Two-dimensional Heat Conduction Problem Research Articles

An n-sided polygonal cell-node-based smoothed finite element method for solving two-dimensional heat conduction problems

An n-sided polygonal cell-node-based smoothed finite element method is proposed to analyze two-dimensional heat conduction problems. Through the gradient smoothing technique, the internal integral of the polygonal element is transformed into the boundary integral based on the smoothing domain, thus reducing the continuity requirement of the trial function, and only requiring the shape function value of the smoothing domain boundary calculated using the Wachspress coordinates. Therefore, when constructing the smoothed finite element form of the heat conduction problems, coordinate mapping is avoided, which greatly improves the computational efficiency. Furthermore, a new division of the cell-based smoothing domain is put forward to improve the accuracy of the solution, where the divisions of the node-based and cell-based smoothing domains are combined in background elements to form cell-node-based smoothing domains. Extensive numerical experiments are carried out to show that the current model performs better in the estimation of the accuracy of the solution, the computational cost, the equivalent energy, and the temperature gradient while maintaining the same precision as the polygonal finite element method (PFEM).

Determination of transient heat transfer by cooling channel in high-pressure die casting using inverse method

Complex shapes of aluminum castings are typically manufactured during the short cycle process known as the high-pressure die casting (HPDC). High productivity is ensured by introducing die cooling through a system of channels, die inserts or jet coolers. Die cooling can also effectively help in reducing internal porosity in cast components. Accurate simulations based on sophisticated numerical models require accurate input data such as material properties, initial and boundary conditions. Although the heat is dominantly dissipated through die cooling, indicating the importance of knowing precise thermal boundary conditions, open literature lacks a detailed information about the spatial distribution of heat transfer coefficient. This study presents an inverse method to determine accurate heat transfer coefficients of a die insert based on temperature measurements in multiple points by 0.5 mm K-type thermocouples and a subsequent solution of the two-dimensional inverse heat conduction problem. The solver was built in the open-source CFD code OpenFOAM and the free library for nonlinear optimization NLopt. The results are presented for the commonly used 10 mm die insert with a hemispherical tip and coolant flow rates ranging from 100 l/h to 200 l/h. Heat transfer coefficients reach values well above 50 kW/m2K in the hemispherical tip, which is followed by a secondary peak and then a gradual drop to values around 1 kW/m2K further downstream.

Application of the adaptive method to determine the process noise in the extended Kalman filter to estimate the parameters of the two dimensional inverse heat transfer problem

To deal with inverse nonlinear heat transfer problems in a two-dimensional heat conduction problem, this paper presents a hybrid approach combining fuzzy logic and the extended Kalman filter (EKF). The proposed algorithm has been applied to the real-time reconstruction of the temperature field, time-varying convective heat transfer coefficient, and time-varying boundary heat flux using temperature data from a set of sensors. The effects of the covariance of the initial estimation error, the time step of data acquisition by the sensors, and the number of installed sensors on the precision and stability of the estimation results has been investigated by numerical experiments. To compare and validate the results, all these parameters were investigated using the EKF method. The results show that the adaptive fuzzy extended Kalman filter (FEKF) method for estimating temperature, convection coefficient, and heat flux at the boundaries is precise and stable. The comparison shows that the performance of the FEKF method in parameter estimation is better than the EKF method, and the highest increase in precision and stability of the results is related to the convective heat transfer coefficient.

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.

A hybrid numerical method for non-linear transient heat conduction problems with temperature-dependent thermal conductivity

This paper introduces a hybrid numerical technique aimed at effectively addressing two-dimensional (2D) non-linear transient heat conduction problems, featuring temperature-dependent thermal conductivity. This scheme involves integrating the Krylov deferred correction (KDC) method in the temporal discretization and the generalized finite difference method (GFDM) in the spatial discretization. The core of the KDC method introduces a new unknown variable in the form of the first-order time derivative of temperature, leading to a non-linear equation after temporal discretization. Subsequently, the corresponding non-linear equation is solved in the spatial domain using the GFDM, supported by the Jacobian-free Newton–Krylov (JFNK) solver and fourth-order Taylor series expansion. The accuracy and stability of the hybrid approach are verified through two numerical experiments, demonstrating the strong performance of the present algorithm known as the KDC-GFDM for the simulations of the interested problems in long-time intervals.

A numerical investigation of 2D transient heat conduction problems in anisotropic FGMs with time-dependent conductivity

A class of two-dimensional (2D) transient heat conduction problems for anisotropic Functionally Graded Materials (FGMs) of space–time dependent conductivity and coefficient of variable separable forms is considered. The problems are reduced to a boundary-only integral equation which can be solved numerically using a standard Boundary Element Method (BEM). Some examples are considered to show the validity of the analysis and to examine the accuracy of the numerical method.

The polygonal finite element method for solving heat conduction problems

In this paper, a general polygonal finite element method (FEM-polygon) based on the Wachspress shape function is used to solve two-dimensional heat conduction problems. The finite element variational form of the heat conduction problem is first established based on the weighted residual method. Then the formulations of the thermal stiffness matrix, thermal damping matrix, and nodal heat vector are derived using the Wachspress coordinates. The global stiffness matrix is further assembled and the well-behaved algebraic equations of the thermal system are established. Finally, intensive numerical tests are conducted to verify the super convergence in temperature and the high precision in equivalent energy and temperature gradient of the present FEM-polygon for 2D heat conduction problems. Besides, the proposed model also performs excellently for traditional triangular and quadrilateral elements.

Thermal conductivity reconstruction method with application in a face milling operation

PurposeThis paper aims to reconstruct the spatially varying orthotropic conductivity based on a two-dimensional inverse heat conduction problem described by a partial differential equation (PDE) model with mixed boundary conditions. The proposed discretization uses a highly accurate technique and allows simple implementations. Also, the authors solve the related inverse problem in such a way that smoothness is enforced on the iterations, showing promising results in synthetic examples and real problems with moving heat source.Design/methodology/approachThe discretization procedure applied to the model for the direct problem uses a pseudospectral collocation strategy in the spatial variables and Crank–Nicolson method for the time-dependent variable. Then, the related inverse problem of recovering the conductivity from temperature measurements is solved by a modified version of Levenberg–Marquardt method (LMM) which uses singular scaling matrices. Problems where data availability is limited are also considered, motivated by a face milling operation problem. Numerical examples are presented to indicate the accuracy and efficiency of the proposed method.FindingsThe paper presents a discretization for the PDEs model aiming on simple implementations and numerical performance. The modified version of LMM introduced using singular scaling matrices shows the capabilities on recovering quantities with precision at a low number of iterations. Numerical results showed good fit between exact and approximate solutions for synthetic noisy data and quite acceptable inverse solutions when experimental data are inverted.Originality/valueThe paper is significant because of the pseudospectral approach, known for its high precision and easy implementation, and usage of singular regularization matrices on LMM iterations, unlike classic implementations of the method, impacting positively on the reconstruction process.

Numerical Analysis of Transient State Heat Transfer by Spectral Method Based on POD Reduced-Order Extrapolation Algorithm

In order to meet the requirements of high accuracy and fast algorithm for numerical heat transfer simulation, an iterative scheme of Proper Orthogonal Decomposition (for short, POD) dimension reduction based on the classical central difference Galerkin spectral method is proposed for solving two-dimensional transient heat conduction problems. The POD dimension reduction spectral method model is constructed by taking the calculation results of classical central difference Galerkin spectral method as sample data. The numerical algorithm characteristics of flow and heat transfer are studied by using a partial differential equation as a mathematical model, and the error estimation is given. Finally, different time intervals are used as parameters to simulate experiments. The results show that the POD method is applicable to transient nonlinear heat conduction problems, and the maximum average relative error of the reconstructed temperature field is 0.89675%. Moreover, the POD method not only has a high calculation accuracy, but also has an average calculation speed as high as 310.25 times that of the central difference Galerkin algorithm. It can be seen that under the condition that the error between the solution of POD dimension reduction extrapolation algorithm and the solution of classical central difference Galerkin spectrum method is small enough, the POD method can greatly reduce the calculation amount, shorten the running time, and ensure a high accuracy of the calculation results, thus verifying the effectiveness and feasibility of the algorithm.

A quasi-reversibility method for solving a two-dimensional time-fractional inverse heat conduction problem

In this paper, we consider a two-dimensional time-fractional inverse heat conduction problem, which is severely ill-posed. A quasi-reversibility method is proposed to solve this problem with disturbed boundary value. We give the selection of regularization parameters of quasi-reversibility method both under a priori and a posteriori rules, and give the proof of error estimates between the exact solution and its regularization approximation. Further more, numerical results are included to verify the effectiveness of the proposed method.

Influence of the Slag Rim on the Heat Transfer Behavior of a Mold

To explore the influence of the slag rim on the heat transfer behavior of a mold, a slag rim was extracted from a stainless steel production factory for high-temperature continuous casting simulation experiments. The 2DIHCP (Two-Dimensional Inverse Heat Conduction Problem) two-dimensional inverse problem calculation model was used to analyze the heat transfer characteristics in the mold, and the temperature and heat flux distribution of the hot surface of the mold copper wall under different working conditions were investigated. The results showed that under a casting speed of 0.5 Â m/min, vibration frequency of 2 Hz, and amplitude of 10 mm, the seepage flow of the mold flux was 9.314 and 6.326 g with and without the slag rim, respectively, the thickness of the slag film was 2.1 and 1.3 mm, respectively, the temperature of the mold hot surface was 118.24 and 134.83°C, respectively, and the maximum heat flux density of the mold was 0.98 and 1.95 MW/m2, respectively. Through data comparison, when there is a slag rim, the amount of slag infiltration decreased by 32%, the thickness of the slag film decreased by 38%, the temperature of the hot surface of the mold increased by 14%, the heat flux density increased by 99%, and the internal crystal morphology of the slag film was complete. This indicates that the slag rim affects the heat transfer behavior in the mold by restricting the inflow of the mold flux and affecting the formation of the slag film.

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.

Comparison of uniform and piecewise-uniform heatings when estimating thermal properties of high-conductivity materials

A d-optimum criterion is applied to the three-layer apparatus used for simultaneously estimating thermal properties (for example, thermal conductivity and effusivity) through the plane source method. The objective is to perform a comparison between uniform heating and piecewise-uniform heating of high-conductivity solid samples. In particular, the latter case is modeled through a two-dimensional heat conduction problem in which a rectangular plate (i.e. the sample) is partially heated at the front boundary through a surface heat flux, while all the other boundaries are kept insulated. The optimal experiment is designed for different set-ups of the experimental apparatus (width of the heated region, number of sensors and their locations). The convergence and the computational efficiency of the estimation iterative procedure are the terms of the comparison, as well as the expected standard deviations of thermal conductivity and effusivity. The results indicate that the use of a piecewise-uniform heating is not completely beneficial for isotropic materials. In fact, if on one hand it may offer standard deviations reduced up to about 40%, on the other hand it would require an experiment about six times longer than that required by a uniform heating to ensure a good convergence of the estimation iterative procedure. Therefore, a major computational effort is required.

A comparative numerical study of a semi-infinite heat conductor subject to double strip heating under non-Fourier models

The present work considers a two-dimensional (2D) heat conduction problem in the semi-infinite domain based on the classical Fourier model and other non-Fourier models, e.g., the Maxwell–Cattaneo–Vernotte (MCV) equation, parabolic, hyperbolic, and modified hyperbolic dual-phase-lag (DPL) equations. Using the integral transform technique, Laplace, and Fourier transforms, we provide a solution of the problem (Green’s function) in Laplace domain. The thermal double-strip problem, allowing the wave interference within the heat conductor, is considered. A numerical technique, based on the Durbin series for inverting Laplace transform and the trapezoidal rule for calculating an integral form of the solution in the double-strip case, is adopted to recover the solution in the physical domain. Finally, discussions for different non-Fourier heat transfer situations are presented. We compare among the speeds of hyperbolic heat transfer models and shed light on the concepts of flux-precedence and temperature-gradient-precedence, hallmarks of the lagging response idea. Otherwise, we emphasize the existence of a relationship between the waves speed and the time instant of interference onset, underlying the five employed heat transfer models.

A polygonal element differential method for solving two-dimensional transient nonlinear heat conduction problems

A novel polygonal element differential method (PEDM) is presented for solving two-dimensional nonlinear transient heat conduction problems for the first time. New shape functions as well as their derivatives with respect to isoparametric coordinates are derived to treat the polygonal elements with an internal node. System equations of the PBEM are formulated in terms of the governing equation and heat flux equilibrium condition, in which, a finite difference scheme is executed for calculating the transient term. Then, the nonlinearity system equations are dealt with by the Newton iterative method. Finally, examples with different structural complexities are designed to examine the property of the proposed method. The results show that the PEDM can effectively solve general two-dimensional transient nonlinear heat conduction problems with excellent accuracy and efficiency.

The MLS-based numerical manifold method for nonlinear transient heat conduction problems in functionally graded materials

In this study, we aim to numerically solve two-dimensional nonlinear transient heat conduction problems of functionally gradient materials (FGMs), where the governing equation is a quasilinear partial differential equation of second order. For this purpose, the weak form of the initial boundary value problem is first established. Then influence domains of nodes in the moving least squares (MLS) instead of finite elements are used as the mathematical patches to construct the mathematical cover of the numerical manifold method (NMM); while the shape functions of MLS-nodes as the weight functions subordinate to the mathematical cover, leading to the moving least squares based numerical manifold method (MLS-NMM). With the weak form, the spatial discretization is carried out using MLS-NMM, whereas temporal discretization follows the Euler backward difference. Finally, a series of numerical experiments are conducted concerning transient heat conduction problems of FGMs with heat source and heat convection, suggesting that the proposed MLS-NMM enjoys advantages of both MLS and NMM in solving nonlinear transient heat conduction of FGMs.

A posteriori regularization method for the two-dimensional inverse heat conduction problem

Abstract In this article, we consider a two-dimensional inverse heat conduction problem that determines the surface temperature distribution from measured data at the fixed location. This problem is severely ill-posed, i.e., the solution does not depend continuously on the data. A quasi-boundary value regularization method in conjunction with the a posteriori parameter choice strategy is proposed to solve the problem. A Hölder-type error estimate between the approximate solution and its exact solution is also given. The error estimate shows that the regularized solution is dependent continuously on the data.

Size determination of interior defects by reconstruction of subsurface virtual heat flux for step heating thermography

Step heating thermography is a non-destructive testing (NDT) technique that inspects interior defects by observing surface temperature rises during a long pulse heat stimulation. Due to the heat diffusion induced blurring of defect shapes, it is a challenging task to accurately determine the sizes of deep and small defects. This work focuses on estimating the size of a subsurface defect by an inversion of temperature images for the step heating thermography, to overcome the information loss during the propagation of thermal waves. We assume a space- and time-dependent virtual heat flux on the defect surface, which is reconstructed from surface temperature measurements by solving a two-dimensional inverse heat conduction problem. The local future time concept is combined with the Tikhonov's regularization to stabilize the inverse solution. A prudent choice of the regularization parameter is carried out through the L-curve. The feasibility of the proposed approach is assessed by numerical simulations and experiments. It is shown that the inversion brings an improvement in defect sizing accuracy for defects with low width-to-depth ratios and an efficient suppression of measurement noises.

Identifying Varying Thermal Diffusivity of Inhomogeneous Materials Based on a Hybrid Physics-Informed Neural Network

Inhomogeneous materials are characterized by uneven material properties with changing dimensions. Effective and precise identification of nonuniform material properties is essential for studying material inhomogeneity. In this study, we propose a hybrid physics-informed neural network (PINN) to identify the inhomogeneous thermal diffusivity only using temperature data collected in a transient heat conduction problem. A PINN and a fully connected neural network are combined in the proposed model. The PINN model is trained to learn the law of physics governed by the transient heat conduction equation. The identification network is trained to uncover the thermal diffusivity variation with changing dimensions. The proposed model successfully identifies two types of inhomogeneity: temperature-dependent and space-dependent thermal diffusivity, and both two-dimensional and three-dimensional heat conduction problems are investigated. The proposed model is examined to be robust to small datasets and noisy training data. In addition to heat conduction problems, the proposed model can be adopted to identify the material properties of various types of inhomogeneous materials.

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.