Two-dimensional Steady-state Heat Conduction Problems Research Articles

Bayesian Inference for Estimating Heat Sources through Temperature Assimilation

Abstract This paper utilizes a Bayesian inference framework to address the two-dimensional steady-state heat conduction problem, focusing on the estimation of unknown distributed heat sources in a thermally-conducting medium with uniform conductivity. The goal is to infer the locations, strength, and shape of heaters by assimilating temperature data in Euclidean space, employing a Fourier series to represent each heater's shape. The Markov Chain Monte Carlo method, incorporating the random-walk Metropolis-Hasting algorithm and parallel tempering, is utilized for posterior distribution exploration in both unbounded and wall-bounded domains. It is found that multiple solutions arise in cases where the number of temperature sensors is less than the number of unknown states. Moreover, smaller heaters introduce greater uncertainty in estimated strength. To address the challenge of estimating the heater's strength and shape simultaneously due to their strong correlation, our method incorporates sharp priors on one to ensure accurate and feasible solutions of the other. The diffusive nature of heat conduction smooths out any deformations in the temperature contours, especially in the presence of multiple heaters positioned near each other, impacting convergence. In wall-bounded domains with Neumann boundary conditions, the inference of heater parameters tends to be more accurate than in unbounded domains.

Modelling 2-D Heat Transfer Problem for Different Material Combination

The analysis of heat transfer problems can be highly complex due to factors such as temperature, position, and time. Most heat transfers are typically two-dimensional as conduction is often negligible in the third dimension. Two-dimensional heat conduction problems can be solved analytically or numerically. In steady-state conditions, the Laplace equation can be applied to solve two-dimensional heat conduction problems analytically, in which the separation of variables method is used to solve the Laplace equation under fixed boundary conditions to determine the temperature at a specific point. The Laplace equation plays a significant role in the solution of heat transfer problems, as it demonstrates the behavior of linear and non-linear equations in the computational fluid dynamics domain. Despite their inability to provide exact results at any point, numerical methods are superior to analytical methods when handling complex geometries with various boundary conditions. This project involves the development of a computational code using MATLAB to solve two-dimensional steady-state heat conduction problems using Gauss-Seidel iterations. Comparing analytical solutions from Excel with numerical solutions from MATLAB and ANSYS, specifically the developed MATLAB code, revealed an accuracy level of 99.902% for the Laplace equation. An analysis of the produced code from MATLAB found that it could solve two-dimensional steady-state heat conduction across different combinations of materials while allowing users to specify initial and boundary conditions to produce a contour plot similar to ANSYS.

An elliptical inhomogeneity under nonuniform heat flux

We use complex variable techniques to study the decoupled two-dimensional steady-state heat conduction and thermoelastic problems associated with an elliptical elastic inhomogeneity perfectly bonded to an infinite matrix subjected to a nonuniform heat flux at infinity. Specifically, the nonuniform remote heat flux takes the form of a linear distribution. It is found that the internal temperature and thermal stresses inside the elliptical inhomogeneity are quadratic functions of the two in-plane coordinates. Explicit closed-form expressions of the analytic functions characterizing the temperature and thermoelastic field in the matrix are derived.

Neutral Inhomogeneities in a Two-dimensional Steady-state Heat Conduction Problem

A steady-state heat conduction problem is considered in a two-dimensional solid body which is filled up composite circular inclusions. The composite circular inclusions consist of a core and a coating both of which are cylindrically orthotropic. In this paper the neutral inhomogeneity is defined as a foreign body (inclusion) which can be introduced into the host body (matrix) without distributing the temperature field in the originally homogeneous body. The perfect thermal contacts are assumed to be between the different components of nonhomogeneous bodies.

Comparative Analysis of Two-Dimensional Steady State Heat Flow Problem of a Rectangular Plate by Analytical and Numerical Approach

Heat transfer problems play a very vital role in many fields of engineering. This present analysis work focuses on analytical and numerical approaches to resolve a two-dimensional steady-state heat conduction problem of a rectangular plate. The temperatures at the inside points of the rectangular plate are calculated analytically and the results are verified numerically. For numerical solution of the problems programming is done in using METLAB software. The material of the rectangular plate used is steel. It will be seen from the literature survey that rigorous analytical solutions are available only for a very few simple boundary conditions and these conditions does not seem to be favourable for complex boundaries. However, majority of engineering problems with simple as well as complex boundary conditions will be solved with the help of numerical approaches. In this analysis work, an attempt is made to obtain solutions using numerical method for a two-dimensional steady state heat conduction problems of a steel rectangular plate. The mathematical formulation of problems is done using Laplace method. The results obtained from the two different methods that is analytical and numerical are then compared with each other. The agreement between the analytical and numerical results is a sign of the accuracy of solution 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.

Isogeometric Boundary Element Method for Two-Dimensional Steady-State Non-Homogeneous Heat Conduction Problem

Isogeometric Boundary Element Method for Two-Dimensional Steady-State Non-Homogeneous Heat Conduction Problem

Meshless Analysis of Nonlocal Boundary Value Problems in Anisotropic and Inhomogeneous Media

In this work, meshless methods based on a radial basis function (RBF) are applied for the solution of two-dimensional steady-state heat conduction problems with nonlocal multi-point boundary conditions (NMBC). These meshless procedures are based on the multiquadric (MQ) RBF and its modified version (i.e., integrated MQ RBF). The meshless method is extended to the NMBC and compared with the standard collocation method (i.e., Kansa’s method). In extended methods, the interior and the boundary solutions are approximated with a sum of RBF series, while in Kansa’s method, a single series of RBF is considered. Three different sorts of solution domains are considered in which Dirichlet or Neumann boundary conditions are specified on some part of the boundary while the unknown function values of the remaining portion of the boundary are related to a discrete set of interior points. The influences of NMBC on the accuracy and condition number of the system matrix associated with the proposed methods are investigated. The sensitivity of the shape parameter is also analyzed in the proposed methods. The performance of the proposed approaches in terms of accuracy and efficiency is confirmed on the benchmark problems.

Two-dimensional steady-state heat conduction problem for heat networks

The method of determining the level of heat losses through structures of heat networks and the two-dimensional steady-state heat conduction problem for multilayer structures of heat networks have been considered, given that each of the layers has its own physical properties and heat transfer coefficients depend on r and z coordinates. The definition of heat losses is an important practical task, since they are one of the main indicators of energy efficiency in the operating heat networks and are included in heat tariffs.

Simulation of two-dimensional steady-state heat conduction problems by a fast singular boundary method

The study presents a fast direct algorithm for solutions of systems arising from singular boundary method (SBM) in two-dimensional (2D) steady-state heat conduction problems. As a strong-form boundary discretization collocation technique, the SBM is mathematically simple, easy-to-implement, and free of mesh. Similar to boundary element method (BEM), the SBM generates a dense coefficient matrix, which requires O(N3) operations and O(N2) memory to solve the system with direct solvers. In this study, a fast direct solver based on recursive skeletonization factorization (RSF) is applied to solve the linear systems in the SBM. Due to the fact that the coefficient matrix is hierarchically block separable, we can construct a multilevel generalized LU decomposition that allows fast application of the inverse of the coefficient matrix. Combing the RSF and the SBM can dramatically reduce the operations and the memory to O(N). The accuracy and effectiveness of the RSF-SBM are tested through some 2D heat conduction problems including isotropic and anisotropic cases.

Boundary function method for boundary identification in two-dimensional steady-state nonlinear heat conduction problems

In this article, a novel meshless boundary function method (BFM) is proposed for solving the boundary identification problem of steady-state nonlinear heat conduction in arbitrary plane domain. Firstly, the original governing equation is transformed to a new one with homogeneous Cauchy boundary conditions by using a homogenization technique. Secondly, the domain type meshless collocation method is employed to solve the new partial different equation in a reduced domain, in which the numerical solution is expanded by a sequence of boundary functions, automatically satisfying the homogeneous boundary conditions on the known boundary. After that, a nonlinear equation corresponding to each angle is formed and then is solved by the Newton iterative method in order to determine the missing boundary shape. Finally, the accuracy and robustness of the proposed BFM are examined through three numerical examples.

Topology optimization of conductive heat transfer problems using parametric L-systems

Generative encodings have the potential of improving the performance of evolutionary algorithms. In this work we apply parametric L-systems, which can be described as developmental recipes, to evolutionary topology optimization of widely studied two-dimensional steady-state heat conduction problems. We translate L-systems into geometries using the turtle interpretation, and evaluate their objective functions, i.e. average and maximum temperatures, using the Finite Volume Method (FVM). The method requires two orders of magnitude fewer function evaluations, and yields better results in 10 out of 12 tested optimization problems (the result is statistically significant), than a reference method using direct encoding. Further, our results indicate that the method yields designs with lower average temperatures than the widely used and well established SIMP (Solid Isotropic Material with Penalization) method in optimization problems where the product of volume fraction and the ratio of high and low conductive material is less or equal to 1. Finally, we demonstrate the ability of the methodology to tackle multi-objective optimization problems with relevant temperature and manufacturing related objectives.

An optimization algorithm for determining a point heat source position in a 2D domain using a hybrid metaheuristic

Abstract In this paper, we study an inverse problem of determining the unknown heat source position of a point-wise heat source in a two-dimensional steady-state heat conduction problem governed by a linear elliptic equation with the Dirichlet boundary conditions. The problem is solved by the hybridization of particle swarm optimization and the gravitational search algorithm with a newly defined mutation operator. Some empirical studies are also performed to gauge the accuracy of the proposed approach.

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.

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.

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.