A three-dimensional inverse geometry problem in identifying irregular boundary configurations

A three-dimensional inverse geometry problem (shape identification problem) in determining the unknown irregular surface configurations by utilizing the conjugate gradient method (CGM) and a general purpose commercial code CFD-RC is successfully developed and examined in this study based on the simulated measured temperature distributions on the bottom surface by infrared thermography. Results obtained by using the technique of CGM to solve the inverse geometry problem are justified based on the numerical experiments. Three test cases are performed to test the validity of the present algorithm by using different types of surface shapes, initial guess and measurement errors. Results show that excellent estimations on the unknown surface geometry can be obtained with any arbitrary initial guesses.     

Geometry estimation for the inner surface in a furnace wall made of functionally graded materials

The aim of this study is to solve an inverse geometry heat conduction problem (shape identification problem) to estimate the unknown geometry of the inner surface in a furnace wall which is made of functionally graded materials (FGMs). The inner surface geometry is estimated from the temperatures of measured points within the furnace wall. The inverse algorithm used in the study is based on the conjugate gradient method (CGM) and the discrepancy principle. The effect of measurement errors and measurement locations on the estimation accuracy is also investigated. Two different examples are discussed. Results show that the unknown geometry of the inner wall surface can be predicted precisely by using the present approach.     

A transient three-dimensional inverse geometry problem in estimating the space and time-dependent irregular boundary shapes

A transient three-dimensional shape identification problem (inverse geometry problem) to determine the unknown irregular and moving boundary configurations by utilizing the steepest descend method (SDM) and a general purpose commercial code CFD-RC is successfully developed and examined in this study based on the simulated measured temperature distributions on the bottom surface by infrared thermography. The advantage of calling CFD-RC as a subroutine in the present inverse calculation lies in that its auto-mesh function enables the handling of this moving boundary problem. Results obtained by using the technique of SDM to solve the inverse geometry problem are justified based on the numerical experiments. Two test cases are performed to test the validity of the present algorithm by using different types of boundary shapes, initial guesses and measurement errors. Results show that reliable estimations on the unknown space and time-dependent boundary geometry can be obtained when the measurement errors are considered.     

Inverse Geometry Design of Radiating Enclosure Filled with Participating Media Using Meshless Method

A new inverse geometry design methodology is presented in this work for designing a two-dimensional radiating enclosure filled with participating media to meet the pre-specified radiative heat flux distribution on a designed boundary wall. Akima cubic interpolation is employed to approximate the shape of the unknown design surface and transform the continuous geometry shape design to the discrete points' position design. To avoid the tedious remeshing of the variable computational domain in the inverse geometry design processes, the direct collocation meshless method is adopted to solve the radiative transfer problem in the enclosure. The geometry shape of the design surface is optimized using the conjugate gradient method, and the zeroth order regularization method is chosen to stabilize the inverse solutions. A test example is taken to verify the new method presented in this work. The inverse design results show that pre-specified design requirement on the boundary wall can be successfully obtained using the new methodology.     

A shape design problem in determining the interfacial surface of two bodies based on the desired system heat flux

A shape design problem (or inverse geometry problem) in determining the geometry of interfacial surface between two conductive bodies in a three-dimensional multiple region domains, based on the desired system heat flux and domain volume, is examined in this study. The design algorithm utilized the Levenberg–Marquardt method (LMM), B-spline surface generation and the commercial software CFD-ACE+. The validity of this shape design analysis is examined using the numerical experiments. Different desired system heat fluxes are considered in the numerical test cases to justify the validity of the present algorithm in solving the three-dimensional shape design problems. Finally, the results show that for the two different cases considered in this work, the maximum increasing in the system heat flux is obtained as 11.3% and 14.1%, respectively. It is also concluded that when the boundary control points of interfacial surface are free to move, maximum system heat flux can be obtained by the present algorithm since it has more degree of freedom in describing the interfacial surface.     

Function estimation of laser-induced heat generation in a gas-saturated powder layer heated by a short-pulsed laser

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to solve the inverse heat conduction problem with a dual-phase-lag equation for estimating the unknown space- and time-dependent laser-induced heat generation in a gas-saturated porous medium exposed to short-pulse laser heating from the temperature measurements taken within the medium. Subsequently, the powder particle temperature distributions in the porous medium can be determined as well. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The effect of measurement errors on the estimation accuracy is also investigated. The inverse solutions are justified based on the numerical experiments in which two different forms of heat generation are estimated. Results show that the unknown laser-induced heat generation can be predicted precisely by using the present approach for the test cases considered in this study.     

An iterative regularization method in estimating the base temperature for non-Fourier fins

An inverse non-Fourier fin problem is examined in the present study by an iterative regularization method, i.e., conjugate gradient method (CGM), in estimating the unknown base temperature of non-Fourier fin based on the boundary temperature measurements. Results obtained in this inverse problem will be justified based on the numerical experiments where three different temperature distributions are to be determined. Results show that the inverse solutions can always be obtained with any arbitrary initial guesses of the base temperature. Moreover, the drawbacks of previous study for this identical inverse problem, such as (1) the inverse solutions become poor when the frequency of base temperature is increased, (2) the estimations depend strongly on the size of grids, (3) the estimations are sensitive to the measurement errors and (4) the uncertainty of using the concept of future time step, can all be avoided by applying this algorithm. Finally, it is concluded that accurate base temperatures can be estimated in the present study.     

An inverse biotechnology problem in estimating the optical diffusion and absorption coefficients of tissue

An inverse algorithm for biotechnology problem utilizing the conjugate gradient method is applied in the present study in determining the unknown spatial-dependent optical diffusion and absorption coefficients of the biological tissue based on irradiance and temperature measurements. The accuracy of this inverse problem is examined by using the simulated exact and inexact irradiance and temperature measurements in the numerical experiments. Results show that the estimation on the spatial-dependent diffusion and absorption coefficients can be obtained with any arbitrary initial guesses on a Pentium IV 1.4 GHz personal computer for the test cases considered in the present study.     

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.     

Two-dimensional inverse problem in estimating heat flux of pin fins

The two-dimensional inverse problem of estimating the unknown heat flux of a pin fin base has been solved using the conjugate gradient method. The advantage of the conjugate gradient method is that no information on the functional form of the unknown quantity is required beforehand. The accuracy of the inverse analysis is examined by using simulated exact and inexact measurements of temperature in an interior location of a pin fin. Numerical results show that good estimations on the heat flux can be obtained for all the test cases considered here. Furthermore, such a technique can be applied to determine the heat flux acting on an internal wall surface, where direct measurements are difficult to make.     

An iterative regularization method in simultaneously estimating the inlet temperature and heat-transfer rate in a forced-convection pipe

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown space- and time-dependent inlet temperature and heat-transfer rate on the external wall of a pipe system using temperature measurements at two different locations. It is assumed that no prior information is available on the functional form of the unknown inlet temperature and heat-transfer rate; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The accuracy of the inverse analysis is examined by using simulated exact and inexact temperature measurements. Results show that an excellent estimation on the space- and time-dependent inlet temperature and heat-transfer rate can be obtained for the test case considered in this study.     

Inverse Natural Convection Problem with Radiation in Rectangular Enclosure

Inverse thermal problem is applied to natural convective flow with radiative heat transfer. The bottom wall temperature in the 2-D cavity domain is estimated by using gas temperature measurements in the flow field. The inverse problem is solved through a minimization of an objective function using the conjugate gradient method with adjoint problem. The effects of functional form of bottom wall temperature profile, the number and the position of measurement points, and the measurement errors are investigated and discussed. The conjugate gradient method is found to work well in estimating the bottom wall temperature, even when natural convection with radiation phenomena is involved.     

A THREE-DIMENSIONAL INVERSE PROBLEM OF ESTIMATING THE VOLUMETRIC HEAT GENERATION FOR A COMPOSITE MATERIAL

A 3-D transient inverse heat conduction problem is solved in this study by using the conjugate gradient method (CGM) and the general-purpose commercial code CFX4.2-based inverse algorithm to estimate the strength of the unknown heat generation in a 3-D irregular composite medium. The advantage of calling the CFX4.2 code as a subroutine in the present inverse calculation is that many difficult but practical 3-D inverse problem can be solved under this construction because the general-purpose commercial code has the ability to solve the direct problem easily. Results obtained by using the CGM to solve this 3-D inverse problems are justified based on the numerical experiments with the simulated exact and inexact measurements. It is concluded that accurate heat generation can be estimated by the CGM except for the final time. The reason and improvement of this singularity are addressed.     

Inverse analysis for estimating the unsteady inlet temperature distribution for two-phase laminar flow in a channel

An inverse thermal problem was considered for two-phase laminar flow in a parallel plate duct. The inlet temperature, which varies temporally as well as spatially, was estimated when measured temperatures were available at downstream of the duct. In the present study, the problem is solved through a minimization of an objective function by using two regularization methods, i.e., the iterative conjugate gradient method (CGM) and the Tikhonov regularization method (TRM). The effects of the functional form of inlet temperature profile, the number of the measurement points and the measurement errors are investigated and discussed. The computational accuracy and efficiency of these two regularization method are compared and discussed.     

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.     

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.     

Flow and Forced-Convection Characteristics of Turbulent Flow Through Parallel Plates with Periodic Transverse Ribs

An iterative regularization method (or conjugate gradient method, CGM) is utilized in the present inverse phonon radiative transport problem in estimating the unknown boundary temperature distributions, based on the phonon intensity measurements, for a double-layer thin-film structure. The CGM in dealing with the present integrodifferential governing equations is not as straightforward as for the normal differential equations; special treatment is needed to overcome the difficulties. Results obtained in this inverse analysis are justified based on numerical experiments in which three different unknown temperature (or phonon intensity) distributions are to be determined. Finally, it is shown that accurate boundary temperatures can always be obtained with the CGM.     

An Inverse Design Problem of Estimating the Optimal Shape of Annular Fins Adhered to a Bare Tube of an Evaporator

The optimum shapes for annular fins adhered to a bare tube are estimated in the present inverse design problem using the conjugate gradient method (CGM), based on the desired fin efficiency and fin volume. The shape of the annular fins adhered to the bare tube must be optimally designed to produce a greater cold-producing effect in the evaporator. One of the advantages of using the present inverse algorithm in this inverse design problem is that it can handle problems with a large number of unknown parameters easily and converge rapidly. The results obtained using the CGM to solve the inverse design problem are justified based on numerical experiments. The study shows that when the Biot number and fin volume are varied, the optimum fin efficiency and optimum fin shape will also change; however, the optimum shape of an annular fin can always be obtained, and its fin efficiency is always better than that of five common annular fins.     

燃烧室内三维温度场的辐射反问题

本文提出了一种在介质辐射特性已知的条件下,由壁面入射辐射热流的测量值反演燃烧室内三维温度场的方法。该方法是在辐射传递方程离散坐标近似的基础上,用求目标函数极小值的共轭梯度法进行反演计算。通过对吸收系数、散射不对称因子、反照率、壁面黑度和燃烧室大小尺寸等参数对反演精度影响的分析,结果表明,即使存在随机测量误差,这些参数对温度场反演精度的影响也不大,本文所提出的方法可较精确地反演燃烧室内三维温度场。     

Control of the boundary heat flux during the heating process of a solid material

In this paper we apply the conjugate gradient method to solve the inverse problem of determining a time-dependent boundary heat flux in order to achieve a given temperature distribution at the final time. The derivation of sensitivity and adjoint equations in conjunction with the conjugate gradient algorithm are given in detail. The zeroth-order Tikhonov regularization is introduced to stabilize the inverse solution. Solutions by finite differences are obtained for various heat flux profiles. It is found that the time-dependent heat flux may be predicted only for a non-dimensional time of the order of 0.1 while the control problem can be satisfactorily solved for an arbitrary period of time.