A New Hybrid Method for Solving Inverse Heat Conduction Problems

Solving the inverse problems, especially in the field of heat transfer, is one of the challenges of engineering due to its importance in industrial applications. It is well-known that inverse heat conduction problems (IHCPs) are severely ill-posed, which means that small disturbances in the input may cause extremely large errors in the solution. This paper introduces an accurate method for solving inverse problems by combining Tikhonov's regularization and the genetic algorithm. Finding the regularization parameter as the decisive parameter is modelled by this method, a few sample problems were solved to investigate the efficiency and accuracy of the proposed method. A linear sum of fundamental solutions with unknown constant coefficients assumed as an approximated solution to the sample IHCP problem and collocation method is used to minimize residues in the collocation points. In this contribution, we use Morozov's discrepancy principle and Quasi-Optimality criterion for defining the objective function, which must be minimized to yield the value of the optimum regularization parameter.

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.

Inverse Approach for the Average Convection Coefficient Induced by a Forced Fluid Flow Over an Annular Fin of Rectangular Profile Using Tip Temperature Measurements

The objective of the present paper is to develop a simple algebraic computational procedure for the estimation of the average convection coefficient of a forced fluid flow over an annular fin of rectangular profile within the platform of inverse heat conduction problems. The data required is the tip temperatures of an annular fin of rectangular profile, which are measured in an experimental setup. Based on nonlinear regression analysis, an empirical correlation equation is constructed for the dimensionless average tip temperature depending upon the dimensionless thermo–geometrical parameter and the radius ratio. When compared against the outcome of a direct heat conduction problem, the good quality of the estimated average convection coefficient for the annular fin of rectangular profile demonstrates the feasibility of the simple algebraic computational procedure.

Influence of Errors in Known Constants and Boundary Conditions on Solutions of Inverse Heat Conduction Problem

This work examines the effects of the known boundary conditions on the accuracy of the solution in one-dimensional inverse heat conduction problems. The failures in many applications of these problems are attributed to inaccuracy of the specified constants and boundary conditions. Since the boundary conditions and material properties in most thermal problems are imposed with uncertainty, the effects of their inaccuracy should be understood prior to the inverse analyses. The deviation from the exact solution has been examined for each case according to the errors in material properties, boundary location, and known boundary conditions. The results show that the effects of such errors are dramatic. Based on these results, the applicability and limitations of the inverse heat conduction analyses have been evaluated and discussed.

To the Solution of Geometric Inverse Heat Conduction Problems

On the basis of A. N. Tikhonov’s regularization theory, a method is developed for solving inverse heat conduction problems of identifying a smooth outer boundary of a two-dimensional region with a known boundary condition. For this, the smooth boundary to be identified is approximated by Schoenberg’s cubic splines, as a result of which its identification is reduced to determining the unknown approximation coefficients. With known boundary and initial conditions, the body temperature will depend only on these coefficients. With the temperature expressed using the Taylor formula for two series terms and substituted into the Tikhonov functional, the problem of determining the increments of the coefficients can be reduced to solving a system of linear equations with respect to these increments. Having chosen a certain regularization parameter and a certain function describing the shape of the outer boundary as an initial approximation, one can implement an iterative process. In this process, the vector of unknown coefficients for the current iteration will be equal to the sum of the vector of coefficients in the previous iteration and the vector of the increments of these coefficients, obtained as a result of solving a system of linear equations. Having obtained a vector of coefficients as a result of a converging iterative process, it is possible to determine the root-mean-square discrepancy between the temperature obtained and the temperature measured as a result of the experiment. It remains to select the regularization parameter in such a way that this discrepancy is within the measurement error. The method itself and the ways of its implementation are the novelty of the material presented in this paper in comparison with other authors’ approaches to the solution of geometric inverse heat conduction problems. When checking the effectiveness of using the method proposed, a number of two-dimensional test problems for bodies with a known location of the outer boundary were solved. An analysis of the influence of random measurement errors on the error in identifying the outer boundary shape is carried out.