Inverse Heat Conduction Problem Research Articles

A Physics Informed Neural Network for Solving the Inverse Heat Transfer Problem in Gas Turbine Rotating Cavities

Abstract Recently, Physics-Informed Neural Networks have displayed great potential in delivering swift and accurate solutions for inverse problems. Here, a Physics-Informed Neural Network (PINN) was used to solve the inverse heat conduction problem in the rotating cavities of a aeroengine high-pressure compressor internal air system. The neural network was designed to receive experimentally captured radially distributed temperature profiles as inputs and predict the associated surface heat fluxes. The correctness of these predicted heat fluxes are assessed by numerically solving the direct heat conduction equation using a 2D Finite-Element model, thereby recovering the original temperature profiles. A comparative analysis is conducted between the predicted temperature profiles and the initial inputs. The physics informed neural network is trained using noise free synthetic data, created from a range of radial temperature curve fit coefficients, and subsequently tested on noisy experimental data at engine representative conditions. The predicted temperature values exhibit some good agreement with their respective actual counterparts. Furthermore, the sensitivity of model hyperparameters are explored to showcase the capability of the proposed approach. The results show that Physics- Informed Neural Networks exhibit reduced susceptibility to experimental uncertainties when addressing inverse problems, in contrast to inverse solution methods, and offer a possible new approach for analysis of experimental data if trained on a sufficiently large dataset.

Identification of Boundary Conditions in a Spherical Heat Conduction Transmission Problem

Although numerous analytical and numerical methods have been developed for inverse heat conduction problems in single-layer materials, few methods address such problems in composite materials. The following paper studies inverse interface problems with unknown boundary conditions by using interior point observations for heat equations with spherical symmetry. The zero degeneracy at the left interval 0<r<R1 leads to solution difficulties in the one-dimensional interface problem. So, we first investigate the well-posedness of the direct (forward) problem in special weighted Sobolev spaces. Then, we formulate three groups of unknown boundary conditions and inverse problems upon internal point measurements for the heat equation with spherical symmetry. Second-order finite difference scheme approaches for direct and inverse problems are developed. Computational test examples illustrate the theoretical statements proposed.

Numerical study of the influence of temperature data error on the determination of frictional heat generation

Subject of research: determination of the friction torque and functions of the specific intensity of heat generation in system of sliding bearings. The purpose of the study: To develop a method for determining the friction torque in sliding bearings using temperature data. Based on computational experiments, study the influence of errors in temperature data on the solution of the inverse problem. Methods and object of research: A system of sliding bearings made of a polymer composite material on which a rotating shaft rests is considered. A mathematical model of the thermal process in the considered system of friction units is presented, taking into account the spatial distribution of temperature and its change over time. The temperature is measured at several points in each bearing. Frictional heat generation is determined by solving the inverse heat conduction problem from the condition that the measured and calculated temperatures are close. To ensure continuous data processing and determine the friction torque during long-term tests, the inverse problem of determining the friction torque from temperature data is solved at successive short time intervals. Then the resulting solutions are «glued together». The main results of the study: With such a restoration, the discrepancy between the specified and restored functions of the specific intensity of heat generation is 10–15 % with an error level in temperature data of 10 %. The developed algorithm for solving the inverse problem of thermal conductivity can be used to determine the friction torques in real systems of self-lubricating sliding bearings.

Flow behavior and heat transfer characteristics of liquid film on vertical hot surface by inclined jet impingement

In this study, the flow behavior and heat transfer characteristics of the liquid film on the hot wall by inclined jet impingement are experimentally studied in detail. The effects of jet parameters, such as jet inclination angle, impingement distance, jet Reynolds number (Rej), and nozzle diameter are explored. The liquid film flow is visualized using a high-speed camera, and the surface temperature and heat flux are obtained by solving the inverse heat conduction problem. The results indicate that the liquid film shape is strongly affected by jet inclination angle but is almost unaffected by other parameters. As the inclination angle increases, the liquid film shape changes from elliptical to fusiform. In addition, the onset, enhancement, and disappearance of boiling cause the expansion and contraction of liquid film. The splashing rate is barely affected by jet parameters and remains within the range of 80–95 % under all conditions. The propagation of wetting fronts exhibits anisotropy. Except for the impingement distance, the position of wetting fronts along y-axis direction displays a high dependence on all parameters. The Rej and nozzle diameter have a significant effect on the heat flux at the impingement point and parallel flow zone, while the jet inclination angle and impingement distance only effect the impingement point. The visualization image proves that the droplet impingement pattern is the main reason for the increase in heat flux at higher impingement distances. Optimizing jet parameters can promote the wall to enter the rapid cooling stage in advance and increase maximum heat flux, thereby improving the maximum cooling capacity of the jet. Finally, an empirical equation is proposed to predict the maximum Nusselt number.

Adaptive fractional physics-informed neural networks for solving forward and inverse problems of anomalous heat conduction in functionally graded materials

In this paper, adaptive fractional physics-informed neural networks (PINNs) are employed to solve the forward and inverse problems of anomalous heat conduction in three-dimensional functionally graded materials. In adaptive fractional PINNs, the finite difference L1 scheme is employed to discretize the time fractional derivatives in anomalous heat conduction problems. By combining the finite difference L1 scheme with automatic differentiation technique, adaptive fractional PINNs minimize a loss function constructed based on the governing equation, initial and boundary conditions to obtain solutions to heat conduction problems. To avoid competition among various loss terms during the training process, an adaptive loss balancing algorithm is adopted to balance the interactions among different terms of the loss functions. In addition, polynomial basis functions are used to expand unknown functions characterizing material parameters or heat sources, aiming to enhance the performance of adaptive fractional PINNs in resolving inverse problems. Five numerical examples involving forward, inverse, and nonlinear problems related to anomalous heat conduction are carried out to demonstrate the feasibility and effectiveness of adaptive fractional PINNs.

An Experimental Study of Boiling Heat Transfer and Quench Front Propagation Velocity During Quenching of a Cylinder Rod in Subcooled Water

In this study, a quenching experiment was conducted at atmospheric pressure to investigate the flow and heat-transfer characteristics of cylindrical rods made from SS, FeCrAl, and Zr-4 under various subcooling degrees (ΔTsub). The inverse heat-conduction problem (IHCP) method and image-processing technique were utilized to determine the surface temperature and heat flux, vapor film thickness, and quench front propagation. The results show that smaller solid kρcp and larger ΔTsub result in relatively more efficient quenching boiling heat transfer, thinner vapor film thickness, and greater quench front propagation velocity. The quench front originates at the bottom of the test specimen and becomes progressively larger in velocity with time. It eventually converges with the downward-propagating quench front in the upper middle of the test specimen. Moreover, at the beginning of quench front propagation, the SS and FeCrAl test specimens have a constant velocity region. However, because the Zr-4 test specimen has a small kρcp, the velocities gradually increase from the onset of quench front generation. Furthermore, the measured average quench front velocities are consistent with the experimental datum from the literature. However, the predicted model proposed by Duffey underestimates the propagation velocity due to ignoring the cooling effect of film boiling.

Determination of the Temperature on the Inner Wall of the Piston Machine Body by Solving the Inverse Problem

In the study, the solution of the inverse heat conduction problem for the cross-section of the piston machine body is presented. The internal surface of the body is unevenly thermally loaded. In its upper part, where the heat flux on the wall is the highest, there is a cooling channel. Based on the temperature values at the measurement points inside the body, unknown temperature values on the inner wall surface were determined. Calculations were performed for time-averaged boundary conditions on the side of hot gases (inner surface of the body), the assumed temperature of the fluid, and the heat transfer coefficient in the cooling channel and on the external surface of the piston machine. A new method was used to solve the problem, in which the solution is sought in the form of a linear combination of basic functions that identically fulfill the heat conduction equation. The stability of the inverse problem was achieved using Tikhonov regularization.

Spectral integrated neural networks (SINNs) for solving forward and inverse dynamic problems

This study introduces an innovative neural network framework named spectral integrated neural networks (SINNs) to address both forward and inverse dynamic problems in three-dimensional space. In the SINNs, the spectral integration technique is utilized for temporal discretization, followed by the application of a fully connected neural network to solve the resulting partial differential equations in the spatial domain. Furthermore, the polynomial basis functions are employed to expand the unknown function, with the goal of improving the performance of SINNs in tackling inverse problems. The performance of the developed framework is evaluated through several dynamic benchmark examples encompassing linear and nonlinear heat conduction problems, linear and nonlinear wave propagation problems, inverse problem of heat conduction, and long-time heat conduction problem. The numerical results demonstrate that the SINNs can effectively and accurately solve forward and inverse problems involving heat conduction and wave propagation. Additionally, the SINNs provide precise and stable solutions for dynamic problems with extended time durations. Compared to commonly used physics-informed neural networks, the SINNs exhibit superior performance with enhanced convergence speed, computational accuracy, and efficiency.

A fast identification method of surface heat flux based on numerical pre-calibration and regularization

Abstract Identification of surface heat flux is a technique of retrieving heat flux from the measured temperature in an object. The inverse heat conduction problem abbreviated as IHCP is ill-posed, and its highly ill-conditioned characteristics lead to unstable numerical calculation. In this paper, we build an identification equation with the theory of linear superposition. The matrix of pulse sensitivity coefficient is established by formula for one-dimensional model. And a finite element pre-calibration approach is proposed to obtain the matrix of pulse sensitivity coefficient for complex three-dimensional structures. Regularization methods including Tikhonov method and algebraic reconstruction technique are utilized to reduce the sensitivity of measurement error. Numerical simulation shows that this method is efficient and accurate.

Estimation of Surface Temperature and Heat Flux over a Hollow Cylinder and Slab using an Inverse Heat Conduction Approach

A non-iterative approximation of the inverse heat conduction problem has been developed through energy balance between the control volumes. The heat flow domain along the surface normal was divided into three control volumes and studied in three cases, specifically on a heated hollow metallic cylinder cooled in crossflow of air, a heating flat plate in still air, and a heated flat plate when cooled in still air with random air blows. The time derivatives of temperature and heat flux estimates at the surface appearing in derived equations were evaluated by approximate expressions. Both estimates were obtained from the derived equations by simultaneous measurement of temperature-time histories at two locations: one at the inner surface and the other anywhere within the control volume along the surface normal. The deviation was checked by real-time simultaneous measurement of temperature-time history at the outer surface along the normal surface. Comparison between the estimated surface temperature-time history using the derived equations and the real-time measurement showed deviations within 0.5 % for the cylinder, within 0.03 % for the plate in still air and within 0.5 % when air blows were given. Heat fluxes estimated using these time histories were correspondingly arrived at consistent and close approximations for all cases. Estimates from the derived equations were compared with reported equations from the literature, and a wind tunnel experiment for the validation of circumferential distribution of heat transfer was conducted, for which they also showed reasonably good agreements.

Analytical solution to the two-dimensional inverse heat conduction problem in an axisymmetric cylindrical coordinate system

The inverse heat conduction problem involves estimating an unknown cooling or heating action based on internal temperature histories in a body. Such a problem is ill-posed because the estimated results are highly sensitive to input data noise. An analytical solution is developed for the two-dimensional inverse heat conduction problem in an axisymmetric cylindrical coordinate system using the Laplace transform technique. On the basis of the known temperatures at multiple measuring points distributed on two cylindrical measuring surfaces along the axial direction inside the solid, the expressions for the surface temperature and heat flux are explicitly obtained in the form of a power series of time and eigenfunctions of the space variable. A comprehensive estimation error assessment is conducted using the internal temperature distribution that is obtained using a direct heat conduction analysis under the prescribed stationary and moving boundary conditions. The constant thermal condition is well reproduced using the present inverse solution, except in the vicinity of the discontinuity point. In addition, the effective heat flux and effective heat transfer coefficient are introduced to obtain the best estimation of the transient heat transfer in the major cooling area of the rotating cylinder.

Calculation of Green's functions for heat conduction in curved anisotropic media

AbstractSolving the inverse problems of heat conduction often requires performing a very large number of Green's function evaluations. This paper addresses the calculation of Green's functions in anisotropic curved media, such as composite lamina, where the principal axes of thermal conductivity follow the curved surface. We start with established exact series solutions for cylindrical convex and concave shapes. These solutions are extended to accommodate geometries of real‐world composite materials that do not form a closed surface like a cylinder. Unfortunately, the exact solutions have the form of an infinite series of sums or integrals and are computationally infeasible for the inverse problems of interest, especially for large radii of curvature. This motivates a perturbation solution that is accurate and computationally efficient at large radii of curvature. In addition, it motivates a phenomenological approximation that is extremely computationally efficient over a broad range of curvatures, but with some sacrifice in accuracy. These solutions are compared with exact solutions, with each other, and with numerical finite difference calculation. We identify regions in parameter space where the different approaches are preferable and where they lead to the same numerical result.

The inverse problem of heat conduction in the case of non-uniqueness: A functional identification approach

Abstract A problem of identification of the set of a thermal-conductivity coefficients for the nonlinear heat equation in the case of non-uniqueness is considered. Classes of inverse heat conduction problems (IHCP) with a non-unique solution are defined. Explicit descriptions of sets of thermal-conductivity coefficients for these classes are obtained. For solving the identification problem the functional identification approach is used. Unlike traditional methods, the proposed algorithm does not utilize approximations of the coefficient with a finite system of basis functions. The results of computational experiments are presented. It is shown that the functional identification approach makes it possible to numerically identify the non-uniqueness of the solution of the inverse problem of heat conduction.

Identification of an N‐valued heterogeneous conductivity profile in an inverse heat conduction problem

Identification of an <i>N</i>‐valued heterogeneous conductivity profile in an inverse heat conduction problem

An accelerated sequential function specification method based on LM gradient for transient inverse heat conduction problem

The sequential function specification method (SFSM) and the Levenberg-Marquardt method (LM) are two typical methods for inverse heat conduction problems (IHCP). However, these two methods have poor convergence and low convergence speed. Therefore, the combination of LM with SFSM is depended on to propose a modified Sequential Levenberg-Marquardt gradient method (SLM). The computation results of three shapes of heat fluxes with noise show the SLM is averagely improved over 16.3% more than SFSM in computation accuracy, but equally reduced over 39.9% less than in computation time. On other hand, when the emphasis is on the effect of the Aitken acceleration method, Tikhonov regularization and polynomial order on SLM, a neural network prediction model of the root mean square error is trained by regularization factor, polynomial order and the number of future time steps. Subsequently, the genetic algorithm optimization method is proposed for the minimum root mean square error so that it can improve the computation accuracy and efficiency of IHCP to the maximum extent. Finally, Second-order polynomial order and small order-of-magnitude regularization factor are highly recommended for parameter selection with multiple choices. The ASLM only needs to determine the optimal number of future time steps.

Noninvasive reconstruction of internal heat source in biological tissue using adaptive simulated annealing algorithm

The heat distribution information of human lesions is of great value for disease analysis, diagnosis, and treatment. It is a typical inverse problem of heat conduction that deriving the distribution of internal heat sources from the temperature distribution on the body surface. This paper transforms such an inverse problem of bio-heat transfer into a direct one, thereby avoiding complex boundary conditions and regularization processes. To noninvasively reconstruct the internal heat source and its corresponding 3D temperature field in biological tissue, the adaptive simulated annealing (ASA) algorithm is used in the simulation module, where the position P(x, y, z) of point heat source in biological tissue and its corresponding temperature T are set as the optimization variables. Under a certain optimized sample, one can obtain the simulated temperature distributing on the surface of the module, then subtract the simulated temperature from the measured temperature of the same surface which was measured using a thermal infrared imager. If the sum of absolute values of the difference is smaller, it indicates that the current sample is closer to the true location and temperature of the heat source. When the values of optimization variables are determined, the corresponding 3D temperature field is also confirmed. The simulation results show the simulated position and temperature of the heat source are very approximate with those of the real experimental module. The method presented in this paper has enormous potential and promising prospects in clinical research and application, such as tumor hyperthermia, disease thermal diagnosis technology, etc.

A Bayesian Spatiotemporal Modeling Approach to the Inverse Heat Conduction Problem

Abstract This study introduces a Bayesian spatiotemporal modeling approach to solve inverse heat conduction problems (IHCPs), employing penalized splines within a spatiotemporal forward model. The complexity and ill-posed nature of IHCPs, characterized by potential nonexistence, nonuniqueness, or instability of solutions, pose significant challenges for traditional methods. Addressing this, our study presents a spatiotemporal forward model that incorporates spatial, temporal, and interaction terms, accurately capturing the intricate dynamics inherent in IHCPs and using this information as a leverage to solve the inverse problem. We adopted a Bayesian inference framework for the subsequent parameter estimation problem and developed a Gibbs sampling algorithm to sample from the posterior distribution of the model's parameters, enhancing the estimation process. Through case studies on a one-dimensional (1D) heat simulation and a pool boiling experiment using multisensor thermocouple data for heat flux reconstruction, we demonstrate the model's superiority over traditional methods. The inclusion of the spatiotemporal interaction term significantly enhances model performance, indicating its potential for broader application in solving IHCPs. The application of this method in both simulated and real-world scenarios highlights its effectiveness in capturing the spatiotemporal complexities of IHCPs. This work contributes to the field by offering a robust methodology for addressing the spatial and temporal complexities inherent in IHCPs, supported by a comprehensive Bayesian inference framework and the use of a Gibbs sampling algorithm for parameter estimation.

Genetic Algorithm as the Solution of Non-Linear Inverse Heat Conduction Problems: A Novel Sequential Approach

Abstract Direct measurement of surface heat flux could be extremely challenging, if not impossible, in numerous applications. In such cases, the use of temperature measurement data from subsurface locations can facilitate the determination of surface heat flux and temperature through the solution of the inverse heat conduction problem (IHCP). Several different techniques have been developed over the years for solving IHCPs, with different levels of complexity and accuracy. The filter coefficient technique has proved to be a promising approach for solving IHCPs. Inspired by the filter coefficient approach, a novel method is presented in this paper for solving one-dimensional IHCPs in a domain with temperature-dependent material properties. A test case is developed in COMSOL Multiphysics where the front side of a slab is subject to known transient heat flux and the temperature distributions within the domain are calculated through numerical simulation. The IHCP solution in the form of filter coefficients is defined and a genetic algorithm (GA) is used for the calculation of filter matrix. The number of significant filter coefficients required to evaluate surface heat flux at each time-step is determined through trial and error and the optimal number is selected for evaluating the solution. The structure of the filter matrix is assessed and discussed. The resulting filter coefficients are then used to evaluate the surface heat flux for several different test cases and the performance of the proposed approach is assessed in detail. The results showed that the presented approach is robust and can result in finding optimal filter coefficients that can accurately estimate various types of surface heat flux profiles using temperature data from a limited number of time steps and in a near real-time fashion.

Restoration of unsteady heat flow from a thermal energy accumulator by solving the inverse heat conduction problem

Restoration of unsteady heat flow from a thermal energy accumulator by solving the inverse heat conduction problem

Oil and Gas Structures: Forecasting the Fire Resistance of Steel Structures with Fire Protection under Hydrocarbon Fire Conditions

The hydrocarbon temperature–time curve is widely used instead of the standard curve to describe the temperature in the environment of structural surfaces exposed to fire in oil and gas chemical facilities and tunnels. This paper presents calculations of the ratio of time to reach critical temperatures at different nominal fire curves for steel structures such as bulkheads and columns with different types of fireproofing. The thermophysical properties of the fireproofing materials were obtained by solving the inverse heat conduction problem using computer simulation. It was found that the time interval for reaching critical temperatures in structures with different types of fireproofing in a hydrocarbon fire decreased, on average, by a factor of 1.2–1.7 compared to the results of standard fire tests. For example, for decks and bulkheads with mineral wool fireproofing, the K-factor of the ratio of the time for reaching the critical temperature of steel under the standard curve to the hydrocarbon curve was 1.30–1.62; for plaster, it was 1.56; for cement boards, it was 1.34; for non-combustible coatings, it was 1.38–2.0; and, for epoxy paints, it was 1.71. The recommended values of the K-factor for fire resistance up to 180 min (incl.) were 1.7 and, after 180 min, 1.2. The obtained dependencies would allow fireproofing manufacturers to predict the insulation thickness for expensive hydrocarbon fire experiments if the results of fire tests under standard (cellulosic) conditions are known.