Two-dimensional Steady-state Heat Conduction Research Articles

Coupled MLPG–FVM simulation of steady state heat conduction in irregular geometry

The two-dimensional steady-state heat conduction in irregular geometry is solved by a MLPG–FVM coupled method. The meshless local Petrove–Galerkin (MLPG) method is applied to the sub-region with skewed wall surface while the finite volume method (FVM) is used in the rest of the domain. The Dirichlet–Dirichlet method is adopted to couple the temperature between MLPG and FVM methods. In MLPG method, the Dirac's Delta function is taken as the test function to avoid the local domain integration which does not need the numerical integration and the solution is independent of the size of the test function. The proposed MLPG–FVM method is validated and proved to be an efficient numerical method for 2-D heat conduction in irregular geometry, which can exert their own advantages of MLPG and FVM.

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

ABSTRACTMeshless 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.

Thermal performance optimization of the underground power cable system by using a modified Jaya algorithm

This paper presents a modified Jaya algorithm for optimizing the material costs and electric-thermal performance of an Underground Power Cable System (UPCS). A High Voltage (HV) underground cable line with three 400 kV AC cables arranged in flat formation in an exemplary case study is considered. When buried underground, three XLPE high voltage cables are situated in thermal backfill layer for ensuring the optimal thermal performance of the cable system. The study discusses the effect of thermal conductivities of soil and cable backfill material on the UPCS total investment costs. The soil thermal conductivity is assumed constant and equal to 0.8 W/(m K). The cable backfills considered in the study are as follows: sand and cement mix, Fluidized Thermal Backfill™ (FTB) and Powercrete™ a product of Heidelberg Cement Group. Constant thermal conductivities of the backfills in the dry state are assumed, respectively, 1.0 W/(m K), 1.54 W/(m K) and 3.0 W/(m K). The cable backfill dimensions and cable conductor area are selected as design variables in the optimization problem. The modified JAYA algorithm is applied to minimize material costs of UPCS under the constraint that the cable conductor temperature shall not exceed its optimum value of 65 °C. The cable temperature is determined from the two-dimensional steady state heat conduction equation discretized using the Finite Element Method (FEM). The performance of the modified Jaya algorithm was compared with classical Jaya and PSO algorithms. The modified Jaya algorithm, for the presented case study, allows one to obtain lower values of the cost function.

A NURBS-based scaled boundary finite element method for the analysis of heat conduction problems with heat fluxes and temperatures on side-faces

The scaled boundary finite element method (SBFEM) combined with isogeometric analysis (IGA) is proposed to solve the two-dimensional steady-state heat conduction problems in complex geometries. The main benefit of SBFEM is that the spatial dimension of analyzed domain is reduced by one and the solution is analytical in the radial direction. In this method, only the boundary of the computational domain requires discretization with finite elements leading to the reduction of computational efforts. However, SBFEM suffers from the finite element method related drawbacks. In the case of the complex geometric shapes, a large number of elements are necessary to obtain the exact representation of geometry in finite element method. Isogeometric analysis is a novel numerical technique based on the non-uniform rational B-splines (NURBS), where the geometry can be exactly represented. Moreover, this technique yields superior numerical accuracy, efficiency and convergence property in comparison to finite element method. In the proposed method, the segments of domain boundary with complex geometries are described with NURBS basis functions in IGA, while the straight segments of boundary are represented with polynomial basis functions as in the conventional SBFEM. Thus, the present approach combines the advantages of both SBFEM and IGA. The heat conduction problems of complex geometry can be more effectively handled with the proposed method considering the prescribed heat fluxes and temperatures on side-faces. The accuracy and efficiency of the proposed formulation are demonstrated by modeling five numerical examples involving the complicated geometry.

SEMI-ANALYTICAL SOLUTION OF THE HEAT CONDUCTION IN A PLATE WITH HEAT GENERATION

In the present work, a formulation for the solution of the two-dimensional steady state heat conduction with heat generation is presented. The classical integral transform technique (CITT) is used to solve the problem in a semi- analytical manner. CITT deals with expansions of the sought solution in terms of infinite orthogonal basis of eigenfunctions, keeping the solution process always within a continuous domain. For the particular problem, the resulting system is composed of a set of uncoupled differential equations which can be solved analytically. However, a truncation error is involved since the infinite series must be truncated to obtain numerical results. For comparison and validation purposes, the second order central finite difference method (FDM) is also implemented. The convergence analysis showed that CITT has a greater performance having no difficulties to obtain accurate results with very few terms in the solution summation. The FDM had convergence troubles specially for the positions near the center and for high concentration of heat generation in the center of the plate.

Influence of Prandtl number on heat transfer of a flat vertical plate

Liquid metals, such as sodium (Na), lead (Pb), and lead-bismuth (Pb-Bi) eutectic (e), are considered as potential coolants for the fast spectrum nuclear reactors of the next generation. So the main objective of this paper is to study the heat transfer and fluid flow characteristics of liquid metal coolants flowing over a nuclear fuel element having uniform volumetric energy generation. Stream function vorticity formulation method was used to solve the full Navier Stokes equations governing the flow. The energy equation was solved using central finite difference method. For the two-dimensional steady state heat conduction and stream-function equation, the discretisation was done in the form suitable to solve using 'line-by-line Gauss-Seidel' solution technique whereas the discretisation of vorticity transport and energy equations was done using Alternating Direction Implicit (ADI) scheme. After discretisation the systems of algebraic equations were solved using 'Thomas algorithm'. The complete work was done by writing a well-validated indigenous computer code using C-language. The parameters considered for the study were: aspect ratio of fuel element, Ar, conduction-convection parameter Ncc, total energy generation parameter Qt, and flow Reynolds number ReH. The results obtained can be used to minimise the maximum temperature in the fuel element (hot spots) and prevent its melting.

An invariant method of fundamental solutions for two-dimensional steady-state anisotropic heat conduction problems

We investigate both theoretically and numerically the so-called invariance property, see e.g. Sun and Ma (2015a,b), of the solution of boundary value problems associated with the anisotropic heat conduction equation (or Laplace–Beltrami’s equation) in two dimensions with respect to elementary transformations of the solution domain, e.g. dilations or contractions. We also show that the standard method of fundamental solutions (MFS) does not satisfy the invariance property. Motivated by these reasons, we introduce, in a natural manner, a modified version of the MFS that remains invariant under elementary transformations of the solution domain and is referred to as the invariant MFS (IMFS). Five two-dimensional examples are thoroughly investigated to assess the numerical accuracy, convergence and stability of the proposed IMFS, in conjunction with the Tikhonov regularization method (Tikhonov and Arsenin, 1986) and Morozov’s discrepancy principle (Morozov, 1966), for Laplace–Beltrami’s equation with perturbed boundary conditions.

Parallel ant colony optimization for the determination of a point heat source position in a 2-D domain

The Ant Colony Optimization (ACO) and its parallel implementations are applied to determine the unknown position of a point heat source in a two-dimensional steady-state heat conduction problem. The heuristic value, the determination of path and the objective function are all studied based on the inverse heat source problem. Some of the standard steps of the ACO are also modified according to the features of the heat conduction problem. In order to accelerate the speed of solving the inverse problem, two kinds of parallel ACO strategies, the fine-grained master-slave strategy and the coarse-grained strategy combined with the above modification are studied in this paper. The results show that the fine-grained strategy has a higher speedup than the coarse-grained strategy whiles the coarse-grained strategy has a higher accuracy rate. This means that the coarse-grained strategy is more proper during the parallel implementation for solving the inverse heat source problem and the parallel implementation of the ACO can improve the computational efficiency.

Virtual Boundary Meshless with Trefftz Method for the Steady-State Heat Conduction Crack Problem

This article presents a virtual boundary meshless with Trefftz method for calculation of the two-dimensional steady-state heat conduction crack problem, which is based on the idea of multidomain combination. The virtual source function is constructed by the radial basis functions approximation in the subdomain not containing the crack. The temperature or heat flux is calculated by crack special solution of Trefftz function in the crack subdomain. Thus, the proposed method has the advantages of both the boundary-type meshless method and the Trefftz method. No nodes or elements are distributed on the crack, because the boundary condition of the crack is automatically satisfied by employing the crack special solution of the Trefftz function. The heat stress intensity factor is simply and directly calculated.

Optimizing of the underground power cable bedding using momentum-type particle swarm optimization method

Thermal performance optimization of underground power cable system is presented in this paper. The analyzed system consists of three underground power cables situated in an in-line arrangement. The HDPE (High-Density Polyethylene) casing pipes, filled with SBM (Sand-Bentonite Mixture), covers the cables to protect them from heavy mechanical loads (e.g. vibrations). The FTB (Fluidized Thermal Backfill) layer is applied to prevent the cables from overheating. Due to the substantial costs of FTB backfill material (in relation to the native soil or dry sand), the cross-sectional area of FTB bedding layer has to be minimized. Furthermore, the maximum cable conductor temperature is expected not to exceed the optimum operating temperature. Therefore, the optimization procedure i.e. momentum-type PSO (Particle Swarm Optimization) is applied. The FEM (Finite Element Method) is used to solve the two-dimensional steady-state heat conduction problem. As a result, temperature distribution is determined for the native soil, FTB bedding, and cables. The performed computations considered the temperature dependent current rating and volumetric heat generation rate from cable conductor. The applied optimization procedure resulted in determination of the optimum cable spacing and cross-sectional area of the rectangular-shaped FTB bedding layer. Moreover, the obtained maximum temperature for the cable core do not exceed the allowable value.

A finite element method for prediction of unknown boundary conditions in two-dimensional steady-state heat conduction problems

This paper presents a finite element method in predicting unknown boundary conditions of homogeneous and two-layered materials subjected to steady-state heat conduction. Firstly, a finite element formulation is introduced to solve a steady-state boundary inverse heat conduction problem of homogeneous material. Effects of bias error of temperature specified on interior nodes and locations of such nodes on accuracy of predicted temperature distribution are examined. Then, modified cubic spline is specified on material boundary to stabilize predicted temperature distribution. Cubic spline functions using different numbers of control points are used in examining their effects on accuracy of predicted temperature distribution and computing time when specifying no bias temperatures. Finally, the formulation with cubic spline function specification is employed in predicting unknown boundary conditions of two-layered materials with thermal conductivity ratio of 0.1, 1, and 10. Concept of coincident nodes is applied in handling physical condition characterized by thermal contact resistance and heat source strength at layer interface. Effect of bias error of temperatures specified on nodes within thicker layer is examined under three interface conditions. Cubic spline function with five control points can predict temperature distributions accurately for all interface conditions when specifying no bias temperatures. RMS errors vary linearly with bias errors for interface conditions with no heat source but are drastically affected by bias error when heat source exists at the interface.

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.

2차원 열전도 문제에 대한 Fast Multipole 경계요소법의 이론과 실행 알고리즘의 분석

This paper presents the fast multipole boundary element method (FM-BEM) as a new BEM solution methodology that overcomes many disadvantages of conventional BEM. In conventional BEM, large-scale problems cannot be treated easily because the computation time increases rapidly with an increase in the number of boundary elements owing to the dense coefficient matrix. Analysis results are obtained to compare FM-BEM with conventional BEM in terms of computation time and accuracy for a simple two-dimensional steady-state heat conduction problem. It is confirmed that the FM-BEM solution methodology greatly enhances the computation speed while maintaining solution accuracy similar to that of conventional BEM. As a result, the theory and implementation algorithm of FM-BEM are discussed in this study.

Regularized collocation Trefftz method for void detection in two-dimensional steady-state heat conduction problems

We propose the use of the collocation Trefftz method for the solution of inverse geometric problems and, in particular, the determination of the boundary of a void. As was the case in the solution of such problems using the method of fundamental solutions, the algorithm for imaging the interior of the medium also makes use of radial polar parametrization of the unknown void shape in two dimensions. The centre of this radial polar parametrization is considered to be unknown. The feasibility of this new method is illustrated by several numerical examples highlighting its advantages and shortcomings.

A Method for Geometry Optimization in a Simple Model of Two-Dimensional Heat Transfer

This investigation is motivated by the problem of optimal design of cooling elements in modern battery systems. We consider a simple model of two-dimensional steady-state heat conduction described by elliptic partial differential equations and involving a one-dimensional cooling element represented by a contour on which interface boundary conditions are specified. The problem consists of finding an optimal shape of the cooling element which will ensure that the solution in a given region is close (in the least squares sense) to some prescribed target distribution. We formulate this problem as PDE-constrained optimization and the locally optimal contour shapes are found using a gradient-based descent algorithm in which the Sobolev shape gradients are obtained using methods of the shape-differential calculus. The main novelty of this work is an accurate and efficient approach to the evaluation of the shape gradients based on a boundary-integral formulation which exploits certain analytical properties of the solution and does not require grids adapted to the contour. This approach is thoroughly validated and optimization results obtained in different test problems exhibit nontrivial shapes of the computed optimal contours.

A complex variable solution of two-dimensional heat conduction of composites reinforced with periodic arrays of cylindrically orthotropic fibers

Problems of two-dimensional steady-state heat conduction for composites with doubly periodic arrays of cylindrically orthotropic fibers are dealt with. A new complex variable method is presented by introducing an appropriate coordinate transformation to convert the governing differential equation into a harmonic one, and the eigenfunction expansions of the field variables in a unit cell are derived. Then by using a generalized variational functional which absorbs the periodicity condition, an eigenfunction expansion–variational method based on a unit cell is developed to solve such problems. A convergence analysis and a comparison with finite element calculations are conducted to demonstrate the correctness and efficiency of the present method. A discussion is made about the effects of the cylindrical orthotropy of the fiber and the existence of the isotropic core in the fiber on the effective conductivity of the composite. An engineering equivalent parameter, which reflects the overall influence of the thermal conductivities of the matrix and fibers as well as the interfacial characteristic on the effective thermal conductivity of the composite, is found. It is shown that the present first-order approximation of the effective thermal conductivity of the composite can be written in a unified formula for different microstructural characteristics and possesses a good engineering accuracy.

Relaxation procedures for an iterative MFS algorithm for two-dimensional steady-state isotropic heat conduction Cauchy problems

We investigate two algorithms involving the relaxation of either the given Dirichlet data (boundary temperatures) or the prescribed Neumann data (normal heat fluxes) on the over-specified boundary in the case of the alternating iterative algorithm of Kozlov et al. [26] applied to two-dimensional steady-state heat conduction Cauchy problems, i.e. Cauchy problems for the Laplace equation. The two mixed, well-posed and direct problems corresponding to each iteration of the numerical procedure are solved using a meshless method, namely the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is chosen according to the generalized cross-validation (GCV) criterion. The iterative MFS algorithms with relaxation are tested for Cauchy problems associated with the Laplace operator in various two-dimensional geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the method.

Multicomponent Energy Conserving Dissipative Particle Dynamics: A General Framework for Mesoscopic Heat Transfer Applications

A multicomponent framework for energy conserving dissipative particle dynamics (DPD) is presented for the first time in both dimensional and dimensionless forms. Explicit definitions for unknown scaling factors that are consistent with DPD convention are found by comparing the present, general dimensionless governing equations to the standard DPD expressions in the literature. When the scaling factors are chosen based on the solvent in a multicomponent system, the system of equations reduces to a set that is easy to handle computationally. A computer code based on this multicomponent framework was validated, under the special case of identical components, for one-dimensional transient and one- and two-dimensional steady-state heat conduction in a random DPD solid. The results, which compare well with existing DPD works and with analytical solutions in one and two dimensions, show the promise of energy conserving DPD for modeling heat transfer at mesoscopic length scales.

Analysis of Two-Dimensional Steady-State Heat Conduction in Anisotropic Solids by Boundary Element Method Using Analog Equation Method and Green's Theorem

This paper is concerned with the application of the boundary element method (BEM) with the analog equation method (AEM), proposed by Katsikadelis and Nerantzaki, and Green's theorem to analyze steady-state heat conduction in anisotropic solids. In this study, the linear differential operator (the Laplacian) of steady-state heat conduction in isotropic solids is extracted from the governing differential equation. The integral equation formulated employs the fundamental solution of the Laplace equation for isotropic solids, and therefore, from the anisotropic part of the governing differential equation, a domain integral appears in the boundary integral equation. This domain integral is transformed into boundary integrals using Green's theorem with a polynomial function. The mathematical formulation of this approach for two-dimensional problems is presented in detail. The proposed solution is applied to two typical examples, and the validity and other numerical properties of the proposed BEM are demonstrated in the discussion of the results obtained.

Analysis of Two-Dimensional Steady-State Heat Conduction in Anisotropic Solids by Boundary Element Method Using Analog Equation Method and Green's Theorem

This paper is concerned with the application of the boundary element method with the analog equation method (AEM), proposed by Katsikadelis and Nerantzaki, and Green's theorem to analyze steady-state heat conduction in anisotropic solids. In this study, the linear differential operator (the Laplacian) of steady-state heat conduction in isotropic solids is extracted from the governing differential equation. The integral equation formulation employs the fundamental solution of the Laplace equation for isotropic solids, and therefore, from the anisotropic part of the governing differential equation, a domain integral arises in the boundary integral equation. This domain integral is transformed into boundary integrals using Green's theorem with a polynomial function. Mathematical formulations of this approach for two-dimensional problems are presented in detail. The proposed solution is applied to two typical examples, and the validity and other numerical properties of the proposed BEM are demonstrated in the discussion of the results obtained.