One-dimensional Inverse Problem Research Articles

Implementation of total variation regularization-based approaches in the solution of linear inverse heat conduction problems concerning the estimation of surface heat fluxes

In this work, we propose the extension of total variation regularization strategies in principle formulated to be implemented in image processing and signal analysis research fields, to the solution of one-dimensional linear inverse heat conduction problems concerning the estimation of surface heat fluxes. Three solution procedures are considered in the current study, and these include the lagged diffusivity fixed point iteration, the iteratively reweighted least-squares, and the split Bregman iteration methods. The performance of such procedures is tested using four cases, and the regularization parameter is selected via the L-curve criterion. The results are compared to the reconstruction obtained via a classical Tikhonov-like strategy. The main outcome is that the staircase effect dominated all the reconstructions. Despite it deteriorated the quality of the solutions in some specific cases, it did not prevent the appropriate fulfillment of the reconstruction task. The results achieved in this communication have been useful to demonstrate the suitability and reliability of extending total variation approaches to the solution of linear inverse heat conduction problems concerning the estimation of surface heat fluxes, as an appropriate and novel alternative to standard procedures.

Experimental investigation of wall heat transfer due to spray combustion in a high-pressure/high-temperature vessel

Combustion chamber wall heat transfer is a major contributor to efficiency losses in diesel engines. In this context, thermal swing materials (adapting to the surrounding gas temperature) have been pinpointed as a promising mitigative solution. In this study, experiments are carried out in a high-pressure/high-temperature vessel to (a) characterise the wall heat transfer process ensuing from wall impingement of a combusting fuel spray, and (b) evaluate insulative improvements provided by a coating that promotes thermal swing. The baseline experimental condition resembles that of Spray A from the Engine Combustion Network, while additional variations are generated by modifying the ambient temperature as well as the injection pressure and duration. Wall heat transfer and wall temperature measurements are time-resolved and accompanied by concurrent high-speed imaging of natural luminosity. An investigation with an uncoated wall is carried out with several sensor locations around the stagnation point, elucidating sensor-to-sensor variability and setup symmetry. Surface heat flux follows three phases: (i) an initial peak, (ii) a slightly lower plateau dependent on the injection duration, and (iii) a slow decline. In addition to the uncoated reference case, the investigation involves a coating made of porous zirconia, an established thermal swing material. With a coated setup, the projection of surface quantities (heat flux and temperature) from the immersed measurement location requires additional numerical analysis of conjugate heat transfer. Starting from the traces measured beneath the coating, the surface quantities are obtained by solving a one-dimensional inverse heat transfer problem. The present measurements are complemented by CFD simulations supplemented with recent rough-wall models. The surface roughness of the coated specimen is indicated to have a significant impact on the wall heat flux, offsetting the expected benefit from the thermal swing material.

COMPARATIVE ANALYSIS OF NUMERICAL METHODS OF THE SOLUTION OF A ONEDIMENSIONAL INVERSE PROBLEM OF ACOUSTICS

Nowadays, a large number of methods for solving inverse problems arising in electrodynamics and acoustics have been developed, but the development of practical systems is necessary to combine a large number of equations that contribute to the substantiation of numerical methods for solving various multidimensional problems. Therefore, the main goal of the work is a comparative analysis of statistical methods for solving the one-dimensional inverse acoustic problem, as well as in the search for acoustic resistance. To achieve this goal, the means of description and comparison, which contributed to the identification of the characteristics of acoustic impedance, were used. Also, the finite-difference solution method, the differential circuit circulation method, and the Landweber iteration method were used. It was established that the inversion method of the difference scheme is expedient to apply in the case when additional information is known accurately enough, and the reconstructed solution is quite smooth. It was determined that if one of these conditions is violated, the method of reversing the difference scheme becomes unstable. The problems of the correctness of the issues for the wave equation with complex velocity in the one-dimensional and spatial cases were investigated. Formulas for solving these problems were obtained – analogs of classical formulas. Numerical computations show the kind of results that may be expected from the method under consideration. The materials of the paper imply the practical significance for the university teachers of the information technology specializations.

Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process

We consider a one-dimensional inverse problem for a partial differential equation of hyperbolic type with sources - the Dirac delta-function and the Heaviside theta-function. The generalized inverse problem is reduced to the inverse problem with data on the characteristics using the method of characteristics and the method of isolation of singularities. At the beginning, the inverse problem of the wave process with data on the characteristics with additional information for the inverse problem without small perturbations is solved by the finite-difference method. Then, for the inverse problem of the wave process with data on the characteristics with additional information with small perturbations, that is, with small changes is used by the finite-difference regularized method, which developed by one of the authors of this article. The convergence of the finite-difference regularized solution to the exact solution of the one-dimensional inverse problem of the wave process on the characteristics is shown, and the theorem on the convergence of the approximate solution to the exact solution is proved. An estimate is obtained for the convergence of the numerical regularized solution to the exact solution, which depends on the grid step, on the perturbations parameter, and on the norm of known functions. From the equivalence of the problems, the one-dimensional inverse problem of the wave process with sources - the Dirac delta-function and the Heaviside theta-function and the one-dimensional inverse problem of the wave process with data on the characteristics, it follows that the solution of the last problem will be the solution of the posed initial problem. An algorithm for solving a finite-difference regularized solution of a generalized one-dimensional inverse problem is constructed.

Experimental investigation on transient boiling heat transfer during quenching of fuel cladding surfaces

In case of postulated accidents in a nuclear reactor, the fuel rods uncovered from coolant can be overheated. Immediate quenching of overheated fuel rods is desired. Quenching experiment is performed at atmospheric pressure with rodlets of the FeCrAl alloy, one of the accident tolerant fuel cladding materials, and the Zircaloy-4, the conventional cladding material. In the experiment 600 °C rodlet specimens are plunged into the subcooled water bath. A high-speed camera is employed to record quenching phenomenon. With the embedded thermocouples the rodlet temperature is measured at the frequency of 50 Hz. One-dimensional inverse heat conduction problem is solved to obtain the surface temperature and heat flux. Influence of surface condition, solid thermal properties and liquid subcooling on transient boiling heat transfer is investigated. In the film boiling regime heat transfer is mainly affected by solid thermal properties and liquid subcooling. Small solid (kρCp)w and large liquid subcooling results in relatively thin vapor film and more efficient film boiling heat transfer. Roughness on the polished level shows no appreciable influence on the minimum film boiling temperature, TMFB, which increases with decreasing contact angle. Surface oxidation is slow on the FeCrAl surface and hence, does not significantly affect the boiling heat transfer. However, on the Zircaloy-4 surface the oxide layer with small (ρkCp)w increases TMFB and decreases the critical heat flux. Large (ρCp)w results in slow quench front propagation, long quench duration and weak subcooling effect. It is found that on each surface the vapor film collapses at the critical thickness which is irrelevant to liquid subcooling. Based on the critical thickness, a new approach for prediction of TMFB is proposed.

Hybrid-Spline Dictionaries for Continuous-Domain Inverse Problems

We study one-dimensional continuous-domain inverse problems with multiple generalized total-variation regularization, which involves the joint use of several regularization operators. Our starting point is a new representer theorem that states that such inverse problems have hybrid-spline solutions with a total sparsity bounded by the number of measurements. We show that such continuous-domain problems can be discretized in an exact way by using a union of B-spline dictionary bases matched to the regularization operators. We then propose a multiresolution algorithm that selects an appropriate grid size that depends on the problem. Finally, we demonstrate the computational feasibility of our algorithm for multiple-order derivative regularization operators.

Pseudospectral method for a one-dimensional fractional inverse problem

In this paper, a method is implemented to a one-dimensional inverse problem with a parabolic differential equation of fractional order in which the fractional derivative is in the Caputo sense. The considered inverse problem involves a time-dependent source control parameter . In order to numerically solve the problem, first, the main problem is converted to a homogeneous problem by Lagrange interpolation. Consequently, a new problem is derived by a practical technique that verifies all the conditions of the main problem. Finally, a system of nonlinear algebraic equations is solved by Newton's method to obtain the unknown coefficients. It is notable that all the needed computations are done in MATHEMATICA. In this work, operational matrices of Bernoulli polynomials are stated and applied to approximate functions. Illustrative examples are included to prove the efficiency and applicability of the proposed methods. In the numerical tests, a low amount of polynomials is needed to acquire a precise estimate solution. For demonstrating the low running time of this method, CPU time for all examples is exhibited.

One-dimensional inverse coefficient problems of anisotropic viscoelasticity

One-dimensional inverse coefficient problems of anisotropic viscoelasticity

Numerical Method for Solving the Parametric Identification Problem for Loaded Differential Equations

In the work, we propose a method of numerical solution to coefficient inverse problems with respect to loaded differential equations with multi point and integral conditions. The method is based on the operation that convolves given integral conditions into point conditions. This method allows reducing the solution to the initial problem to a Cauchy problem with respect to systems of ordinary differential and of linear algebraic equations. The approach is extended to a class of one-dimensional inverse problems for loaded parabolic equations.

Application of repulsive particle swarm optimization for inverse heat conduction problem – Parameter estimations of unknown plane heat source

The major objective of the present study is to demonstrate the performance and efficiency of the Repulsive Particle Swarm Optimization (RPSO) method as an inverse analysis solver by extending its application to the inverse heat conduction problem. As the first outcome, an estimation of unknown parameters of a time-varying plane heat source in a one-dimensional inverse heat conduction problem was considered. The overall performance of the RPSO method was examined based on the effects of the forms of unknown heat source, the number of parameters, the measurement errors and the population sizes on the estimation accuracy. In addition, the final results were compared with those of the Levenberg-Marquardt Method (LMM), which is widely used for the inverse heat conduction problem, to investigate the advantages and disadvantages of the RPSO method.The present results prove that the RPSO method has more robust characteristics and yields rather confidential inverse estimations than the LMM for the inverse heat conduction problem although it requires more computational cost.

Modeling the one-dimensional inverse heat transfer problem using a Haar wavelet collocation approach

In this paper, a numerical method to solving the one-dimensional inverse heat transfer problem in Cartesian and Cylindrical coordinates, which is combination of the Haar wavelet collocation and Tikhonov regularization approach, has been used. The audit consisted in comparing the heat transfer in reality with numerical result as a noisy data, in order to identify errors of 1%–5% has been presented. In this study, the Haar functions, in addition to estimating the unknown functions, are also used to reduce output noises. Based on the obtained results, two main advantages of the repeated method have been proven, first, the precision of this method in estimating the unknown boundary condition and the second, processing speed due to the lack of need for wavelet functions to be collocated at low intervals. This suggested that this method, also has a high speed. According to the obtained results, it can be asserted that the present method by applying a small error in the input data, is preserved its stability.

Revisiting the Temperature Gradient Method

Abstract Cooling curve analysis has emerged as a preferred method to characterize the cooling power of quenching media. This methodology is based on measuring the local cooling curve in a laboratory-scale probe using thermocouples. Several methods have been proposed to analyze cooling curves. Among them, the temperature gradient method (TGM) developed by Professor Liščić uses cooling curves measured at the surface and 1.5 mm below the surface of a cylindrical probe (the Liščić-Nanmac probe) to calculate the surface heat flux during quenching. In this work, we measured the thermal response at a location near the surface of cylindrical probes, fabricated with AISI 304 stainless steel, with two different geometries (conical- and hemispherical-end) during quenching from 850°C with water at 60°C, flowing parallel to the probe’s longitudinal axis. Together with the probe geometry, the experimental matrix included two values of water velocity (0.2 and 0.6 m/s). The data was used to estimate the surface heat flux and thermal response at the probe surface by solving a one-dimensional inverse heat conduction problem (IHCP) without phase change. The TGM was then applied to re-estimate the surface heat flux using the cooling curves at the subsurface (measured) and at the surface (estimated by solving the IHCP). The surface thermal gradient computed solving the IHCP is higher and, therefore, the surface heat flux estimated solving the IHCP is also higher than the value calculated with the TGM. The hemispherical-end probe delays the rupture of the vapor film, producing low values of the surface temperature at the time of maximum heat extraction; this results in extremely high values of the heat-transfer coefficient, which precludes the use of this geometry in conjunction with the TGM. The thermal profiles are parabolic, which restricts the maximum depth at which a subsurface thermocouple may be placed to use the TGM confidently.

Effect of Coating by a Carbon Nanostructure on Heat Transfer with Unsteady Film Boiling

We present the results of an experimental study of heat transfer regimes during cooling in water of high-temperature, heated steel balls with a technically smooth and modified surface. The modification consisted of the application of a finely dispersed carbon coating to the surface followed by electron beam treatment. When the samples are placed in subcooled water, an intensive heat transfer regime occurs on the both samples; it appeared upon film boiling with heat fluxes on the surface up to 6 MW/m2. The heat flux values based on the initial cooling thermograms were obtained from a one-dimensional inverse heat conduction problem. The carboncoating leads to a decrease in the surface temperature, which corresponds to the transition to the intensive cooling mode, whereas the intensive cooling mode itself is identical for samples with the different surface treatments. Experimental results confirm the approximate model proposed for the conditions for the occurrence of the intensive cooling regime during film boiling of a subcooled liquid.

Global Solvability of the One-dimensional Inverse Problem for the Integro-differential Equation of Acoustics

Global Solvability of the One-dimensional Inverse Problem for the Integro-differential Equation of Acoustics

FEATURES OF THE APPLICATION OF THE IQLAB PROGRAM FOR SOLVING THE INVERSE HEAT CONDUCTION PROBLEM FOR CHROMIUM-NICKEL CYLINDRICAL THERMOSONDES

The paper presents an experimental-computational study of the results of using the IQLab program to solve inverse heat conduction problem and restore the surface temperature of cylindrical thermosondes from heat-resistant chromium-nickel alloys while cooling them in liquid media. The purpose of this paper is to verify the correct operation of the IQLab program when restoring the surface temperature of thermosondes with 1-3 thermocouples. The IQLab program is also designed to solve one-dimensional nonlinear direct lines and inverse heat conduction problems with constant initial and boundary conditions specified as a function of time in a tabular form with a constant and variable time step. A finite-difference method is used to solve the heat equation. Experiments were carried out on samples D = 10-50 mm in liquids with different cooling capacities such as aqueous solutions of NaCl and Yukon-E polymer, rapeseed oil and I-20A mineral oil. For the calculation we used the readings of thermocouples installed at internal points of cylindrical thermosondes. The advantages of solving inverse heat conduction problems with the IQLab program include the possibility of restoring the surface temperature for cylindrical samples with a diameter of 10 mm to 50 mm with practical accuracy according to the indications of a single thermocouple located in the geometrical center of the thermosonde, which simplifies the manufacture of the probe. For larger dimensions with a diameter D ≥ 50 mm, it is necessary to install control intermediate thermocouples and perform additional tests. The solution of inverse heat conduction problems and restoration of the surface temperature of the sample makes it possible to calculate other important characteristics of the cooling process: the heat flux density and the heat transfer coefficient.

A class of multistep numerical difference schemes applied in inverse heat conduction problem with a control parameter

ABSTRACTAs this paper is concerned with a class of multistep numerical difference techniques to solve one-dimensional parabolic inverse problems with source control parameter , we apply the linear multistep method combining with Lagrange interpolation to develop three different numerical difference schemes. The problem of numerical differentiation with noisy scattered data is mildly ill-posed, the smoothing spline model based on Tikhonov regularization method is developed to compute numerical derivative contaminated by noise error. Simultaneously, the truncation error estimations and the convergence conclusions are proposed for the above difference methods respectively. The results of numerical tests with different noise levels are given to show that the presented algorithms are accurate and effective.

Solution of an Inverse Stefan Problem in Analyzing the Freezing of Groundwater in a Rock Mass

A one-dimensional inverse Stefan problem is formulated in cylindrical coordinates. The problem is of interest in the analysis of freezing a mass of rock and forming an iced rock barrier around a single freezing column in building shafts as well as solving the problem of controlling the process of thawing. Additional conditions in the formulation of the inverse problem are the data of experimental measurements of temperatures in thermal boreholes located at a certain given distance from a freezing column and the magnitudes of temperatures and of the cold-carrier flow rate in freezing columns. An analytical expression is presented for the functional that determines the measure of proximity of the obtained solution to the exact one. A numerical method has been developed for solving the posed inverse problem by minimizing the functional.

A Computational Source Modeling of Brain Activity: An Inverse Problem

Source determination in an inverse problem from the over-specified data plays a crucial role in cognitive modeling. In this paper, an accurate and fast method is proposed for solving the one-dimensional inverse problem concerning diffusion equation with source control parameter. The proposed method is based on applying a compact finite difference scheme for spatial components and solving the system that arised from this scheme by multigrid.

The Projection Method for Identification of the Characteristics of Inhomogeneous Solids

A projection method for solving the inverse problems of identification of the characteristics of inhomogeneous solids is proposed on the basis of acoustic sounding. Examples of using the method for solution of one-dimensional inverse problems for rods and round plates are given. The results of computational experiments on the identification of functions of various types are presented.

Two-probe calibration integral equation method for nonlinear inverse heat conduction problem of surface heat flux estimation

In this paper, a new two-probe surface heat flux calibration integral equation method (CIEM) is proposed for resolving nonlinear inverse heat conduction problem possessing temperature-dependent thermophysical properties and an unknown time-varying back surface boundary condition. Compared to the one-probe CIEM, the two-probe CIEM can be applied to a more general class of one-dimensional inverse heat conduction problems that can be used for accurately predicting the front surface heat flux without any restrictions on the back surface boundary condition. Temperature-dependent thermophysical properties and probe locations are not specified a priori but are implicitly accounted through a calibration campaign. The final mathematical formulation involves resolving a Volterra integral equation of the first kind for the unknown surface heat flux. A first kind Chebyshev expansion possessing undetermined coefficients, is used for approximating the introduced property transform function. Next, three well-defined calibration tests are used for estimating the undetermined coefficients associated with the Chebyshev expansion. A time sequential study of the coefficients is shown to yield the optimal truncation of the series expansion. A future-time method is applied for stabilizing the ill-posed first kind Volterra integral equation. Phase plane and cross-correlation analyses are employed for estimating the optimal regularization parameter (i.e., the future-time parameter) from a spectrum of chosen values. The effectiveness of the proposed approach is verified by numerical simulation using stainless steel as the host material. The calibration-based or “parameter free” inverse approaches significantly reduce time and costs in the resolution of the surface condition as there is no need to experimentally determine the thermophysical properties, probe locations, and probe characteristics. These system properties are implicitly included in the proposed calibration procedure. Many applications, especially in aerospace sciences where exotic materials are often used, do not have replicable properties among samples. Conventional inverse methods rely on accurately specifying the input parameters for forming the predictions. These “parameter required” methods are solely based on numerical algorithms and thus their predictions are strongly relied on accurate input of parameters and measured probe data.