Steady Heat Conduction Problem Research Articles

Precise integration boundary element method for solving nonlinear transient heat conduction problems with temperature dependent thermal conductivity and heat capacity

The precise integration boundary element method (PIBEM) is used to solve nonlinear transient heat conduction problems with temperature dependent thermal conductivity and heat capacity in this article. At first, a boundary-domain integral equation can be obtained by using the fundamental solution for steady linear heat conduction problems. Secondly, the domain integrals are converted into boundary integrals by using the radial integration method to get a pure boundary integral equation, which can retain the advantage of dimension reduction of the boundary element method. Then, after discretization, a first-order nonlinear differential equation system in time is obtained, and the precise integration method with predictor-corrector technique is used to solve the system of nonlinear equations. In the end, several numerical examples are presented to verify the accuracy and stability of the presented method. The results obtained by the presented method are less affected by the time step size and satisfactory results can be obtained even for a large time step size.

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.

ACA-accelerated hybrid boundary node method applied to multi-domain steady heat conduction problems

This paper presents a fast implementation of the hybrid boundary node method (HBNM) to solve the multi-domain steady heat conduction problems using the Adaptive Cross Approximation (ACA). It is well known that the matrices arising from the discretization of boundary integral equation are fully populated and require special compression techniques for the efficient treatment. The ACA using only few of the original entries for the approximation of the whole matrix is a purely algebraic algorithm which is kernel-independent and easy to be applied directly for various problems. Accelerated by the ACA, several multi-domain steady heat conduction problems have been solved by the HBNM. The results demonstrate the potentials of the ACA-HBNM for solving multi-domain steady heat conduction problems.

Heat Conduction Plate Layout Optimization Using Physics-Driven Convolutional Neural Networks

Optimizing heat conduction layout is essential during engineering design, especially for sensible thermal products. However, when the optimization algorithm iteratively evaluates different loading cases, the traditional numerical simulation methods usually lead to a substantial computational cost. To effectively reduce the computational effort, data-driven approaches are used to train a surrogate model as a mapping between the prescribed external loads and various geometry. However, the existing model is trained by data-driven methods, which require intensive training samples from numerical simulations and do not effectively solve the problem. Choosing the steady heat conduction problems as examples, this paper proposes a physics-driven convolutional neural networks (PD-CNNs) method to infer the physical field solutions for randomly varied loading cases. After that, the particle swarm optimization (PSO) algorithm is used to optimize the sizes, and the positions of the hole masks in the prescribed design domain and the average temperature value of the entire heat conduction field is minimized. The goal of reducing heat transfer is achieved. Compared with the existing data-driven approaches, the proposed PD-CNN optimization framework predicts field solutions that are highly consistent with conventional simulation results. However, the proposed method generates the solution space without pre-obtained training data. We obtained thermal intensity results for holes 1, hole 2, hole 3, and hole 4 with 0.3948, 0.007, 0.0044, and 0.3939, respectively, by optimization PD-CNN model.

A new method for the evaluation of the vacuum boundary in circular and D-shaped Tokamaks

A new method for the evaluation of the vacuum boundary in a Tokamak is introduced. It is a two-step iterative algorithm that converges to the actual value of the boundary location only after a few iterations. The key point in this method is to solve a well-posed boundary value problem that satisfies a set of exact boundary conditions. A Cauchy problem in steady heat-conduction problem is used to explain the method first. Then, inverse evaluation of the vacuum boundary location in circular as well as D-shaped Tokamaks are considered. For circular Tokamak, the method does not require any regularization. The algorithm is quite simple and very effective. Numerical examples are presented to study the performance of the algorithm in the presence of noise.

Analytic Solution to Steady Heat Conduction with Transverse Radiation Heat Transfer

A unique analytic solution to the problem of steady heat conduction with radiative heat transfer normal to the conduction heat flow is presented. The solution is unique, as it does not assume zero temperature of the surroundings and addresses all possible tip boundary conditions for in-space applications. The temperature profiles in the direction of heat conduction are produced for a constant-temperature boundary condition at the base and three different boundary conditions at the tip: adiabatic, constant temperature, and radiative heat transfer. Approximate solutions to the implicit exact solution are also developed. An extended solution that includes convection for the adiabatic-tip boundary condition is also presented and compares very well to experimental data.

Robust RRE technique for increasing the order of accuracy of SPH numerical solutions

This study presents the use of a post-processing technique called repeated Richardson extrapolation (RRE) to improve the accuracy of numerical solutions of local and global variables obtained using the smoothed particle hydrodynamics (SPH) method. The investigation focuses on both the steady and unsteady one-dimensional heat conduction problems with Dirichlet boundary conditions, but this technique is applicable to multidimensional and other mathematical models. By using all the variables of the real type and quadruple precision (extended precision or Real*16) we were able to, for example, reduce the discretization error from 1.67E−08 to 3.46E−33 with four extrapolations, limited only by the round-off error and, consequently, determining benchmark solutions for the variable of interest ψ(1/2) using the SPH method. The increase in CPU time and memory usage owing to post-processing was almost null. RRE has proven to be robust in determining up to a sixteenth order of accuracy in meshless discretization for the spatial domain.

Residual-based error estimation and adaptivity for stabilized immersed isogeometric analysis using truncated hierarchical B-splines

Abstract We propose an adaptive mesh refinement strategy for immersed isogeometric analysis, with application to steady heat conduction and viscous flow problems. The proposed strategy is based on residual-based error estimation, which has been tailored to the immersed setting by the incorporation of appropriately scaled stabilization and boundary terms. Element-wise error indicators are elaborated for the Laplace and Stokes problems, and a THB-spline-based local mesh refinement strategy is proposed. The error estimation and adaptivity procedure are applied to a series of benchmark problems, demonstrating the suitability of the technique for a range of smooth and non-smooth problems. The adaptivity strategy is also integrated into a scan-based analysis workflow, capable of generating error-controlled results from scan data without the need for extensive user interactions or interventions.

Application of the Complex Variable Step Method and the Boundary Element Method for Sensitivity Analysis of Steady Heat Conduction Problems

The paper concerns the problems related to applying the complex variable step method for the sensitivity analysis of the steady temperature field in the solid body domain due to the perturbations of the geometrical and physical parameters. The optimization problem using the approach proposed is also discussed. At the stage of numerical modelling, the boundary element method is used. The first part of the paper is devoted to the shape sensitivity. The results obtained are compared with the solution resulting from the implicit approach of sensitivity analysis. In the second part, the practical problem concerning optimizing the geometry of continuous casting mould cross-section is considered. The project variable vector contains the cooling pipes' radius and the volume flow rate of the cooling water. The numerical results and the conclusions are presented in the final part of the paper.

Analytically-integrated radial integration PBEM for solving three-dimensional steady heat conduction problems

Recently, a polygonal boundary element method (PBEM) has been developed for solving heat conduction problems. In the PBEM, the radial integration method (RIM) is employed to shift the domain and surface integrals in the boundary-domain integral equation into equivalent line integrals, and all the radial integrals are computed by Gaussian quadrature, which may obtain an accurate solution. However, the computation costs much. Therefore, analytical expressions of radial integrals are derived in this paper, concerning four kinds of varying thermal conductivities and two kinds of heat generation functions, aiming at improving the efficiency. Then all the radial integrals in the boundary-domain integral equation can be analytically calculated. Three examples are employed to examine the analytically-integrated PBEM for solving steady heat conduction problems. The results show that the proposed method is accurate, and it is more efficient than traditional PBEM.

A Fully Meshless Approach to the Numerical Simulation of Heat Conduction Problems over Arbitrary 3D Geometries

One of the goals of new CAE (Computer Aided Engineering) software is to reduce both time and costs of the design process without compromising accuracy. This result can be achieved, for instance, by promoting a “plug and play” philosophy, based on the adoption of automatic mesh generation algorithms. This in turn brings about some drawbacks, among others an unavoidable loss of accuracy due to the lack of specificity of the produced discretization. Alternatively it is possible to rely on the so called “meshless” methods, which skip the mesh generation process altogether. The purpose of this paper is to present a fully meshless approach, based on Radial Basis Function generated Finite Differences (RBF-FD), for the numerical solution of generic elliptic PDEs, with particular reference to time-dependent and steady 3D heat conduction problems. The absence of connectivity information, which is a peculiar feature of this meshless approach, is leveraged in order to develop an efficient procedure that accepts as input any given geometry defined by a stereolithography surface (.stl file format). In order to assess its performance, the aforementioned strategy is tested over multiple geometries, selected for their complexity and engineering relevance, highlighting excellent results both in terms of accuracy and computational efficiency. In order to account for future extensibility and performance, both node generation and domain discretization routines are entirely developed using Julia, an emerging programming language that is rapidly establishing itself as the new standard for scientific computing.

A novel Boundary Element model for solving stationary inhomogeneous heat conduction problems

A new formulation of BEM applied to steady heat conduction problems, governed by the inhomogeneous Laplace Equation will be presented. The thermal property is isotropic, but varies according to a function known throughout the domain. DIBEM and DST are used together to generate the numerical model. Being used in the interpolation process, two distinct RBF will be used to evaluate the formulation performance. In order to show the generality and capacity of the proposed model, the heat flux variation in a machine tool temperature sensor is calculated. The comparison of the quality of the results achieved is made using FEM.

Subdivision Surface-Based Isogeometric Boundary Element Method for Steady Heat Conduction Problems with Variable Coefficient

The present work couples isogeometric analysis (IGA) and boundary element methods (BEM) for three dimensional steady heat conduction problems with variable coefficients. The Computer-Aided Design (CAD) geometries are built by subdivision surfaces, and meantime the basis functions of subdivision surfaces are employed to discretize the boundary integral equations for heat conduction analysis. Moreover, the radial integration method is adopted to transform the additional domain integrals caused by variable coefficients to the boundary integrals. Several numerical examples are provided to demonstrate the correctness and advantages of the proposed algorithm in the integration of CAD and numerical analysis.

An efficient and accurate hybrid weak-form meshless method for transient nonlinear heterogeneous heat conduction problems

Our purpose is to establish a numerical method meeting the requirements of efficiency, stability, accuracy and easy-using. Faced with these demands, in this paper, a hybrid method combining the advantages of 2 weak-form meshless methods is proposed for solving steady and transient heat conduction problems with temperature-dependent thermophysical properties in heterogeneous media. In the process of the calculation for an arbitrary model discretized by the hybrid method, two different weak-form methods are used for the nodes categorized as: (1) boundary nodes, (2) near boundary interior nodes and (3) interior nodes. A Galerkin free element method (GFREM) is proposed for the interior nodes of the Cartesian grid; and the local radial point interpolation method (LRPIM) is used for the other two types of nodes. The GFREM using the Cartesian grid can greatly improve the accuracy and efficiency of the hybrid scheme. Special decomposition technology to determine the irregular integral domain for LRPIM can increase the flexibility for complex models. In addition, the full implicit finite difference scheme and Newton–Raphson method are used for solving transient nonlinear problems. And non-crossing interface integral domain and support domain are employed for heterogeneous media. Compared with other methods, numerical results show better accuracy, stability and efficiency from the numerical experiments.

An improved implementation of triple reciprocity boundary element method for three-dimensional steady state heat conduction problems

The domain integrals caused by heat generation appearing in the boundary integral equation (BIE) of steady state heat conduction problems can be converted into boundary integrals by triple reciprocity method (TRM). However, the current triple reciprocity approximation (TRA) of domain functions is time-consuming and its accuracy is not stable in different geometric models, due to the need to solve a combination equation, whose degree of freedom is twice the number of boundary nodes plus the number of domain interpolation points. Therefore, a new formula of TRA is proposed in the current study to save computing time and storage space and improve accuracy. In the new proposed TRA formula, the combination equation is transferred into three equations that are solved in sequence. Four numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method. Results show that although the new TRA formulation is equivalent to the original one, the approximation results are quite different, especially for non-thin structures. Particularly, the improved formula of TRA can save computing time and storage space, and get better accuracy. The improved triple reciprocity BEM is a useful approach to analyze the three-dimensional steady heat conduction problems more efficiently and accurately.

Element-free Galerkin scaled boundary method based on moving Kriging interpolation for steady heat conduction analysis

This paper develops an element-free Galerkin scaled boundary method (EFG-SBM) for solving steady heat conduction problems, in which the circumferential direction is constructed by the moving Kriging (MK) interpolation based on the EFG approach. Because the MK interpolation satisfies the Kronecker delta property, it is more convenient in enforcing the essential boundary conditions than the traditional MLS-based EFG-SBM. As a newly boundary-type meshless method, EFG-SBM possesses advantages of EFG and scaled boundary finite element method (SBFEM). This method inherits the semi-analytical property of SBFEM by introducing the normalized radial coordinate system, in which the governing differential equations are weakened in the circumferential direction and solved analytically in the radial direction. Unlike the traditional SBFEM, the preprocessing and postprocessing processes of EFG-SBM are simplified and the more accuracy can be obtained because of higher continuity of the MK shape functions. Computation of EFG-SBM can be reduced since only the boundary needs to be discretized compared with the EFG approach. This proposed method is verified via four heat conduction examples including problems with thermal crack considering the prescribed heat flux and temperature on the side-face and unbounded domain. The numerical solutions show that EFG-SBM has higher accuracy and better convergence than the traditional SBFEM. An accurate smooth heat flux can be obtained directly without necessity of using the recovery procedure.

Solvability and construction of solution to the de la Vallee – Poussin problem for the second-order matrix Lyapunov equation with a parameter

The paper considers the issues of constructive analysis of the de la Vallee – Poussin boundary-value problem for the second-order linear matrix differential Lyapunov equation with a parameter and variable coefficients. The initial problem is reduced to an equivalent integral problem, and to study its solvability a modification of the generalized contraction mapping principle is used. A connection between the approach used and the Green’s function method is established. The coefficient sufficient conditions for the unique solvability of this problem are obtained. Using the Lyapunov – Poincaré small parameter method, an algorithm for constructing a solution has been developed. The convergence and the rate of convergence of this algorithm have been investigated, and a constructive estimation of the region of solution localization is given. To illustrate the application of the results obtained, the linear problem of steady heat conduction for a cylindrical wall, as well as a two-dimensional matrix model problem is considered. With the help of the developed general algorithm, analytical approximate solutions of these problems have been constructed and on the basis of their exact solutions a comparative numerical analysis has been carried out.

Variational estimates of the parameters of a thermal explosion of a stationary medium in an arbitrary domain

The variational formulation of the nonlinear problem of steady heat conduction in an arbitrary configuration domain is applied to investigate the conditions for the existence of a steady-state temperature distribution in a stationary medium with the intensity of volumetric energy release rising with increasing temperature. Based on the relations of the time-independent theory of thermal explosion, a variational model of this phenomenon is constructed, which makes it possible to obtain estimates of the critical values of the parameters that determine the temperature state of the medium preceding the thermal explosion. The examples of a comparative analysis of such values for a solid and a hollow cylinder of finite length are given.

A radial integration boundary element method for solving transient heat conduction problems with heat sources and variable thermal conductivity

ABSTRACTA new radial integration boundary element method (RIBEM) for solving transient heat conduction problems with heat sources and variable thermal conductivity is presented in this article. The Green’s function for the Laplace equation is served as the fundamental solution to derive the boundary-domain integral equation. The transient terms are first discretized before applying the weighted residual technique that is different from the previous RIBEM for solving a transient heat conduction problem. Due to the strategy for dealing with the transient terms, temperature, rather than transient terms, is approximated by the radial basis function; this leads to similar mathematical formulations as those in RIBEM for steady heat conduction problems. Therefore, the present method is very easy to code and be implemented, and the strategy enables the assembling process of system equations to be very simple. Another advantage of the new RIBEM is that only 1D boundary line integrals are involved in both 2D and 3D problems. To the best of the authors’ knowledge, it is the first time to completely transform domain integrals to boundary line integrals for a 3D problem. Several 2D and 3D numerical examples are provided to show the effectiveness, accuracy, and potential of the present RIBEM.

Steady heat transfer analysis of 2D plates using the numerical manifold method

In the numerical manifold method (NMM), the mathematical cover system can be inconsistent with the physical boundary, which can alleviate the meshing cost to some extent. In the present paper, the NMM is developed to solve two-dimensional steady heat conduction problems with the square mathematical elements. The corresponding formulations are derived and the proposed approach is validated through a typical example.