Determination of the time-dependent reaction coefficient and the heat flux in a nonlinear inverse heat conduction problem

ABSTRACT

Diffusion processes with reaction generated by a nonlinear source are commonly encountered in practical applications related to ignition, pyrolysis and polymerization. In such processes, determining the intensity of reaction in time is of crucial importance for control and monitoring purposes. Therefore, this paper is devoted to such an identification problem of determining the time-dependent coefficient of a nonlinear heat source together with the unknown heat flux at an inaccessible boundary of a one-dimensional slab from temperature measurements at two sensor locations in the context of nonlinear transient heat conduction. Local existence and uniqueness results for the inverse coefficient problem are proved when the first three derivatives of the nonlinear source term are Lipschitz continuous functions. Furthermore, the conjugate gradient method (CGM) for separately reconstructing the reaction coefficient and the heat flux is developed. The ill-posedness is overcome by using the discrepancy principle to stop the iteration procedure of CGM when the input data is contaminated with noise. Numerical results show that the inverse solutions are accurate and stable.     

Numerical approximation of solution of nonhomogeneous backward heat conduction problem in bounded region

In this paper we consider a numerical approximation of solution of nonhomogeneous backward heat conduction problem (BHCP) in bounded region based on Tikhonov regularization method. Error estimate at t=0$t=0$ for this method is provided. According to the error estimate, a selection of regularization parameter is given. Meanwhile, a numerical implementation is described and the numerical results show that our algorithm is effective.     

应用QPSO算法求解二维热传导反问题

热传导反问题在国内研究起步较晚,研究方法有很多,但通常方法很难较好地接近全局最优。在经典的微粒群优化算法(PSO)的基础上,通过研究基于量子行为的微粒群优化算法(QPSO)提出了应用基于量子行为的微粒群优化算法进行二维热传导参数优化,具体介绍依据目标函数如何利用上述的算法去寻找最优参数组合。在具体应用中为了提高算法的收敛性和稳定性对算法进行了改进,并进行了大量实验,结果显示在解决热传导反问题优化问题中,基于QPSO算法的性能优越,证明QPSO在热传导领域具有很大的实际应用价值。     

QPSO算法与PSO算法求解二维热传导反问题

热传导反问题在国内研究起步较晚,研究方法有很多,但通常方法很难较好地接近全局最优.在介绍经典的微粒群优化算法(PSO)的基础上,研究基于量子行为的微粒群优化算法(QPSO)的二维热传导参数优化方法,具体介绍依据目标函数如何利用上述的算法去寻找最优参数组合.为了提高算法的收敛性和稳定性,在具体应用中对算法进行了改进,并进行了大量实验,结果显示在解决热传导反问题优化问题中,基于QPSO算法的性能比经典PSO算法更加优越,证明QPSO在热传导领域具有很大的实际应用价值.     

全变差正则化在抛物型方程初始条件反问题的应用

当反问题反演的函数不连续时,一般的正则化算法反演效果不令人满意,用全变差正则化方法对抛物型方程初始条件反问题进行求解,并进行了数值分析和数值模拟,结果显示数值解与真解吻合较好,表明该方法对于不连续函数求解具有高效、稳定等优点.     

Reconstruction of heat transfer coefficients using the boundary element method

This paper describes the reconstruction of the heat transfer coefficient (space, Problem I, or time dependent, Problem II) in one-dimensional transient inverse heat conduction problems from surface temperature or average temperature measurements. Since the inverse problem posed does not involve internal temperature measurements, this means that non-destructive testing of materials can be performed. In the formulation, convective boundary conditions relate the boundary temperature to the heat flux. Numerical results obtained using the boundary element method are presented and discussed.     

建筑室内边界对流换热量的反向计算模型

针对目前反向计算模型还无法实现对建筑室内边界对流换热量进行反向计算这一制约性差距,采用温度贡献率方法,将边界对流换热量与室内测点温度之间表示成因果关系的温度贡献因子矩阵,基于计算流体力学,将最小二乘与Tikhonov正则化方法相结合,建立依据室内数个测点的离散温度求解边界对流换热量的反问题数学模型。应用三维通风空腔和某建筑内一间办公室进行实验验证,模型求解值与实测值的均方根差均小于80%,结果表明反向计算模型可以准确对室内边界对流换热量进行求解。     

A method of fundamental solutions for radially symmetric and axisymmetric backward heat conduction problems

We propose and investigate an application of the method of fundamental solutions (MFS) to the radially symmetric and axisymmetric backward heat conduction problem (BHCP) in a solid or hollow cylinder. In the BHCP, the initial temperature is to be determined from the temperature measurements at a later time. This is an inverse and ill-posed problem, and we employ and generalize the MFS regularization approach [B.T. Johansson and D. Lesnic, A method of fundamental solutions for transient heat conduction, Eng. Anal. Boundary Elements 32 (2008), pp. 697–703] for the time-dependent heat equation to obtain a stable and accurate numerical approximation with small computational cost.     

A method of fundamental solutions for two-dimensional heat conduction

We investigate an application of the method of fundamental solutions (MFS) to heat conduction in two-dimensional bodies, where the thermal diffusivity is piecewise constant. We extend the MFS proposed in Johansson and Lesnic [A method of fundamental solutions for transient heat conduction, Eng. Anal. Bound. Elem. 32 (2008), pp. 697–703] for one-dimensional heat conduction with the sources placed outside the space domain of interest, to the two-dimensional setting. Theoretical properties of the method, as well as numerical investigations, are included, showing that accurate results can be obtained efficiently with small computational cost.     

Numerical solution of a one-dimensional inverse problem by the discontinuous Galerkin method

In this paper, we combine a Tikhonov regularization with a discontinuous Galerkin method to solve an inverse problem in one-dimension. We show that the regularization is simpler than in the case of the inversion using continuous finite elements. We numerically demonstrate that there exist optimal step sizes and polynomial degrees for inversion using the DG method. Numerical results are compared with those obtained by applying the standard finite element method with B-splines as a basis.     

基于自然元法和精细积分法的功能梯度材料瞬态热传导问题求解

为促进无网格法分析技术在热传导分析中的应用,提出空间离散采用自然单元法、时间离散采用精细积分法求解功能梯度材料瞬态热传导问题的数值计算方法.在计算过程中,取高斯点的材料参数模拟功能梯度材料特性的变化.温度场采用自然邻接点插值形函数进行离散插值.数值算例验证该数值算法的正确性和有效性.     

A numerical study of variable coefficient elliptic Cauchy problem via projection method

In this paper, we investigate a numerical method for the solution of an inverse problem of recovering lacking data on some part of the boundary of a domain from the Cauchy data on other part for a variable coefficient elliptic Cauchy problem. In the process, the Cauchy problem is transformed into the problem of solving a compact linear operator equation. As a remedy to the ill-posedness of the problem, we use a projection method which allows regularization solely by discretization. The discretization level plays the role of regularization parameter in the case of projection method. The balancing principle is used for the choice of an appropriate discretization level. Several numerical examples show that the method produces a stable good approximate solution.     

A quasi-reversibility regularization method for the Cauchy problem of the Helmholtz equation

In this paper, a Cauchy problem for the Helmholtz equation is considered. It is known that such a problem is severely ill-posed, i.e. the solution does not depend continuously on the given Cauchy data. We propose a quasi-reversibility regularization method to solve it. Convergence estimates are established under two different a priori assumptions for an exact solution. Numerical results obtained by two different schemes show that our proposed methods work well.     

A mollification method for a Cauchy problem for the Helmholtz equation

The Cauchy problem for the Helmholtz equation is considered. This problem is severely ill-posed, that is, the solution does not depend continuously on the data. To solve the problem numerically a mollification method is proposed. Convergence on error estimates between the exact solution and its approximation are obtained. Some numerical examples are given to show that the method works effectively.     

A spacetime discontinuous Galerkin method for hyperbolic heat conduction

Non-Fourier conduction models remedy the paradox of infinite signal speed in the traditional parabolic heat equation. For applications involving very short length or time scales, hyperbolic conduction models better represent the physical thermal transport processes. This paper reviews the Maxwell-Cattaneo-Vernotte modification of the Fourier conduction law and describes its implementation within a spacetime discontinuous Galerkin (SDG) finite element method that admits jumps in the primary variables across element boundaries with arbitrary orientation in space and time. A causal, advancing-front meshing procedure enables a patch-wise solution procedure with linear complexity in the number of spacetime elements. An h-adaptive scheme and a special SDG shock-capturing operator accurately resolve sharp solution features in both space and time. Numerical results for one spatial dimension demonstrate the convergence properties of the SDG method as well as the effectiveness of the shock-capturing method. Simulations in two spatial dimensions demonstrate the proposed method’s ability to accurately resolve continuous and discontinuous thermal waves in problems where rapid and localized heating of the conducting medium takes place.     

求解热传导系数反问题的量子行为粒子群算法

量子行为粒子群优化算法（QPSO）和Tikhonov正则化方法用来求解热传导反问题,近似估计平板随时间变化的热传导系数。由于热传导系数的函数形式是未知的,所以问题可以归结为函数估计问题。求解过程是基于最小二乘模型的,采用的是嵌在平板中的传感器所测量得到的温度,优化过程由QPSO算法来求解。给出了由L曲线方法选择正则参数的详细过程。提出算法的有效性经过了数值实验的验证。传感器的位置和数量对结果的影响也做了研究。给出了与共轭梯度法的比较。     

A meshless average source boundary node method for steady-state heat conduction in general anisotropic media

The average source boundary node method (ASBNM) is a recent boundary-type meshless method, which uses only the boundary nodes in the solution procedure without involving any element or integration notion, that is truly meshless and easy to implement. This paper documents the first attempt to extend the ASBNM for solving the steady-state heat conduction problems in general anisotropic media. Noteworthily, for boundary-type meshless/meshfree methods which depend on the boundary integral equations, whatever their forms are, a key but difficult issue is to accurately and efficiently determine the diagonal coefficients of influence matrices. In this study, we develop a new scheme to evaluate the diagonal coefficients via the pure boundary node implementation based on coupling a new regularized boundary integral equation with direct unknowns of considered problems and the average source technique (AST). Seven two- and three-dimensional benchmark examples are tested in comparison with some existing methods. Numerical results demonstrate that the present ASBNM is superior in the light of overall accuracy, efficiency, stability and convergence rates, especially for the solution of the boundary quantities.     

Richardson extrapolation applied to boundary element method results in a Dirichlet problem for the Laplace equation

Richardson extrapolation is used to improve the accuracy of the numerical solutions for the normal boundary flux and for the interior potential resulting from the boundary element method. The boundary integral equations arise from a direct boundary integral formulation for solving a Dirichlet problem for the Laplace equation. The Richardson extrapolation is used in two different applications: (i) to improve the accuracy of the collocation solution for the normal boundary flux and, separately, (ii) to improve the solution for the potential in the domain interior. The main innovative aspects of this work are that the orders of dominant error terms are estimated numerically, and that these estimates are then used to develop an a posteriori technique that predicts if the Richardson extrapolation results for applications (i) and (ii) are reliable. Numerical results from test problems are presented to demonstrate the technique.