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.

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.

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.

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.

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.

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.

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.

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.

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.

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.