Two-dimensional Transient Heat Conduction Problems Research Articles

A numerical investigation of 2D transient heat conduction problems in anisotropic FGMs with time-dependent conductivity

A class of two-dimensional (2D) transient heat conduction problems for anisotropic Functionally Graded Materials (FGMs) of space–time dependent conductivity and coefficient of variable separable forms is considered. The problems are reduced to a boundary-only integral equation which can be solved numerically using a standard Boundary Element Method (BEM). Some examples are considered to show the validity of the analysis and to examine the accuracy of the numerical method.

Numerical Analysis of Transient State Heat Transfer by Spectral Method Based on POD Reduced-Order Extrapolation Algorithm

In order to meet the requirements of high accuracy and fast algorithm for numerical heat transfer simulation, an iterative scheme of Proper Orthogonal Decomposition (for short, POD) dimension reduction based on the classical central difference Galerkin spectral method is proposed for solving two-dimensional transient heat conduction problems. The POD dimension reduction spectral method model is constructed by taking the calculation results of classical central difference Galerkin spectral method as sample data. The numerical algorithm characteristics of flow and heat transfer are studied by using a partial differential equation as a mathematical model, and the error estimation is given. Finally, different time intervals are used as parameters to simulate experiments. The results show that the POD method is applicable to transient nonlinear heat conduction problems, and the maximum average relative error of the reconstructed temperature field is 0.89675%. Moreover, the POD method not only has a high calculation accuracy, but also has an average calculation speed as high as 310.25 times that of the central difference Galerkin algorithm. It can be seen that under the condition that the error between the solution of POD dimension reduction extrapolation algorithm and the solution of classical central difference Galerkin spectrum method is small enough, the POD method can greatly reduce the calculation amount, shorten the running time, and ensure a high accuracy of the calculation results, thus verifying the effectiveness and feasibility of the algorithm.

A polygonal element differential method for solving two-dimensional transient nonlinear heat conduction problems

A novel polygonal element differential method (PEDM) is presented for solving two-dimensional nonlinear transient heat conduction problems for the first time. New shape functions as well as their derivatives with respect to isoparametric coordinates are derived to treat the polygonal elements with an internal node. System equations of the PBEM are formulated in terms of the governing equation and heat flux equilibrium condition, in which, a finite difference scheme is executed for calculating the transient term. Then, the nonlinearity system equations are dealt with by the Newton iterative method. Finally, examples with different structural complexities are designed to examine the property of the proposed method. The results show that the PEDM can effectively solve general two-dimensional transient nonlinear heat conduction problems with excellent accuracy and efficiency.

Explicit Sensitivity Coefficients for Estimation of Temperature-Dependent Thermophysical Properties in Inverse Transient Heat Conduction Problems

Explicit expressions are obtained for sensitivity coefficients to separately estimate temperature-dependent thermophysical properties, such as specific heat and thermal conductivity, in two-dimensional inverse transient heat conduction problems for bodies with irregular shape from temperature measurement readings of a single sensor inside the body. The proposed sensitivity analysis scheme allows for the computation of all sensitivity coefficients in only one direct problem solution at each iteration with no need to solve the sensitivity and adjoint problems. In this method, a boundary-fitted grid generation (elliptic) method is used to mesh the irregular shape of the heat conducting body. Explicit expressions are obtained to calculate the sensitivity coefficients efficiently and the conjugate gradient method as an iterative gradient-based optimization method is used to minimize the objective function and reach the solution. A test case with different initial guesses and sensor locations is presented to investigate the proposed inverse analysis.

A dual interpolation precise integration boundary face method to solve two-dimensional transient heat conduction problems

In this paper, we present a new numerical algorithm combining the dual interpolation boundary face method (DiBFM) with the precise integration method for solving the 2D transient heat conduction problem. In this new combined approach, the transient heat conduction problem is transformed from an initial boundary value problem to an initial value problem through a dual interpolation boundary face approach. This approach merges the conforming and nonconforming elements in the BFM implementation. Potentials and fluxes are approximated by the dual interpolation elements which include source and virtual points. Employing the moving-least-square approximation help to construct the constraint equations relating to virtual points. Then the analytical solution of the problem can be expressed by the matrix exponential function (MEF), which can be computed accurately through a precise integration method (PIM). The proposed numerical algorithm has been successfully implemented. Several numerical examples are given to illustrate the numerical accuracy and stability of the proposed method compared with the traditional precise integration boundary face method.

The scaled boundary finite element method based on the hybrid quadtree mesh for solving transient heat conduction problems

In this paper, a hybrid quadtree mesh is utilized to solve two-dimensional transient heat conduction problems with cracks or inclusions. The current approach developed based on the scaled boundary finite element method (SBFEM) is able to alleviate the hanging node issues typically encountered in the conventional finite element method when using quadtree mesh. Moreover, the present method does not require fundamental solutions, unlike the boundary element method. The formulas of temperature and temperature gradient fields are derived systematically for the insulated crack and the inclusions of different materials with temperature and heat flux boundary conditions. Several examples are presented to demonstrate the validity and stability of SBFEM when solving models with complex geometry and cracks or inclusions.

Sequential particle filter estimation of a time-dependent heat transfer coefficient in a multidimensional nonlinear inverse heat conduction problem

In the applied mathematical modelling of heat transfer systems, the heat transfer coefficient (HTC) is one of the most important parameters. This paper proposes a combination of the Method of Fundamental Solutions (MFS) with particle filter Sequential Importance Resampling (PF-SIR) to estimate the time-dependent HTC in two-dimensional transient inverse heat conduction problems from non-standard boundary integral measurements. These measurements ensure the unique solvability of the boundary coefficient identification problem. Numerical results show high performance on several test cases with both linear and nonlinear Robin boundary conditions, supporting the synergy between the MFS and simulation-based particle filter sequential analysis methods.

Generalized Solution for Two-Dimensional Transient Heat Conduction Problems With Partial Heating Near a Corner

A generalized solution for a two-dimensional (2D) transient heat conduction problem with a partial-heating boundary condition in rectangular coordinates is developed. The solution accommodates three kinds of boundary conditions: prescribed temperature, prescribed heat flux and convective. Also, the possibility of combining prescribed heat flux and convective heating/cooling on the same boundary is addressed. The means of dealing with these conditions involves adjusting the convection coefficient. Large convective coefficients such as 1010 effectively produce a prescribed-temperature boundary condition and small ones such as 10−10 produce 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 (SOV) can be less efficient at the heated surface and another method (non-SOV) is more efficient there. Then, the use of the complementary transient part of the solution at early times is presented as a unique way to compute the steady-state solution. The solution method builds upon previous work done in generating analytical solutions in 2D problems with partial heating. But the generalized solution proposed here contains the possibility of hundreds or even thousands of individual solutions. An indexed numbering system is used in order to highlight these individual solutions. Heating along a variable length on the nonhomogeneous boundary is featured as part of the geometry and examples of the solution output are included in the results.

Identification of thermal conductivity for orthotropic FGMs by DT-DRBEM and L-M algorithm

ABSTRACTThe differential transformation dual reciprocity boundary element method (DT-DRBEM) combined with the Levenberg–Marquardt (L-M) algorithm is proposed to identify the thermal conductivity for orthotropic functionally gradient materials (FGMs). The original governing equation of two-dimensional transient heat conduction problem is transformed into a time-independent and recursive form equation by the differential transformation method (DTM). The analog equation method is used to convert the partial differential equation to a standard thermal diffusion equation. Then, the DRBEM is applied to solve the direct problem and the measured temperature is obtained. After that, the L-M algorithm is used to minimize the objective function and recover the desired thermal conductivity. The DT-DRBEM is compared with the finite difference DRBEM (FD-DRBEM). The L-M algorithm is also compared with the conjugate gradient method (CGM). The influences of different initial guesses, number of measurement points and random errors on the inverse results are also investigated. The initial guesses have little effect on the number of iteration. With the increase of measurement points and with the decrease of measurement errors, the results are more accurate.

The numerical manifold method for 2D transient heat conduction problems in functionally graded materials

Benefiting from the use of two cover systems, that is, the mathematical cover and the physical cover, the numerical manifold method (NMM) is capable of solving both continuous and discontinuous problems in the same platform. Presently, the NMM is further developed to tackle two-dimensional transient heat conduction problems in the functionally graded materials (FGMs). Firstly, the governing equation, the associated boundary conditions and the initial condition are presented. Then, the fundamentals of the NMM are briefly reviewed. Following, the NMM discrete formulations are derived based on the Galerkin-form weighted residual method and then solved with the backward difference scheme. Finally, for verification, three numerical examples with increasing complexity are tested on uniform mathematical covers composed of square mathematical elements, and our results well demonstrate the advantages of the proposed method in discretization and accuracy; besides, the effects of material gradient on the thermal behavior of FGMs are also examined.

Firefly algorithm combined with Newton method to identify boundary conditions for transient heat conduction problems

ABSTRACTFirefly algorithm combined with Newton method (FA–NM) is proposed for identifying time-dependent boundary conditions of two-dimensional transient heat conduction problems with a heat source. The dual reciprocity boundary element method (DRBEM) is applied to solve the direct problem. The improved firefly algorithm has not only a good global search ability, but also a good local search ability. FA–NM can acquire accurate results with much less iterations. Furthermore, different measurement points and noises are also considered. A small number of measurement points cannot obtain precise results. With the increase in the measurement points to a certain value, the results are more accurate. When the measurement noises are not very large, FA–NM can get desirable results. With the decrease in measurement noises, the results are in better agreement with the exact solutions. Numerical results also indicate that the more front the time substep is, the more accurate the estimated boundary conditions are.

Coupling of Adaptive Element Free Galerkin Method with Variational Multiscale Method for Two-Dimensional Transient Heat Conduction Problem

AbstractIn this paper, the coupling of adaptive element free Galerkin method (EFGM) with variational multiscale method is developed to solve two-dimensional transient heat conduction problem. The EFGM is chosen to combine with the adaptive analysis since this method only use a set of particles to generate the discrete equations instead of mesh construction as in finite element method (FEM). This property makes additional nodes can be easily inserted at the high gradient regions to improve the numerical results. The refinement criteria and refinement strategy will be clearly shown for the adaptive analysis procedure. The performance of this coupling method is first validated by solving two-dimensional large localized gradient problem with available theoretical solutions. Next, this method will also be used to solve two-dimensional transient heat conduction problem. The accuracy and efficiency of the method is demonstrated by comparing the results between conventional EFGM and the proposed method. The results obtained show that this method is able to produce higher accuracy results compared with the standard element free Galerkin method.

A new inverse analysis method based on a relaxation factor optimization technique for solving transient nonlinear inverse heat conduction problems

The relaxation factor is a key parameter in gradient-based inversion and optimization methods, as well as in solving nonlinear equations using iterative techniques. In gradient-based inversion methods, the relaxation factor directly affects the inversion efficiency and the convergence stability. In general, the bigger the relaxation factor is, the faster the inversion process is. However, divergences may occur if the relaxation factor is too big. Therefore, there should be an optimal value of the relaxation factor at each iteration, guaranteeing a high inversion efficiency and a good convergence stability. In the present work, an optimization technique is proposed, using which the relaxation factor is adaptively updated at each iteration, rather than a constant during the whole iteration process. Based on this, a new inverse analysis method is developed for solving multi-dimensional transient nonlinear inverse heat conduction problems. One- and two-dimensional transient nonlinear inverse heat conduction problems are involved, and the instability issues occurred in the previous works are reconsidered. The results show that the new inverse analysis method in the present work has the same high accuracy, the same good robustness, and a higher inversion efficiency, compared with the previous least-squares method. Most importantly, the new method is more stable by innovatively optimizing and adaptively updating the relaxation factor at each iteration.

Robust Inverse Approach for Two-Dimensional Transient Nonlinear Heat Conduction Problems

Inverse heat conduction analysis provides an efficient approach for estimating the thermophysical properties of materials, the boundary conditions, or the initial conditions. In this paper, two-dimensional transient nonlinear inverse heat conduction problems are investigated, for estimating time- and space-dependent boundary heat flux, as well as the temperature-dependent thermal conductivities. Modifications are carried out to extend the previous one-dimensional inversion algorithm to solve the two-dimensional transient nonlinear heat conduction problem, to overcome the frequently occurring divergence issues, and to improve the stability of the inversion algorithm. Boundary-only measurements are used as additional information, and a dimensionless objective function is adopted. In the direct problem, formulations for solutions to the two-dimensional transient nonlinear heat conduction problem are derived and validated. Numerical examples show that the inversion algorithm is effective, efficient, accurate, and robust, for recovering multiple parameters, with and without a functional form.

Cavity detection in a heat conductor using linear sampling method

In this paper, solution of inverse problems in heat conduction transient fields is investigated. For this purpose, a new time-domain version of linear sampling method (TDLSM) is developed for cavity detection in a heat conductor. The linear sampling method (LSM) is an effective approach to image the geometrical features of unknown targets. Although this method has been used in the context of inverse scattering problems such as solid, acoustics, and electromagnetism, there is no specific attempt to apply this method to identification of cavities in heat conductors. This study emphasizes the implementation of the LSM in the time-domain fields using finite element method. A set of numerical simulations on two-dimensional transient heat conduction problems is presented to highlight many effective features of the proposed TDLSM fast qualitative identification method.

Application of the finite pointset method to non-stationary heat conduction problems

This paper proposes the use of the finite pointset method for the numerical solution of two-dimensional transient heat conduction problems. The strong formulation of the parabolic partial differential equation is directly used instead of the corresponding weak form. Moreover, a numerical comparison between the finite pointset method and the corresponding analytical solutions is reported.

A new complex variable meshless method for transient heat conduction problems

In this paper, based on the improved complex variable moving least-square (ICVMLS) approximation, a new complex variable meshless method (CVMM) for two-dimensional (2D) transient heat conduction problems is presented. The variational method is employed to obtain the discrete equations, and the essential boundary conditions are imposed by the penalty method. As the transient heat conduction problems are related to time, the Crank-Nicolson difference scheme for two-point boundary value problems is selected for the time discretization. Then the corresponding formulae of the CVMM for 2D heat conduction problems are obtained. In order to demonstrate the applicability of the proposed method, numerical examples are given to show the high convergence rate, good accuracy, and high efficiency of the CVMM presented in this paper.

Generalized Ballistic-Diffusive Formulation and Hybrid SN-PN Solution of the Boltzmann Transport Equation for Phonons for Nonequilibrium Heat Conduction

A generalized form of the ballistic-diffusive equations (BDEs) for approximate solution of the Boltzmann Transport equation (BTE) for phonons is formulated. The formulation presented here is new and general in the sense that, unlike previously published formulations of the BDE, it does not require a priori knowledge of the specific heat capacity of the material. Furthermore, it does not introduce artifacts such as media and ballistic temperatures. As a consequence, the boundary conditions have clear physical meaning. In formulating the BDE, the phonon intensity is split into two components: ballistic and diffusive. The ballistic component is traditionally determined using a viewfactor formulation, while the diffusive component is solved by invoking spherical harmonics expansions. Use of the viewfactor approach for the ballistic component is prohibitive for complex large-scale geometries. Instead, in this work, the ballistic equation is solved using two different established methods that are appropriate for use in complex geometries, namely the discrete ordinates method (DOM) and the control angle discrete ordinates method (CADOM). Results of each method for solving the BDE are compared against benchmark Monte Carlo results, as well as solutions of the BTE using standalone DOM and CADOM for two different two-dimensional transient heat conduction problems at various Knudsen numbers. It is found that standalone CADOM (for BTE) and hybrid CADOM-P1 (for BDE) yield the best accuracy. The hybrid CADOM-P1 is found to be the best method in terms of computational efficiency.

Hybrid discrete ordinates—spherical harmonics solution to the Boltzmann Transport Equation for phonons for non-equilibrium heat conduction

The Boltzmann Transport Equation (BTE) for phonons has found prolific use for the prediction of non-equilibrium heat conduction phenomena in semiconductor materials. This article presents a new hybrid formulation and associated numerical procedures for solution of the BTE for phonons. In this formulation, the phonon intensity is first split into two components: ballistic and diffusive. The governing equation for the ballistic component is solved using two different established methods that are appropriate for use in complex geometries, namely the discrete ordinates method (DOM), and the control angle discrete ordinates method (CADOM). The diffusive component, on the other hand, is determined by invoking the first-order spherical harmonics (or P 1) approximation, which results in a Helmholtz equation with Robin boundary conditions. Both governing equations, referred to commonly as the ballistic-diffusive equations (BDE), are solved using the unstructured finite-volume procedure. Results of the hybrid method are compared against benchmark Monte Carlo results, as well as solutions of the BTE using standalone DOM and CADOM for two two-dimensional transient heat conduction problems at various Knudsen numbers. Subsequently, the method is explored for a large-scale three-dimensional geometry in order to assess convergence and computational cost. It is found that the proposed hybrid method is accurate at all Knudsen numbers. From an efficiency standpoint, the hybrid method is found to be superior to direct solution of the BTE both for steady state as well as for unsteady non-equilibrium heat conduction calculations with the computational gains increasing with increase in problem size.

Transient Heat Conduction in Functionally Graded Materials by LT-MFS

: The LT-MFS approach is proposed to solve two-dimensional transient heat conduction problem in functionally graded materials (FGMs). First, a Laplace transform approach is used to move the time variable. Then, a fundamental solution in Laplace space for FGMs is constructed. Next, the solution in Laplace space is approximated by the linear combination of fundamental solutions. Further, Stefest’s algorithm is employed to convert the results in Laplace space back into the time–space domain. Finally, the method is tested on several benchmark examples. The results demonstrate well the efficiency and accuracy of the proposed method.