Three-dimensional Heat Conduction Problem Research Articles

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.

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.

Transient three-dimensional heat conduction in heterogeneous media: Integral transforms and single domain formulation

A hybrid numerical-analytical methodology based on integral transforms is employed for solving transient three-dimensional heat conduction problems in heterogeneous media comprising arbitrarily space variable thermophysical properties, including configurations with multiple subdomains of different materials and geometries. The Generalized Integral Transform Technique (GITT) is used to solve for both the temperature distribution and the corresponding eigenvalue problem accounting for the spatially variable coefficients. A previously introduced single domain reformulation strategy is adopted when multiple subdomains are involved, so as to rewrite the heat conduction equation in terms of coefficients with abrupt variations in the spatial coordinates. The hybrid numerical-analytical approach is demonstrated for three classes of problems in which the thermophysical properties undergo significant variations throughout the domain, such as: (i) FGMs (Functionally Graded Materials) with three-dimensional space variable thermophysical properties; (ii) composite medium with inclusions of different geometries and thermophysical properties; (iii) a multi-scale/multi-material heat conduction problem in a IGBT (Insulated Gate Bipolar Transistor) module. In each considered application, both the temperature eigenfunction expansions and the associated eigenvalue problem solutions are critically analyzed in terms of convergence rates and co-verified with purely numerical solutions.

Inverse Estimation of Heat Transfer Coefficient and Reference Temperature in Jet Impingement

Abstract Applications of impinging jets are wide-ranging from cooling to heating in industrial as well as domestic field. Most of the reported heat transfer distribution data to and from impinging jets have been found from steady-state measurements. This study utilizes the solution to three-dimensional (3D) inverse heat conduction problem to estimate transient temperatures on the impingement side. Then, the temperature gradient is determined near the impingement wall (∼0.01 mm inside) with which transient heat flux is estimated on the impingement side. Instead of steady-state values, transient heat flux and corresponding wall temperatures are utilized in a thin foil technique to find out heat transfer coefficient and reference temperature simultaneously. The scope of the present technique is examined through its application to impinging jets with various configurations such as laminar jet, turbulent jet, hot jet, cold jet, and multiple jets. In all cases, estimations are reasonably close. The application of this inverse technique can be extended to any configuration of jet impingement irrespective of geometry of nozzle (circular/rectangular), the orientation of nozzle (orthogonal/inclined), the temperature of a jet (hot/cold), Reynolds numbers (laminar/turbulent), the nozzle-to-plate spacing (any Z/d), and roughness of the plate surface. The effect of plate thickness on the accuracy of the present technique is also studied. Up to 5 mm thick plates can be used in impinging jet applications without worrying much on accuracy. The use of the present technique significantly reduces the experimental cost and time since it works on transient data of just a few seconds.

Local least–squares element differential method for solving heat conduction problems in composite structures

In this article, a completely new numerical method called the Local Least-Squares Element Differential Method (LSEDM), is proposed for solving general engineering problems governed by second order partial differential equations. The method is a type of strong-form finite element method. In this method, a set of differential formulations of the isoparametric elements with respect to global coordinates are employed to collocate the governing differential equations and Neumann boundary conditions of the considered problem to generate the system of equations for internal nodes and boundary nodes of the collocation element. For each outer boundary or element interface, one equation is generated using the Neumann boundary condition and thus a number of equations can be generated for each node associated with a number of element interfaces. The least-squares technique is used to cast these interface equations into one equation by optimizing the local physical variable at the least-squares formulation. Thus, the solution system has as many equations as the total number of nodes of the present heat conduction problem. The proposed LSEDM can ultimately guarantee the conservativeness of the heat flux across element surfaces and can effectively improve the solution stability of the element differential method in solving problems with hugely different material properties, which is a challenging issue in meshfree methods. Numerical examples on two- and three-dimensional heat conduction problems are given to demonstrate the stability and efficiency of the proposed method.

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.

Investigation of Numerical Evaluation Improvement for Three-Dimensional Infinite Cylindrical Heat Conduction Problems

Abstract To estimate the thermal properties from transient data, a model is needed to produce numerical values with sufficient precision. Iterative regression or other estimation procedures must be applied to evaluate the model again and again. From this perspective, infinite or semi-infinite heat conduction problems are a challenge. Since the analytical solution usually contains improper integrals that need to be computed numerically, computer-evaluation speed is a serious issue. To improve the computation speed with precision maintained, an analytical method has been applied to three-dimensional (3D) cylindrical geometries. In this method, the numerical evaluation time is improved by replacing the integral-containing solution by a suitable finite body series solution. The precision of the series solution may be controlled to a high level and the required computer time may be minimized by a suitable choice of the extent of the finite body. The practical applications for 3D geometries include the line-source method for obtaining thermal properties, the estimation of thermal properties by the laser-flash method, and the estimation of aquifer properties or petroleum-field properties from well-test measurements. This paper is an extension of earlier works on one-dimensional (1D) and two-dimensional (2D) cylindrical geometries. In this paper, the computer-evaluation time for the finite geometry 3D solutions is shown to be hundreds of times faster than the infinite or semi-infinite solution with the precision maintained.

Modeling of non-isothermic viscoelastic-plastic behavior of flexible reinforced plates

Based on a step-by-step procedure over time, a numerical-analytical model of the nonisothermal viscoelastic-plastic behavior of crisscrossed-reinforced material is constructed. The components of the composition are isotropic; their viscoelastic deformation is described by the Maxwell - Boltzmann model of the body, and instantaneous inelastic behavior is described by the relations of the theory of elastoplastic deformation with isotropic hardening. In this case, the temperature dependence of the loading function and material constants is taken into account. The conditions of the viscoelastic deformation, the beginning of unloading, neutral and active viscoelastic-plastic loading of the thermosensitive materials of the composition are formulated. Relationship between the mechanical and thermophysical characteristics of the composite material is taken into account. The structural relationships of the thermophysical component of the problem are presented. The developed mathematical model is focused on the use of explicit schemes for the numerical integration of viscoelastic-plastic and thermophysical problems. The initial-boundary problem of nonisothermal viscoelastic-plastic bending deformation of fibrous plates is formulated. The poor resistance of such thin-walled constructions to transverse shear is described in the framework of Ambardzumyan's theory. The geometric nonlinearity of the problem is taken into account in the Karman approximation. In the transverse direction, the temperature is approximated by a second-order polynomial. To reduce the three-dimensional heat conduction problem to two-dimensional relations, the method of additional boundary conditions is used. The dynamic bending of a flat-crisscrossed-reinforced fiberglass plate under the influence of an air blast wave is investigated. It is shown that in the absence of external thermal heating in the process of oscillations, this structure heats by only 2-3ºС. In such cases, the calculation can reasonably be carried out without taking into account the temperature effect. In the presence of intense thermal loading of the structure, the heat sensitivity of the materials of the composition must be taken into account. It has been demonstrated that in the presence of a temperature field inhomogeneous over the thickness of the plate, the shape and magnitude of the residual deflection significantly depend on which face surface an external dynamic load is applied to.

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.

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 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 novel boundary-type meshfree method for solving three-dimensional single-domain steady-state heat conduction problem

The virtual boundary meshfree Galerkin method (VBMGM) is further developed to compute three-dimensional single-domain steady-state heat conduction problem. Using VBMGM, three-dimensional problem is reduced into two-dimensional boundary, whose element information can be obtained by the meshing software, namely ANSYS software. It greatly facilitates the programing implementation of the proposed method that ANSYS Parametric Design Language (APDL) is demonstrated for the element information acquisition of the curved surface. The virtual source function on the virtual boundary element method (VBEM) is interpolated by the radial basis function interpolation (RBFI) of the meshfree method. The calculation equation of three-dimensional single-domain steady-state heat conduction problem by VBMGM is constructed by the Galerkin method of the weighted residual method. Therefore, it has the advantages of the VBEM, the meshfree method and the Galerkin method that the proposed method is employed to computed three-dimensional single-domain steady-state heat conduction problem. It is beneficial to further popularize the proposed method to solve other three-dimensional heat conduction problems that the detailed numerical discrete formula of the proposed method for three-dimensional single-domain steady-state heat conduction problem is obtained. The numerical results of two numerical examples are computed and compared with the other numerical method and the extract solutions. The accuracy and stability of VBMGM for three-dimensional single-domain steady-state heat conduction problem are verified.

An improved implementation of triple reciprocity boundary element method for three-dimensional steady state heat conduction problems

The domain integrals caused by heat generation appearing in the boundary integral equation (BIE) of steady state heat conduction problems can be converted into boundary integrals by triple reciprocity method (TRM). However, the current triple reciprocity approximation (TRA) of domain functions is time-consuming and its accuracy is not stable in different geometric models, due to the need to solve a combination equation, whose degree of freedom is twice the number of boundary nodes plus the number of domain interpolation points. Therefore, a new formula of TRA is proposed in the current study to save computing time and storage space and improve accuracy. In the new proposed TRA formula, the combination equation is transferred into three equations that are solved in sequence. Four numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method. Results show that although the new TRA formulation is equivalent to the original one, the approximation results are quite different, especially for non-thin structures. Particularly, the improved formula of TRA can save computing time and storage space, and get better accuracy. The improved triple reciprocity BEM is a useful approach to analyze the three-dimensional steady heat conduction problems more efficiently and accurately.

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.

Localized method of fundamental solutions for large-scale modelling of three-dimensional anisotropic heat conduction problems – Theory and MATLAB code

The method of fundamental solutions (MFS) belongs to the family of meshless boundary collocation methods and now has been successfully tried for many kinds of engineering problems. The traditional MFS based on the “global” boundary discretization, however, leads to dense and non-symmetric coefficient matrices that, although smaller in sizes, require huge computational cost to compute the system of equations using direct solvers. Such an approach will be arduous, time consuming and computationally expensive for analyzing large-scale problems. In the present work, a localized version of the MFS, named as the localized MFS (LMFS), is proposed for large-scale modelling of three-dimensional (3D) anisotropic heat conduction problems. In the LMFS, the computational domain can be divided into small subdomains with a simple geometry such as circle and/or sphere. To each of the subdomains, the MFS formulation is applied and the unknown coefficients on the local simple geometric boundary are approximated by the moving least square (MLS) method. The satisfactions of governing equations at interior points and boundary conditions at boundary nodes lead to a sparse and banded system matrix. Numerical examples with up to 1,000,000 unknowns are solved successfully using the developed LMFS code. The advantages, disadvantages and potential applications of the proposed method, as compared with the traditional MFS and boundary element method (BEM), are discussed. Finally, a fast, reliable and self-contained MATLAB code is provided in the part of Supplementary Materials of the paper.

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.

Fast explicit dynamics finite element algorithm for transient heat transfer

This paper presents a novel methodology for fast simulation and analysis of transient heat transfer problems. The proposed methodology is suitable for real-time applications owing to (i) establishing the solution method from the viewpoint of computationally efficient explicit dynamics, (ii) proposing an element-level thermal load computation to eliminate the need for assembling global thermal stiffness matrix, leading to (iii) an explicit formulation of nodal temperature computation to eliminate the need for iterations anywhere in the algorithm, (iv) pre-computing the constant matrices and simulation parameters to facilitate online calculation, and (v) utilising computationally efficient finite elements to efficiently obtain thermal responses in the spatial domain, all of which lead to a significant reduction in computation time for fast run-time simulation. The proposed fast explicit dynamics finite element algorithm (FED-FEM) employs nonlinear thermal material properties, such as temperature-dependent thermal conductivity and specific heat capacity, and nonlinear thermal boundary conditions, such as heat convection and radiation, to account for the nonlinear characteristics of transient heat transfer problems. Simulations and comparison analyses demonstrate that not only can the proposed methodology handle isotropic, orthotropic, anisotropic and temperature-dependent thermal properties but also satisfy the standard patch tests and achieve good agreement with those of the commercial finite element analysis packages for numerical accuracy, for three-dimensional (3-D) heat conduction, convection, radiation, and thermal gradient concentration problems. Furthermore, the proposed FED-FEM algorithm is computationally efficient and only consumes a small computation time, capable of achieving real-time computational performance, leading to a novel methodology suitable for real-time simulation and analysis of transient heat transfer problems.

Investigation of the non-stationary temperature field of a two-layer cylinder at different angular velocities of motion of normally distributed stream of heat

A three-dimensional non-stationary heat conduction problem for a two-layer cylinder heated by normally distributed heat stream moving on the external surface along the cylinder guide is considered. On the basis of the solution obtained through the Green's functions for the corresponding heat conductivity problem the temperature fields for different values of angular velocity are investigated. Cite as: B. V. Protsiuk, V. M. Syniuta, “Investigation of the non-stationary temperature field of a two-layer cylinder at different angular velocities of motion of normally distributed stream of heat,” Prykl. Probl. Mekh. Mat. , Issue 16, 91–97 (2018) (in Ukrainian), http://doi.org/10.15407/apmm2018.16.91-97