A Lie-group shooting method for reconstructing a past time-dependent heat source

We consider an inverse problem for the reconstruction of a past unknown time-dependent heat source H(t), t < t_f, in a heat conduction equation T_t(x,t) = T_xx(x,t) + H(t) with the aid of an extra measurement of temperature gradient on the left-boundary, where a final time condition is measured at the present terminal time t_f. This inverse problem is quite difficult to be solved numerically owing to a twofold ill-posedness, as a combination of the backward heat conduction problem and the inverse heat source identification problem, which is abbreviated as inverse heat source/backward heat conduction problem (IHSBHCP). The new method proposed here, namely the Lie-group shooting method (LGSM), is examined through the tests by several numerical examples. Although the recovery of an unknown heat source is carried out under a presently measured temperature at a final time and under a large measurement noise both imposed on the final time data and all boundary data, the LGSM still works effectively and accurately. The accuracy in the reconstruction of H(t) is almost uneffected by different levels of noise.     

Heat transfer over a nonlinearly stretching sheet with non-uniform heat source and variable wall temperature

In this paper we study the flow and heat transfer characteristics of a viscous fluid over a nonlinearly stretching sheet in the presence of non-uniform heat source and variable wall temperature. A similarity transformation is used to transform the governing partial differential equations to a system of nonlinear ordinary differential equations. An efficient numerical shooting technique with a fourth-order Runge–Kutta scheme is used to obtain the solution of the boundary value problem. The effects of various parameters (such as the power law index n, the Prandtl number Pr, the wall temperature parameter λ, the space dependent heat source parameter A¹ and the temperature dependent heat source parameter B¹) on the heat transfer characteristics are analyzed. The numerical results for the heat transfer coefficient (the Nusselt number) are presented for several sets of values of the parameters and are discussed. The results reveal many interesting behaviors that warrant further study on the effects of non-uniform heat source and the variable wall temperature on the heat transfer phenomena at the nonlinear stretching sheet.     

Recovering both the space-dependent heat source and the initial temperature by using a fast convergent iterative method

In this paper, we solve two types of inverse heat source problems: one recovers an unknown space-dependent heat source without using initial value, and another recovers both the unknown space-dependent heat source and the initial value. Upon inserting the adjoint Trefftz test functions into Green’s second identity, we can retrieve the unknown space-dependent heat source by an expansion method whose expansion coefficients are derived in closed form. We assess the stability of the closed-form expansion coefficients method by using the condition numbers of coefficients matrices. Then, numerical examples are performed, which demonstrates that the closed-form expansion coefficient method is effective and stable even when it imposes a large noise on the final time data. Next, we develop a coupled iterative scheme to recover the unknown heat source and initial value simultaneously, under two over specified temperature data at two different times. A simple regularization technique is derived to overcome the highly ill-posed behavior of the second inverse problem, of which the convergence rate and stability are examined. This results in quite accurate numerical results against large noise.     

A parameter estimation approach to solve the inverse problem of point heat sources identification

This paper deals with an inverse problem that consists of the identification of multiple line heat sources placed in a homogeneous domain. In the inverse problem under investigation the location and strength of the line heat sources are unknown. The estimation procedure is based on the boundary element method. As the discrete problem is non-linear if the location of the line heat sources is unknown, an iterative procedure has to be applied to find out the location of the sources. The proposed approach has been tested for steady and transient experiments. In the present study we propose an original approach to solve the steady problem. As in the steady heat conduction case we have a limited number of unknown for each source, a “parameter estimation” approach can be applied to estimate the sources. Using the techniques of parameter estimation, we can also estimate the confidence interval of the estimated locations, which permits to design an optimal experiment. We intend to present some numerical and experimental 2D results.     

A self-adaptive LGSM to recover initial condition or heat source of one-dimensional heat conduction equation by using only minimal boundary thermal data

The present study is concerned with the recovery of an unknown initial condition for a one-dimensional heat conduction equation by using only the usual two boundary conditions of the direct problem for heat equation. The algorithm assumes a function for the unknown initial condition and derives an inverse problem for estimating a spatially-dependent heat source F(x) in T_t(x, t) = T_xx(x, t) + F(x). A self-adaptive Lie-group shooting method, namely a Lie-group adaptive method (LGAM), is developed to find F(x), and then by integrations or by solving a linear system we can extract the information for unknown initial condition. The new method possesses twofold advantages in that no a priori information of unknown functions is required and no extra data are needed. The accuracy and efficiency of present method are confirmed by comparing the estimated results with some exact solutions.     

An inverse problem of finding the time-dependent thermal conductivity from boundary data

We consider the inverse problem of determining the time-dependent thermal conductivity and the transient temperature satisfying the heat equation with initial data, Dirichlet boundary conditions, and the heat flux as overdetermination condition. This formulation ensures that the inverse problem has a unique solution. However, the problem is still ill-posed since small errors in the input data cause large errors in the output solution. The finite difference method is employed as a direct solver for the inverse problem. The inverse problem is recast as a nonlinear least-squares minimization subject to physical positivity bound on the unknown thermal conductivity. Numerically, this is effectively solved using the lsqnonlin routine from the MATLAB toolbox. We investigate the accuracy and stability of results on a few test numerical examples.     

Investigation of heat transfer characteristics in plate-fin heat sink

The present study applies the inverse method in conjunction with the experimental temperature data to investigate the accuracy of the heat transfer coefficient on the fin in the plate-fin heat sink for various fin spacings. The commercial software is applied to solve the governing differential equations with the RNG k–? model in order to obtain the heat transfer and fluid flow characteristics. Under the assumption of the non-uniform heat transfer coefficient, the entire fin is divided into several sub-fin regions before performing the inverse scheme. The average heat transfer coefficient in each sub-fin region is assumed to be unknown. Later, the present inverse scheme in conjunction with the experimental temperature data is applied to determine the heat transfer coefficient and fin efficiency. In order to determine a more reliable heat transfer coefficient, a comparison between the present inverse and numerical results and those obtained from the existing correlations will be made. The numerical fin temperatures will also be compared with the experimental data.     

A modified genetic algorithm for solving the inverse heat transfer problem of estimating plan heat source

This work considers a new approach for solving the inverse heat conduction problem of estimating unknown plan heat source. It is shown that the physical heat transfer problem can be formulated as an optimization problem with differential equation constraints. A modified genetic algorithm is developed for solving the resulting optimization problem. The proposed algorithm provides a global optimum instead of a local optimum of the inverse heat transfer problem with highly-improved convergence performance. Some numerical results are presented to demonstrate the accuracy and efficiency of the proposed method.     

The BEM for point heat source estimation: application to multiple static sources and moving sources

This paper deals with an inverse problem, which consists of the identification of point heat sources in a homogeneous solid in transient heat conduction. The location and strength of the line heat sources are both unknown. For a single source we examine the case of a source which moves in the system during the experiment. The two-dimensional and three-dimensional linear heat conduction problems are considered here. The identification procedure is based on a boundary integral formulation using transient fundamental solutions. The discretized problem is non-linear if the location of the line heat sources is unknown. In order to solve the problem we use an iterative procedure to minimize a quadratic norm. The proposed numerical approach is applied to experimental 2D examples using measurements provided by an infrared scanner for surface temperatures and heat fluxes. A numerical example is presented for the 3D application.     

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.     

Study on heat conduction of functionally graded plate with the variable gradient parameters under the H(t) heat source

Abstract

A hybrid numerical method for heat conduction of functionally graded plate with the variable gradient parameters under the H(t) heat source was studied. A weighted residual equation of heat conduction was considered under thermal boundary conditions. In order to calculate temperature distribution of functionally graded plate with variable gradient parameters, the Fourier transform and inverse Fourier transform were applied and the temperature field was obtained under the H(t) heat source. Results show that the influences of the gradient parameters on temperature distribution are dramatic. But with the increase of gradient parameters, the influences of parameters on the temperature distribution are gradually reduced. When the gradient parameters reach a certain critical value, the temperature does not change anymore. By comparing the temperature distribution of the upper and lower surfaces, it is seen that the temperature presents a gentle downward trend with the increase of the heat source distance, while the temperature does not change with the time in farther distance from heat source. Also, the results show that the influence of the heat source has only partial and limited influence on the temperature, which is in accordance with St. Venant’s Principle. The law of the temperature distribution of the lower surface varies with the gradient parameters, which is also discussed, an optimal gradient parameter with the thermal insulation effect of the functionally graded plate is obtained.     

A regularized integral equation method for the inverse geometry heat conduction problem

This paper addresses a new technique for solving the inverse geometry heat conduction problem of the Laplace equation in a two-dimensional rectangle, which, named regularized integral equation method (RIEM), consists of three parts. First of all, the Fourier series expansion technique is used to calculate the temperature field u(x, y). Second, we consider a Lavrentiev regularization by adding a term αg(x) to obtain a second kind Fredholm integral equation. The termwise separable property of the kernel function allows us to transform the inverse geometry heat conduction problem into a two-point boundary value problem and therefore, an analytical regularized solution is derived in the final part by using orthogonality. Principally, the RIEM possesses the following advantages: it does not need any guess of the initial profile, it does not need any iteration and a regularized closed-form solution can be obtained. The uniform convergence and error estimate of the regularized solution u^α(x, y) are proved and a boundary geometry p(x) is solved by half-interval method. Several numerical examples present the effectiveness of our novel approach in providing excellent estimates of unknown boundary shapes from given data.     

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.     

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.     

To recover heat source G(x) + H(t) by using the homogenized function and solving rectangular differencing equations

ABSTRACT

For recovering an unknown heat source F(x, t) = G(x) + H(t) in the heat conduction equation, we develop a homogenized function method and the expansion methods by polynomials or eigenfunctions, which can solve the inverse heat source recovery problem by using collocation technique. Because the initial condition/boundary conditions/supplementary condition are satisfied automatically and a rectangular differencing technique is developed, a middle-scale linear system is sufficient to determine the expansion coefficients. After deriving a multiscale postconditioning matrix, the present methods converge very quickly, and are accurate and stable against large noise, as verified by numerical tests.     

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.     

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.     

Identification of spacewise and time dependent source terms in 1D heat conduction equation from temperature measurement at a final time

Inverse problems of identifying the unknown spacewise and time dependent heat sources F(x) and H(t) of the variable coefficient heat conduction equation u_t = (k(x)u_x)_x + F(x)H(t) from supplementary temperature measurement (u_T(x)?u(x, T_f)) at a given single instant of time T_f > 0, are investigated. For both inverse source problems, defined to be as ISPF and ISPH respectively, explicit formulas for the Fréchet gradients of corresponding cost functionals are derived. Fourier analysis of these problems shows that although ISPF has a unique solution, ISPH may not have a unique solution. The conjugate gradient method (CGM) with the explicit gradient formula for the cost functional J₁(F) is then applied for numerical solution of ISPF. New collocation algorithm, based on the piecewise linear approximation of the unknown source H(t), is proposed for the numerical solution of the integral equation corresponding to ISPH. The proposed two numerical algorithms are examined through numerical examples for reconstruction of continuous and discontinuous heat sources F(x) and H(t). Computational results, with noise free and noisy data, show efficiency and high accuracy of the proposed algorithms.     

Multiple transient point heat sources identification in heat diffusion: application to experimental 2D problems

This paper deals with an inverse problem, which consists of the experimental identification of line heat sources in a homogeneous solid in transient heat conduction. The location and strength of the line heat sources are both unknown. For a single source we examine the case of a source which moves in the system during the experiment. The identification procedure is based on a boundary integral formulation using transient fundamental solutions. The discretized problem is non-linear if the location of the line heat sources is unknown. In order to solve the problem we use an iterative procedure to minimize a cost function comparing the modelled heat source term and the measurements. The proposed numerical approach is applied to experimental 2D examples using measurements provided by an infrared scanner for surface temperatures and heat fluxes. In some particular examples, internal thermocouples can be used. A time regularization procedure associated to future time-steps is used to correctly solve the ill-posed problem.     

INVERSE FORCED CONVECTION PROBLEM OF SIMULTANEOUS ESTIMATION OF TWO BOUNDARY HEAT FLUXES IN IRREGULARLY SHAPED CHANNELS

This article deals with the use of the conjugate gradient method of function estimation for the simultaneous identification of two unknown boundary heat fluxes in channels with laminar flows. The irregularly shaped channel in the physical domain is transformed into a parallel plate channel in the computational domain by using an elliptic scheme of numerical grid generation. The direct problem, as well as the auxiliary problems and the gradient equations, required for the solution of the inverse problem with the conjugate gradient method are formulated in terms of generalized boundary-fitted coordinates. Therefore, the solution approach presented here can be readily applied to forced convection boundary inverse problems in channels of any shape. Direct and auxiliary problems are solved with finite volumes. The numerical solution for the direct problem is validated by comparing the results obtained here with benchmark solutions for smoothly expanding channels. Simulated temperature measurements containing random errors are used in the inverse analysis for strict cases involving functional forms with discontinuities and sharp corners for the unknown functions. The estimation of three different types of inverse problems are addressed in the paper: (i) time-dependent heat fluxes; (ii) spatially dependent heat fluxes; and (iii) time and spatially dependent heat fluxes.