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.     

Direct and inverse mixed convections in an enclosure with ventilation ports

The numerical study presented in this work describes the direct and inverse mixed convection problems in a slot-ventilated enclosure subjected to an unknown heat flux on one side. Particularly, the interaction of internal natural convection with the cold ventilated flow leads to various flow fields depending on the Richardson number, Reynolds number, and the functional form of the imposed boundary heat flux. Fluid and heat transport structures across the enclosure are visualized by the streamlines and heatlines, respectively. Subsequently, an iterative conjugate gradient method is applied such that the gradient of the cost function is introduced when the appropriate sensitivity and adjoint problems are defined for a domain of arbitrary geometries. In this approach, no a priori information is needed about the unknown boundary heat fluxes to be determined. The accuracy of the heat flux profile solutions is shown to depend strongly on the values of Reynolds number and flux functional forms. Effects of measurement errors on the accuracy of estimation are also investigated. The present work is significant for the flow control simultaneously involving the natural convection and forced convection.     

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.     

Estimation of the time-dependent heat flux using the temperature distribution at a point by conjugate gradient method

In this paper, the conjugate gradient method coupled with adjoint problem is used in order to solve the inverse heat conduction problem and estimation of the time-dependent heat flux using the temperature distribution at a point. Also, the effects of noisy data and position of measured temperature on final solution are studied. The numerical solution of the governing equations is obtained by employing a finite-difference technique. For solving this problem the general coordinate method is used. We solve the inverse heat conduction problem of estimating the transient heat flux, applied on part of the boundary of an irregular region. The irregular region in the physical domain (r,z) is transformed into a rectangle in the computational domain (ξ,η). The present formulation is general and can be applied to the solution of boundary inverse heat conduction problems over any region that can be mapped into a rectangle. The obtained results for few selected examples show the good accuracy of the presented method. Also the solutions have good stability even if the input data includes noise and that the results are nearly independent of sensor position.     

Three-Dimensional Transient Optimal Boundary Heating of Functionally Graded Plates

An optimal control procedure for estimating the heat fluxes on the boundaries of functionally graded (FG) thick plates to reach the desired domain temperature distributions in a specified time interval of heating is presented. The conjugate gradient method (CGM) is employed for optimization, and the differential quadrature method as an accurate and numerically efficient method in conjunction with the forward finite-difference method are applied to solve the three-dimensional transient heat transfer, sensitivity, and adjoint problems. The validity of the presented optimal control problem is demonstrated by solving different numerical examples. Results show that excellent estimation on the boundary heat fluxes can be obtained with arbitrary initial guesses of these functions.     

Inverse radiation problem of temperature field in three-dimensional rectangular furnaces

The discrete ordinates method is used to developed a solution to an inverse radiation problem of temperature field in rectangular furnaces. It is assumed that, with the exception of the inhomogeneous temperature field, all aspects of the radiation transport problem are known. A method is developed to determine the inhomogeneous temperature field from specified incident radiation heat fluxes at the centers of boundary walls. The inverse problem is solved using conjugate gradient method that minimize the error between the incident radiation heat fluxes calculated and the experimental data. The results of temperature estimation show that the temperature field can be estimated accurately, even with noisy data.     

Reconstruction of a space and time dependent heat source from finite measurement data

This paper deals with an inverse problem of determining a heat source function in heat conduction equations when the solution is known in a discrete point set. Being different from other ordinary inverse source problems which are often dependent on only one variable, the unknown coefficient in this paper not only depends on the space variable x, but also depends on the time t. On the basis of the optimal control framework, the inverse problem is transformed into an optimization problem. The existence and necessary condition of the minimizer for the cost functional are established. The convergence of the minimizer as the mesh parameters tend to zero is also proved. The conjugate gradient method is applied to the inverse problem and some typical numerical experiments are performed in the paper. The numerical results show that the proposed method is stable and the unknown heat source is recovered very well.     

Inverse problem of unsteady conjugated forced convection in parallel plate channels

The heat transfer phenomena of the unsteady laminar forced convection in parallel plate channels with wall conduction effects are still not very well understood. An inverse algorithm based on the conjugate gradient method is proposed to estimate the boundary conditions of these problems, and the minimization of object function is used to reduce the estimated error. The estimation of applied heat flux is found to be highly dependent of temperature sensor location and uncertainty, plate thickness, and heating way. The results show that the predicted boundary conditions by the present inverse method are consistent with the initially specified ones.     

Inverse Boundary Design Problem of Turbulent Forced Convection Between Parallel Plates With Surface Radiation Exchange

The inverse methodology is employed to estimate the unknown heat flux distribution over the heater surface of a channel formed by two parallel plates with forced convection and surface radiation exchange, from the knowledge of the desired temperature and heat flux distributions over a given design surface. The energy and radiative transfer equations are solved by the finite-volume method and the net radiation method, respectively. The conjugate gradient method is used for minimization of an objective function, which is expressed by the sum of square residuals between estimated and desired heat fluxes over the design surface. The performance and accuracy of the present method for solving inverse problems are evaluated by some numerical experiments.     

Simultaneously estimating the contact heat and mass transfer coefficients in a double-layer hollow cylinder with interface resistance

Based on the conjugate gradient method, this study presents a means of solving the inverse boundary value problem of coupled heat and moisture transport in a double-layer hollow cylinder. While knowing the temperature and moisture history at the measuring positions, the unknown time-dependent contact heat and mass transfer coefficients can be simultaneously determined. It is assumed that no prior information is available on the functional form of the unknown coefficients. The accuracy of this inverse heat and moisture transport problem is examined by using the simulated exact and inexact temperature and moisture measurements in the numerical experiments. Results show that excellent estimation on the time-dependent contact heat and mass transfer coefficients can be simultaneously obtained with any arbitrary initial guesses.     

Inverse boundary design of square enclosures with natural convection

An optimization technique is applied to design of heat transfer systems in which the natural convection is important. The inverse methodology is employed to estimate the unknown strengths of heaters on the heater surface of a square cavity with free convection from the knowledge of the desired temperature and heat flux distributions over a given design surface. The direct and the sensitivity problems are solved by finite volume method. The conjugate gradient method is used for minimization of an objective function, which is expressed by the sum of square residuals between estimated and desired heat fluxes over the design surface. The performance and accuracy of the present method for solving inverse convection heat transfer problems is evaluated by comparing the results with a benchmark problem and a numerical experiment.     

Estimation of unknown heat source function in inverse heat conduction problems using quantum-behaved particle swarm optimization

The estimation of temporal dependent heat source in transient heat conduction problem is investigated. A stochastic method known as quantum-behaved particle swarm optimization (QPSO) is used to estimate the heat source without a priori information on its functional form, which is classified as the function estimation by inverse calculation. Because of the ill-posedness of this kind of inverse problems, Tikhonov regularization method is applied in this paper to stabilize the solution. Numerical experiments indicate the validity and stability of the QPSO method. Comparison with the conjugate gradient method (CGM) is also presented in this paper.     

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.     

Determining boundary heat flux profiles in an enclosure containing solid conducting block

A numerical implementation of estimating boundary heat fluxes in enclosures is proposed in the present work. Particularly, the flow field is dynamically coupled with the heat convection in the fluid and the heat conduction in the solid domain. An iterative conjugate gradient method is applied such that the gradient of the cost function is introduced when the appropriate sensitivity and adjoint problems are defined. In this approach, no a priori information is needed about the unknown function to be determined. Numerical solutions are obtained for the case of a square enclosure centrally-inserted with a solid block and subjected to an unknown heat flux on one side and to known conditions on the remaining sides. Fluid and heat transports are visualized by the streamlines and heatlines respectively, which are evidently affected by the thermal Rayleigh number, solid body size and thermal conductivity of solid phase, and the functional form of the imposed heat flux. The accuracy of the heat flux profile estimations is shown to depend strongly on the thermal Rayleigh number, body size and relative thermal conductivity of the solid material. Effects of functional form of the unknowns, sensors number and position, and measurement errors on the accuracy of estimation are also investigated. The present work is significant for the flow control simultaneously involving the heat conduction and convection.     

Estimation of the Boundary Heat Flux in Grinding via the Conjugate Gradient Method

The unknown boundary surface heat flux in workpieces during grinding is estimated by the application of inverse heat transfer analysis. The conjugate gradient method of function estimation is used for the minimization procedure. Simulated temperature measurements are used in the inverse analysis for typical practical cases, in order to show that results more accurate than those available in the literature are obtained with the present solution approach. Actual experimental data are also used in the computations to estimate the surface heat flux.     

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.     

Transient heat flux function estimation utilizing the variable metric method

A new method is presented to solve inverse heat conduction problems (IHCP's). The method belongs to the whole domain type of IHCP and utilizes the variable metric method (VMM) instead of the conjugate gradient method (CGM) in order to minimize the function of sum of square errors. Appropriate formulations for sensitivity coefficients, objective function and its gradient are set up in a manner suitable for computer programming of VMM. Numerically simulated data are utilized to assess the effectiveness of the method in comparison with the conjugate gradient method in estimation of space and time varying heat flux. Results indicate that the presented method is quite faster and more accurate than the conjugate gradient method in estimation of unknowns in the whole domain inverse problems.     

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.     

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.     

An inverse problem in simultaneous estimating the Biot numbers of heat and moisture transfer for a porous material

A conjugate gradient method based inverse algorithm is applied in the present study in simultaneous determining the unknown time-dependent Biot numbers of heat and moisture transfer for a porous material based on interior measurements of temperature and moisture.It is assumed that no prior information is available on the functional form of the unknown Biot numbers in the present study, thus, it is classified as the function estimation in inverse calculation.The accuracy of this inverse heat and moisture transfer problem is examined by using the simulated exact and inexact temperature and moisture measurements in the numerical experiments. Results show that the estimation on the time-dependent Biot numbers can be obtained with any arbitrary initial guesses on a Pentium IV 1.4 GHz personal computer.