Heat Conduction Problem Research Articles

A merging approach for hole identification with the NMM and WOA-BP cooperative neural network in heat conduction problem

Defect identification is an important issue in structural health monitoring. Herein, originated from inverse techniques, a merging approach is established by the numerical manifold method (NMM) and whale optimization algorithm-back propagation (WOA-BP) cooperative neural network to identify hole defects in heat conduction problems. On the one hand, the NMM can simulate varying hole configurations on a fixed mathematical cover, which eases the generation of “big data” for the training of neural network to a large extent. On the other hand, the WOA, a global optimization algorithm, is adopted to optimize the initial weights and thresholds of the BP neural network to alleviate its frequently encountered local optimum phenomenon. The boundary temperatures of sampling points by the NMM and the associated hole geometries are used for the learning of WOA-BP neural network, which is then applied to predict the hole defects. Numerical examples concerning the detection of circular/ elliptical holes demonstrate that the proposed method possesses higher accuracy and satisfying robustness in holes prediction compared with standard BP network under the same condition. The present work provides a convenient pathway and great potential in application of structural health monitoring.

Bayesian Inference for Estimating Heat Sources Through Temperature Assimilation

Abstract This paper utilizes a Bayesian inference framework to address the two-dimensional (2D) steady-state heat conduction problem, focusing on the estimation of unknown distributed heat sources in a thermally conducting medium with uniform conductivity. The goal is to infer the locations, strength, and shape of heaters by assimilating temperature data in Euclidean space, employing a Fourier series to represent each heater's shape. The Markov Chain Monte Carlo (MCMC) method, incorporating the random-walk Metropolis–Hasting (MH) algorithm and parallel tempering, is utilized for posterior distribution exploration in both unbounded and wall-bounded domains. It is found that multiple solutions arise in cases where the number of temperature sensors is less than the number of unknown states. Moreover, smaller heaters introduce greater uncertainty in estimated strength. To address the challenge of estimating the heater's strength and shape simultaneously due to their strong correlation, our method incorporates sharp priors on one to ensure accurate and feasible solutions of the other. The diffusive nature of heat conduction smooths out any deformations in the temperature contours, especially in the presence of multiple heaters positioned near each other, impacting convergence. In wall-bounded domains with Neumann boundary conditions, the inference of heater parameters tends to be more accurate than in unbounded domains.

A Special Corner Element for Solving Heat Conduction Problems

A hybrid Trefftz finite element method is proposed for solving heat conduction problems in a domain with concave or convex angles. In this method, two temperature fields are independently assumed, one of which is the intra-element field defined inside the element and the other is the frame field defined on the boundary. The corner effects embed in the truncated complete solutions which are used to construct the internal interpolation function for approximating the intra-element field. It is able to accurately analyze the problems with particular corners. The key characteristics lie in the finite element formulation which is evaluated along the element boundary. Compared with the conventional FEM, the corner element does not lose any accuracy by using a small number of elements. Numerical examples demonstrate that the proposed method exhibits low sensitivity to mesh distortion. Compared with ABAQUS, the proposed method also exhibits higher accuracy and a faster convergence rate.

Analysis of the accuracy of the inverse marching method used to determine thermal stresses in cylindrical pressure components

This paper analyses the inverse marching method used to determine the thermal stresses on the inner surface of a thick-walled cylindrical element not weakened by holes in the transient state. The heat conduction problem was considered one-dimensional, i.e. it was assumed that heat is transferred only in the radial direction. The method is based on measuring the temperature inside the pipeline wall at a single point and assuming that the pipeline is thermally insulated. The paper undertook an evaluation of the influence of the measuring point's distance from the inner surface, the number of control volumes into which the inverted area was divided and the length of the time step on the accuracy of the calculated temper-ature, heat transfer coefficient and thermal stresses on the inner surface of the pressure element. Verification was performed by comparing the calculation results obtained from the direct analytical method perturbed by random errors with those obtained from the numerical inverse step method.

Two-phase heat conduction problems with Sturm-type boundary conditions

AbstractIn this work, the solution is justified using the method of separation of variables for initial-boundary value problems of the heat conduction equation with piecewise constant thermal conductivity coefficient under Sturm-type boundary conditions (separated boundary conditions). Various cases are considered. A system of eigenfunctions of the corresponding spectral problem is constructed and it is proved that this system of eigenfunctions forms a Riesz basis. The theorem of the existence of a unique classical solution to the problem is proved.

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.

Synthetic iterative scheme for thermal applications in hotspot systems with large temperature variance

A synthetic iterative scheme is developed for thermal applications in hotspot systems with large temperature variance. Different from previous work with linearized equilibrium state and small temperature difference assumption, the phonon equilibrium distribution shows a nonlinear relationship with temperature and mean free path changes with the spatial temperature when the temperature difference of system is large, so that the same phonon mode may suffer different transport processes in different geometric regions. In order to efficiently capture nonlinear and multiscale thermal behaviors, the Newton method is used and a macroscopic iteration is introduced for preprocessing based on the iterative solutions of the stationary phonon BTE. Macroscopic and mesoscopic physical evolution processes are connected by the heat flux, which is no longer calculated by classical Fourier’s law but obtained by taking the moment of phonon distribution function. These two processes exchange information from different scales, such that the present scheme could efficiently deal with heat conduction problems from ballistic to diffusive regime. Numerical tests show that the present scheme could efficiently capture the multiscale heat conduction in hotspot systems with large temperature variances. In addition, a comparison is made between the solutions of the present scheme and effective Fourier’s law by several heat dissipations problems under different sizes or selective phonon excitation. Numerical results show that compared to the classical Fourier’s law, the results of the effective Fourier’s law could be closer to the BTE solutions by adjusting effective coefficients. However, it is still difficult to capture some local nonlinear phenomena in complex geometries.

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.

A fast methodology for identifying thermal parameters based on improved POD and particle swarm optimization and its applications

The identification method based on the traditional Proper Orthogonal Decomposition (POD) reduced-order model has the problem of low efficiency, due to the large amount of both data and computation, when dealing with complicated problems with a large number of spatially distributed nodes. To deal with this issue, an improved POD reduced-order model is proposed in this work. The improved POD reduced-order model only requires the establishment of a three-dimensional (3D) database of training samples varied with both time and measurement locations. Therefore, the amount of data and computation is independent of the total number of spatially distributed nodes, which enables the amount of data and computation to be greatly reduced. To identify thermal parameters in heat conduction problems, a database of transient temperature field is constructed with different training parameters and space nodes by using polygonal boundary element method, and a set of POD basis vectors is obtained by the POD reduced-order model. Then, a surrogate model combined with the improved particle swarm optimization (PSO) is employed for the identification of thermal parameters. Three different inverse heat conduction problems are designed and analyzed to verify the performance of the improved methodology. The results show that the efficiency of the modified method is superior to the traditional POD method with comparable accuracy. The more of the number of spatially distributed nodes, the more obvious advantages of the efficiency. Furthermore, this method has been tested on noisy data, proving its reliability in dealing with problems arising from measurement errors.

A thermodynamics-consistent spatiotemporally-nonlocal model for microstructure-dependent heat conduction

The spatiotemporally-nonlocal phenomena in heat conduction become significant but challenging for metamaterials with artificial microstructures. However, the microstructure-dependent heat conduction phenomena are captured under the hypothesis of spatiotemporally local equilibrium. To capture the microstructure-dependent heat conduction phenomena, a generalized nonlocal irreversible thermodynamics is proposed by removing both the temporally-local and spatially-local equilibrium hypotheses from the classical irreversible thermodynamics. The generalized nonlocal irreversible thermodynamics has intrinsic length and time parameters and thus can provide a thermodynamics basis for the spatiotemporally-nonlocal law of heat conduction. To remove the temporally-local equilibrium hypothesis, the generalized entropy is assumed to depend not only on the internal energy but also on its first-order and high-order time derivatives. To remove the spatially local equilibrium hypothesis, the thermodynamics flux field in the dissipation function is assumed to relate not only to the thermodynamics force at the reference point but also to the thermodynamics force of the neighboring points. With the developed theoretical framework, the thermodynamics-consistent spatiotemporally-nonlocal models can then be developed for heat transfer problems. Two examples are provided to illustrate the applications of steady-state and transient heat conduction problems.

A simple, effective and high-precision boundary meshfree method for solving 2D anisotropic heat conduction problems with complex boundaries

A simple, effective and high-precision boundary meshfree method called virtual boundary meshfree Galerkin method (VBMGM) is used to tackle 2D anisotropic heat conduction problems with complex boundaries. Temperature and heat flux are expressed by virtual boundary element method. The virtual source function is constructed through the utilization of radial basis function interpolation. Calculation model diagram and discrete model diagram of real boundaries, and schematic diagram of VBMGM are demonstrated. Using Galekin method and considering boundary conditions, the integral equation and the discrete formula of VBMGM are given in detail. The benefits of the Galerkin, meshfree, and boundary element methods are all presented in VBMGM. Seven numerical examples of general anisotropic heat conduction problems (including three numerical examples with complex boundaries and four numerical examples with mixed boundary conditions) are computed and contrasted with precise solutions and different numerical methods. The computation time of each example is given. The number of degrees of freedom used in many examples is half or less than that of the numerical method being compared. The suggested method has been demonstrated to be effective and high-precision for solving the 2D anisotropic heat conduction problems with complex boundaries.

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.

Deep learning-based topology optimization for multi-axis machining

This paper presents a novel framework that integrates topology optimization (TO) and deep learning (DL) to generate high-performance structures suitable for multi-axis machining. Within the proposed framework, DL is built on the pix2pix network, with the conditional channel used to determine the tool shape and feed direction in multi-axis machining. This DL model will be trained using our own generated dataset on TO for multi-axis machining. Then, users can customize tool dimensions and machining orientations of the multi-axis machining operation and specify the design boundary and loading conditions as input. The DL model will rapidly generate a near-optimized structure, which subsequently serves as the starting point for further optimization. Ultimately, a topology-optimized structure that meets the tailored requirements is apt for multi-axis machining and can be finalized with only a few iterations. 2D and 3D numerical examples for heat conduction problems are studied to prove the effectiveness of the proposed method, validating improved structural performance and optimization efficiency compared to conventional TO for multi-axis machining.

A weakly coupled system of p-Laplace type in a heat conduction problem

Abstract Given is a bounded domain Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} , and a vector-valued function defined on ∂ ⁡ Ω {\partial\Omega} (depicting temperature distributions from different sources), our objective is to study the mathematical model of a physical problem of enclosing ∂ ⁡ Ω {\partial\Omega} with a specific volume of insulating material to reduce heat loss in a stationary scenario. Mathematically, this task involves identifying a vector-valued function 𝐮 = ( u 1 , … , u m ) {\mathbf{u}=(u^{1},\dots,u^{m})} ( m ≥ 1 {m\geq 1} ) that represents the temperature within Ω and gives rise to a free boundary, somehow reminiscent of, but not equivalent to, the Bernoulli free boundary problem.

Застосування методу двобічних наближень до розв’язання першої крайової задачі для одновимірного рівняння теплопровідності з експоненціально нелінійним коефіцієнтом теплопровідності

The problem of heat conduction in nonlinear media reduces to solving boundary value problems for nonlinear heat conduction equations, where the coefficients of the equation or the function of the power of heat sources depend on temperature according to some law. Among the numerical methods for solving problems for nonlinear equations of mathematical physics, one can distinguish finite difference methods, finite element methods, variational and projection methods, as well as iterative methods. Among the latter group, the two-sided approximation method is particularly attractive due to its ability to provide a convenient estimate for the error of the approximate solution and to prove the existence of a solution to the original problem. The theory of linear partially ordered spaces was developed by L. V. Kantorovich in the second half of the 1930s. Further development of this theory is associated with the works of M. A. Krasnosel’skii, H. Amann, V. I. Opojtsev, N. S. Kurpel, B. A. Shuvar, A. I. Kolosov etc. The aim of this article is to develop a two-sided approximation method based on the use of Green's functions for solving the first boundary value problem for a one-dimensional nonlinear heat conduction equation and to investigate its performance when solving test problems. To achieve this goal, the unknown function was replaced, and the boundary value problem was reduced to a Hammerstein integral equation, which was considered as a nonlinear operator equation in a partially ordered Banach space. Conditions for the existence of a unique positive solution to the problem and conditions for two-sided convergence of successive approximations to it were obtained. The developed method was implemented in software and tested on solving test problems. The results of the computational experiment are illustrated with graphical and tabular information

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.