The estimation of the strength of the heat source in the heat conduction problems

A general method is proposed to determine the strength of the heat source in the Fourier and non-Fourier heat conduction problems. A finite difference method, the concept of the future time and a modified Newton–Raphson method are adopted in the problem. The undetermined heat source at each time step is formulated as an unknown variable in a set of equations 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. Three examples are used to demonstrate the characteristics of the proposed method. The validity of the proposed method is confirmed by the numerical results. The results show that the proposed method is an accurate and stable method to determine the strength of the heat source in the inverse hyperbolic heat conduction problems. Furthermore, the result shows that more future times are needed in the hyperbolic equation than that of parabolic equation. Moreover, the robustness and the accuracy of the estimated results in the non-Fourier problem are not as well as those of the Fourier problem.     

Inverse determination of thermal conductivity using semi-discretization method

In this work a semi-discretization method is presented for the inverse determination of spatially- and temperature-dependent thermal conductivity in a one-dimensional heat conduction domain without internal temperature measurements. The temperature distribution is approximated as a polynomial function of position using boundary data. The derivatives of temperature in the differential heat conduction equation are taken derivative of the approximated temperature function, and the derivative of thermal conductivity is obtained by finite difference technique. The heat conduction equation is then converted into a system of discretized linear equations. The unknown thermal conductivity is estimated by directly solving the linear equations. The numerical procedures do not require prior information of functional form of thermal conductivity. The close agreement between estimated results and exact solutions of the illustrated examples shows the applicability of the proposed method in estimating spatially- and temperature-dependent thermal conductivity in inverse heat conduction problem.     

Investigation of the effect of the measurement with the abrupt noise and the number of approximated terms in conductional boundary estimation problem

A direct error vector analysis of inverse heat conduction problem is presented to detect the measured noise. Based on the reverse matrix approach that the inverse problem is solved directly in a linear domain, and the error vector is formulated from the difference between the measured temperature and the estimated temperature. There is no prior knowledge on the exact solution while the error vector is constructed. The error vector is used to investigate the consistence of the measured data in the domain and lead to detect the noise data. Furthermore, the proper number of the undetermined variable is able to suggest based on the mean value of the error vector and the value of the condition number of the reverse matrix. In the first example, a test problem with the measurement noise is presented. The estimated result is influent by the noise globally. The result shows that the value of error vector is changed significantly at the coordinate of the measurement noise appeared. In other words, the error vector analysis is able to identify the noise data. In the second example, the proper number of series expansion terms is investigated. From the result, it shows that the number of expansion terms with the small mean value and condition number can better approximate to the unknown condition. It means that the proposed method is able to suggest a proper number of expansion terms when the function of the recovered boundary is unknown.     

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.     

A numerical technique for solving IHCPs using Tikhonov regularization method

This study is intended to provide a numerical algorithm for solving a one-dimensional inverse heat conduction problem. The given heat conduction equation, the boundary conditions, and the initial condition are presented in a dimensionless form. The numerical approach is developed based on the use of the solution to the auxiliary problem as a basis function. To regularize the resultant ill-conditioned linear system of equations, we apply the Tikhonov regularization method to obtain the stable numerical approximation to the solution.     

A complex variable boundary element method for axisymmetric heat conduction in a nonhomogeneous solid

A complex variable boundary element method is proposed for solving numerically the axisymmetric steady-state problem of heat conduction in a nonhomogeneous isotropic solid. To assess the validity and the accuracy of the method, it is applied to solve specific cases of the problem. The numerical solutions obtained agree well with known solutions.     

基于时域精细积分算法的瞬态传热多宗量反演

基于有限元法和精细积分算法,提出了一种求解瞬态热传导多宗量反演问题的新方法．采用有限元法和精细积分算法分别对空间、时间变量进行离散,可以得到正演问题高精度的半解析数值模型,由此建立了多宗量反演的计算模式,并给出敏度分析的计算公式．对一维和二维的热物性参数、热源项、边界条件等进行了单宗量和多宗量的反演求解,初步考虑了初值和噪音等对反演结果的影响,数值算例验证了该方法的有效性．     

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.     

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.     

An inverse hyperbolic heat conduction problem in estimating surface heat flux of a living skin tissue

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to solve the inverse hyperbolic heat conduction problem in estimating the unknown time-dependent surface heat flux in a living skin tissue from the temperature measurements taken within the tissue. The inverse solutions will be justified based on the numerical experiments in which three different heat flux distributions are to be determined. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The influence of measurement errors upon the precision of the estimated results is also investigated. Results show that an excellent estimation on the time-dependent surface heat flux can be obtained for the test cases considered in this study.     

Goursat problem for two-dimensional second-order hyperbolic operator-differential equations with variable domains

We develop a modification of the energy inequality method and use it to prove the well-posedness of the Goursat problem for linear second-order hyperbolic differential equations with operator coefficients whose domains depend on the two-dimensional time. An energy inequality for strong solutions of this Goursat problem is derived with the help of abstract smoothing operators, and we prove that the range of the problem is dense by using properties of a regularizing Cauchy problem whose inverse operator is a family of smoothing operators of a new type. We give an example of a well-posed boundary value problem for a two-dimensional complete second-order hyperbolic partial differential equation with Goursat conditions and with a boundary condition depending on the two-dimensional time.     

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( …     

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.     

A spline collocation method for parabolic pseudodifferential equations

The purpose of this paper is to examine a boundary element collocation method for some parabolic pseudodifferential equations. The basic model problem for our investigation is the two-dimensional heat conduction problem with vanishing initial condition and a given Neumann or Dirichlet type boundary condition. Certain choices of the representation formula for the heat potential yield boundary integral equations of the first kind, namely the single layer and the hypersingular heat operator equations. Both of these operators, in particular, are covered by the class of parabolic pseudodifferential operators under consideration. Moreover, the spatial domain is allowed to have a general smooth boundary curve. As trial functions the tensor products of the smoothest spline functions of odd degree (space) and continuous piecewise linear splines (time) are used. Stability and convergence of the method is proved in some appropriate anisotropic Sobolev spaces.     

一个抛物型方程侧边值问题的正则逼近解在一类Sobolev空间中的最优误差界

逆热传导问题(IHCP)是严重不适定问题,即问题的解(如果存在)不连续依赖于数据．但目前关于逆热传导问题的已有结果主要是针对标准逆热传导问题．文中给出了出现在实际问题中的一个抛物型方程侧边值问题,即一个含有对流项的非标准型逆热传导问题的正则逼近解一类Sobolev空间中的最优误差界．     

Improved bounds for some nonstandard problems in generalized heat conduction

We consider the Maxwell-Cattaneo system of equations for generalized heat conduction where the temperature and heat flux satisfy a nonstandard auxiliary condition which prescribes a combination of their values initially and at a later time. We obtain L₂ bounds for the temperature and heat flux by means of Lagrange identities. These bounds extend the range of validity for the parameter in the nonstandard condition under a constraint on the coefficients in the differential equations.     

BOUNDARY ELEMENT ANALYSIS FOR A CLASS OF MULTI-DIMENSIONAL LINEAR PARABOLIC EQUATIONS

In this paper,by me as of beundary element method,we try to deal with the initial -boundary value problem for a class of linear parunolic equations,which is a linear heat conduction equation. We tresent a boundary integral equation for the solution to the problem and its variational formalation The well-posedness of the variational formulation is proved. And the error estimates for the approsutate solutions are provided. The results of this paper are more general than those of[1]     

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.     

Some sufficient conditions for the non-negativity preservation property in the discrete heat conduction model

In the course of the numerical approximation of mathematical models there is often a need to solve a system of linear equations with a tridiagonal or a block-tridiagonal matrices. Usually it is efficient to solve these systems using a special algorithm (tridiagonal matrix algorithm or TDMA) which takes advantage of the structure. The main result of this work is to formulate a sufficient condition for the numerical method to preserve the non-negativity for the special algorithm for structured meshes. We show that a different condition can be obtained for such cases where there is no way to fulfill this condition. Moreover, as an example, the numerical solution of the two-dimensional heat conduction equation on a rectangular domain is investigated by applying Dirichlet boundary condition and Neumann boundary condition on different parts of the boundary of the domain. For space discretization, we apply the linear finite element method, and for time discretization, the well-known Θ-method. The theoretical results of the paper are verified by several numerical experiments.