Local RBF-DQ method for two-dimensional transient heat conduction problems

The meshless local radial basis function-based differential quadrature (RBF-DQ) method is applied on two-dimensional heat conduction for different irregular geometries. This method is the combination of differential quadrature approximation of derivatives and function approximation of radial basis function. Four different geometries with regular and irregular boundaries are considered, and numerical results are compared with those gained by finite element (FE) solution achieved by COMSOL commercial code. Outcomes prove that current technique is in very good agreement with FEM and this fact that RBF-DQ method is an accurate and flexible method in solution of heat conduction problems.     

Heat conduction analysis for irregular functionally graded material geometries using the meshless weighted least-square method with temperature-dependent material properties

Abstract

This article presents the heat conduction analysis for irregular functionally graded material (FGM) with temperature-dependent material properties. For irregular FGM geometries, the meshless weighted least-square (MWLS) method is easy to model, implement, and interpolate those irregularly distributed field variables. To solve the heat conduction problem coupled with temperature-dependent FGM, the Laplace’s equilibrium equation and boundary condition become nonlinear. Thus, the Kirchhoff transformation is employed to convert the nonlinear problem to linear solution. MWLS method as a pure meshless analysis is then used to solve the linear equation of FGM geometries. Next, the temperature field is obtained by the inverse Kirchhoff transformation. Finally, the accuracy and effectiveness of the method were demonstrated by several numerical cases.     

On preconditioned BiCGSTAB solver for MLPG method applied to heat conduction in complex geometry

Meshless local Petrov–Galerkin (MLPG) method is a promising meshfree method for continuum problems in complex domains, especially for large deformation, moving boundary and phase change problems. For large-scale problems, iterative methods for solving the discretized equations are more suitable than direct methods. Krylov subspace solvers of conjugate gradient type are the most preferred iterative solvers. The convergence rate of these methods depends on preconditioner used. Recently, proposed schedule relaxation Jacobi (SRJ) method can be used as a stand-alone solver and as a preconditioner. In the present work, the SRJ method is tested as a stand-alone solver and as a preconditioner for BiCGSTAB solver in the MLPG method, and its performance has been compared with successive overrelaxation (k) preconditioner. Two-dimensional linear steady-state heat conduction in complex shape geometry has been used as the model test problem.     

Analysis of Conduction and Radiation Heat Transfer in a Differentially Heated 2‐D Square Enclosure

Radiative heat transfer with and without conduction in a differentially heated 2‐D square enclosure is analyzed. The enclosure with diffuse gray boundaries contains radiating and/or conducting gray homogeneous medium. Radiatively, the medium is absorbing, emitting and scattering. On the south boundary, four types of discrete heated regions, viz., the full boundary, the left one‐third, left two third and middle one third, are considered. In the absence of conduction, distributions of heat flux along the south boundary are studied for the effect of extinction coefficient. In the presence of conduction, distributions of radiation, conduction and total heat fluxes along the south boundary are analyzed for the effects of extinction coefficient, scattering albedo, conduction–radiation parameter, and south boundary emissivity. Effects of these parameters on centerline temperature distribution are also studied. To assess the performance of three commonly used radiative transfer methods, in all cases, the radiative transfer equation is solved using the discrete ordinate method (DOM), the conventional discrete ordinate method (CDOM) and the finite volume method (FVM). In the combined mode problem, with volumetric radiative information known from one of the three methods, viz., DOM, CDOM, and FVM, the energy equation is solved using the finite difference method (FDM). In all cases, the results from FDM‐DOM, FDM‐CDOM, and FDM‐FVM are in good agreement. Computationally, all three sets of methods are equally efficient.     

Estimation of the time-dependent heat flux using the temperature distribution at a point by conjugate gradient method

In this paper, the conjugate gradient method coupled with adjoint problem is used in order to solve the inverse heat conduction problem and estimation of the time-dependent heat flux using the temperature distribution at a point. Also, the effects of noisy data and position of measured temperature on final solution are studied. The numerical solution of the governing equations is obtained by employing a finite-difference technique. For solving this problem the general coordinate method is used. We solve the inverse heat conduction problem of estimating the transient heat flux, applied on part of the boundary of an irregular region. The irregular region in the physical domain (r,z) is transformed into a rectangle in the computational domain (ξ,η). The present formulation is general and can be applied to the solution of boundary inverse heat conduction problems over any region that can be mapped into a rectangle. The obtained results for few selected examples show the good accuracy of the presented method. Also the solutions have good stability even if the input data includes noise and that the results are nearly independent of sensor position.     

Matrix free meshless method for transient heat conduction problems

In the paper, the element free Galerkin method (EFGM) is applied to calculate two-dimensional unsteady state heat conduction problems. As is well known, most of the meshless methods have higher computational cost than that of finite element method (FEM). In order to overcome this shortcoming especially for transient heat conduction problems, mass lumping procedure is adopted in EFGM, which can decrease the computational cost evidently. Moreover, this technique which can simplify the solution procedure makes the essential boundary conditions enforced directly. The results obtained by EFGM combining mass lumping technique are compared with those obtained by finite element method as well as analytical solutions, which shows that the solutions of the present method are in good agreement with FEM’s and analytical solutions.     

具有辐射边界的三维非规则域内稳态温度场分析

研究了具有辐射边界的空间非规则域内稳态导热问题,求解方法为在球极坐标系内分离变量,获得级数形式的解后,采用边界离散法确定级数项的待定系数,算例表明,边界离形方法不仅可以解决非正交边界问题,而且也可以处理诸如辐射边界的非线性边值问题。     

Computational fluid dynamics (CFD) modeling of heat transfer in a polymeric membrane using finite volume method

The efficiency, robustness and reliability of recent numerical methods for finding solutions to flow problems have given rise to the implementation of computational fluid dynamics (CFD) as a broadly used analysis method for engineering problems like membrane separation system. The CFD modeling in this study observes steady and unsteady (transient) heat flux and temperature profiles in a polymeric (cellulose acetate) membrane. This study is novel due to the implementation of user defined scalar (UDS) diffusion equation by using user-defined functions (UDFs) infinite volume method (FVM). Some details of the FVM used by the solver are carefully discussed when implementing terms in the governing equation and boundary conditions (BC). The contours of temperature due to high-temperature gradient are reported for steady and unsteady problems.     

Weighted Least-Squares Collocation Method (WLSCM) for 2-D and 3-D Heat Conduction Problems in Irregular Domains

In this article, the weighted least-squares collocation method (WLSCM) is adopted to deal with two- and three-dimensional heat conduction problems in irregular domains. A radial basis function (RBF) is selected to construct the approximation function. To improve the accuracy and stability, some auxiliary points are increased within the domain of interest. Only inner nodes are used to construct the approximation function, and the equilibrium equations are satisfied not only at collocation points but also at auxiliary points, so the equations should be solved in a least-square sense. A 2-D case that has an analytical solution is simulated by the proposed method and the outcome verifies that the present method can obtain desired accuracy and efficiency. Then the current method is adopted to compute one 2-D and two 3-D cases of engineering heat conduction problems in irregular complex domains. The results show that the present method can deal effectively with the heat conduction problems of both 2-D and 3-D irregular domains.     

Local RBF meshless scheme for coupled radiative and conductive heat transfer

ABSTRACT

A local radial basis function meshless (LRBFM) method is developed to solve coupled radiative and conductive heat transfer problems in multidimensional participating media, in which compact support radial basis functions (RBFs) augmented on a polynomial basis are employed to construct the trial function, and the radiative transfer equation (RTE) and energy conservation equation are discretized directly at nodes by the collocation method. LRBFM belongs to a class of truly meshless methods which require no mesh or grid, and can be readily implemented in a set of uniform or irregular node distributions with no node connectivity. Performances of the LRBFM is compared to numerical results reported in the literature via a variety of coupled radiative and conductive heat transfer problems in 1D and 2D geometries. It is demonstrated that the local radial basis function meshless method provides high accuracy and great efficiency to solve coupled radiative and conductive heat transfer problems in multidimensional participating media with uniform and irregular node distribution, especially for coupled heat transfer problems in irregular geometry with Cartesian coordinates. In addition, it is extremely simple to implement.     

Lattice Boltzmann Method and Modified Discrete Ordinate Method Applied to Radiative Transport in a Spherical Medium with and without Conduction

This article deals with the application of the modified discrete ordinate method (MDOM) to calculate volumetric radiative information with and without conduction in a concentric spherical enclosure containing a participating medium. With radiative information known from the MDOM, the energy equation of the combined mode transient conduction and radiation heat transfer is formulated and solved using the lattice Boltzmann method (LBM). Without conduction, for pure radiation case, two benchmark problems, representing nonradiative and radiative equilibrium situations are taken up. In the case of non-radiative equilibrium, an isothermal medium is bounded by cold walls and medium is the source of radiation, while in the case of radiative equilibrium, nonisothermal medium is confined between a hot and a cold wall, and the hot (inner sphere) wall is the radiation source. Depending upon the problem, heat flux, energy flow rate, emissive power, and temperature distributions in the medium are calculated for different values of parameters such as the extinction coefficient, the scattering albedo, the conduction-radiation parameter, the boundary emissivity, and the radius ratio. To validate the MDOM and the LBM-MDOM formulations, problems are also solved using the finite volume method (FVM) and the finite-difference method (FDM)–FVM approach, in which the FVM is used to calculate the volumetric radiation and the energy equation is also solved using the FDM. Results of the MDOM, LBM–MDOM, FVM and FDM–FVM are also benchmarked against those available in the literature. MDOM and LBM–MDOM have been found to provide accurate results.     

The finite volume method approach to the collapsed dimension method in analyzing steady/transient radiative transfer problems in participating media

With the finite volume formulation (FVM) approach applied to the collapsed dimension method (CDM), this article deals with the application of the CDM to analyze radiative heat transfer problems in a participating medium subjected to a continuous diffuse or a continuous/short-pulse collimated boundary radiative loading. The planar medium contained between diffuse gray boundaries is absorbing, emitting and anisotropically scattering. With three categories of thermal boundary radiative loadings, for the four types of problems considered, the CDM results are compared for a wide range of radiative parameters with that of the FVM.     

Transient heat conduction analysis for distance-field-based irregular geometries using the meshless weighted least-square method

The paper analyzes the transient heat conduction problem with the irregular geometry using the meshless weighted least-square method (MWLS). The MWLS as a meshless method is fully independent of mesh, a discrete function is used to construct a series of linear equations, which avoided the troublesome task of numerical integration. First, irregular geometries are represented by the signed distance field. Then sampling the distance field, discrete nodes are obtained for MWLS analysis. The effectiveness and accuracy of the approach are illustrated by several numerical examples. Numerical cases show that a good agreement is achieved between the results obtained from the proposed meshless method and available analytical solutions or commercial software ANSYS.     

A New High-Precision Boundary-Type Meshless Method for Calculation of Multidomain Steady-State Heat Conduction Problems

This article presents the idea for calculating 2-D steady-state heat conduction problems with multidomain combination by employing the virtual boundary meshless least-square method. Being different from the conventional virtual boundary-element method (VBEM), this method incorporates the point interpolation method (PIM) with the compactly supported radial basis function (CSRBF) to approximately construct the virtual source function of the VBEM. Thus, the proposed method has the advantages of both the boundary-type meshless method and the virtual boundary element method. Since the configuration of the virtual boundary requires a certain preparation, the integration along the virtual boundary can be carried out over the smooth simple curve that can be structured beforehand (for 2-D problems) to reduce the complexity and difficulty of calculus without loss of accuracy, while the “vertex question” existing in the BEM can be avoided. Numerical examples show that the proposed method is more precise than several other numerical methods while selecting fewer degrees of freedom. In addition, its numerical stability is also verified by computing several cases.     

Singular boundary method for steady-state heat conduction in three dimensional general anisotropic media

The singular boundary method (SBM) is a recent strong-form meshless boundary collocation method. Like the method of fundamental solutions (MFS), the SBM uses the fundamental solution of the governing differential equation of interest as the basis function and is mathematically simple, truly meshless, accurate, and easy-to-program. Unlike the MFS, the SBM, however, uses the concept of the origin intensity factor to isolate the singularity of the fundamental solutions and overcomes the fictitious boundary issue which has long perplexed the MFS. This study makes the first attempt to apply the SBM to steady-state heat conduction in three-dimensional (3D) anisotropic materials. Five benchmark numerical examples demonstrate that the SBM is accurate, convergent, stable, and computationally efficient in solving this kind of problems.     

A finite volume method on distorted quadrilateral meshes for discretization of the energy equation's conduction term

A finite volume method (FVM) on distorted meshes for discretizing the energy equation's conduction term is presented. In this method, it is possible to compose the computational mesh of general quadrilateral elements (cells), namely, the cells are not required to be rectangular. The gradient of temperature on the cell's surface is computed to be second‐order accurate. Therefore, the error of numerical results by this method is smaller than using the traditional multilateral element method (MEM). The error does not depend on the degree of mesh distortion. The formulation based only on Taylor's theorem is straightforward. These are advantageous features to revise the fluid flow computation programs (based on FVM) that neglected the heat conduction term of the energy equation. The test calculations show that the convergence tendency of the numerical error using this method with the distorted mesh is the same as using an ordinary 2‐node central difference scheme on a constant‐interval rectangular mesh. By this method a conduction term was added to the energy equation of a SALE [ 1 ] program which had neglected that term originally, and z numerical calculation of a fluid flow with a heat transfer problem was performed. The numerical result of the present method with the distorted mesh well agrees with the analytical solution and the result of REM with a rectangular mesh. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20375     

A Lattice Boltzmann Formulation for the Analysis of Radiative Heat Transfer Problems in a Participating Medium

Use of the lattice Boltzmann method (LBM) has been extended to analyze radiative transport problems in an absorbing, emitting, and scattering medium. In terms of collision and streaming, the present approach of the LBM for radiative heat transfer is similar to those being used in fluid dynamics and heat transfer for the analyses of conduction and convection problems. However, to mitigate the effect of the isotropy in the polar direction, in the present LBM approach, lattices with more number of directions than those being used for the 2-D system have been employed. The LBM formulation has been validated by solving benchmark radiative equilibrium problems in 1-D and 2-D Cartesian geometry. Temperature and heat flux distributions have been obtained for a wide range of extinction coefficients. The LBM results have been compared against the results obtained from the finite-volume method (FVM). Good comparison has been obtained. The numbers of iterations and CPU times for the LBM and the FVM have also been compared. The number of iterations in the LBM has been found to be much more than the FVM. However, computationally, the LBM has been found to be much faster than the FVM.     

A distance-function-based Cartesian (DIFCA) grid method for irregular geometries

A distance-function-based Cartesian grid (DIFCA) method is presented for conduction heat transfer in irregular geometries. The irregular geometries are identified by distance functions. The finite-volume method is used to discretize the heat conduction equation. Non-zero departure from regular geometries terms are added to the discretization equations for the control volumes bisected by irregular boundaries. With these additional departure terms, the existing Cartesian finite-volume solver can be modified easily to model heat conduction in irregular geometries. Given boundary temperatures, given boundary fluxes and convective heat transfer at irregular boundaries are considered. Non-zero heat generation is also modeled. The proposed procedure is validated against eight test cases where good agreements are achieved.     

A new meshless “fragile points method” and a local variational iteration method for general transient heat conduction in anisotropic nonhomogeneous media. Part I: Theory and implementation

Abstract

A new and effective computational approach is presented for analyzing transient heat conduction problems. The approach consists of a meshless Fragile Points Method (FPM) being utilized for spatial discretization, and a Local Variational Iteration (LVI) scheme for time discretization. Anisotropy and nonhomogeneity do not give rise to any difficulties in the present implementation. The meshless FPM is based on a Galerkin weak-form formulation and thus leads to symmetric matrices. Local, very simple, polynomial and discontinuous trial and test functions are employed. In the meshless FPM, Interior Penalty Numerical Fluxes are introduced to ensure the consistency of the method. The LVIM in the time domain is a combination of the Variational Iteration Method (VIM) and a collocation method in each finitely large time interval. The present methodology represents a considerable improvement to the current state of science in computational transient heat conduction in anisotropic nonhomogeneous media. In this first part of the two-paper series, we focus on the theoretical formulation and implementation of the proposed methodology. Numerical results and validation are then presented in Part II.     

Error and variance of solution to the stochastic heat conduction problem by multiquadric collocation method

Error and variance of the solution to the heat conduction problem having stochastic initial and boundary conditions are determined by a formulation based on a meshless method known as the multiquadric collocation method. This formulation expresses the solution in terms of initial and boundary conditions. Inspection of solutions to two test problems reveals that a large value of the shape parameter, which is the free parameter of multiquadrics, should not be used for a stochastic problem because it may lead to a solution that is too sensitive to uncertainties in boundary and initial conditions.