Three-dimensional Heat Conduction Problem Research Articles

Detection of surface moving heat source using experimental temperature measurements on the opposite surface and inverse techniques

The identification of mobile sources is a widely researched area in multiple scientific and technical domains. Detecting mobile heat sources is an essential task in different fields, including fire safety, environmental monitoring, material science, and energy systems. In this investigation, we present an innovative solution to solve the inverse problem of detecting mobile heat sources, utilizing a three-dimensional transient inverse heat conduction problem (IHCP) formulation. The goal is to identify a moving heat source from thermographic infrared measurements of the rear face. The proposed method includes a finite volume approach with model reduction techniques to improve computational efficiency. Additionally, we use the conjugate gradient method to solve the inverse problem. We verified the proposed approach by conducting experiments on a metal plate with a moving heat source from a laser and comparing recalculated temperature of the front face with thermocouple measurements. Our results indicate that our method can detect the location and trajectory of the mobile heat source with high accuracy and efficiency. This study also highlighted the limits of source speed that can be detected by this method. This technique could be considered as an excellent option for real-time monitoring of mobile heat sources in practical applications.

MLS-based numerical manifold method based on IPIM for 3D transient heat conduction of FGMs

In this study, the moving least squares based numerical manifold method (MLS-NMM) is presented to solve three-dimensional (3D) transient heat conduction problems of functionally graded materials (FGMs), where the increment dimensional precise integration method (IPIM) is applied to integrate the ordinary differential equations derived from the semi-discrete form. In the 3D MLS-NMM, the influence domains of the MLS-nodes are taken as the mathematical patches which are used to construct the mathematical cover (MC), and the shape functions of the MLS-nodes are employed as the weight functions subordinate to the MC. MLS-NMM is utilized for spatial discretization of the weak form of the problem. The IPIM overcomes the shortcomings of traditional backward difference scheme dependent on the time step and greatly improves the computing efficiency. Meanwhile, a mass lumping technique is proposed for the 3D MLS-NMM. Furthermore, a number of numerical experiments on transient thermal conduction are conducted, suggesting that the 3D MLS-NMM based on IPIM and the proposed mass lumping has excellent accuracy, absolutely stable and high computing efficiency.

MLS-based numerical manifold method for steady-state three-dimensional heat conduction problems of functionally graded materials

In this study, the moving least squares based numerical manifold method (MLS-NMM) is firstly applied to discretize three-dimensional (3D) steady heat conduction problems of functionally graded materials (FGMs). In the 3D MLS-NMM, the influence domains of nodes in the moving least squares (MLS) are used as the mathematical patches to construct the mathematical cover (MC); while the shape functions of MLS-nodes as the weight functions subordinate to the MC. Compared with the traditional NMM using finite elements to form the MC, the cutting operations in generating the physical cover are unnecessary and the computation complexity is decreased significantly. Based on the Galerkin method, the discrete form of 3D heat conduction is derived. Finally, a series of numerical experiments concerning steady heat conduction problems are performed, suggesting that the proposed MLS-NMM enjoys advantages of both MLS and NMM in solving steady heat conduction.

A new general analytical PBEM for solving three-dimensional transient nonlinear heat conduction problems with spatially-varying heat generation

In this paper, a polygonal boundary element method (PBEM) is presented, to solve the transient nonlinear heat conduction problems with spatially variable heat generation and temperature-dependent thermal physical properties. A new general method is developed to analytically compute the domain integrals regarding arbitrary nonlinearly-varying thermal conductivity, specific heat, and density, by which the computations of numerous Gaussian points in the numerical method can be circumvented and the efficiency of the PBEM can be significantly improved. The finite difference scheme is employed to evaluate the transient term, and the Newton iterative method is used to search the exact solution of the nonlinear system equation. Four numerical examples are designed to examine the property of the proposed method. The results indicate that the proposed analytical PBEM can accurately solve the three-dimensional transient nonlinear heat conduction problems with heat generation.

Inverse Estimation of Effective Thermal Conductivity of Multilayer Materials Considering Thermal Contact Resistance

Abstract Multilayer materials have been widely used in engineering applications, attributed to excellent thermo-mechanical performances. However, the thermal or mechanical properties of multilayer materials remain elusive, owing to contact behaviors etc. factors. In order to address this issue, an innovative method is employed to estimate the effective thermal conductivity (ETC) of multilayer materials considering thermal contact resistance (TCR) between layers, and the equivalence performance is investigated by solving three-dimensional inverse heat conduction problems. First, the equivalence method is validated by available experimental data of a multilayered insulation composite material. Then, the precision of different equivalence methods is compared, and the results indicate that the anisotropic equivalence method has higher accuracy than the isotropic and orthotropic equivalences for the five-layer material in the present work. Finally, the robustness and stability of the anisotropic equivalence method are evaluated in detail. The present work provides a new alternative method for predicting the effective thermal conductivity of multilayer materials.

Mathematical model of slab heating in a furnace with walking beams

Slab heating before hot rolling is necessary for obtaining required metal ductility. The most effective for this purpose are furnaces with walking beams that provide heat supply to all sides of the slab. However, the areas of slabs lower surfaces, contacting with water-cooled beams, are shielded from the radiation of the furnace lower heating zones and take the heat from the beams.To study the inhomogeneity of the slab temperature field and its dependence on the peculiarities of their transport system design, a mathematical model of slab heating in a furnace with walking beams was developed and programmatically implemented, based on numerical solution of a three-dimensional heat conduction problem with piecewise defined boundary conditions on the lower surface. For the open areas of the slab bottom surface, the same boundary conditions were set as on the top surface; and for the areas of contact with the beams, modified boundary conditions were set, taking into account the duration of this contact. For the numerical solution of the system of difference equations, the line-by-line method was applied, which allows us to obtain a system with a three-diagonal matrix of coefficients. The calculations carried out in the approximation of adiabaticity of the contact areas of the slab with the beams during the contact period allowed us to obtain temperature fields for different slab sections. As a result, a significant irregularity of temperature field of the slab lower surface was revealed, affecting the irregularity of temperature field of the entire slab. The developed program for calculating and visualizing the results can be used to study the temperature field of the slab under various heating modes if there is experimental information that allows one to clarify the tuning parameters of the model.

The 3-D problem of temperature and thermal flux distribution around defects with temperature-dependent material properties

The analytical solution of 3-D heat conduction problem, including the temperature and thermal flux fields, is one of the important problems that have not been completely solved in solid mechanics. Considering the temperature dependence of material parameters makes the problem more difficult. In this paper, we first reduce the 3-D temperature-dependent heat conduction problem to the solution of 3-D Laplace equation by introducing the intermediate function. Then, the generalized ternary function is proposed, and the general solution of 3-D Laplace equation is given. Finally, the analytical solutions of three specific problems are obtained and the corresponding temperature-thermal flux fields are discussed. The results show that the thermal flux field of 3-D temperature dependent problem is the same as the classical constant thermal conductivity approach result, while the temperature field is different from the classical result. Thermal flux at a planar defect boundary has r?1/2 singularity, and its intensity is proportional to the fourth root of defect width. On the other hand, when blocked by a planar defect, the thermal flux distribution will re-adjusted so that it overflows at the same rate from all parts of the planar defect boundary.

Identifying Varying Thermal Diffusivity of Inhomogeneous Materials Based on a Hybrid Physics-Informed Neural Network

Inhomogeneous materials are characterized by uneven material properties with changing dimensions. Effective and precise identification of nonuniform material properties is essential for studying material inhomogeneity. In this study, we propose a hybrid physics-informed neural network (PINN) to identify the inhomogeneous thermal diffusivity only using temperature data collected in a transient heat conduction problem. A PINN and a fully connected neural network are combined in the proposed model. The PINN model is trained to learn the law of physics governed by the transient heat conduction equation. The identification network is trained to uncover the thermal diffusivity variation with changing dimensions. The proposed model successfully identifies two types of inhomogeneity: temperature-dependent and space-dependent thermal diffusivity, and both two-dimensional and three-dimensional heat conduction problems are investigated. The proposed model is examined to be robust to small datasets and noisy training data. In addition to heat conduction problems, the proposed model can be adopted to identify the material properties of various types of inhomogeneous materials.

A general analytical PBEM for solving three-dimensional transient inhomogeneous heat conduction problems with spatially varying heat generation

In the present work, a polygonal boundary element method (PBEM) for solving transient inhomogeneous heat conduction problems with spatially-varying heat generation is developed for the first time. A new general analytical method is proposed and employed in the present PBEM, by which the radial integrals associated with arbitrary spatially-varying density, specific heat, and thermal conductivity can be analytically computed, and the efficiency of the PBEM for solving the transient inhomogeneous problems can be significantly improved. The transient term is dealt with by a finite difference scheme. Three examples are designed to investigate the performance of the proposed method, and the results show that the present method could accurately and efficiently solve transient heat conduction problems with arbitrarily-varying thermal physical properties.

SOFTWARE FOR SIMULATION OF NON-STATIONARY HEAT TRANSFER IN ANISOTROPIC SOLIDS

The software «AnisotropicHeatTransfer3D» for the numerical solution of three-dimensional non-stationary heat conduction problems in anisotropic solids of complex domain by meshless method [1] was developed. The numerical solution of the non-stationary anisotropic heat conduction problem in the software «AnisotropicHeatTransfer3D» is based on a combination of the dual reciprocity method [2] with anisotropic radial basis functions [3] and the method of fundamental solutions [4]. The dual reciprocity method with anisotropic radial basis functions is used to obtain particular solution, and the method of fundamental solutions is used to obtain a homogeneous solution of boundary-value problem.

Investigation of the transient heat transfer to a supersonic air jet impinging on a high-temperature plate based on a discrimination-experiment method.

A discrimination-experiment method is developed to investigate the transient heat transfer of air jet impingement by discretizing the solid domain into mutually adiabatic test cylinders. This method can not only reduce the influence of the transverse heat transfer of a solid domain on the heat transfer characteristics of the jet but can also simplify the two-dimensional or three-dimensional heat conduction problem into a one-dimensional problem. Moreover, the discrimination-experiment method eliminates the embedment of thermocouples into the solid domains, further improving the accuracy and reliability of the proposed method. The transient heat transfer characteristics of a supersonic air jet impinging on a high-temperature target (860°C) and the effects of thermo physical parameters, such as the density, specific heat capacity, thermal conductivity and nozzle-to-target distance (H/D = 3, 4, and 5) are analyzed in detail using the discrimination-experiment method. The results provide important guidance for the thermal design of supersonic air jet impingement.

Analytically-integrated radial integration PBEM for solving three-dimensional steady heat conduction problems

Recently, a polygonal boundary element method (PBEM) has been developed for solving heat conduction problems. In the PBEM, the radial integration method (RIM) is employed to shift the domain and surface integrals in the boundary-domain integral equation into equivalent line integrals, and all the radial integrals are computed by Gaussian quadrature, which may obtain an accurate solution. However, the computation costs much. Therefore, analytical expressions of radial integrals are derived in this paper, concerning four kinds of varying thermal conductivities and two kinds of heat generation functions, aiming at improving the efficiency. Then all the radial integrals in the boundary-domain integral equation can be analytically calculated. Three examples are employed to examine the analytically-integrated PBEM for solving steady heat conduction problems. The results show that the proposed method is accurate, and it is more efficient than traditional PBEM.

Numerical modelling in problems of thermal control for three-layer structures with defects

This article describes solutions to the direct and inverse problems of the three-dimensional non-stationary heat conduction problem in a three-layer structure, using the finite element method for the direct problem and the gradient descent method for the inverse problem. A comparison of the FEM-solution and the analytical solution for a solid with a simple geometry is presented. Here are presented solutions of the direct and inverse three-dimensional non-stationary heat conductivity problem for a free three-stage turbine. The accuracy of the found and exact solutions is compared.

Three-dimensional transient heat conduction problems in FGMs via IG-DRBEM

In this paper, the isogeometric dual reciprocity boundary element method (IG-DRBEM) is firstly applied to solve three-dimensional (3D) transient heat conduction problems in functionally graded materials (FGMs) with heat sources. Using the fundamental solution of the 3D potential problem, the integral equation of boundary-domain is established by the weighted residual method. The normalized temperature is introduced, and the domain integral is transformed into the boundary integral by the dual reciprocity method (DRM). Finally, the theoretical framework of the isogeometric BEM for FGMs problems is established. Furthermore, the effects of different approximation functions, time steps, element refinements and the distance from the inner point to the boundary on the results are discussed in detail via several examples, which verify the validity of the presented method. In order to promote the development of IGBEM, the program code of this article can be obtained by sending email to yubochina@hfut.edu.cn.

The dimension splitting interpolating element-free Galerkin method for solving three-dimensional transient heat conduction problems

In this paper, based on dimension splitting method and the improved interpolating moving least-squares (IMLS) method, a dimension splitting interpolating element-free Galerkin (DSIEFG) method for three-dimensional (3D) transient heat conduction problems is proposed. The main idea of the DSIEFG method is to split a 3D problem domain into a series of related two-dimensional (2D) subdomains. The improved IMLS method is used to construct shape function on 2D subdomains. Finite difference method is used to couple these discretized equations on 2D subdomains and employed in the time domain. Compared with the improved element-free Galerkin (IEFG) method, the advantage of the DSIEFG method is that the essential boundary conditions can be applied directly, which can improve computational accuracy and efficiency. Six examples are chosen to verify the effectiveness and efficiency of the DSIEFG method. The results of DSIEFG are compared with the numerical solutions of the IEFG method, and it is shown that the efficiency and precision of the DSIEFG method are greater than ones of the IEFG method for 3D transient heat conduction problems.

IG-DRBEM of three-dimensional transient heat conduction problems

In this paper, the isogeometric dual reciprocity boundary element method (IG-DRBEM) is proposed to solve three-dimensional transient heat conduction problems. It is well known that the error of traditional BEM mainly comes from element dispersion, and the introduction of isogeometric ideas makes BEM become a veritable high-precision numerical method. At present, most of the problems solved by isogeometric BEM (IGBEM) are time-independent. The reason is similar to the traditional BEM, which cannot avoid solving domain integrals when solving time-dependent problems. In this paper, based on the potential fundamental solution the boundary-domain integral equation is obtained by the weighted residual method, where the classic dual reciprocity method is adopted to transform domain integrals into boundary integrals. Meanwhile, a two-level time integration scheme is used to solve the discretized differential equations. In addition, the adaptive integration scheme, the radial integral transform method and the power series expansion method are adopted to solve the boundary regular, nearly singular and singular integrals. Several classical numerical examples show that the presented method has good numerical stability and high precision by considering different factors such as the approximation function, the time step, the number of interior points and so on.

A Meshless Method of Solving Three-Dimensional Nonstationary Heat Conduction Problems in Anisotropic Materials

A Meshless Method of Solving Three-Dimensional Nonstationary Heat Conduction Problems in Anisotropic Materials

Numerical and experimental studies of the hypersonic flow around a cube at incidence

In order to improve predictions of the on-ground casualty risk associated with the uncontrolled atmospheric re-entry of satellites from Low Earth Orbit, there is significant research interest in the development of engineering models of hypersonic heating rates to faceted shapes. A key part of developing such models is generating accurate datasets of the heat fluxes experienced by faceted shapes at various orientations in hypersonic flows. In this work, we use wind tunnel experiments and CFD simulations to study the hypersonic flow around a cuboid geometry at 5° incidence in a Mach 5 flow at Reynolds numbers of 79.5×103, 109×103 and 148×103. The wind tunnel data are obtained in the University of Manchester's High SuperSonic Tunnel and consist of schlieren images and temperature histories collected using infrared thermography. These temperature histories are used to calculate experimental heat fluxes by solving a three-dimensional inverse heat conduction problem. CFD simulations around the same geometry at equivalent free-stream conditions are calculated with the DLR-TAU code. The experimental and CFD results show good agreement both in terms of heat fluxes as well as flow structure. Notable flow structures include wedge-shaped regions of high heat flux which emanate from the windward corners of the cube. Analysis of numerical Q-criterion contours show that these high heat flux regions are caused by vortex structures generated by the expansion at the cube corner. Analysis of the numerical skin friction coefficient shows that even at incidence there is no breakaway separation from the expansion edges of the cube and the flow remains attached throughout. We show that although there is little change in the average heat flux experienced by a cube at 5° incidence to the free-stream compared to one at 0° incidence, there are significant changes in the heat flux contours over the cubes at these two incidences. Finally, we calculate a number of heating shape factors which can easily be implemented in satellite re-entry and demise prediction analysis tools.

PaScaL_TDMA: A library of parallel and scalable solvers for massive tridiagonal systems

The aim of this study is to devise an efficient and scalable computational procedure to solve the many tridiagonal systems in multi-dimensional partial differential equations. The modified Thomas algorithm and a newly designed communication scheme were used to reduce the communication overhead encountered while solving the many tridiagonal systems. Benchmark test results reveal an advantage of the proposed procedures compared to global all-to-all communication methods — a significantly reduced communication time that becomes more prominent for larger problem sizes and greater number of cores. The proposed computational procedures are fully implemented in an open-source library called Parallel and Scalable Library for TDMA (PaScaL_TDMA). Considering a three-dimensional heat conduction problem as a practical example, we obtain good strong and weak scalability results up to 262,144 computing cores on the KISTI Nurion cluster system, which, to the best of our knowledge, is the largest parallel simulation for solving tridiagonal systems. The potential of this library for large-scale substantive problems in physics is also demonstrated through direct numerical simulations of the Rayleigh–Bénard convection problem, which yielded excellent scalability and accurate results. Program summaryProgram Title: PaScaL_TDMACPC Library link to program files:http://dx.doi.org/10.17632/49z6fh94z3.1Developer’s repository link:https://github.com/MPMC-LabLicensing provisions: MITProgramming language: Fortran90Nature of problem: Tridiagonal systems for solving multi-dimensional partial differential equations.Solution method: The divide-and-conquer method is used to solve partitioned tridiagonal systems of equations in the distributed memory system. The partitioned tridiagonal systems of equations are transformed into modified sub-matrices using the modified Thomas algorithm [1]. Reduced tridiagonal systems are constructed by collecting the first and last rows of the modified submatrices from each computing core. The newly designed communication scheme based on MPI_Alltoallw accelerates to collect rows and construct the reduced tridiagonal systems. The solutions of the reduced tridiagonal systems are obtained via the sequential Thomas algorithm. Thereafter, the remaining unknowns in the modified sub-matrices are solved using solutions of the reduced tridiagonal systems.

Effect of Temperature Field on Mechanical Properties of Direct Laser Deposited Ti-6Al-4V Alloy

The mechanical and service properties of Ti-6Al-4V alloy parts produced by direct laser deposition (DLD) depend on the thermal cycle parameters. The temperature field during deposition is significantly affected not only by the parameters of the process, but also by the interpass dwell time and length of the deposited layers. The aim of the article is to establish the relationship between mechanical properties of deposited Ti-6Al-4V samples and DLD thermal cycle parameters. Numerical simulation was used in order to establish relationship between temperature field parameter and the process parameters. The nonlinear three-dimensional heat conduction problem was solved by the finite element method. It is shown that an increase in the dwell time between passes from 5 to 10 seconds leads to a significant decrease in the inter-pass temperature and an increase in the cooling rate. This leads to the metastable structure formation of the deposited layers of Ti-6Al-4V that consists mainly of a nonequilibrium a’-phase which hardness is higher than 390 HV. Without dwell time an equilibrium a + (3 structure with hardness of 360 HV and higher elongation is formed.