非线性热传导反问题参数辨识

基于一种时域正演精细算法,引入Bregman距离加权函数作为正则项,应用Tikhonov正则化方法,对非线性热传导反问题进行求解。所建正/反演数值模型在便于敏度分析的同时,能够对非线性内热源强度、导温系数和边界条件等多个热学参数进行有效组合识别。该文给出了相关的数值算例,并对信息误差以及不同正则项的计算效率作了探讨,得到满意的计算结果。数值结果表明所提的求解策略在求解非线性热传导反问题时,不仅能够对相关的热学参数进行有效的组合识别,而且具有较高的计算精度、较好的稳定性和一定的抗噪性,采用加权的Bregman距离函数作正则项可以提高计算效率。     

各向异性材料平面问题基本解析解的特征方程解法

该文提出了一种利用特征方程解法构造基本解析解的新方法,并将其应用到各向异性材料平面问题,成功构造了完备且独立的系列基本解析解.构造各向异性材料平面问题控制微分方程的算子矩阵,通过其行列式计算可得到问题特征通解所需满足的特征方程,将所求得特征通解代入到微分方程算子矩阵所对应的伴随矩阵,可推导得出各向异性材料平面问题的基本解析解.根据基本解析解独立性的论证,可得到系列独立且完备各向异性材料平面问题基本解析解.利用特征方程解法求解基本解析解思路简单、并且容易找到独立且完备的解析解,其结果可以成为相关数值计算方法的基础.     

动态载荷时域识别的联合去噪修正和正则化预优迭代方法

系统响应可表示为单位脉冲响应函数与激励载荷的卷积,将其离散化一组线性方程组,则载荷识别问题即转化为求解线性方程组的反问题。针对响应中带有噪音时载荷识别的困难,提出了联合奇异熵去噪修正和正则化预优的共轭梯度迭代识别方法。一方面对含噪信号进行基于奇异熵的去噪处理,提高反问题求解中输入数据的精度。另一方面利用正则化方法对共轭梯度迭代算法进行预优,改善反问题的非适定性。由于从输入的响应数据去噪和正则化算法两方面同时改善动态载荷识别反问题的求解,因此可以有效地抑制噪声,提高识别精度。通过数值算例分析,表明在不同的噪声水平干扰下,其识别精度均优于常规的正则化方法,能够实现有效稳定地识别动态载荷。最后通过实验研究进一步验证了该方法的正确性和有效性。     

基于l1正则化无迹卡尔曼滤波的结构损伤方法

采用传统卡尔曼滤波类算法对结构进行损伤识别时,损伤识别反问题的不适定性使得识别结果易受噪声干扰,甚至算法不收敛。为此,该文提出了一种结合l1范数正则化的无迹卡尔曼滤波损伤识别算法。根据结构出现局部损伤时其损伤参数分布具有稀疏性的特点,通过伪测量方法,将l1范数正则化引入到无迹卡尔曼滤波框架中,在改善反问题求解不适定性的同时,能有效地提高结构局部损伤识别能力。梁、桁架结构的数值分析与实验研究表明,该文方法可以对损伤的位置与程度进行准确识别,且具有良好的鲁棒性。     

基于有限元-正则化法进行结构弹性模量识别

在对工程结构进行各项受力分析时,准确输入弹性模量是确保分析结果可靠的前提。该文利用位移相对弹性模量容易观测的特点,首先定义了通过结构若干点实测位移识别弹模的数学模型,然后基于工程反问题求解方法梯度正则化法进行求解。并通过线性变换使求解过程中的Jacobi矩阵对角线元素归一,从而提高了求解速度和精度。编制了通用有限元计算程序,通过数值模拟算例验证了该方法的可行性,讨论了方法应用的初值选择、模型误差、附加位移选择等问题。     

求解病态问题的一种新的正则化子与正则化算法

根据紧算子的奇异系统理论,提出了一种新的正则化子,进而建立了一类新的求解病态问题的正则化方法。证明了正则解的收敛性并得到了其最优的渐近收敛阶,数值算例说明文中建立的正则化算法是可行而有效的。     

一类积分方程的数值解法

研究了形如∫^t 0H(T,τ)f(τ)dτ=g(t)一类积分方程的数值求解，从讨论病态性质入手，基于吉洪诺夫的正则化思想，构造了正则化算了,而给出了求解这类积分方程稳定的数值方法，并给出了一些数值算例。     

Tikhonov正则化在Zernike多项式拟合中的应用

Zernike多项式系数的求解问题是一个典型的离散不适定问题,最小二乘法,格拉姆-斯密特正交化法和Householder变换法均无法求得稳定的数值解.本文对导致该问题解的不稳定性的原因进行了分析,并采用Tikhonov正则化法对Zernike多项式系数进行求解,利用L曲线准则确定了正则参数.数值仿真结果表明,Tikhonov正则化法有效的保证了解的稳定性,利用该方法得到的拟合面形很好的反映了面形的真实情况.     

实验观测数据的最优正则平滑方法

为了滤除测量噪声 ,提出了一种对实验观测数据进行最优化正则平滑的数据处理方法 .文中阐述了方法的基本原理 ,并就稳定泛函和正则参数的选择等关键问题作了分析和论述 .通过一个数学模拟实例对正则化平滑方法的效果进行了验证 ,这种正则化平滑方法在数学物理反问题求解等领域具有独特的优点     

基于Tikhonov正则化迭代求解的结构损伤识别方法

求解以结构物理与模态信息所构成的线性方程组,而获得结构的损伤位置和损伤程度,是进行结构损伤检测的一种常用做法。然而,在噪声影响下,其求解往往会出现振荡发散的情况,导致损伤检测结果不准确。Tikhonov正则化方法广泛应用于噪声条件下的线性系统求解,该方法执行的关键是选择合理的正则化矩阵及正则化参数。提出了一种迭代化的Tikhonov正则化方法,通过迭代的方式重构正则化矩阵,在充分抑制噪声的同时,保留了真实的损伤信息。同时,提出了奇异值二分法,自适应地调整正则化参数,避免了传统"L-曲线"方法选取正则化参数时需要进行大量试算等诸多问题。选取一海洋平台结构对提出方法的有效性进行验证,并与传统Tikhonov正则化方法进行对比,结果表明:提出的迭代型Tikhonov正则化方法具有更好的损伤识别结果。     

Desingularized meshless method for solving Laplace equation with over-specified boundary conditions using regularization techniques

The desingularized meshless method (DMM) has been successfully used to solve boundary-value problems with specified boundary conditions (a direct problem) numerically. In this paper, the DMM is applied to deal with the problems with over-specified boundary conditions. The accompanied ill-posed problem in the inverse problem is remedied by using the Tikhonov regularization method and the truncated singular value decomposition method. The numerical evidences are given to verify the accuracy of the solutions after comparing with the results of analytical solutions through several numerical examples. The comparisons of results using Tikhonov method and truncated singular value decomposition method are also discussed in the examples.     

A method of fundamental solutions for inverse heat conduction problems in an anisotropic medium

Recently, Hon and Wei proposed a method of fundamental solutions for solving isotropic inverse heat conduction problems (IHCP). It provides an efficient global approximation scheme in both spatial and time domains. In this paper, we try to extend the inherently meshless and integration-free method to solve 2D IHCP in an anisotropic medium. First, we acquire the fundamental solution of the governing equation through variables transformation. Then the truncated singular value decomposition and the L-curve criterion are applied to solve the resulting matrix equation which is highly ill-conditioned. Results for several numerical examples are presented to demonstrate the efficiency of the method proposed. The relationship between the accuracy of the numerical solutions and the value of the parameter T is also investigated.     

The method of fundamental solutions for inverse boundary value problems associated with the steady‐state heat conduction in anisotropic media

In this paper, the method of fundamental solutions is applied to solve some inverse boundary value problems associated with the steady‐state heat conduction in an anisotropic medium. Since the resulting matrix equation is severely ill‐conditioned, a regularized solution is obtained by employing the truncated singular value decomposition, while the optimal regularization parameter is chosen according to the L‐curve criterion. Numerical results are presented for both two‐ and three‐dimensional problems, as well as exact and noisy data. The convergence and stability of the proposed numerical scheme with respect to increasing the number of source points and the distance between the fictitious and physical boundaries, and decreasing the amount of noise added into the input data, respectively, are analysed. A sensitivity analysis with respect to the measure of the accessible part of the boundary and the distance between the internal measurement points and the boundary is also performed. The numerical results obtained show that the proposed numerical method is accurate, convergent, stable and computationally efficient, and hence it could be considered as a competitive alternative to existing methods for solving inverse problems in anisotropic steady‐state heat conduction. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical solutions of singular integral equations having Cauchy-type singular kernel by means of expansion method

This paper is concerned with numerical solutions of singular integral equations with Cauchy-type singular kernel. It is well-known that this type of singular integral equations appears in the analysis of crack problems using the continuously distributed dislocation method. In addition, it also appears in the analysis of notch problems using the body force method. In the present analysis, the unknown function of densities of dislocations and body forces are approximated by the product of the fundamental density functions and polynomials. The accuracy of stress intensity factors and stress concentration factors obtained by the present method is verified through the comparison with the exact solution and the reliable numerical solution obtained by other researchers. The present method is found to give good convergency of the numerical results for notch problem as well as internal and edge crack problems.     

Solving potential problems by a boundary-type meshless method—the boundary point method based on BIE

In this paper, a novel boundary-type meshless method, the boundary point method (BPM), is developed via an approximation procedure based on the idea of Young et al. [Novel meshless method for solving the potential problems with arbitrary domain. J Comput Phys 2005;209:290–321] and the boundary integral equations (BIE) for solving two- and three-dimensional potential problems. In the BPM, the boundary of the solution domain is discretized by unequally spaced boundary nodes, with each node having a territory (the point is usually located at the centre of the territory) where the field variables are defined. The BPM has both the merits of the boundary element method (BEM) and the method of fundamental solution (MFS), both of these methods use fundamental solutions which are the two-point functions determined by the source and the observation points only. In addition to the singular properties, the fundamental solutions have the feature that the greater the distance between the two points, the smaller the values of the fundamental solutions will be. In particular, the greater the distances, the smaller the variations of the fundamental solutions. By making use of this feature, most of the off-diagonal coefficients of the system matrix will be computed by one-point scheme in the BPM, which is similar to the one in the MFS. In the BPM, the ‘moving elements’ are introduced by organizing the relevant adjacent nodes tentatively, so that the source points are placed on the real boundary of the solution domain where the resulting weak singular, singular and hypersingular kernel functions of the diagonal coefficients of the system matrix can be evaluated readily by well-developed techniques that are available in the BEM. Thus difficulties encountered in the MFS are removed because of the coincidence of the two points. When the observation point is close to the source point, the integrals of kernel functions can be evaluated by Gauss quadrature over territories.In this paper, the singular and hypersingular equations in the indirect and direct formulations of the BPM are presented corresponding to the relevant BIE for potential problems, where the indirect formulations can be considered as a special form of the MFS. Numerical examples demonstrate the accuracy of solutions of the proposed BPM for potential problems with mixed boundary conditions where good agreements with exact solutions are observed.     

Analytic formulations for calculating nearly singular integrals in two-dimensional BEM

There exist the nearly singular integrals in the boundary integral equations when a source point is close to an integration element but not on the element, such as the field problems with thin domains. In this paper, the analytic formulations are achieved to calculate the nearly weakly singular, strongly singular and hyper-singular integrals on the straight elements for the two-dimensional (2D) boundary element methods (BEM). The algorithm is performed after the BIE are discretized by a set of boundary elements. The singular factor, which is expressed by the minimum relative distance from the source point to the closer element, is separated from the nearly singular integrands by the use of integration by parts. Thus, it results in exact integrations of the nearly singular integrals for the straight elements, instead of the numerical integration. The analytic algorithm is also used to calculate nearly singular integrals on the curved element by subdividing it into several linear or sub-parametric elements only when the nearly singular integrals need to be determined. The approach can achieve high accuracy in cases of the curved elements without increasing other computational efforts. As an application, the technique is employed to analyze the 2D elasticity problems, including the thin-walled structures. Some numerical results demonstrate the accuracy and effectiveness of the algorithm.     

Conforming boundary elements in plane elasticity for shape design sensitivity

The structural design sensitivity analysis of a two-dimensional continuum using conforming (continuous) boundary elements is investigated. Implicit differentiation of the discretized boundary integral equations is performed to obtain design sensitivities in an efficient manner by avoiding the factorization of the perturbed matrices. A singular formulation of the boundary element method is used. Implicit differentiation of the boundary integral equations produces terms that contain derivatives of the fundamental solutions employed in the analysis. The behaviour of the singularity of these derivatives of the boundary element kernel functions with respect to the design variables is investigated. A rigid body motion technique is presented to obtain the singular terms in the resulting sensitivity matrices, thus avoiding the problems associated with their numerical integration. A formulation for obtaining the design sensitivities of the continua under body forces of the gravitational and centrifugal types is also presented. The design sensitivity results are seen to be of the same order of accuracy as the boundary element analysis results. Numerical data comparing the performance of conforming and non-conforming formulations in the calculation of design sensitivities are also presented. The accuracy of the present results is demonstrated through comparisons with existing analytical results.     

Dynamic analysis of 3-D structures by a transformed boundary element method

In this work, an advanced implementation of the direct boundary element method applicable to periodic (steadystate) and transient dynamic problems involving three-dimensional structures of arbitrary shape and connectivity is presented. Interior, exterior and halfspace type of problems can all be solved by the present method. The discussion first focuses on the formulation of the method, followed by material pertaining to the fundamental singular solutions and to the isoparametric boundary elements used for discretizing the surface of the problem. Subsequently, numerical integration techniques and the solution algorithm are introduced. This methodology has been incorporated in a versatile, general purpose computer program. Finally, the stability and high accuracy of this dynamic analysis technique are established through comparisons with available analytical and numerical results.     

A meshless singular boundary method for three‐dimensional elasticity problems

This study documents the first attempt to extend the singular boundary method, a novel meshless boundary collocation method, for the solution of 3D elasticity problems. The singular boundary method involves a coupling between the regularized BEM and the method of fundamental solutions. The main idea here is to fully inherit the dimensionality and stability advantages of the former and the meshless and integration‐free attributes of the later. This makes it particularly attractive for problems in complex geometries and three dimensions. Four benchmark 3D problems in linear elasticity are well studied to demonstrate the feasibility and accuracy of the proposed method. The advantages, disadvantages, and potential applications of the proposed method, as compared with the FEM, BEM, and method of fundamental solutions, are also examined and discussed. Copyright © 2015 John Wiley & Sons, Ltd.     

Regularized solutions with a singular point for the inverse biharmonic boundary value problem by the method of fundamental solutions

Two numerical methods for the Cauchy problem of the biharmonic equation are proposed. The solution of the problem does not continuously depend on given Cauchy data since the problem is ill-posed. A small noise contained in the Cauchy data sensitively affects on the accuracy of the solution. Our problem is directly discretized by the method of fundamental solutions (MFS) to derive an ill-conditioned matrix equation. As another method, our problem is decomposed into two Cauchy problems of the Laplace and the Poisson equations, which are discretized by the MFS and the method of particular solutions (MPS), respectively. The Tikhonov regularization and the truncated singular value decomposition are applied to the matrix equation to stabilize a numerical solution of the problem for the given Cauchy data with high noises. The L-curve and the generalized cross-validation determine a suitable regularization parameter for obtaining an accurate solution. Based on numerical experiments, it is concluded that the numerical method proposed in this paper is effective for the problem that has an irregular domain and the Cauchy data with high noises. Furthermore, our latter method can successfully solve the problem whose solution has a singular point outside the computational domain.