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 (2D) 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 (MCMC) method, incorporating the random-walk Metropolis–Hasting (MH) 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.

A thermodynamics-consistent spatiotemporally-nonlocal model for microstructure-dependent heat conduction

The spatiotemporally-nonlocal phenomena in heat conduction become significant but challenging for metamaterials with artificial microstructures. However, the microstructure-dependent heat conduction phenomena are captured under the hypothesis of spatiotemporally local equilibrium. To capture the microstructure-dependent heat conduction phenomena, a generalized nonlocal irreversible thermodynamics is proposed by removing both the temporally-local and spatially-local equilibrium hypotheses from the classical irreversible thermodynamics. The generalized nonlocal irreversible thermodynamics has intrinsic length and time parameters and thus can provide a thermodynamics basis for the spatiotemporally-nonlocal law of heat conduction. To remove the temporally-local equilibrium hypothesis, the generalized entropy is assumed to depend not only on the internal energy but also on its first-order and high-order time derivatives. To remove the spatially local equilibrium hypothesis, the thermodynamics flux field in the dissipation function is assumed to relate not only to the thermodynamics force at the reference point but also to the thermodynamics force of the neighboring points. With the developed theoretical framework, the thermodynamics-consistent spatiotemporally-nonlocal models can then be developed for heat transfer problems. Two examples are provided to illustrate the applications of steady-state and transient heat conduction problems.

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.

Solving stationary inverse heat conduction in a thin plate

We consider a steady state heat conduction problem in a thin plate. In the application, it is used to connect two cylindrical containers and fix their relative positions. At the same time it serves to measure the temperature on the inner cylinder. We derive a two dimensional mathematical model, and use it to approximate the heat conduction in the thin plate. Since the plate has sharp edges on the sides the resulting problem is described by a degenerate elliptic equation. To find the temperature in the interior part from the exterior measurements, we formulate the problem as a Cauchy problem for stationary heat equation. We also reformulate the Cauchy problem as an operator equation, with a compact operator, and apply the Landweber iteration method to solve the equation. The case of the degenerate elliptic equation has not been previously studied in this context. For numerical computation, we consider the case where noisy data is present and analyse the convergence.

Numerical instability and its treatment for steady state heat conduction problems in exchanger tubes using the dual boundary element method

In this paper, we employed the boundary element method (BEM) to determine the heat conduction shape factor (CSF) for the steady-state heat conduction problem in the heat exchanger tube. Various doubly-connected geometric shapes were considered. Numerical results of the conduction shape factor matched well with the theoretical data. Numerical instability was found for a special size. Degenerate scale occurred for all cases of regular N-gons in determining the conduction shape factor and matched the data as analytically predicted. Numerical results showed that a degenerate scale is only dependent on the outer boundary instead of the inner boundary. Three regularization techniques, the dual BEM, the CLEEF(UT), and the CLEEF(LM), were provided to avoid numerical instability. It was found that the larger number of sides for the N-gon shape is given, the lower the CSF is obtained. As N approaches infinity, it converges to the minimum value of the solution for an annulus.

Deep Learning Method Based on Physics-Informed Neural Network for 3D Anisotropic Steady-State Heat Conduction Problems

This paper uses the physical information neural network (PINN) model to solve a 3D anisotropic steady-state heat conduction problem based on deep learning techniques. The model embeds the problem’s governing equations and boundary conditions into the neural network and treats the neural network’s output as the numerical solution of the partial differential equation. Then, the network is trained using the Adam optimizer on the training set. The output progressively converges toward the accurate solution of the equation. In the first numerical example, we demonstrate the convergence of the PINN by discussing the effect of the neural network’s number of layers, each hidden layer’s number of neurons, the initial learning rate and decay rate, the size of the training set, the mini-batch size, the amount of training points on the boundary, and the training steps on the relative error of the numerical solution, respectively. The numerical solutions are presented for three different examples. Thus, the effectiveness of the method is verified.

MLS-based numerical manifold method for steady-state three-dimensional heat conduction problems of functionally graded materials

In this study, the moving least squares based numerical manifold method (MLS-NMM) is firstly applied to discretize three-dimensional (3D) steady heat conduction problems of functionally graded materials (FGMs). In the 3D MLS-NMM, the influence domains of nodes in the moving least squares (MLS) are used as the mathematical patches to construct the mathematical cover (MC); while the shape functions of MLS-nodes as the weight functions subordinate to the MC. Compared with the traditional NMM using finite elements to form the MC, the cutting operations in generating the physical cover are unnecessary and the computation complexity is decreased significantly. Based on the Galerkin method, the discrete form of 3D heat conduction is derived. Finally, a series of numerical experiments concerning steady heat conduction problems are performed, suggesting that the proposed MLS-NMM enjoys advantages of both MLS and NMM in solving steady heat conduction.

Steady State Heat Conduction in Exchanger Tubes by Using the Meshfree Boundary Integral Equation Method: Conduction Shape Factor and Degenerate Scale

AbstractRegarding the steady-state heat conduction problem in exchanger tubes, the meshfree boundary integral equation method is employed to determine the conduction shape factor in this paper. Different from the conventional boundary element method, the present method is free of mesh generation. After using the parametric function to represent the boundary contour and adopting the Gaussian quadrature, only collocating points on the boundary is required to obtain the linear algebraic equations. By introducing the local exact solution, the singular integral in the sense of the Cauchy principal value can be novelly determined. In addition, the boundary layer effect due to the nearly singular integral in the boundary integral equation can be dealt with. Two cases of different boundary conditions are considered. One is the isothermal condition on both the inner and outer surfaces. The other is the isothermal condition on the inner surface and the convection condition on the outer surface. Besides, numerical instability is found and the nonuniqueness solution due to the degenerate scale is examined by calculating the conduction shape factor and the temperature on the outer surface.

Numerical verification of a nonlocal discrete model for anisotropic heat conduction problems

This paper presents the numerical study of a recently proposed nonlocal discrete model for anisotropic heat conduction problems. Both transient and steady-state heat conduction problems are studied. The solution schemes and applications of the temperature and flux boundary conditions are discussed in detail. Various benchmark problems are employed to demonstrate the convergence characteristic and the prediction accuracy of the proposed model. The proposed model is then applied to study the heat conduction in a bi-continuous composite microstructure. It is observed that the proposed model has an expected linear convergence rate and can accurately predict the solution with respect to finite element results. As demonstrated using the bi-continuous microstructure, the proposed model offers significant convenience to image-based modeling and simulation by directly take the digitized material voxels as discrete material particles without generating quality mesh.

A multiscale scaled boundary finite element method solving steady-state heat conduction problem with heterogeneous materials

A new numerical model is proposed by bridging small and large scales via the numerical base functions constructed by Scaled Boundary Finite Element Method (SBFEM) for the multiscale heterogeneous analysis of steady-state heat conduction problems. Utilizing the advantages of polygonal Scaled Boundary Finite Element (SBFE) and image-based quadtree gridding, the construction of base functions can be conveniently and efficiently conducted particularly for the heterogeneous media with complex geometries or singularity of heat flux at the small-scale, and the solution accuracy at the large-scale can be improved by increasing nodes of coarse element only without increasing new nodes inside. Consequently, the SBFEM based solutions at the large-scale can be solved with a reduced order and sufficient accuracy, and the solutions at the small-scale can be achieved using SBFEM based base functions. Numerical examples are provided and the effectiveness of the proposed approach is stressed at both the large and small scales.

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.

Multi-domain physics-informed neural network for solving forward and inverse problems of steady-state heat conduction in multilayer media

In this paper, a novel deep learning technique, called multi-domain physics-informed neural network (M-PINN), is presented to solve forward and inverse problems of steady-state heat conduction in multilayer media. By adopting the domain decomposition technique, the multilayer media is first divided into several sub-domains. Then, the fully connected neural network is employed to approximate the temperature field on each sub-domain. Finally, a large total network framework is formed by combining subnetworks of all the mediums and using continuity conditions on interfaces. By training the total network, we can obtain the temperature distribution over the whole computational domain, including the interface between every two mediums. In the proposed method, the boundary conditions are introduced into the loss function, and the governing equation is used as a constrain item, which ensures the accuracy and stability of numerical approximation. As a meshless collocation technology, the M-PINN does not require tedious procedures such as meshing and numerical integration, and can freely address forward and inverse problems of thin body and coating structure. Several numerical examples are given to illustrate the efficiency and performance of the new method. Results indicate that the Swish and the Sigmoid functions are two better activation functions for such problems. As the number of nodes increases, the number of hidden layers does not need to be increased. Even for the thin film at nanoscale, the M-PINN still obtains accurate results. Moreover, the proposed scheme shows better performance than the traditional boundary element method in solving nonlinear heat conduction problems.

DHEM: a deep heat energy method for steady-state heat conduction problems

DHEM: a deep heat energy method for steady-state heat conduction problems

Isogeometric and NURBS-enhanced boundary element formulations for steady-state heat conduction with volumetric heat source and nonlinear boundary conditions

Boundary Element Method (BEM) is a numerical tool that is applied to many different types of engineering problems. BEM possesses several advantages over other numerical methods such as boundary-only discretization and its semi-analytical nature. Application of Isogeometric Analysis (IGA) to BEM is a recently proposed computational method where the main purpose is to use geometrical basis functions as the shape functions of field variables. Key features of combining these two techniques are the exact representation of the geometry, elimination or suppression of the meshing procedure, reduction of the problem dimension and obtaining highly accurate results especially for the flux values on the boundaries compared to conventional BEM. In this study, steady-state heat conduction problems with nonlinear boundary condition and surface heat source are analyzed where geometries are formed by NURBS, and field variables are described with Lagrangian (constant and quadratic) basis functions and NURBS basis functions. Convergence rates and CPU time of different types of element representations are discussed which eventually reveals the performance and capabilities of IGABEM.

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.

A regularized diffuse domain-lattice Boltzmann model for heat transfer in complex geometries with temperature Dirichlet boundary condition

In this paper, a regularized diffuse domain-lattice Boltzmann model for heat transfer with the Dirichlet boundary condition in complex geometries is proposed. In this model the complex computational domain is embedded into a larger and regular domain. The level set function is used to capture the complex interface. A diffuse domain formulation for the temperature field is established on the larger domain. To remove the ill-conditioned property of the original diffuse domain equation, we develop a new regularization technique to tackle this issue. We also give a multiple-relaxation-time lattice Boltzmann model to solve the regularized diffuse domain equation. The convergence rate of present model is investigated by the steady state and time-decay heat conduction problems. Then the natural convection problems in a square enclosure with a heated circular cylinder and in a horizontal concentric annulus are simulated. Both the uniform and non-uniform wall temperature cases are considered. Moreover, an alternative method to calculate the average Nusselt number on the complex boundary is employed. The numerical results show a good agreement with the previous solutions in previous literatures. The effect of non-uniform wall temperature on the heat transfer rate is also investigated. At last, the forced convection across the staggered tube banks is also simulated to verify the potential capability of the present model in solving practical heat transfer problem with the complex geometries.

Numerical Analysis of Temperature Distribution in Tandem Welding Process

In this study, the numerical solutions for the steady-state heat conduction problem with uniform heat source, the steady-state heat conduction problem with convective heat transfer and the transient heat conduction problem have been developed using finite difference method. These numerical solutions have been validated with analytical solutions. After observing the good agreements between numerical solutions and analytical solutions, these three different problems combined to simulate the tandem welding process. The first objective of this study is to present a numerical simulator for the transient heat conduction problem that includes non-uniform moving heat sources and convective heat transfer term. This numerical simulator contains explicit and implicit time discretization methods. In this simulator, it is possible to change the grid sizes, time step sizes, total simulation time, distance between electrodes, magnitude of the sources' power, speed of the sources, etc. Secondly, the temperature distribution of single and twin wire welding processes have been compared using proposed numerical simulator to investigate the premature solidification of liquid metal in low-temperature zone of molten pool. Thirdly, experimental study was carried out using Fluke Thermal Imager to validate numerical results. It was obtained that the maximum temperature of numerical result is very close to the maximum temperature of experimental result with 0.248 % error. Finally, the all Matlab codes related to developed numerical simulator have been added to Appendix to facilitate other researchers’ work.

Design and thermal conductivity of 3D artificial cross-linked random fiber networks

We propose a novel procedure to design three-dimensional (3D) printable cross-linked random fiber networks. Our procedure describes how to implement, in a novel way, a modified version of random-walk based algorithm using a combination of different free and open-source software programs. Second, we describe in detail a method based on an open-source finite-element software, to analyse the steady-state heat conduction and to compute the effective thermal conductivity for cross-linked fibrous structures immersed in vacuum. We apply this method to examine how the orientations of the fibers and the fiber volume fraction influence the effective thermal conductivity of 3D artificial cross-linked random glass fiber networks. Our numerical results show explicitly the direct link between the effective thermal conductivity and the percolating conduction paths through a 3D cross-linked fiber network. The proposed procedure for creating the 3D drawing and the one for volume meshing as well as the method for solving the steady-state heat conduction problem described in this paper can be extended to other 3D complex structures with closed surfaces.

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.

An ANN-based data-predictive approach for comparative study between CFD finite difference and finite volume method

In computational fluid dynamics (CFD), there is a transformation of methods over the years for building commercially coded software. Each method has predicted its own set of importance, but the exportation and prediction of data are some of the crucial elements for post-processing and validating results. In the present investigation, a detailed comparative analysis is performed over finite difference method (FDM) and finite volume method (FVM) method for the 1D steady-state heat conduction problem over a 1-m-long plate. The comparison was made between solution creation and validation between FDM and FVM for the analytical and computational scheme. The convergence-dependent study is performed as multi-objective optimization to predict how artificial neural network (ANN) can be used to verify and validate the solution of CFD.