Regularization methods in electrical impedance tomography technique for the two-phase flow visualization

The electrical impedance tomography (EIT) technique for the two-phase flow visualization requires the solution of a nonlinear inverse problem, which is typically ill-posed and subject to noisy data. During the solution procedure, hence, regularization methods are often introduced. This work presents a comparative study on several regularization methods to seek a suitable one for the visualization of liquid/vapor phase distribution. The images reconstructed by first order difference and implicitly scaled Levenberg-Marquardt regularizations give relatively good results even under some errors in measurement data. Also some computational issues relating EIT inverse problem are discussed.     

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.     

On the treatment of non-optimal regularization parameter influence on temperature distribution reconstruction accuracy in participating medium

The choice of the regularization parameter plays a very important role in the inverse radiation problem of temperature distribution in participating medium and in practice the regularization parameter is not easy to determine accurately, which can directly affect the reconstruction accuracy and introduce errors into reconstruction results. This paper presents the alleviation of non-optimal regularization parameter influence on the temperature distribution reconstruction accuracy in participating medium using coupled methods, i.e., two kinds of regularization method (least square QR decomposition (LSQR) method and truncated singular value decomposition (TSVD) method) coupled with genetic algorithm (GA). The radiative heat transfer was described by the backward Monte Carlo method for its efficiency. Two kinds of temperature distributions with one peak and two peaks are considered. The results show that GA can still improve the accuracy of solutions even though the optimal regularization parameters are used in the coupled methods (LSQR-GA and TSVD-GA). GA can also reduce the temperature reconstruction errors due to the non-optimal choice of the regularization parameter and improve the accuracy of the reconstruction results in the coupled methods. Moreover, the coupled methods can even reach the same or better solutions accuracy for some samples with non-optimal regularization parameter, compared with the accuracy of solutions obtained by the single LSQR method or TSVD method with the optimal regularization parameter. This study demonstrates that the coupled method can alleviate non-optimal regularization parameter influence and obtain more accurate results for the inverse radiation problem of temperature distribution in participating medium.     

Estimation of boundary conditions in the presence of unknown moving boundary caused by ablation

Ablative materials can sustain very high temperatures in which surface thermochemical processes are significant enough to cause surface recession. Existence of moving boundary over a wide range of temperatures, temperature-dependent thermophysical properties of ablators, and no prior knowledge about the location of the moving surface augment the difficulty for predicting the exposed heat flux at the receding surface of ablators. In this paper, the conjugate gradient method is proposed to estimate the unknown surface recession and time-varying net surface heat flux for these kinds of problems. The first order Tikhonov regularization is employed to stabilize the inverse solution. Considering the complicated phenomena that are taking place, it is shown via simulated experiment that unknown quantities can be obtained with reasonable accuracy using this method despite existing noises in the measurement data.     

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.     

MULTIPLE TRANSIENT POINT HEAT SOURCES IDENTIFICATION IN HEAT DIFFUSION: APPLICATION TO NUMERICAL TWO- AND THREE-DIMENSIONAL PROBLEMS

This article deals with an inverse problem, which consists of the location and strength identification of multiple-point heat sources in transient heat conduction. The identification procedure is based on a boundary integral formulation using space and time Green functions. The discretized problem is nonlinear if the location of the point heat sources is unknown. In order to reduce the sensitivity of the solution to errors, we use the future time step procedure associated to a Tikhonov regularization procedure. The proposed numerical approach is applied to numerical two- and three-dimensional examples.     

A spectral method for the estimation of a thermomechanical heat source from infrared temperature measurements

The inverse problem of 2D time-dependent heat source reconstruction is solved. The scientific objectives are the quantification of thermal effects associated to the mechanical deformation of materials during tensile tests. The experiment provides infrared measurements of the specimen’s surface temperature and the inverse algorithm aims at providing a volumic heat source that is free of errors due to heat diffusion. This algorithm is based on an analytical solution of the direct problem in the Laplace-Fourier domain. The solution proposed here is compared to a previously used method [1] based on an adjoint formulation and a regularization of Tikhonov type. This allows to check the validity of the results.     

On the iterative regularization of inverse heat conduction problems by conjugate gradient method

The convergence and regularization properties of the conjugate gradient algorithm applied to the inverse heat conduction problem are considered for a time-dependent boundary heat flux. An analysis based on both numerical and analytical results clearly shows that the convergence process of the algorithm is strongly frequency-dependent and provides in this way a very efficient regularization mechanism against the destabilizing effect of random errors in the input data.     

Application of Wavelet in Highly Ill-Posed Inverse Heat Conduction Problem

In this work, the prefiltering of the sensor data is taken into consideration when solving an inverse heat conduction problem. The temperature data obtained from each sensor is considered as a discrete signal, and discrete wavelet transform in a multi-resolution filter bank structure is utilized for the signal analysis, after which wavelet denoising algorithm is applied to remove noise from data signal. Subsequently, noisy and denoised temperatures are separately used as input data to an inverse heat conduction problem for comparison. The inverse heat conduction problem considered in this article is an inverse volumetric heat source problem, and it is solved using the conjugate gradient method along with the associated adjoint problem used to obtain the gradient of the objective function. Three sets of results in two case studies are compared (i.e., the result obtained from non-noisy data, noisy data, and denoised data). In the case of noisy data, iterative regularization is used to regularize the solution. The root mean square error of the estimated heat source from denoised data is reduced approximately by a factor of seven to nine as compared to those obtained from noisy data.     

NUMERICAL SOLUTION OF 2-D NONLINEAR INVERSE HEAT CONDUCTION PROBLEMS USING FINITE-ELEMENT TECHNIQUES

A general method is presented for solving different classes of nonlinear inverse heat conduction problems (IHCP) for two-dimensional, arbitrarily shaped bodies. It is based on the systematic use of a finite-element library. It is shown that, following this approach, the conjugate gradient method can be easily implemented. The method offers a very wide field of practical applications in inverse thermal analysis, while reducing very significantly the amount of work which remains specific for each particular IHPC. Two numerical experiments illustrate the influence of data errors and the iterative regularization principle.     

A regularization method for the inverse design of solidification processes with natural convection

An inverse algorithm is developed for the design of the solidification processing systems. The algorithm entails the use of the Tikhonov regularization method, along with the L-curve method to select an optimal regularization parameter. Both the direct solution of moving boundary problems and the inverse design formulation are presented. The design algorithm is applied to determine the optimal boundary heat flux distribution to obtain a unidirectional solid–liquid interface in a 2-D cavity. The inverse calculation is also performed with a prescribed sinuous solid–liquid interface. To this end, a whole time-domain method and a sequential method are used and evaluated. The L-curve based regularization method is found to be reasonably accurate for the purpose of designing solidification processing systems. We also found that the sequential method with appropriately selected time domains is comparative to the whole time-domain method.     

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.     

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.     

A robust and fast algorithm for three-dimensional transient inverse heat conduction problems

Despite numerous studies of inverse heat conduction problems (IHCP) over the last several decades, their solutions still suffer from the mathematical difficulties and the bottleneck of currently available numerical methods for large-scale problems. In this paper, we present a robust and efficient algorithm for the solution of a specific type of three-dimensional (3D) IHCP commonly involved in various engineering applications. The solution method incorporates the Tikhonov regularization for tackling the severe ill-posedness and the conjugate gradient (CG) method for solving the resulting minimization problems. A model function approach is used to significantly reduce the effort needed to find the optimal Tikhonov regularization parameter. The proposed solution method requires no a priori knowledge of the measurement noise and is much more computationally efficient than the traditional Tikhonov regularization-based inversion approaches. Thus, it can be used for the efficient solution of large-scale practical problems. Two simulation case studies of practical significance are presented to validate and assess the performance of the proposed method. Finally, the solution method is successfully applied to the reconstruction of instantaneous heat fluxes from experimentally measured temperature data.     

An Inverse Problem Method for Overall Heat Transfer Coefficient Estimation in a Partially Filled Rotating Cylinder

The objective of this article is to study the estimation of an overall heat transfer coefficient in a partially filled rotating cylinder. Herein is an inverse analysis for estimating the overall heat transfer coefficient in an arbitrary cross-section of the aforementioned system from the temperatures measured on the shell. The material employs the finite-volume method to solve the direct problem. The hybrid effective algorithm applied here contains the local optimization algorithm to estimate the unknown parameter by minimizing the objective function. The data measured here are simulated by adding random errors to the exact solution. An investigation is made of the impact of the measurement errors on the accuracy of the inverse analysis. Two-optimization algorithms in determining the overall heat transfer coefficient are used. It is determined that the Conjugate Gradient Method is better than the Levenberg-Marquardt Method because the former produces greater accuracy for the same measurement errors. The resulting observation indicates that good agreement exists between the exact value and estimated result for both algorithms.     

Convection Coefficient Estimation by the Truncated Singular Value Decomposition Method Applied to the Associated Inverse Thermal Problem

This paper presents a regularization technique suited to the inverse problem associated with the reconstruction of the internal convection coefficient from external temperature measurements. More precisely, the error function between measured and prospective temperatures, obtained from a prospective convection coefficient, is minimized by applying the Newton method. The truncated singular value decomposition method is used to regularize the corresponding Hessian matrix. Numerical simulations were performed for three different test cases and three different randomly perturbed initial guesses in order to validate the proposed method. Results confirm that the convection coefficient can be reconstructed within reasonable precision.     

Using genetic algorithms for the determination of an heat transfer coefficient in three-phase inverse Stefan problem

In the paper, an example is presented of the application of a genetic algorithm to a design inverse Stefan problem. The problem consists in the reconstruction of the function which describes the heat transfer coefficient, where the positions of phase change moving interfaces are well-known. In numerical calculations, the Tikhonov regularization, a genetic algorithm and a generalized alternating phase truncation method were used. The featured examples of calculations show a very good approximation of the exact solution.     

Inverse estimation of the unknown base heat flux in irregular fins made of functionally graded materials

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle has been successfully applied to an irregular fin made of functionally graded materials to estimate the unknown base heat flux distributions by using temperatures at some measurement locations. The inverse results, in which three different base heat flux distributions are to be determined, have proven current method's capability to accurately estimate arbitrary fin-base heat flux distributions even measurement errors have been taken into account. The temperature data calculated from the direct problem are used to simulate the measured temperature. The influence of measurement errors upon the precision of the estimated results is also investigated. This method does not need any prior information on the unknown quantity, and results show that excellent estimations can be obtained for the test cases considered in this study.     

Explicit boundary heat flux reconstruction employing temperature measurements regularized via truncated eigenfunction expansions

The present work is aimed at developing and validating an explicit inverse problem solution for boundary heat flux reconstruction in thermally thin plates, supposing a large amount of temperature measurements is available, both in space and time, such as typically obtained by modern infrared thermography systems with fine spatial resolution and high frequency. In order to handle the magnification of the measurement errors when applying the derived explicit inversion formula, a regularized representation is proposed for the measured temperatures, consisting of truncated eigenfunction expansions in which only the most important modes, based on the discrepancy principle, are employed. Two applications are considered, the first one consists of a spatially uniform but time varying heat flux with fast transitions, and the second one consists of a spatially varying heat flux concentrated in two small spots. Good results are obtained with very little computational effort, indicating the feasibility of the proposed approach. This work adds to the available standard inverse problems tools, either as an alternative approach or as a companion in obtaining a starting approximate solution for iterative methods.     

Shape identification for inverse geometry heat conduction problems by FEM without iteration

The boundary geometry shape is identified by the finite element method (FEM) without iteration and mesh reconstruction for two-dimensional (2-D) and three-dimensional (3-D) inverse heat conduction problems. First, the direct heat conduction problem with the exact domain is solved by the FEM and the temperatures of measurement points are obtained. Then, by introducing a virtual boundary, a virtual domain is formed. By minimizing the difference between the temperatures of measurement points in the exact domain and those in the virtual domain, the temperatures of the points on the virtual boundary are calculated based on the least square error method and the Tikhonov regularization. Finally, the objective geometry shape can be estimated by the method of searching the isothermal curve or isothermal surface for 2-D or 3-D problems, respectively. In the process, no iterative calculation is needed. The proposed method has a tremendous advantage in reducing the computational time for the inverse geometry problems. Numerical examples are presented to test the validity of the proposed approach. Meanwhile, the influences of measurement noise, virtual boundary, measurement point number, and measurement point position on the boundary geometry prediction are also investigated in the examples. The solutions show that the method is accurate and efficient to identify the unknown boundary geometry configurations for 2-D and 3-D heat conduction problems.