Three-dimensional Transient 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.

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.

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.

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.

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

In this study, the interpolating element-free Galerkin (IEFG) method for solving three-dimensional (3D) transient heat conduction problem is presented. By using the improved interpolating moving least-squares (IIMLS) method to form the shape function, and using the weak form of 3D transient heat conduction problems to obtain the discretized equations, the formulae of the IEFG method are obtained. The shape function of the IIMLS method satisfies the property of Kronecker delta function, and then the IEFG method can apply essential boundary conditions directly, which can result in higher computational speed and accuracy. Some examples are given to discuss the convergence and advantages of the IEFG method. By analyzing the numerical results obtained by IEFG method and improved EFG method, we conclude that IEFG method has clear advantages in computational speed and accuracy.

Analyzing three-dimensional transient heat conduction problems with the dimension splitting reproducing kernel particle method

In this paper, to improve the computational speed of the reproducing kernel particle method (RKPM), the dimension splitting reproducing kernel particle method (DSRKPM) is presented for solving three-dimensional (3D) transient heat conduction problems. The idea of the proposed method is transforming the 3D problem into a series of two-dimensional (2D) ones by using the finite difference method in the splitting direction. Since the shape function of the RKPM for 2D problem is much simpler than the one of 3D problem, the DSRKPM has higher computational speed than the RKPM. To demonstrate the applicability of the proposed method, four example problems are presented, and each of them is solved by the DSRKPM and the RKPM, respectively. And the numerical solutions show that the DSRKPM not only greatly improves the computational efficiency, but also has a higher computational accuracy.

Three-dimensional transient inverse heat conduction problems in the enhanced pool boiling heat transfer

In this paper, the solution methods of three-dimensional transient inverse heat conduction problems (3D IHCP) arising from pool boiling applications that need to reconstruct Neumann boundary conditions are summarized. Due to their mathematical ill-posedness, it is still non-trivial to solve these 3D IHCP at a reasonable cost. Firstly, we introduce the optimization-based regularization methods, such as Tikhonov regularization and iterative regularization. In general, it requires to solve a large number of optimization problems constrained by 3D partial differential equations (PDE). In this case, an improved Tikhonov regularization method is developed to quickly obtain optimal Tikhonov regularization parameter and thereby can accelerate the entire inverse calculation process. Furthermore, an iterative regularization method that is based on the conjugate gradient method is proposed to handle the high-resolution measurement data and many pointwise measurement data. In pool boiling experiments, they can be obtained by an infrared camera and mirco-thermocouples, respectively. Secondly, a completely different approach, a direct two-step strategy is presented to solve the same 3D transient ICHP on thin-plates that commonly occur in engineering applications. The main idea of the approach is to transform the inverse problem into a forward problem that is easier to solve by replacing the unknown Neumann boundary condition with an approximated Dirichlet boundary condition. The method has great advantage in efficiency compared with the optimization-based method that often requires to solve dozens or hundreds of direct, adjoint and sensitivity PDE problems, and therefore has the potential to be developed as efficient online heat-flux soft sensors for many practical applications. Thirdly, to further speed up the numerical solution procedure we develop a time-space dependent adaptive mesh refinement strategy due to the fact that the heat-flux and temperature dynamics related to bubble structures rapidly change in space and time on the boiling surface. A few classical engineering benchmark problems are considered to validate the aforementioned solution approaches, and the proposed approaches are also successfully applied to reconstruct the transient heat-flux distributions on the boiling surface from the noisy experimental data obtained in a series of real pool boiling experiments along the entire boiling curve. On the one hand, the corresponding research works reviewed can be used to explore the heat-transfer mechanism and establish a more accurate heat transfer model enabling the fast, robust and accurate reconstruction of local heat fluxes, which can be considered as realistic boundary conditions on the boiling surface to further study the two-phase flow and heat transfer process of boiling fluid above the heater. On the other hand, it can be also applied to explore the local details of the enhanced pool boiling heat transfer for the design of new heaters such as micro/nano structured surfaces, provide predictions related to the boiling process and conduct optimal experimental design in a systematic way. In the future work, a large number of 3D IHCP for different test regularization parameters will be calculated through a high-throughput parallel computational strategy. Besides, for thick heaters the direct two-step strategy can be used to solve the 3D ICHP accurately with an initial heat-flux guess obtained by the proposed optimization-based inversion approach. This way, the computational efficiency can be further improved. Furthermore, other new inversion methods, for example, which are based on artificial neural network method are being developed to solve 3D transient IHCP for more complex structured heater surfaces such as the micro/nano structures.

The Hot Optimal Transportation Meshfree (HOTM) method for materials under extreme dynamic thermomechanical conditions

We present a monolithic incremental Lagrangian framework based on meshfree methods, the Hot Optimal Transportation Meshfree (HOTM) method, for a robust and efficient solution of the dynamic response of materials under extreme thermomechanical conditions, possibly involving extremely large deformation, phase transition and multiphase mixing. The HOTM method combines the Optimal Transportation Meshfree (OTM) method and the variational thermomechanical constitutive updates. The variational structure of a dynamic system with general internal dissipative mechanisms is discretized in time by applying the Optimal Transportation theory, while material points and nodes are introduced for the spatial discretization. A phase-aware constitutive model is developed to describe the history-dependent material behavior in various phases due to melting, vaporization and solidification. The fully discretized conservation equations of linear momentum and energy are solved simultaneously using an operator splitting algorithm to predict the deformation, temperature and internal variables of the computational domain. The convergence property of the meshfree solution of the energy conservation equation is studied in a three-dimensional transient heat conduction problem by comparing to the analytical solutions. Accuracy of the HOTM method is also assessed in the example of upsetting a metallic billet up to a compression ratio of 95% under various external heating and cooling strategies. The scope and robustness of the HOTM method are demonstrated in the application of the laser cladding technology.

Generalized Solution for Rectangular Three-Dimensional Transient Heat Conduction Problems with Partial Heating

The development of a generalized solution is presented for a three-dimensional transient heat conduction problem in a rectangular parallelepiped. To make the method as general as possible, one face of the body is subjected to a nonhomogeneous boundary condition over part of the surface. The solution accommodates three kinds of boundary conditions: prescribed temperature, prescribed heat flux, and convective. The means of dealing with these conditions involves adjusting the convection coefficient. Large Biot numbers such as effectively produce a prescribed-temperature boundary condition, and small ones such as are used for generating an insulated boundary condition. This paper also presents three different methods to develop the computationally difficult steady-state component of the solution, as separation of variables can be inefficient at the heated surface. The generalized solution proposed here provides thousands of individual solutions with one analytical formulation. Boundary conditions of prescribed temperature, prescribed heat flux, and convection, as well as a semi-infinite boundary condition, can be simulated. Partial heating at the corner of a parallelepiped is featured as an example, which can be extended to any location on the heated surface using superposition, including subtractive superposition.

The Method of Fundamental Solutions for Two- and Three-Dimensional Transient Heat Conduction Problems Involving Point and Curved Line Heat Sources

The objective of this paper is to formulate the method of fundamental solutions (MFS) for solving transient heat conduction problems involving concentrated heat sources with time- and space-dependent intensities. The heat sources can be concentrated on points or arbitrary curved paths. In the proposed MFS formulation, a linear combination of time-dependent fundamental solutions and particular solutions is considered for approximating the solution. The particular solutions are expressed as time- and space-dependent integrals, which are computed effectively without considering any internal points or internal cells and without using any kind of time transformation. Several 2D and 3D numerical examples are analyzed. The finite element method (FEM) with a fine mesh and with a small size time step is used to find an accurate reference solution. In comparison to the FEM solution, the proposed MFS formulation achieves accurate results with a small number of source points and a large size time step.

A general approach for solving three-dimensional transient nonlinear inverse heat conduction problems in irregular complex structures

A novel general approach is proposed for solving three-dimensional transient nonlinear inverse heat conduction problems in irregular complex structures. The complex-variable-differentiation method (CVDM) is introduced into the commercial finite element method (FEM) software ABAQUS, for solving three-dimensional inverse heat conduction problems for the first time. First, the complex variable finite elements are set up, and the complex variable FEM is innovatively developed through the user element subroutine (UEL) in ABAQUS. Then, the key parameters in the Levenberg-Marquardt algorithm for solving inverse problems – sensitivity matrix coefficients – are accurately evaluated by utilizing the CVDM and ABAQUS standard solvers with the UEL. Finally, several numerical tests are carried out to verify the performances of the new approach for solving the three-dimensional transient nonlinear inverse heat conduction problem in an irregular complex structure.

Three-dimensional transient heat conduction analysis by boundary knot method

This paper makes the first attempt to apply the boundary knot method (BKM), in conjunction with dual reciprocity technique, for the solution of three-dimensional transient heat conduction problems. The BKM is a meshless, integration-free, easy-to-program boundary-only numerical technique for high-dimensional problems. The first step of our strategy is to use the finite difference method for temporal derivative to convert the transient heat conduction equation into a nonhomogeneous modified Helmholtz equation. And then the corresponding nonhomogeneous problem is solved using the proposed BKM strategy in conjunction with dual reciprocity technique. Four benchmark numerical examples are investigated in detail, and the numerical results show that the present scheme has the merits of high accuracy, wide applicability, good stability, and rapid convergence and is appealing to solve 3D transient heat conduction problems.

Evaluation of heat partition in machining CFRP using inverse method

An inverse heat conduction method was used to determine the energy balance at the cutting zone in edge trimming of carbon fiber-reinforced polymer composites. Three-dimensional transient heat conduction problem was modeled and solved independently in Abaqus for both the workpiece and cutting tool. Temperatures at specific locations were also measured during cutting using thermocouples and infrared thermography. Minimizing the difference between measured and calculated temperatures allowed the estimation of the heat flux applied in each problem. The total electric power consumed in machining was also measured. The heat partition was determined from the measured and calculated energies to be 0.07, 0.56 and 0.37 for the workpiece, tool and chips, respectively. The temperature distribution in the workpiece indicated that heat penetration is shallow due to poor thermal conductivity. It was also found that the extent of estimated machining thermal damage in the workpiece is within 0.35 mm below the machined surface.

Estimation of the time-dependent convective boundary condition in a horizontal pipe with thermal stratification based on inverse heat conduction problem

The time-dependent convective boundary condition on the inner wall are the essential causes of flow and heat transfer that induce thermal fatigue in pipes with thermal stratification. A three-dimensional transient inverse heat conduction problem (IHCP) was developed in this work for estimating simultaneously the multi-variables of the time-dependent convective boundary condition. Conjugate gradient method (CGM) is chosen as the optimization method. The known parameter in the IHCP is the measuring temperature on the outer wall. Experiments under different seven operating conditions have been taken for the validation of the method. Based on the estimated results, the influence of volume flow rate of cold water on the conductive heat transfer coefficient near the inner wall in the test section and the temperature distribution of the pipe wall were discussed.

The precise integration method for semi-discretized equation in the dual reciprocity method to solve three-dimensional transient heat conduction problems

A dual reciprocity precise integration method (DRPIM) is presented to solve 3-dimensional transient heat conduction problems. In the presented method, the transient heat conduction problem, which is an initial boundary value problem, is transformed into an initial value problem (IVP) through a dual reciprocity scheme. Then an analytical solution, which is described by terms of the matrix exponential function (MEF), to this IVP is applied. A precise integration method (PIM) is finally applied to compute the MEF accurately. The accuracy and stability of the presented method are demonstrated by three numerical examples.

Inverse identification of boundary conditions in a scramjet combustor with a regenerative cooling system

Accurately determining boundary conditions in a scramjet combustor is of great importance for modeling the coupled process of fuel burning, fluid flow and heat transfer in the scramjet combustor, and for design and optimization of the cooling system. In this paper, a new methodology is proposed for determining boundary conditions at inaccessible surfaces of a scramjet combustor with a regenerative cooling system, which are identified by solving a three-dimensional transient inverse heat conduction problem. The finite element method in the ABAQUS is employed to solve the direct heat conduction problem in the scramjet combustor with the regenerative cooling system. A gradient-based method is used to solve the inverse problem attributed to its high accuracy and efficiency, and temperatures at accessible positions provide additional information for the inverse analysis. Examples are given to examine the performances for identifying boundary conditions in a scramjet combustor with a regenerative cooling system.

A triple reciprocity method in Laplace transform boundary element method for three-dimensional transient heat conduction problems

In this paper, a boundary element method employing the Laplace transform is developed to solve the three-dimensional transient heat conduction problems. The fundamental solution for the modified Helmholtz equation is adopted in order to derive the basic integral equations. Due to the effects of initial temperature and heat generation, the domain integrals appearing in the integral equation will degrade the advantages of boundary element method. Therefore, a new triple reciprocity formulation in Laplace domain is proposed to convert the domain integrals into boundary integrals. The higher order fundamental solutions required in the triple reciprocity method can be obtained from the original fundamental solution by multiplying a constant term, which leads to a much simpler formulation in the Laplace domain. Then, the inverse transformation can be used to obtain the solutions in the time domain. Several numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method. In summary, the triple reciprocity formulation is a useful approach to capture the transient heat conduction responses.