the Bayesian method has been widely used in inverse heat conduction problems(IHCPs).However,in the real science and engineering IHCPs,the likelihood function of the Bayesian method is commonly computationally expensive or analytically unavailable.In this study,in order to circumvent this intractable likelihood function,the approximate Bayesian computation(ABC)is expanded to the IHCPs.In ABC,the high dimensional observations in the intractable likelihood function are equalized by their low dimensional summary statistics.Thus,the performance of the ABC depends on the selection of summary statistics.In this study,a machine learning-based ABC(ML-ABC)is proposed to address the complicated selections of the summary statistics.The Auto-Encoder(AE)is a powerful Machine Learning(ML)framework which can compress the observations into very low dimensional summary statistics with little information loss.In addition,in order to accelerate the calculation of the proposed framework,another neural network(NN)is utilized to construct the mapping between the unknowns and the summary statistics.With this mapping,given arbitrary unknowns,the summary statistics can be obtained efficiently without solving the time-consuming forward problem with numerical method.Furthermore,an adaptive nested sampling method(ANSM)is developed to further improve the efficiency of sampling.The performance of the proposed method is demonstrated with two IHCP cases.

A Numerical Method on Inverse Determination of Heat Transfer Coefficient Based on Thermographic Temperature Measurement

The heat transfer coefficient in a multidimensional heat conduction problem is obtained from the solution of the inverse heat conduction problem based on the thermographic temperature measurement. The modified one-dimensional correction method （MODCM）, along with the finite volume method, is employed for both two- and three-dimensional inverse problems. A series of numerical experiments are conducted in order to verify the effectiveness of the method. In addition, the effect of the temperature measurement error, the ending criterion of the iteration, etc. on the result of the inverse problem is investigated. It is proved that the method is a simple, stable and accurate one that can solve successfully the inverse heat conduction problem.

Spatial interfacial heat transfer and surface characteristics during gravity casting of A356 alloy

As one of the key boundary conditions during casting solidification process, the interfacial heat transfer coefficient (IHTC) affects the temperature variation and distribution. Based on the improved nonlinear estimation method (NEM), thermal measurements near both bottom and lateral metal-mold interfaces throughout A356 gravity casting process were carried out and applied to solving the inverse heat conduction problem (IHCP). Finite element method (FEM) is employed for modeling transient thermal fields implementing a developed NEM interface program to quantify transient IHTCs. It is found that IHTCs at the lateral interface become stable after the volumetric shrinkage of casting while those of the bottom interface reach the steady period once a surface layer has solidified. The stable value of bottom IHTCs is 750 W/(m^2·℃), which is approximately 3 times that at the lateral interface. Further analysis of the interplay between spatial IHTCs and observed surface morphology reveals that spatial heat transfer across casting-mold interfaces is the direct result of different interface evolution during solidification process.

Numerical Methods for Inverse Problem of Heat Conduction with Unknown Boundary Based on Variational Principles with Variable Domain

In this article a variable-domain variational approach to the entitled problem is presented.A pair of comple- mentary variational principles with a variable domain in terms of temperature and heat-streamfunction are first established.Based on them,two methods of solution—generalized Ritz method and variable-domain FEM— both capable of handling problems with unknown boundaries,are suggested.Then,three sample numerical examples have been tested.The computational process is quite stable,and the results are encouraging.This variational approach can be extended straightforwardly to 3-D inverse problems as well as to other problems in mathematical physics.

Inverse analysis of thermal conductivities in transient non-homogeneous and non-linear heat conductions using BEM based on complex variable differentiation method

This paper presents a new inverse analysis approach to sensitivity analysis and material property identification in transient non-homogeneous and non-linear heat conduction Boundary Element Method (BEM) analysis based on Complex Variable Differentiation Method (CVDM). In this approach, the material properties are taken as the optimization variables, and the sensitivity coefficients are computed by CVDM. The advantages of using CVDM are that the computation of partial derivatives of an implicit function is reduced to function calculation in a complex domain, and the parameter sensitivity coefficients can be determined in a more accurate way than the traditional Finite Difference Method (FDM). Based on BEM and CVDM in evaluation of the sensitivity matrix of heat flux, the parameter such as thermal conductivity can be accurately identified. Six numerical examples are given to demonstrate the potential of the proposed approach. The results indicate that the presented method is efficient for identifying the thermal conductivity with single or multiple parameters.

A Novel Variational Formulation of Inverse Problem of Heat Conduction with Free Boundary on an Image Plane

By introducing an image plane, the inverse heat conduction problem with free boundary is transformed into one with completely known boundaryt which is much simpler to handle.As a by-product, the classical Kirchhoff’s transformation for accounting for variable conductivity is rederived and an invariance property of the inverse problem solution with respect to variable conductivity is indicated. Then a pair of complementary extremum principles are established on the image plane, providing a sound theoretical foundation for the Ritz’s method and finite element method (FEM).An example solved by FEM is also given.

A mixed Newton-Tikhonov method for nonlinear ill-posed problems

Newton type methods are one kind of the efficient methods to solve nonlinear ill-posed problems, which have attracted extensive attention. However, computational cost of Newton type methods is high because practical problems are complicated. We propose a mixed Newton-Tikhonov method, i.e., one step Newton-Tikhonov method with several other steps of simplified Newton-Tikhonov method. Convergence and stability of this method are proved under some conditions. Numerical experiments show that the proposed method has obvious advantages over the classical Newton method in terms of computational costs.

OPTIMAL ERROR BOUND IN A SOBOLEV SPACE OF REGULARIZED APPROXIMATION SOLUTIONS FOR A SIDEWAYS PARABOLIC EQUATION

The inverse heat conduction problem （IHCP） is a severely ill-posed problem in the sense that the solution （ if it exists） does not depend continuously on the data. But now the results on inverse heat conduction problem are mainly devoted to the standard inverse heat conduction problem. Some optimal error bounds in a Sobolev space of regularized approximation solutions for a sideways parabolic equation, i. e. , a non-standard inverse heat conduction problem with convection term which appears in some applied subject are given.

The Method of Fundamental Solution for a Radially Symmetric Heat Conduction Problem with Variable Coefficient

We consider an inverse heat conduction problem with variable coefficient on an annulus domain.In many practice applications,we cannot know the initial temperature during heat process,therefore we consider a non-characteristic Cauchy problem for the heat equation.The method of fundamental solutions is applied to solve this problem.Due to ill-posedness of this problem,we first discretize the problem and then regularize it in the form of discrete equation.Numerical tests are conducted for showing the effectiveness of the proposed method.

Finite Difference Approach for Estimating the Thermal Conductivity by 6-point Crank-Nicolson Scheme

Based on inverse heat conduction theory, a theoretical model using 6-point Crank-Nicolson finite difference scheme was used to calculate the thermal conductivity from temperature distribution, which can be measured experimentally. The method is a direct approach of second-order and the key advantage of the present method is that it is not required a priori knowledge of the functional form of the unknown thermal conductivity in the calculation and the thermal parameters are estimated only according to the known temperature distribution. Two cases were numerically calculated and the influence of experimental deviation on the precision of this method was discussed. The comparison of numerical and analytical results showed good agreement.

Semi-discretization Difference Approximation for a Cauchy Problem of Heat Equation in Two-dimensional Space

In this paper we consider a semi-descretization difference scheme for solving a Cauchy problem of heat equation in two-dimensional setting. Some error estimates are proved for the semi-descretization difference regularization method which cannot be fitted into the framework of regularization theory presented by Engl, Hanke and Neubauer. Numerical results show that the proposed method works well.

Experimental Investigation of Transient Heat Transferin the Once-Through Boiler Water Separator

The transient heat transfer between working fluid and inner wall of 600 MW supercritical pressureboiler's water separator model during start-up was experimentally explored. The test conditions forwater steam two-phase were as follows: pressure P = 3 ~ 11 MPa, mass now M = 0.2 ~ 0.35 kg/s,and mass quality x = O.2 ~ 0.8; for subcooled water P = 3 ~ 11 MPa, M = 0.2 ~ 0.35 kg/s, andsubcooled degree △θ = 10℃. By using an accurate and tractable inverse heat conduction solution,the correlations of single phase and twouphase transient heat transfer coefficients were obtained. Theexperimental results demonstrated the heat transfer enhanced for subcooled water and water steammixture compared with fully developed turbulent flow in tube. The increasing heat transfer effect maybe attributed to the deformation or destruction f the boundary layer.

A Modified Landweber Iteration for General Sideways Parabolic Equations

Abstract In this paper, we introduce a modified Landweber iteration to solve the sideways parabolic equation, which is an inverse heat conduction problem （IHCP） in the quarter plane and is severely ill-posed. We shall show that our method is of optimal order under both a priori and a posteriori stopping rule. Furthermore, if we use the discrepancy principle we can avoid the selection of the a priori bound. Numerical examples show that the computation effect is satisfactory.