Steady-state Heat Conduction Problems Research Articles

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 meshless average source boundary node method for steady-state heat conduction in general anisotropic media

The average source boundary node method (ASBNM) is a recent boundary-type meshless method, which uses only the boundary nodes in the solution procedure without involving any element or integration notion, that is truly meshless and easy to implement. This paper documents the first attempt to extend the ASBNM for solving the steady-state heat conduction problems in general anisotropic media. Noteworthily, for boundary-type meshless/meshfree methods which depend on the boundary integral equations, whatever their forms are, a key but difficult issue is to accurately and efficiently determine the diagonal coefficients of influence matrices. In this study, we develop a new scheme to evaluate the diagonal coefficients via the pure boundary node implementation based on coupling a new regularized boundary integral equation with direct unknowns of considered problems and the average source technique (AST). Seven two- and three-dimensional benchmark examples are tested in comparison with some existing methods. Numerical results demonstrate that the present ASBNM is superior in the light of overall accuracy, efficiency, stability and convergence rates, especially for the solution of the boundary quantities.

Virtual boundary element method in conjunction with conjugate gradient algorithm for three-dimensional inverse heat conduction problems

ABSTRACTIn this article, the virtual boundary element method (VBEM) in conjunction with conjugate gradient algorithm (CGA) is employed to treat three-dimensional inverse problems of steady-state heat conduction. On the one hand, the VBEM may face numerical instability if a virtual boundary is improperly selected. The numerical accuracy is very sensitive to the choice of the virtual boundary. The condition number of the system matrix is high for the larger distance between the physical boundary and the fictitious boundary. On the contrary, it is difficult to remove the source singularity. On the other hand, the VBEM will encounter ill-conditioned problem when this method is used to analyze inverse problems. This study combines the VBEM and the CGA to model three-dimensional heat conduction inverse problem. The introduction of the CGA effectively overcomes the above shortcomings, and makes the location of the virtual boundary more free. Furthermore, the CGA, as a regularization method, successfully solves the ill-conditioned equation of three-dimensional heat conduction inverse problem. Numerical examples demonstrate the validity and accuracy of the proposed method.

Numerical analysis of heat conduction problems on 3D general-shaped domains by means of a RBF Collocation Meshless Method

A Collocation Meshless Method based on Radial Basis Function (RBF) interpo-lation is employed to solve steady state heat conduction problems on 3D domains of arbitrary shape. The set of points required by the numerical method is generated through a novel and simple technique which automatically produces a distribution with variable point density and which adapts to each specific geometry. Numerical results are systematically compared to the corresponding analytical solutions considering several combinations of parameters; convergence tests have also been carried out. The favorable properties that will be outlined suggest that this approach can be an effective and exible tool in the numerical simulation of heat conduction problems with complex 3D geometries.

Neural network-based discretization of nonlinear differential equations

This article presents a discretization scheme for a nonlinear differential equation using a regression analysis technique. As with many other numerical solvers such as finite difference methods and finite element methods, the presented scheme discretizes a simulation field into a finite number of points. Although other solvers “directly and mathematically” discretize a differential equation governing the field, the presented scheme “indirectly and statistically” constructs the discretized equation using regression analyses. The regression model learns a pre-prepared training data including dependent-variable values at neighboring points in the simulation field. Each trained regression model expresses the relation between the variables and returns a variable value for a point (output) referring to the variables for its surrounding points (input). In other words, the regression model performs a role as the discretized equation of the variable. The presented approach can be applied to many kinds of nonlinear problems. This study employs artificial neural networks and polynomial functions as the regression models. The main aim here is to assess whether the neural network-based spatial discretization approach can solve a nonlinear problem. In this work, I apply the presented scheme to nonlinear steady-state heat conduction problems. The computational accuracy of the presented technique was compared with a standard finite difference method and a homogenization method that is one of representative multiscale modeling approaches.

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.

Inverse problem of simultaneously estimating the thermal conductivity and boundary shape

ABSTRACTAn inverse analysis is used to simultaneously estimate the thermal conductivity and the boundary shape in steady-state heat conduction problems. The numerical scheme consists of a body-fitted grid generation technique to mesh the heat conducting body and solve the heat conduction equation – a novel, efficient, and easy to implement sensitivity analysis scheme to compute the sensitivity coefficients, and the conjugate gradient method as an optimization method to minimize the mismatch between the computed temperature distribution on some part of the body boundary and the measured temperatures. Using the proposed scheme, all sensitivity coefficients can be obtained in one solution of the direct heat conduction problem, irrespective of the large number of unknown parameters for the boundary shape. The obtained results reveal the accuracy, efficiency, and robustness of the proposed algorithm.

A Comparison of Fourier Spectral Iterative Perturbation Method and Finite Element Method in Solving Phase-Field Equilibrium Equations

AbstractThis paper systematically compares the numerical implementation and computational cost between the Fourier spectral iterative perturbation method (FSIPM) and the finite element method (FEM) in solving partial differential equilibrium equations with inhomogeneous material coefficients and eigen-fields (e.g., stress-free strain and spontaneous electric polarization) involved in phase-field models. Four benchmark numerical examples, including inhomogeneous elastic, electrostatic, and steady-state heat conduction problems demonstrate that (1) the FSIPM rigorously requires uniform hexahedral (3D) and quadrilateral (2D) mesh and periodic boundary conditions for numerical implementation while the FEM permits arbitrary mesh and boundary conditions; (2) the FSIPM solutions are comparable to their FEM counterparts, and both of them agree with the analytic solutions, (3) the FSIPM is much faster in solving equilibrium equations than the FEM to achieve the accurate solutions, thus exhibiting a greater potential for large-scale 3D computations.

Estimation of linearly temperature-dependent thermal conductivity using an inverse analysis

A linearly temperature-dependent thermal conductivity is estimated in steady state heat conduction problems using an inverse analysis. A body fitted grid generation technique is employed to mesh the two-dimensional body and solve the direct heat conduction problem. An efficient, accurate, and easy to implement method is presented to compute the sensitivity coefficients through derived expressions. The main feature of the sensitivity analysis is that all sensitivities can be obtained in one solve, irrespective of the number of unknown parameters. The conjugate gradient method along with the discrepancy principle is used in the inverse analysis to minimize the objective function and achieve the desired solution. The ability to efficiently and accurately recover the non-constant thermal conductivity is demonstrated through a number of benchmark problems.

Comparative Assessment of the Finite Difference, Finite Element, and Finite Volume Methods for a Benchmark One-Dimensional Steady-State Heat Conduction Problem

The finite difference (FD), finite element (FE), and finite volume (FV) methods are critically assessed by comparing the solutions produced by the three methods for a simple one-dimensional steady-state heat conduction problem with heat generation. Three issues are assessed: (1) accuracy of temperature, (2) accuracy of heat flux, and (3) satisfaction of global energy conservation. It is found that if the order of accuracy of the numerical discretization schemes is the same (central difference for FD and FV, linear basis functions for FE), the accuracy of the temperature produced by the three methods is similar, except close to the boundaries where the FV method outshines the other two methods. Consequently, the FV method is found to predict more accurate heat fluxes at the boundaries compared to the other two methods and is found to be the only method that guarantees both local and global conservation of energy irrespective of mesh size. The FD and FE methods both violate energy conservation, and the degree to which energy conservation is violated is found to be mesh size dependent. Furthermore, it is shown that in the case of prescribed heat flux (Neumann) and Newton cooling (Robin) boundary conditions, the accuracy of the FD method depends in large part on how the boundary condition is implemented. If the boundary condition and the governing equation are both satisfied at the boundary, the predicted temperatures are more accurate than in the case where only the boundary condition is satisfied.

An algorithm for solving steady-state heat conduction in arbitrarily complex composite planar walls with temperature-dependent thermal conductivities

An algorithm for solving steady-state heat conduction problems in arbitrarily complex composite walls is presented. Per se, steady-state heat conduction across a wall can easily be solved by hand. Yet, in practical applications the wall structure is often complex enough to deter such an approach if a finer yet simple analysis of the thermal bridges is of interest. Moreover, if high-temperature applications are involved, the additional complexity of including time-dependent thermal conductivity must be considered. Thus, a general methodology for solving arbitrary topology walls, involving any kind of thermal resistances in series and in parallel is discussed. While such a problem is formally simple to solve for a given wall following the theory, its algorithmic generalization is not. A method is provided, involving a program written in python language. The focus of the work is mainly on the algorithmic point of view: a simple way for the assessment of the wall topology and for the resolution of the heat conduction problem originating is sought. Temperature-dependent thermal conductivity of the materials is addressed, resulting in the need of evaluating the heat fluxes and the average temperature at each thermal resistance.

Numerical analysis of heat conduction problems on irregular domains by means of a collocation meshless method

A Least Squares Collocation Meshless Method based on Radial Basis Function (RBF) interpolation is used to solve steady state heat conduction problems on 2D polygonal domains using MATLAB® environment. The point distribution process required by the numerical method can be fully automated, taking account of boundary conditions and geometry of the problem to get higher point distribution density where needed. Several convergence tests have been carried out comparing the numerical results to the corresponding analytical solutions to outline the properties of this numerical approach, considering various combinations of parameters. These tests showed favorable convergence properties in the simple cases considered: along with the geometry flexibility, these features confirm that this peculiar numerical approach can be an effective tool in the numerical simulation of heat conduction problems.

Validation of accuracy and stability of numerical simulation for 2-D heat transfer system by an entropy production approach

The entropy production in 2-D heat transfer system has been analyzed systematically by using the finite volume method, to develop new criteria for the numerical simulation in case of multidimensional systems, with the aid of the CFD codes. The steady-state heat conduction problem has been investigated for entropy production, and the entropy production profile has been calculated based upon the current approach. From results for 2-D heat conduction, it can be found that the stability of entropy production profile exhibits a better agreement with the exact solution accordingly, and the current approach is effective for measuring the accuracy and stability of numerical simulations for heat transfer problems.

Application of the meshless generalized finite difference method to inverse heat source problems

The generalized finite difference method (GFDM) is a relatively new domain-type meshless method for the numerical solution of certain boundary value problems. This paper documents the first attempt to apply the method for recovering the heat source in steady-state heat conduction problems. In order to guarantee the uniqueness of the solution, the heat source here is assumed to satisfy a second-order partial differential equation, and thereby transforming the problem into a fourth-order partial differential equation, which can be solved conveniently and stably by using the GFDM. Numerical analysis are presented on three benchmark test problems with both smooth and piecewise smooth geometries. The stability and sensitivity of the scheme with respect to the amount of noise added into the input data are analyzed. The numerical results obtained show that the proposed algorithm is accurate, computationally efficient and numerically stable for the numerical solution of inverse heat source problems.

Study on random walk and its application to solution of heat conduction equation by Monte Carlo method

In this paper optimized and fast numerical method based on Monte Carlo simulation are introduced to achieve several algorithms for solving the steady state and transient heat conduction problems with any prescribed convection boundary conditions and interior thermal source distribution. The methods are mesh-less and thus can solve problems with complex geometries, non-uniform heat source or porous bodies with no inherent limitation. In the proposed approach the temperature of any point in the studying body can be found independently from other points without solution of related system of linear algebraic equations. As part of this study, an effort was undertaken to summarize various random walk methods in Monte Carlo. The Monte Carlo methods that are introduced in this paper use Semi Floating Random Walk (SFRW) and Full Floating Random Walk (FFRW) to estimate temperature in the domain of the definition. Also, Fixed Random Walk (FRW) method is applied to solve transient heat conduction equation. Each method has its own unique features that will be discussed during this work. It was observed that FFRW method is the fastest method to steady state conditions and also FRW method has capability to run both steady state and transient heat conduction equation. It was showed that the results are in good agreement with those of the finite difference method.

A coupled thermo-mechanical model based on the combined finite-discrete element method for simulating thermal cracking of rock

Based on a combined finite-discrete element method (FDEM), this study builds a coupled thermo-mechanical model (termed FDEM-TM) to simulate thermal cracking of rock. The coupled thermo-mechanical model consists of two major parts. In the first part the temperature distribution of the system is analyzed based on the heat conduction equation. In the second part the thermal stress caused by temperature change is added to perform mechanical fracture calculation. Based on these two parts, we can model rock fracture driven by thermo-mechanical coupling. Three examples with analytic solutions are used to verify the correctness of the model in dealing with the problems of steady-state heat conduction, unsteady-state conduction, and thermal-mechanic coupling, respectively. In addition, an example of thermal cracking is also given and compared with the experimental results. The simulation results are in excellent agreement with the analytical solutions or experimental results, verifying the correctness of the coupled thermo-mechanical model to simulate thermal cracking. The proposed method provides a new tool for thermal-mechanical coupling problems in geothermal exploitation.

An ACA-SBM for some 2D steady-state heat conduction problems

In this paper, an accelerated singular boundary method (SBM) incorporating adaptive cross approximation (ACA) is developed for the steady-state heat conduction problems. The SBM, a recently developed boundary collocation method, employs the fundamental solutions of the governing operators as the kernel functions, and desingularizes the source singularity with a concept of origin intensity factor. However, the SBM suffers fully-populated influence matrix which results in prohibitively expensive operation counts and memory requirements as the number of degrees of freedom increases. In this paper, the ACA is applied to accelerate the SBM meanwhile reducing the memory requirement. Furthermore, the ACA-SBM is robust to different fundamental solutions, which enables it to deal with different heat conduction problems. The effectiveness, feasibility and robustness of the proposed method are numerically tested on different heat conduction problems including isotropic homogeneous, anisotropic homogeneous and non-homogeneous media with quadratic material variation of thermal conductivity, highlighting the accuracy as well as the significant reduction in memory storage and analysis time in comparison with the traditional SBM.

An Analytical Model for Steady-State and Transient Temperature Fields in 3-D Integrated Circuits

This paper proposes a noniterative analytical thermal model for 3-D integrated circuits (3-D-ICs) based on the solution of the governing energy equations using Green’s function, separation of variables, and superposition methods. The model is applicable to both transient and steady-state heat conduction problems. Any finite number of dies equally sized in plane can be taken into consideration in current analysis. In addition, the analytical solution presented here is very general, so that it takes into account the different interface boundary conditions between the dies. Importantly, the thermal properties (i.e., $k$ , $\rho $ , $c$ ) and thickness of different dies can be different, and the spatial and time-varying volumetric heat generation can be applied in each of the different dies. The results compared with the finite-element simulations show that the temperature fields predicted by the proposed model can offer an accurate and efficient solution to compute the temperature fields in the representative 3-D-ICs, with a maximum deviation between the two being around 2.0% in the transient heat conduction. In the steady-state case, the maximum deviation can reach even less than 1.0%.

Parameter estimation in heat conduction using a two-dimensional inverse analysis

ABSTRACTThis article is concerned with a two-dimensional inverse steady-state heat conduction problem. The aim of this study is to estimate the thermal conductivity, the heat transfer coefficient, and the heat flux in irregular bodies (both separately and simultaneously) using a two-dimensional inverse analysis. The numerical procedure consists of an elliptic grid generation technique to generate a mesh over the irregular body and solve for the heat conduction equation. This article describes a novel sensitivity analysis scheme to compute the sensitivity of the temperatures to variation of the thermal conductivity, the heat transfer coefficient, and the heat flux. This sensitivity analysis scheme allows for the solution of inverse problem without requiring solution of adjoint equation even for a large number of unknown variables. The conjugate gradient method (CGM) is used to minimize the difference between the computed temperature on part of the boundary and the simulated measured temperature distribution. The obtained results reveal that the proposed algorithm is very accurate and efficient.

A new meshless method for solving steady-state nonlinear heat conduction problems in arbitrary plane domain

For solving steady-state nonlinear heat conduction problems (HCPs) in arbitrary plane domain enclosed by a complex boundary shape, the meshless methods are convenient to programming, of which the polynomial expansion method seems the simplest one. However, it is seldom used as a major medium to solve nonlinear partial differential equations owing to its highly ill-conditioned behavior. First, we decompose the steady-state nonlinear heat conduction equation into a linear portion on the left-hand side, and leave other terms on the right-hand side as a non-homogeneous portion. Then, we need to solve a steady-state linear heat conduction problem in each iteration, for which we propose a multiple-scale Pascal triangle method to generate the linear algebraic equations, where the multiple scales are automatically decided by the collocation points. We can find that these scales can largely reduce the condition number of the coefficient matrix in each linear system, such that the iteration process is convergent very quickly, and the numerical solutions obtained are very accurate and stable against very large noise to over 50%, no matter for the Dirichlet and mixed-type boundary value problems. Numerical results confirm the validity of the present multiple-scale polynomial expansion method for solving steady-state nonlinear HCPs in arbitrary plane domains.