Direct and inverse solutions of the two-dimensional hyperbolic heat conduction problems

A sequential method is proposed to estimate boundary condition of the two-dimensional hyperbolic heat conduction problems. An inverse solution is deduced from a finite difference method, the concept of the future time and a modified Newton–Raphson method. The undetermined boundary condition at each time step is denoted as an unknown variable in a set of non-linear equations, which are formulated from the measured temperature and the calculated temperature. Then, an iterative process is used to solve the set of equations. No selected function is needed to represent the undetermined function in advance. The example problem is used to demonstrate the characteristics of the proposed method. In the example, a well-known problem is used to demonstrate the validity of the proposed direct method and then the inverse solutions are evaluated. In the second example, the larger value of the relaxation time is implemented in the direct solutions and the inverse solutions. The close agreement between the exact values and the estimated results is made to confirm the validity and accuracy of the proposed method. The results show that the proposed method is an accurate and stable method to determine the boundary conditions in the two-dimensional inverse hyperbolic heat conduction problems.     

非线性热传方程的相似解

文[1]研究了非线性热传导方程的波动解,即相似变量ξ为波动变量(ξ=a+bx+ct)的情形,并规定热传导系数也只是ξ的函数。本文抛弃了这些限制条件,从更加普遍的角度去研究非线性热传导方程的相似性解。     

Enriched finite element spaces for transient conduction heat transfer

This paper presents an alternative approach via finite elements to treat numerically the thermal shocks in heat transfer finite element analysis. The method consists in using the standard enriched finite element approaches with time-interpolation. It will be applied here to the transient conduction heat equation where the classical Galerkin method is shown to be unstable. The proposed method consists in adding and eliminating bubbles to the finite element space and then to interpolate the solution to the real time step. This modification is equivalent to the addition of a stabilizing term tuned by a local time-dependent stability parameter, which ensures an oscillating-free solution. To validate this approach, the numerical results obtained in classical 2D and 3D benchmark problems are compared with the Galerkin and the analytical solutions.     

一类热传导方程的整体解及解的逐点估计

研究长柱体区域中的热传导方程.通过构造辅助函数,利用Hopf极值原理,得到整体解的充分条件,并给出解的逐点估计与空间衰减估计.     

三维热传导型半导体问题的交替方向有限元方法

1　引　　言三维热传导型半导体器件瞬态问题的数学模型由四个非线性偏微分方程描述[1 ,2 ] ,记 Ω为 Ω=[0 ,1 ] 3的边界 ,三维问题-Δψ =α( p -e+ N( x) ) ,　　 ( x,t)∈Ω× [0 ,T] ,( 1 .1 ) e t= . ( De( x) e-μe( x) e ψ) -R( e,p,T) ,　 ( x,t)∈Ω× ( 0 ,T] ,( 1 .2 ) p t= . ( Dp( x) p +μp( x) p ψ) -R( e,p,T) ,　 ( x,t)∈Ω× ( 0 ,T] ,( 1 .3 )ρ( x) T t-ΔT =[( Dp( x) p +μp( x) p ψ) -( De( x) e-μe( x) e ψ) ] . ψ,　　　　　　 ( x,t)∈Ω× ( 0 ,T] . ( 1 .4 )ψ( x,t) =e( x,t) =p( …     

A local iteration scheme for nonlinear two-dimensional steady-state heat conduction: a BEM approach

An efficient algorithm is proposed to solve the steady-state nonlinear heat conduction equation using the boundary element method (BEM). Nonlinearity of the heat conduction equation arises from nonlinear boundary conditions and temperature dependence of thermal conductivity. Using Kirchhoff's transformation, the case of temperature dependence of thermal conductivity can be transformed to the nonlinear boundary conditions case. Applying the BEM technique, the resulting matrix equation becomes nonlinear. The nonlinearity, however, only involves the boundary nodes that have nonlinearboundary conditions. The proposed local iterative scheme reduces the entire BEM matrix equation to a smaller matrix equation whose rank is the same as the number of boundary nodes with nonlinear boundary conditions. The Newton-Raphson iteration scheme is used to solve the reduced nonlinear matrix equation. The local iterative scheme is first applied to two one-dimensional problems (analytical solutions are possible) with different nonlinear boundary conditions. It is then applied to a two-region problem. Finally, the local iterative scheme is applied to two cavity problems in which radiation plays a role in the heat transfer.     

Analytical solutions to the heat conduction problems for a cylinder and a ball based on determining the temperature perturbation front

Analytical solutions to the heat conduction problems for a cylinder and a ball are obtained by the integral method of heat balance. To improve the accuracy of the solutions, the temperature function is approximated by polynomials of high degrees. Their coefficients are determined via introducing additional boundary conditions, which are found from the governing differential equation and the basic boundary conditions, including those specified at the temperature perturbation front. It is shown that the additional boundary conditions, even in the second approximation, lead to a considerable improvement in the solution accuracy.     

A note on a non-standard problem for an equation with a delay term

In this note we investigate the continuous dependence of the solutions for a theory of heat conduction with a delay term. We use energy arguments to obtain the continuous dependence results and spectral arguments to prove the non-uniqueness result. The extension to the thermoelastic problem is also pointed out.     

有限厚平板局部表面非均布局部加热的热传导解*

在工程技术上常遇到如下问题;一个有一定厚度的平板,在其一个表面的某个局部区域内加热,热流沿边界面非均匀分布,且随时间变化。这类问题在已有的热传导专着中无现成的解析解可供使用。本文给出了此问题的解析解,其计算结果与实验对照,符合得很好。     

基于MFS的PDE-约束优化法求解热传导逆问题

在本文中,我们给出了一种有效的无网格方法来求解逆热传导问题,含有Neumann边界条件情形.所得到的PDE-约束优化法是一种在空间与时间域上的全局近似方法,其中将控制方程的基本解作为基函数.由于初始测量数据包含有噪声误差,则所得线性方程组的系数矩阵通常是病态的,文中利用广义交叉验证(GCV)的Tikhonov正则化方法来获得更加稳定的数值解.通过数值结果表明,本文给出的方法是精确、有效、鲁棒的.     

Dynamic adaptation method in gasdynamic simulations with nonlinear heat conduction

A dynamic adaptation method is applied to gas dynamics problems with nonlinear heat conduction. The adaptation function is determined by the condition that the energy equation is quasi-stationary and the grid point distribution is quasi-uniform. The dynamic adaptation method with the adaptation function thus determined and a front-tracking technique are used to solve the model problem of a piston moving in a heat-conducting gas. It is shown that the results significantly depend on the thermal conductivity chosen. The numerical results obtained on a 40-node grid are compared with self-similar solutions to this problem.     

Image solutions for boundary value problems without sources

In this paper, we employ the image method to solve boundary value problems in domains containing circular or spherical shaped boundaries free of sources. two and threeD problems as well as symmetric and anti-symmetric cases are considered. By treating the image method as a special case of method of fundamental solutions, only at most four unknown strengths, distributed at the center, two locations of frozen images and one free constant, need to be determined. Besides, the optimal locations of sources are determined. For the symmetric and anti-symmetric cases, only two coefficients are required to match the two boundary conditions. The convergence rate versus number of image group is numerically performed. The differences of the image solutions between 2D and 3D problems are addressed. It is found that the 2D solution in terms of the bipolar coordinates is mathematically equivalent to that of the simplest MFS with only two sources and one free constant. Finally, several examples are demonstrated to see the validity of the image method for boundary value problems.     

Hybrid differential transform and finite difference method to solve the nonlinear heat conduction problem

In this paper, the differential transform is employed to discuss the behaviors of nonlinear heat conduction problem. A hybrid method of differential transform and finite difference approach is proposed to solve the transient responses of a nonlinear heat conduction problem. Different parameters of the equation and boundary conditions are considered to verify the feasibility of the proposed method to such problems. Simulation results are illustrated and discussed in comparison with the linear case. The results show that the hybrid method can achieve good results for such problems.     

Axially symmetric problems of stationary heat conduction and thermoelasticity for a body with thermally active or thermally insulated disk inclusion (crack)

We present solutions of axially symmetric problems of stationary heat conduction and thermoelasticity for a body with a thin thermally active disk inclusion (where the temperature or heat flow is given) and also with a thermally insulated inclusion. The heat conduction problems are reduced to integral equations, and exact solutions are obtained in the case where their right-hand sides are polynomials of arbitrary degree. We determine the components of the stress tensor and displacement vector as well as, in the case of cracks, the stress intensity factors.     

Hybrid pseudospectral–finite difference method for solving a 3D heat conduction equation in a submicroscale thin film

This research aims to develop a time‐dependent pseudospectral‐finite difference scheme for solving a 3D dual‐phase‐lagging heat transport equation in a submicroscale thin film. The scheme uses periodic pseudospectral discretization in space and a fully second‐order finite difference discretization in time. The three consecutive time steps model is then solved explicitly, by using a preconditioned conjugate gradient method. The scheme is illustrated by an example which is used to investigate the heat transfer in a gold submicroscale thin film. Comparisons are made with available literature. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

热方程侧边值问题的正则化方法及频域中的修改“核”思想

讨论和分析几种著名的求解热方程侧边值问题的正则化方法.在频域中发现了这些方法之间的有趣联系,同时通过对已有文献中相关方法及问题(不同方法应用于同一个问题和一种方法应用于不同问题)的比较分析,提出频域中的修改"核"思想.反之,基于该思想,可以方便地衡量已有文献中的部分正则化方法的好坏和构建某类线性不适定问题的新的正则化方法.以一个二维逆热传导问题和一个反向热传导问题为例说明了修改核思想的部分应用.     

The Method of Fundamental Solution for a Radially Symmetric Heat Conduction Problem with Variable Coefficient

We consider an inverse heat conduction problem with variable coefficient on an annulus domain. In many practice applications, we cannot know the initial temperature during heat process, therefore we consider a non-characteristic Cauchy problem for the heat equation. The method of fundamental solutions is applied to solve this problem. Due to ill-posedness of this problem, we first discretize the problem and then regularize it in the form of discrete equation. Numerical tests are conducted for showing the effectiveness of the proposed method.     

Use of 3D-transient analytical solution based on Green’s function to reduce computational time in inverse heat conduction problems

Inverse problems can be found in many areas of science and engineering and can be applied in different ways. Two examples can be cited: thermal properties estimation or heat flux function estimation in some engineering thermal process. Different techniques for the solution of inverse heat conduction problem (IHCP) can be found in literature. However, any inverse or optimization technique has a basic and common characteristic: the need to solve the direct problem solution several times. This characteristic is the cause of the great computational time consumed. In heat conduction problem, the time consumed is, usually, due to the use of numerical solutions of multidimensional models with refined mesh. In this case, if analytical solutions are available the computational time can be reduced drastically. This study presents the development and application of a 3D-transient analytical solution based on Green’s function. The inverse problem is due to the thermal properties estimation of conductors. The method is based on experimental determination of thermal conductivity and diffusivity using partially heated surface method without heat flux transducer. Originally developed to use numerical solution, this technique can, using analytical solution, estimate thermal properties faster and with better accuracy.     

快速求解参数化偏微分方程的缩减基有限元方法及其在核工程中的应用

基于求解偏微分方程的高保真数值模拟已经广泛应用于科学研究和工程设计.然而,即使借助超级计算机的并行计算能力,经典的有限元方法和其它数值方法在面对需要多次求解或需要快速甚至实时求解的问题时仍然面临效率的挑战.针对可用参数化微分方程表示的问题,缩减基有限元方法利用少数代表性的经典有限元解构造基函数,同时通过仿射分解使得系统矩阵和载荷向量的组装变为简单的代数叠加,因此该方法可以大幅度地提高这类问题的求解效率.本文介绍了这种方法的原理,并以固体热传导和中子扩散的快速求解为例,展示了这种方法的优良特性.结果表明,在线阶段的求解效率可以实现两到三个数量级的提升.基于高保真模拟的缩减基模型是将高性能计算应用于工程优化设计、应急指挥以及复杂问题的反分析等工作的有效手段.     

Spatial behavior for solutions in heat conduction with two delays

In this note, we investigate the spatial behavior of the solutions of the equation proposed to describe a theory for the heat conduction with two delay terms. We obtain an alternative of the Phragmén-Lindelöf type, which means that the solutions either decay or blow-up at infinity, both options in an exponential way. We also describe how to obtain an upper bound for the amplitude term. This is the first contribution on spatial behavior for partial differential equations involving two delay terms. We use energy arguments. The main point of the contribution is the use of an exponentially weighted energy function.