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This paper aims to develop an efficient numerical method for nonlinear transient heat conduction problems with local radiation boundary conditions and nonlinear heat sources.

Based on the physical characteristic of the transient heat conduction and the distribution characteristic of the Green’s function, a quasi-superposition principle is presented for the transient heat conduction problems with local nonlinearities. Then, an efficient method is developed, which indicates that the solution of the original nonlinear problem can be derived by solving some nonlinear problems with small structures and a linear problem with the original structure. These problems are independent of each other and can be solved simultaneously by the parallel computing technique.

Within a small time step, the nonlinear thermal loads can only induce significant temperature responses of the regions near the positions of the nonlinear thermal loads, whereas the temperature responses of the remaining regions are very close to zero. According to the above physical characteristic, the original nonlinear problem can be transformed into some nonlinear problems with small structures and a linear problem with the original structure.

An efficient and accurate numerical method is presented for transient heat conduction problems with local nonlinearities, and some numerical examples demonstrate the high efficiency and accuracy of the proposed method.

The purpose of this paper is to present a new approach of meshless local B-spline based finite difference (FD) method for solving two dimensional transient heat conduction problems.

In the present method, any governing equations are discretized by B-spline approximation which is implemented in the spirit of FD technique using a local B-spline collocation scheme. The key aspect of the method is that any derivative is stated as neighbouring nodal values based on B-spline interpolants. The set of neighbouring nodes are allowed to be randomly distributed thus enhanced flexibility in the numerical simulation can be obtained. The method requires no mesh connectivity at all for either field variable approximation or integration. Time integration is performed by using the Crank-Nicolson implicit time stepping technique.

Several heat conduction problems in complex domains which represent for extended surfaces in industrial applications are examined to demonstrate the effectiveness of the present approach. Comparison of the obtained results with solutions from other numerical method available in literature is given. Excellent agreement with reference numerical method has been found.

The method is presented for 2D problems. Nevertheless, it would be also applicable for 3D problems.

A transient two dimensional heat conduction in complex domains which represent for extended surfaces in industrial applications is presented.

The presented new meshless local method is simple and accurate, while it is also suitable for analysis in domains of arbitrary geometries.

This paper aims to propose an efficient and accurate method to analyse the transient heat conduction in a periodic structure with moving heat sources.

The moving heat source is modelled as a localised Gaussian distribution in space. Based on the spatial distribution, the physical feature of transient heat conduction and the periodic property of structure, a special feature of temperature responses caused by the moving heat source is illustrated. Then, combined with the superposition principle of linear system, within a small time-step, computation of results corresponding to the whole structure excited by the Gaussian heat source is transformed into that of some small-scale structures. Lastly, the precise integration method (PIM) is used to solve the temperature responses of each small-scale structure efficiently and accurately.

Within a reasonable time-step, the heat source applied on a unit cell can only cause the temperature responses of a limited number of adjacent unit cells. According to the above feature and the periodic property of a structure, the contributions caused by the moving heat source for the most of time-steps are repeatable, and the temperature responses of the entire periodic structure can be obtained by some small-scale structures.

A novel numerical method is proposed for analysing moving heat source problems, and the numerical examples demonstrate that the proposed method is much more efficient than the traditional methods, even for larger-scale problems and multiple moving heat source problems.

An eigenvalue method is presented for solving the transient heat conduction problem with time‐dependent or time‐independent boundary conditions. The spatial domain is divided into finite elements and at each finite element node, a closed‐form expression for the temperature as a function of time can be obtained. Three test problems which have exact solutions were solved in order to examine the merits of the eigenvalue method. It was found that this method yields accurate results even with a coarse mesh. It provides exact solution in the time domain and therefore has none of the time‐step restrictions of the conventional numerical techniques. The temperature field at any given time can be obtained directly from the initial condition and no time‐marching is necessary. For problems where the steady‐state solution is known, only a few dominant eigenvalues and their corresponding eigenvectors need to be computed. These features lead to great savings in computation time, especially for problems with long time duration. Furthermore, the availability of the closed form expressions for the temperature field makes the present method very attractive for coupled problems such as solid—fluid and thermal—structure interactions.

The purpose of this paper is to examine the transient heat conduction in a two-dimensional anisotropic substrate coated with a thin layer of thermal barrier coating (TBC). Nowadays, materials with anisotropic properties have been extensively applied in various engineering applications for enhanced strength. However, under an extreme operating environment of high temperature, the strength of the materials may largely decline. As a common practice in engineering, TBC are usually applied to thermally insulate the substrates so as to allow for higher operating temperature. This research provides engineers a numerical approach for properly designing the TBC to protect the anisotropic substrate.

For this investigation, a finite difference scheme using the domain mapping technique, transforming the anisotropic domain into isotropic one, is employed. The analysis considers three respective boundary conditions, namely Dirichelete condition, Neumann condition, and also forced convection, and studies the effect of various variables on the heat conduction in the coated system. Additionally, formulas for the steady-state temperature drop across the coating layer at the center are analytically derived. By comparing the numerical results with the analytical solutions, the veracity of the formulas is verified.

A few interesting phenomena are observed from the numerical results. First, the rotation of the substrate's principal axes affects the temperature on the TBC front surface in a more obvious manner for the Neumann condition than that for convection. Second, the temperature profile of the Dirichelete condition rises faster than the other cases, although all their profiles present a similar pattern. Third, the transient temperature drop across the TBC under the convection condition presents a complicated pattern, depending on the TBC thickness. Finally, the increase of TBC thickness under the Dirichelete condition may provide better insulation than the other cases. In this paper, approximate analytical formulations for the steady-state temperature drop across the TBC are also presented. Numerical results by the finite difference method indicate excellent agreements with the analytical solutions.

In the past, the finite element method (FEM) is usually applied for analyzing the heat conduction problem of TBC. However, one serious deficiency of applying the FEM to the TBC problem lies in the demand for a vast amount of elements (or cells) when the TBC thickness is far smaller than the substrate dimension. For ultra-thin coating, an enormous amount of elements are required that may lead to an extremely heavy computational burden. The paper presents an innovative finite difference approach that can be applied to analyze the heat conduction across the TBC coated on an anisotropic substrate. On the interface between the TBC and the substrate, a special heat equilibrium condition and the compatibility condition of identical temperature on the adjacent materials are used to propose three new models to predict the temperature drop across the TBC.

The transient temperature distribution in a rotating cylindrical shell which is heated by an incident time varying heat flux (nuclear pulse) as well as a constant heat flux, is determined numerically by a finite difference method. The shell is cooled by combined convection and thermal radiation. The effects of cooling and rotation on the temperature distribution as well as the time‐ and space‐dependence are shown. Rotation provides a sinusoidal temperature variation in time for a fixed surface and circumferential position. Increased rotation reduces the maximum temperature in the shell and also provides a more uniform temperature distribution in the circumferential direction.

The spectral boundary element method for solving two‐dimensional transient heat conduction problems is developed. This method is combined with the fast Fourier transform (FFT) to convert the solution between the time and frequency domains. The fundamental solutions in the frequency domain, required for the present method, are discussed. The resulting line integrations in the frequency domain are discretized using constant boundary elements and used in a Fortran boundary element program. Three examples are used to illustrate the accuracy and effectiveness of the method in both the frequency and time domains. First, the frequency domain solution procedure is verified using the steady‐state example of a semi‐infinite half space with a heat flux applied to a patch of the surface. This spectral boundary element method is then applied to the problem of a circular hole in an infinite solid subjected to a time‐varying heat flux, and solutions in both the frequency and time domains are presented. Finally, the method is used to solve the circular hole problem with a convection boundary condition. The accurary of these results leads to the conclusion that the spectral boundary element method in conjunction with the FFT is a viable option for transient problems. In addition, this spectral approach naturally produces frequence domain information which is itself of interest.

This paper presents a computationally efficient numerical solution scheme to solve transient heat conduction problems using the boundary element method (BEM) without volume discretization. Traditionally, a transient solution using BEM is very computer intensive due to the excessive numerical integration requirements at each time increment. In the present work a numerical solution scheme based on the separation of time and space integrals in the boundary integral equation through the use of an appropriate series expansion of the integrand (incomplete gamma function) is presented. The space integrals are evaluated only once in the beginning and within each time increment the additional integrals are obtained from the previously evaluated space integrals by a simple calculation. Three‐dimensional applications are provided to compare the proposed strategy with that used traditionally. The CPU requirements are reduced substantially. The solution scheme presented allows for dynamically changing the time step size as the solution evolves. This feature is not practical in the traditional schemes based on boundary discretization only.

An adaptive mesh refinement procedure that uses nodeless variables and quadratic interpolation functions is presented for analysing transient thermal problems. A temperature based finite element scheme with Crank‐Nicolson time marching is used to obtain the thermal solution. The strategies used for mesh adaptation, computing refinement indicators, and time marching are described. Examples in one and two dimensions are presented and comparisons are made with exact solutions. The effectiveness of this procedure for transient thermal analysis is reflected in good solution accuracy, reduction in number of elements used, and computational efficiency.

The purpose of this paper is to investigate the effect of numerical boundary condition implementation on local error and convergence in L2-norm of a finite volume discretization of the transient heat conduction equation subject to several boundary conditions, and for cases with volumetric heat generation, using both fully implicit and Crank-Nicolson time discretizations. The goal is to determine which combination of numerical boundary condition implementation and time discretization produces the most accurate solutions with the least computational effort.

The paper studies several benchmark cases including constant temperature, convective heating, constant heat flux, time-varying heat flux, and volumetric heating, and compares the convergence rates and local to analytical or semi-analytical solutions.

The Crank-Nicolson method coupled with second-order expression for the boundary derivatives produces the most accurate solutions on the coarsest meshes with the least computation times. The Crank-Nicolson method allows up to 16X larger time step for similar accuracy, with nearly negligible additional computational effort compared with the implicit method.

The findings can be used by researchers writing similar codes for quantitative guidance concerning the effect of various numerical boundary condition approximations for a large class of boundary condition types for two common time discretization methods.

The paper provides a comprehensive study of accuracy and convergence of the finite volume discretization for a wide range of benchmark cases and common time discretization methods.