Method For Heat Conduction Problems Research Articles

Three-dimensional numerical manifold method for heat conduction problems with a simplex integral on the boundary

Three-dimensional numerical manifold method for heat conduction problems with a simplex integral on the boundary

Development of numerical scheme using boundary element method for heat conduction problems

This paper deals with the boundary element method solution of set of parabolic differential equations of heat conducting problems with continuous and discontinuous boundary conditions. Time-dependent terms have been used as a non-homogeneous forcing function by changing a parabolic differential equation into a Poisson's equation. The suggested approach for solving the Poisson equation has been modified which decreases the number of algebraic equations and reduces the computational time. The constant coefficient matrix and LU decomposition provide an economical means for the solution of the concern problem. The discretization of the boundary of parabolic differential equations with respect to (w.r.t.) time is one of the main features of this article. Direct determination of unknown at interior points is also trademark of the proposed method. The method has been supported by examples, which include discontinuous boundary conditions and non-homogeneous forcing terms to highlight its effectiveness and accuracy.

Research on the Fictitious Source Points of the Hybrid Fundamental Solution-Based Finite Element Method for Heat Conduction Problems

Research on the Fictitious Source Points of the Hybrid Fundamental Solution-Based Finite Element Method for Heat Conduction Problems

Data-driven M-VCUT topology optimization method for heat conduction problem of cellular structure with multiple microstructure prototypes

Cellular structures are increasingly used in practical engineering due to their excellent physical properties, such as lightweight, high strength, impact resistance, and high ductility. In this work, a data-driven topology optimization method is developed for the steady-state heat conduction problem of cellular structures with multiple microstructure prototypes. Cellular structure is formed by splicing a series of real microstructures and each of them is produced by the superposition of several basic microstructures with different configurations. In addition, each basic microstructure is represented by cutting a corresponding level set function with a plane or an iso-surface. The mapping relationships among equivalent conductivity matrices, relative densities, and cutting heights are established and stored in an offline database. The optimization iterative process is performed only on the coarse-scale mesh, which can save lots of computational resources. Based on the optimized distribution of design variables and the offline database, the full-scale cellular structure can be reconstructed and the smooth connectivity between adjacent microstructures is guaranteed. Finally, both 2D and 3D numerical examples are carried out for verifying the validity and effectiveness of the proposed method.

Two-dimensional numerical manifold method for heat conduction problems

The numerical manifold method (NMM) is a calculation method based on Galerkin's variation and contains a dual covering system. Therefore, the NMM can easily deal with the construction of high-order manifold elements and is convenient for adaptive analysis. This study employed the NMM to simulate steady-state and transient heat conduction problems. The system equations were derived and a penalty function method was applied to deal with the boundary conditions. By simulating calculation examples for a one-dimensional bar, cuboid rock specimens, and thick-walled cylinders, the process of solving steady-state and transient heat-conduction problems and the influence law of the temperature pattern are demonstrated. Moreover, the convergence and effectiveness of the NMM in handling two-dimensional (2D) steady-state and 2D transient heat conduction problems was verified.

Data-Driven M-Vcut Topology Optimization Method for Heat Conduction Problem of Cellular Structure with Multiple Microstructure Prototypes

Data-Driven M-Vcut Topology Optimization Method for Heat Conduction Problem of Cellular Structure with Multiple Microstructure Prototypes

An adapted plane waves method for heat conduction problems

In this paper we construct a new set of basis functions for the numerical solution of nonhomogeneous heat conduction problems with Dirichlet boundary conditions and null initial data. These functions can be seen as Newtonian potentials of plane waves for the heat equation and satisfy a null initial condition. Density results for adapted waves will be established and several numerical simulations will be presented in order to discuss the accuracy and feasibility of the proposed method. An application of the method for heat problems with non null initial temperature will also be discussed.

Stabilised MLS in MLPG method for heat conduction problem

PurposeThe purpose of this paper is to assess the performance of the stabilised moving least squares (MLS) scheme in the meshless local Petrov–Galerkin (MLPG) method for heat conduction method.Design/methodology/approachIn the current work, the authors extend the stabilised MLS approach to the MLPG method for heat conduction problem. Its performance has been compared with the MLPG method based on the standard MLS and local coordinate MLS. The patch tests of MLS and modified MLS schemes have been presented along with the one- and two-dimensional examples for MLPG method of the heat conduction problem.FindingsIn the stabilised MLS, the condition number of moment matrix is independent of the nodal spacing and it is nearly constant in the global domain for all grid sizes. The shifted polynomials based MLS and stabilised MLS approaches are more robust than the standard MLS scheme in the MLPG method analysis of heat conduction problems.Originality/valueThe MLPG method based on the stabilised MLS scheme.

Multiscale computational method for heat conduction problems of composite structures with diverse periodic configurations in different subdomains

This study develops a novel multiscale computational method for heat conduction problems of composite structures with diverse periodic configurations in different subdomains. Firstly, the second-order two-scale (SOTS) solutions for these multiscale problems are successfully obtained based on multiscale asymptotic analysis. Then, the error analysis of SOTS solutions in the pointwise sense is given to illustrate the importance of developing the SOTS solutions. Furthermore, the error estimate for the SOTS approximate solutions in the integral sense is presented. In addition, a SOTS numerical algorithm is proposed to effectively solve these problems based on finite element method (FEM). Finally, some numerical examples verify the feasibility and effectiveness of the SOTS numerical algorithm we proposed. In this paper, a unified two-scale computational framework is established for heat conduction problems of composite structures with diverse periodic configurations in different subdomains.

High-order three-scale computational method for heat conduction problems of axisymmetric composite structures with multiple spatial scales

This study develops a novel high-order three-scale (HOTS) computational method for heat conduction problems of axisymmetric composite structures with multiple spatial scales. The heterogeneities of the composites are taken into account by periodic distributions of unit cells on the mesoscale and microscale. Firstly, the multiscale asymptotic analysis for these multiscale problems is given detailedly. Based on the above-mentioned analysis, the new unified micro-meso-macro HOTS approximate solutions are successfully constructed for these multiscale problems. Two classes of auxiliary cell functions are established on the mesoscale and microscale. Then, the error analyses for the conventional two-scale solutions, low-order three-scale (LOTS) solutions and HOTS solutions are obtained in the pointwise sense, which illustrate the necessity of developing HOTS solutions for simulating the heat conduction behaviors of composite structures with multiple periodic configurations. Furthermore, the corresponding HOTS numerical algorithm based on finite element method (FEM) is brought forward in details. Finally, some numerical examples are presented to verify the feasibility and validity of our HOTS computational method. In this paper, a unified three-scale computational framework is established for heat conduction problems of axisymmetric composite structures with multiple spatial scales.

Two numerical methods for heat conduction problems with phase change

A fixed-space-step method and a fixed-time-step method are presented, respectively, for solving the Stefan problems with time-dependent boundary conditions. The evolution of the moving interface and the temperature distribution in the phase change domain are simulated numerically by using two methods for melting in the half-plane and outward spherical solidification. Numerical experiment results show that the numerical results obtained from the two methods are in good agreement for the different test examples, and the two methods can be applied to solve Stefan problems in engineering practice.

Radial integration boundary element method for heat conduction problems with convective heat transfer boundary

ABSTRACTA new boundary domain integral equation with convective heat transfer boundary is presented to solve variable coefficient heat conduction problems. Green’s function for the Laplace equation is used to derive the basic integral equation with varying heat conductivities, and as a result, domain integrals are included in the derived integral equations. The existing domain integral is converted into an equivalent boundary integral using the radial integration method by expressing the normalized temperature as a series of radial basis functions. This treatment results in a pure boundary element analysis algorithm and requires no internal cells to evaluate the domain integral. Numerical examples are presented to demonstrate the accuracy and efficiency of the present method.

Steady- and Transient-State Analyses of Fully Ceramic Microencapsulated Fuel with Randomly Dispersed Tristructural Isotropic Particles via Two-Temperature Homogenized Model—I: Theory and Method

As a type of accident-tolerant fuel, fully ceramic microencapsulated (FCM) fuel was proposed after the Fukushima accident in Japan. The FCM fuel consists of tristructural isotropic particles randomly dispersed in a silicon carbide (SiC) matrix. For a fuel element with such high heterogeneity, we have proposed a two-temperature homogenized model using the particle transport Monte Carlo method for the heat conduction problem. This model distinguishes between fuel-kernel and SiC matrix temperatures. Moreover, the obtained temperature profiles are more realistic than those of other models. In Part I of the paper, homogenized parameters for theFCM fuel in which tristructural isotropic particles are randomly dispersed in the fine lattice stochastic structure are obtained by (1) matching steady-state analytic solutions of the model with the results of particle transport Monte Carlo method for heat conduction problems, and (2) preserving total enthalpies in fuel kernels and SiC matrix. The homogenized parameters have two desirable properties: (1) they are insensitive to boundary conditions such as coolant bulk temperatures and thickness of cladding, and (2) they are independent of operating power density. By performing the Monte Carlo calculations with the temperature-dependent thermal properties of the constituent materials of the FCM fuel, temperature-dependent homogenized parameters are obtained.

Singular boundary method for heat conduction problems with certain spatially varying conductivity

The singular boundary method (SBM) is a recent boundary-type meshless collocation method, in which the solution of a given problem is expanded as a linear combination of the fundamental solutions in terms of the source points. The method circumvents the fictitious boundary long perplexing the method of fundamental solution (MFS) by the introduction of the concept of origin intensity factors (OIFs). This paper documents the first attempt to extend the method to heat conduction problems in nonhomogeneous materials. We derive the fundamental solutions of heat conduction problems with the thermal conductivity of the quadratic, exponential and trigonometric material variations in three directions. Furthermore, we firstly theoretically derive the value of the OIF for the natural logarithm function, which is later extended to the OIFs for a group of fundamental solutions. The feasibility, accuracy and stability of the presented SBM formulation are confirmed for both two- (2D) and three-dimensional (3D) examples.

AN INTERPOLATING ELEMENT-FREE GALERKIN METHOD FOR STEADY-STATE HEAT CONDUCTION PROBLEMS

In this paper, an interpolating element-free Galerkin (IEFG) method is presented for steady-state heat conduction problems with heat generation and spatially varying conductivity. The shape function in the moving least-squares (MLS) approximation does not satisfy the property of Kronecker delta function, so an interpolating moving least-squares (IMLS) method is discussed, then combining the shape function constructed by the IMLS method and Galerkin weak form of the two-dimensional steady-state heat conduction problems, the IEFG method for heat conduction problems is presented, and the corresponding formulae are obtained. The main advantage of this approach over the conventional meshless methods is that essential boundary conditions can be applied directly. Numerical results show that the IEFG method not only has high computational accuracy, but also enhances computational efficiency greatly.

Three-dimensional single-channel thermal analysis of fully ceramic microencapsulated fuel via two-temperature homogenized model

The fully ceramic microencapsulated (FCM) fuel, one of the accident tolerant fuel (ATF) concepts, consists of TRISO particles randomly dispersed in SiC matrix. This high heterogeneity in compositions leads to difficulty in explicit thermal calculation of such a fuel. For thermal analysis of a fuel element of very high temperature reactors (VHTRs) which has a similar configuration to FCM fuel, two-temperature homogenized model was recently proposed by the authors. The model was developed using particle transport Monte Carlo method for heat conduction problems. It gives more realistic temperature profiles, and provides the fuel-kernel and graphite temperatures separately.In this paper, we apply the two-temperature homogenized model to three-dimensional single-channel thermal analysis of the FCM fuel element for steady- and transient-states using 2-D FEM/1-D FDM hybrid method. In the analyses, we assume that the power distribution is uniform in radial direction at steady-state and that in axial direction it is in the form of cosine function for simplicity. As transient scenarios, we consider (i) coolant inlet temperature transient, (ii) inlet mass flow rate transient, and (iii) power transient. The results of analyses are compared to those of conventional UO2 fuel having the same geometric dimension and operating conditions.

Heterogeneous Heat Conduction Problems by an Improved Element-Free Galerkin Method

In this article an improved element-free Galerkin method is proposed to solve heat conduction problems with heterogeneous media. Because the method almost possesses interpolation property, the implementation of essential boundary condition is as simple as that in the finite-element method. In order to validate the proposed method, several heat conduction problems with different degrees of heterogeneity are presented. In these test problems, we focus on the influence of nodal distribution to the proposed method for heat conduction problems with heterogeneous media. It is shown that, for different degrees of heterogeneity, regardless of matter whether the node is located on the interface, accurate solutions can be obtained by the proposed method for heterogeneous heat conduction problems without a source term.

A Well-Condition Asymptotic Waveform Evaluation Method for Heat Conduction Problems

The well condition asymptotic waveform evaluation (WCAWE) is presented to solve heat conduction problem with different boundary conditions. The method introduced by R. D. Slone and his colleague to solve electromagnetic problems in the frequency domain. Specially, the novelty of this paper is: This is the first time WCAWE method is presented for thermal analysis, the method is presented for time-dependent problems. The general formulation procedure is given and various examples are solved to illustrate the capabilities of the proposed scheme. The results obtain in this work by using WCAWE method showed that, the WCAWE method successfully able to approximate the initial delay. Therefore, WCAWE method is able to remove the limitation of time domain AWE

Analytical Method for Heat Conduction Problem with Internal Heat Source in Irregular Domains

In engineering field, analytical methods can only be applied to classic heat transfer problems with regular geometric boundaries. It is difficult to apply analytical methods in solving the mathematical and physical equations in nonorthogonal boundary of irregular domains. In this paper, we presented a method by conformal mapping from the solution for heat conduction problems in regular domains to solve problems in irregular domains.

Fuzzy and interval finite element method for heat conduction problem

Traditional finite element method is a well-established method to solve various problems of science and engineering. Different authors have used various methods to solve governing differential equation of heat conduction problem. In this study, heat conduction in a circular rod has been considered which is made up of two different materials viz. aluminum and copper. In earlier studies parameters in the differential equation have been taken as fixed (crisp) numbers which actually may not. Those parameters are found in general by some measurements or experiments. So the material properties are actually uncertain and may be considered to vary in an interval or as fuzzy and in that case complex interval arithmetic or fuzzy arithmetic has to be considered in the analysis. As such the problem is discretized into finite number of elements which depend on interval/fuzzy parameters. Representation of interval/fuzzy numbers may give the clear picture of uncertainty. Hence interval/fuzzy arithmetic is applied in the finite element method to solve a steady state heat conduction problem. Application of fuzzy finite element method in the said problem gives fuzzy system of linear equations in general. Here new methods have also been proposed to handle such type of fuzzy system of linear equations. Corresponding results are computed and has been reported here.