Numerically solving twofold ill-posed inverse problems of heat equation by the adjoint Trefftz method

The inverse problem endowing with multiple unknown functions gradually becomes an important topic in the field of numerical heat transfer, and one fundamental problem is how to use limited minimal data to solve the inverse problem. With this in mind, in the present article we search the solution of a general inverse heat conduction problem when two boundary data on the space-time boundary are missing and recover two unknown temperature functions with the help of a few extra measurements of temperature data polluted by random noise. This twofold ill-posed inverse heat conduction problem is more difficult than the backward heat conduction problem and the sideways heat conduction problem, both with one unknown function to be recovered. Based on a stable adjoint Trefftz method, we develop a global boundary integral equation method, which together with the compatibility conditions and some measured data can be used to retrieve two unknown temperature functions. Several numerical examples demonstrate that the present method is effective and stable, even for those of strongly ill-posed ones under quite large noises.     

Inverse estimation of spatially and temporally varying heating boundary conditions of a two-dimensional object

In many dynamic heat transfer situations, the temperature at the heated boundary is not directly measurable and can be obtained by solving an inverse heat conduction problem (IHCP) based on measured temperature or/and heat flux at the accessible boundary. In this study, IHCP in a two-dimensional rectangular object is solved by using the conjugate gradient method (CGM) with temperature and heat flux measured at the boundary opposite to the heated boundary. The inverse problem is formulated in such a way that the heat flux at heated boundary is chosen as the unknown function to be recovered, and the temperature at the heated boundary is computed as a byproduct of the IHCP solution. The measurement data, i.e., the temperature and heat flux at the opposite boundary, are obtained by numerically solving a direct problem where the heated boundary of the object is subjected to spatially and temporally varying heat flux. The robustness of the formulated IHCP algorithm is tested for different profiles of heat fluxes along with different random errors of the measured heat flux at the opposite boundary. The effects of the uncertainties of the thermophysical properties and back-surface temperature measurement on inverse solutions are also examined.     

The determination temperature-dependent thermal conductivity as inverse steady heat conduction problem

The paper deals with the non-iterative inverse determination of the temperature-dependent thermal conductivity in 2-D steady-state heat conduction problem. The thermal conductivity is modeled as a polynomial function of temperature with the unknown coefficients. The identification of the thermal conductivity is obtained by using the boundary data and additionally from the knowledge of temperature inside the domain. The method of fundamental solutions is used to solve the 2-D heat conduction problem. The golden section search is used to find the optimal place for pseudo-boundary on which are placed the singularities in the frame of method of fundamental solutions.     

Application of the method of fundamental solutions and radial basis functions for inverse heat source problem in case of steady-state

The paper deals with the inverse determination of heat sources in steady 2-D heat conduction problem. The problem is described by Poisson equation in which the function of the right hand side is unknown. The identification of the strength of a heat source is given by using the boundary condition and a known value of temperature in chosen points placed inside the domain. For the solution of the inverse problem of identification of the heat source the method of fundamental solution with radial basis functions is proposed. The accurate results have been obtained for five test problems where the analytical solutions were available.     

Inverse Radiation Problem of Boundary Incident Radiation Heat Flux in Semitransparent Planar Slab with Semitransparent Boundaries

INTRODUCTI0NInverseradiati0nproblemshavedefinedasubjectofinterestf0rthepast3Oyears0nsoandthereex-istsac0nsiderablebody0fknowledgesurroundingthesubjectthathasbeenextensivelyreviewedinaseries0fpapersbyM.C.rmick[1-4].Theyarecon-cernedwiththedeterminati0noftheradiativepr0p-ertiesandthetemperaturedistributionsofmediaus-ingvari0ustypesofradiationmeasurements.Despitetherelativelylargeinterestexpressedininverseradia-tionproblems,mostoftheworkfocusedontheinverseestimati0noftemperaturedistributions…     

A decentralized fuzzy inference method for solving the two-dimensional steady inverse heat conduction problem of estimating boundary condition

This paper addresses a new technique for solving the two-dimensional steady inverse heat conduction problem, which named decentralized fuzzy inference (DFI) method. First of all, a group of decentralized fuzzy inference units are designed, and the fuzzy inference for each fuzzy inference unit is conducted which bases on the difference between the measured and the computed temperature at each measuring location. The computed temperatures are obtained by solving the direct heat conduction problem with the finite difference method. And then, inference results of fuzzy inference units are weighted to yield compensation values of the unknown boundary temperatures. The unknown boundary temperatures are estimated by updating guess temperatures continuously with compensation values. Numerical experiments are carried out with different initial guesses, the number of measuring points and measurement errors. Comparing results of DFI method and Levenberg–Marquardt (L–M) method, we can conclude that DFI method is valid.     

Recovering A Heat Source and Initial Value by a Lie-Group Differential Algebraic Equations Method

We consider an inverse problem of a nonlinear heat conduction equation for recovering unknown space-dependent heat source and initial condition under Cauchy-type boundary conditions, which is known as a sideways heat equation. With the aid of two extra measurements of temperature and heat flux which are being polluted by noisy disturbances, we can develop a Lie-group differential algebraic equations (LGDAE) method to solve the resulting differential algebraic equations, and to quickly recover the unknown heat source and initial condition simultaneously. Also, we provide a simple LGDAE method, without needing extra measurement of heat flux, to recover the above two unknown functions. The estimated results are quite promising and robust enough against large random noise.     

Inverse shape design for heat conduction problems via the ball spine algorithm

ABSTRACT

In this article, a novel iterative physical-based method is introduced for solving inverse heat conduction problems. The method extends the ball spine algorithm concept, originally developed for inverse fluid flow problems, to inverse heat conduction problems by employing a subtle physical-sense rule. The inverse problem is described as a heat source embedded within a solid medium with known temperature distribution. The object is to find a body configuration satisfying a prescribed heat flux originated from a heat source along the outer surface. Performance of the proposed method is evaluated by solving many 2-D inverse heat conduction problems in which known heat flux distribution along the unknown surface is directly related to the Biot number and surface temperature distribution arbitrarily determined by the user. Results show that the proposed method has a truly low computational cost accompanied with a high convergence rate.     

Two-Dimensional Inverse Heat Transfer Analysis of Functionally Graded Materials in Estimating Time-Dependent Surface Heat Flux

Two-dimensional transient inverse heat conduction problem (IHCP) of functionally graded materials (FGMs) is studied herein. A combination of the finite element (FE) and differential quadrature (DQ) methods as a simple, accurate, and efficient numerical method for FGMs transient heat transfer analysis is employed for solving the direct problem. In order to estimate the unknown boundary heat flux in solving the inverse problem, conjugate gradient method (CGM) in conjunction with adjoint problem is used. The results obtained show good accuracy for the estimation of boundary heat fluxes. The effects of measurement errors on the inverse solutions are also discussed.     

Identification of boundary conditions for non-Fourier heat conduction problems by differential transformation DRBEM and improved cuckoo search algorithm

Abstract

The differential transformation method is combined with the dual reciprocity boundary element method to solve the non-Fourier heat conduction problems in functionally gradient materials. The cuckoo search algorithm is improved by the Broyden–Fletcher–Goldfarb–Shanno algorithm to identify the boundary conditions for the heat conduction problems. The polynomial function related to coordinate and time is proposed to approximate the unknown boundary conditions. Numerical examples discuss the influences of measurement point numbers and measurement errors on inverse solutions. Numerical results demonstrate the effectiveness and accuracy of the proposed method.     

A SERIAL SOLUTION FOR THE 2-D INVERSE HEAT CONDUCTION PROBLEM FOR ESTIMATING MULTIPLE HEAT FLUX COMPONENTS

A serial algorithm for the inverse heat conduction problem (IHCP) has been developed to estimate the individual flux components, one by one, at the unknown boundary, based on the function specification method. The sensitivity coefficient defined in this algorithm brings out the influence of the heat flux components independent of each other. The objective function minimizes the difference in the measured temperature and the contribution of the individual flux component to the thermal field at the sensor location. The serial algorithm developed here could be used with data from both overspecified and underspecified sensors with respect to the number of flux components. The method was tested for delineating independent heat fluxes at the boundary of a two-dimensional solid for both space- and time-varying heat fluxes. Simulated thermal histories obtained from direct solution were used as inputs for the inverse problem for characterizing the new algorithm.

Three types of analyses were done on the results of the IHCP, focused on (1) the convergence of error in estimated temperatures at the different sensor locations, (2) overall error in estimated temperatures for the whole domain, and (3) the total heat energy transferred across the boundary. It is shown that the optimum configuration of independent unknown fluxes is given by the one with minimum energy estimates across the boundary, for both cases.     

A finite element analysis for inverse heat conduction problems

This paper presents an efficient inverse analysis technique based on a sensitivity coefficient algorithm to estimate the unknown boundary conditions of multidimensional steady and transient heat conduction problems. Sensitivity coefficients were used to represent the temperature response of a system under unit loading conditions. The proposed method, coupled with the sensitivity analysis in the finite element formulation, is capable of estimating both the unknown temperature and heat flux on the surface provided that temperature data are given at discrete points in the interior of a solid body. Inverse heat conduction problems are referred to as ill-posed because minor inaccuracy or error in temperature measurements cause a drastic effect on the predicted surface temperature and heat flux. To verify the accuracy and validity of the new method, two-dimensional steady and transient problems are considered. Their surface temperature and heat flux are evaluated. From a comparison with the exact solution, the effects of measurement accuracy, number and location of measuring points, a time step, and regularization terms are discussed. © 1998 Scripta Technica. Heat Trans Jpn Res, 26(6): 345–359, 1997     

Structured Multiblock Grid Solution of Two-Dimensional Transient Inverse Heat Conduction Problems in Cartesian Coordinate System

ABSTRACT

In this study a structured multiblock grid is used to solve two-dimensional transient inverse heat conduction problems. The multiblock method is implemented for geometric decomposition of the physical domain into regions with blocked interfaces. The finite-element method is employed for direct solution of the transient heat conduction equation in a Cartesian coordinate system. Inverse algorithms used in this research are iterative Levenberg-Marquardt and adjoint conjugate gradient techniques for parameter and function estimations. The measured transient temperature data needed in the inverse solution are given by exact or noisy data. Simultaneous estimation of unknown linear/nonlinear time-varying strengths of two heat sources in two joined surfaces with equal and different heights is obtained for the solution of the inverse problems, and the results of the present study for unknown heat source functions are compared to those of exact functions. This study is an attempt to challenge the goal of combining a multiblock technique with inverse analysis methods. In fact, the structured multiblock grid is capable of providing accurate solutions of inverse heat conduction problems (IHCPs) in industrial configurations, including composite structures. In addition, the multiblock IHCP solver is suitable to estimate unknown parameters and functions in these structures.     

On estimating thermal diffusivity using analytical inverse solution for unsteady one-dimensional heat conduction

A modified procedure for calculating the thermal diffusivity of solids based on temperature measurements at two points and the semi-infinite boundary condition is presented. The method makes use of a solution to the unsteady one-dimensional inverse heat conduction problem for the semi-infinite solid. The procedure gives accurate results based on temperature changes produced by an arbitrary fluctuating heat flux source at the boundary.     

A simple direct estimation of temperature-dependent thermal conductivity with kirchhoff transformation

A direct method is proposed to estimate the temperature-dependent thermal conductivity without internal measurements. In the proposed method, the steady-state nonlinear heat conduction equation is transformed into the Laplace equation via the Kirchhoff transformation. The thermal conductivity is modeled as a linear combination of known functions with unknown coefficients, which are directly determined from the imposed heat flux and measured temperatures at the boundary. Several inverse heat conduction problems are successfully introduced to confirm the validity of the proposed method.     

Implementation of Geometrical Domain Decomposition Method for Solution of Axisymmetric Transient Inverse Heat Conduction Problems

The objective of this article is to study the performance of iterative parameter and function estimation techniques to solve simultaneously two unknown functions (quadratic in time, and linear in time and space) using transient inverse heat conduction method in conjunction with a geometrical domain decomposition approach, in cylindrical coordinates. For geometrical decomposition of physical domain, a multi-block method has been used. The numerical scheme for the solution of the governing partial differential equations is the finite element method. The results of the present study for a configuration composed of two joined disks with different heights are compared to those of exact heat source and temperature boundary condition using inverse analysis. Good agreement between the estimated results and exact functions has been observed for parameter estimation techniques in contrast to those of function estimation approach. In summary, the results show that the function estimation technique is sensitive to the location of measurement points, but is useful to estimate unknown functions without a priori knowledge of the functions' spatial and/or temporal distributions. However, the function estimation technique suffers from a drawback: its implementation and data extraction are less straightforward than parameter estimation method. Finally, it is shown that the use of geometrical domain decomposition offers the possibility of developing a robust inverse analysis code for general purpose heat conduction problems.     

An inverse problem in estimating the base heat flux of an annular fin based on the hyperbolic model of heat conduction

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to solve the inverse hyperbolic heat conduction problem in estimating the unknown time-dependent base heat flux of an annular fin from the knowledge of temperature measurements taken within the fin. The inverse solutions will be justified based on the numerical experiments in which two specific cases to determine the unknown base heat flux are examined. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The influence of measurement errors upon the precision of the estimated results is also investigated. Results show that an excellent estimation on the time-dependent base heat flux can be obtained for the test cases considered in this study.     

Fourier analysis of conjugate gradient method applied to inverse heat conduction problems

The convergence and regularization mechanism of the conjugate gradient algorithm applied to inverse heat conduction problems are studied within the context of a Fourier analysis, for a square enclosure subjected to an unknown time-varying heat flux on one side, and to known boundary conditions on the remaining sides. Analytic solutions are derived for the Fourier components of the unknown flux over a given time interval. The convergence rate of the algorithm is thereby shown to depend essentially on the time frequency of the data. Numerical solutions are also presented to describe in details the convergence process and solution regularization power of the conjugate gradient method, when the unknown heat flux contains many frequency components and the measurement data are noisy. It is found that an unknown time-dependent heat flux may be satisfactorily recovered using a single sensor even when the temperature field becomes two-dimensional, and that the sensor should be placed in a symmetric manner for better results.     

The Inverse Reconstruction of the Heat Transfer Coefficient for the Free Surface Water Jet

This article presents an application of inverse algorithm for reconstruction of heat transfer coefficient (HTC) for a water jet impinging a flat surface. Such an approach, allows for decoupling complex fluid flow from heat conduction in a solid impinged by jet. The approach starts with parameterization of a functional form of unknown boundary temperature and heat flux occurring at the fluid–solid interface. Later, Newton's law of cooling is used to force temporal invariability of HTC. Unknown coefficients of HTC distribution are determined from a least square fit between measured and computed temperatures. Temperatures entering the objective function are recorded by an infrared camera at the surface opposite to impinged one.     

Inverse Hyperbolic Heat Conduction in Fins with Arbitrary Profiles

This study aims to estimate unknown base temperature distribution in different non-Fourier fins. The Cattaneo–Vernotte (CV) heat model is used to predict the heat conduction behavior in these fins. This inverse problem is solved by the function-estimation version of the Adjoint conjugate gradient method (ACGM) based on boundary temperature measurements. The ACGM includes direct, sensitivity, and adjoint problems. For each of these problems, a one-dimensional general formulation of the non-Fourier model for longitudinal fins with arbitrary profile is driven and solved by an implicit finite difference method. In this study, three different profiles are considered: triangular, convex parabolic, and concave parabolic. For each of them, two different base temperature distributions are estimated using an inverse method. Moreover, the effects of sensor positions at the fin tip and a specific place in-between are considered on the base temperature estimation. A close agreement between the exact values and the estimated results is found, confirming the validity and accuracy of the proposed method. The results show that the ACGM is an accurate and stable method to determine the thermal boundary conditions in different non-Fourier fin problems.