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。     

基于贝叶斯正则化BP神经网络的铝平板超高速撞击损伤模式识别

损伤模式识别是空间碎片撞击航天器在轨感知技术中的一个重要功能模块,是目前研究的重点和难点。采用超高速撞击声发射技术,以铝合金平板为研究对象,通过大量超高速撞击实验获取实验信号,结合虚拟波阵面的精确源定位技术、时频分析技术及小波分解技术。从超高速撞击声发射信号中提取并优选与损伤模式直接相关的时频参数,建立了基于贝叶斯正则化BP神经网络的损伤模式识别方法,识别了铝合金板面受撞击形成的成坑/穿孔两种主要损伤模式。     

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。     

脉冲源反演介质参数的近似波前法

在时域内对弹性波动方程退化的非均匀介质声波方程，引入背景场参数与扰动参数，并化为积分方程形式；针对脉冲源情况，根据射线理论中的传递方程和程函方程，对非均匀介质中的波场形式引入一种波前近似形式，得到波散射点满足散射关系曲线及散射波幅值与介质参数扰动比的代数关系方程式；为求解非均匀介质中散射波场及反演介质参数提供了一种方法，通过对一个完整算例全部过程的模拟，验证了此方法的正确性。     

考虑剪切非线性影响的复合材料连续损伤模型及损伤参数识别

在复合材料正交各向异性连续损伤力学的基础上,引入由应变描述的连续综合变量,损伤演化方案采用综合变量的指数函数。考虑复合材料剪切非线性,通过非线性损伤因子在全局损伤出现之前将其引入本构方程。编写了VUMAT实施损伤模型,介绍连续损伤模型中材料参数的取值方法。对无法通过试验获得的损伤参数,采用遗传算法原理,定义个体适应度为试验及模拟载荷-位移曲线误差,通过编写MATLAB程序调用ABAQUS有限元模型实施参数反演方法。选择了常用铺层结构的复合材料带孔层合板进行仿真参数识别分析,验证了基于遗传算法参数识别方法的可行性。     

基于参数识别的结构损伤概率诊断方法

为解决观测噪声等不确定性因素影响下确定性损伤识别方法的损伤误判问题,提出了结构损伤识别的概率可靠度法。通过正则化修正改善参数识别问题的不适定性,利用数据摄动和蒙特卡罗法研究观测噪声对参数识别结果的影响并获得各待识别参数的统计特征,在此基础上基于概率统计理论推导了损伤概率的计算公式。根据损伤识别试验验证了方法的有效性和准确性,分析并比较了概率可靠度法和确定性方法的损伤识别结果。试验研究表明,损伤识别的概率可靠度法能够显著改善传统确定性损伤识别方法的损伤误判问题。     

基于尾波干涉法反演结构微小损伤

飞机在飞行中普遍存在振动及噪声,会对飞机的结构产生很大的损伤,如果不及时发现将会对飞行安全造成很大的威胁.尾波干涉法是近几年提出的一种结构损伤探测的新方法,相比于直达波,尾波在介质中来回传播,对变化区域进行多次采样,可以把介质的微小变化进行放大.基于尾波干涉和敏感核理论建立仿真平台,对结构的微小变化进行定位成像仿真,通过反演计算成功地对结构的损伤程度及位置进行成像,验证了算法及程序的有效性,为尾波干涉法在工程实践中的推广及应用奠定了基础.     

具有非均匀损伤带状区域中波的传播

对弹性波在二次曲线变化的非均匀损伤介质中的传播理论进行了研究。通过非均匀损伤区两端界面处力的平衡条件和位移连续性条件对方程式进行求解，利用勒让德多项式，求出了二次曲线变化的非均匀损伤介质中波动方程的解析形式解。对带状渐增和渐减的非均匀损伤介质中波的传播进行了实例计算分析，讨论了弹性波在二次曲线变化的非均匀损伤介质中传播的一般性质。     

传递矩阵方法反演介质参数

描述了一种居层状介质中，从一束源激发产生发射记录上获取速度，密度的反学。通过把测点的所有波分解为两类，一类是第ｎ层反射所得的波，另一类是前ｎ－１层反射所得的波。在假设已知前ｎ层介质的情况下，把反演第ｎ＋１层的问题公为一个线性回归。一些数值例子表明该反演方法较好地恢复了产生标准资料的那些参数。     

结构损伤识别中的若干正则化问题研究

针对结构损伤识别计算中由于矩阵奇异带来的识别结果不稳定问题,利用列主元QR分解、截断奇异值分解这两种正则化技术研究识别计算中的迭代方法,给出了这两种技术改善识别结果稳定性的证明;同时研究了对得到合理识别结果有较大影响的两个计算参数:实施正则化技术的阀值以及限制迭代增量的步长限值。该文以IASC-ASCE的BENCHMARK结构为算例,分别采用列主元QR分解、截断奇异值分解,识别出BENCHMARK结构第I阶段损伤问题第一种工况下第二种损伤模式的位置,识别的损伤程度最大误差分别为7.33%、6.36%,验证了研究的正则化技术的有效性。通过算例对阀值与步长限值进行研究和讨论,发现列主元QR分解宜应用在步长限值α较大(α≥0.5)的情况下,截断奇异值分解则既可应用在步长限值α较大、也可应用在α较小(α≥0.25)的情况下。     

Adaptive error estimation technique of the Trefftz method for solving the over-specified boundary value problem

In this paper, numerical solutions are investigated based on the Trefftz method for an over-specified boundary value problem contaminated with artificial noise. The main difficulty of the inverse problem is that divergent results occur when the boundary condition on over-specified boundary is contaminated by artificial random errors. The mechanism of the unreasonable result stems from its ill-posed influence matrix. The accompanied ill-posed problem is remedied by using the Tikhonov regularization technique and the linear regularization method, respectively. This remedy will regularize the influence matrix. The optimal parameter λ of the Tikhonov technique and the linear regularization method can be determined by adopting the adaptive error estimation technique. From this study, convergent numerical solutions of the Trefftz method adopting the optimal parameter can be obtained. To show the accuracy of the numerical solutions, we take the examples as numerical examination. The numerical examination verifies the validity of the adaptive error estimation technique. The comparison of the Tikhonov regularization technique and the linear regularization method was also discussed in the examples.     

Numerical solution of two-dimensional inverse force function in the wave equation with nonlocal boundary conditions

In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the inverse time-dependent force function in the wave equation on regular and irregular domains. The SMRPI is developed for identifying the force function which satisfies in the wave equation subject to the integral overspecification over a portion of the spatial domain or to the overspecification at a point in the spatial domain. This method is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. Since the problem is known to be ill-posed, Thikhonov regularization strategy is employed to solve effectively the discrete ill-posed resultant linear system. Three numerical examples are tested to show that numerical results are accurate for exact data and stable with noisy data.     

Optimally Generalized Regularization Methods for Solving Linear Inverse Problems

In order to solve ill-posed linear inverse problems, we modify the Tikhonov regularization method by proposing three different preconditioners, such that the resultant linear systems are equivalent to the original one, without dropping out the regularized term on the right-hand side. As a consequence, the new regularization methods can retain both the regularization effect and the accuracy of solution. The preconditioned coefficient matrix is arranged to be equilibrated or diagonally dominated to derive the optimal scales in the introduced preconditioning matrix. Then we apply the iterative scheme to find the solution of ill-posed linear inverse problem. Two theorems are proved that the iterative sequences are monotonically convergent to the true solution. The presently proposed optimally generalized regularization methods are able to overcome the ill-posedness of linear inverse problems, and provide rather accurate numerical solution.     

超声逆散射成像问题中正则化参数研究

研究了二维理想情况下,基于精确场描述的超声逆散射成像问题,先用矩量法将波动方程化为离散形式,分别用BI和DBI算法进行迭代重建。影响整个算法的一个关键因素是散射场方程的正则化求解,具有明显的不适定性。文章基于L曲线法,提出以解的范数和残差变化量的加权形式作为确定正则化参数的依据,在迭代过程根据问题不适定性程度,自适应地调整搜索范围。仿真结果表明,该算法可快速地找到最优正则化参数。     

Determination of the unknown source term in a space-fractional diffusion equation

This paper studies an inverse problem of determining the unknown source term in a space-fractional diffusion equation. Three types of spectral regularization method are proposed to deal with the ill-posed problem and the corresponding error estimates are obtained with an a priori strategy to find the regularization parameter. We verify the efficiency of the proposed numerical method with some numerical experiments.     

Wave-front recovery from two orthogonal sheared interferograms

We present a new technique for using the information of two orthogonal lateral-shear interferograms to estimate an aspheric wave front. The wave-front estimation from sheared inteferometric data may be considered an ill-posed problem in the sense of Hadamard. We apply Thikonov regularization theory to estimate the wave front that has produced the lateral sheared interferograms as the minimizer of a positive definite-quadratic cost functional. The introduction of the regularization term permits one to find a well-defined and stable solution to the inverse shearing problem over the wave-front aperture as well as to reduce wave-front noise as desired.     

Landweber iteration regularization method for identifying unknown source on a columnar symmetric domain

In this paper, an inverse source problem for the Helium Production–Diffusion Equation on a columnar symmetric domain is investigated. Based on an a priori assumption, the optimal error bound analysis and a conditional stability result are given. This problem is ill-posed and Landweber iteration regularization method is used to deal with this problem. Convergence estimates are presented under the priori and the posteriori regularization choice rules. For the a priori and the a posteriori regularization parameters choice rules, the convergence error estimates are all order optimal. Numerical examples are given to show that the regularization method is effective and stable for dealing with this ill-posed problem.     

数值求解Symm积分方程的正则化MINRES方法

Symm积分方程在位势理论中具有重要应用,它是Hadamard意义下的不适定问题。离散该方程将产生对称线性不适定系统。基于GCV准则,并应用截断奇异值分解,本文提出数值求解Symm积分方程的正则化MINRES方法。与Tikhonov正则化方法相比,在数据出现噪声的情况下,新方法能有效地求得Symm积分方程的数值解。     

Direct collocation method for identifying the initial conditions in the inverse wave problem using radial basis functions

A direct collocation method associated with explicit time integration using radial basis functions is proposed for identifying the initial conditions in the inverse problem of wave propagation. Optimum weights for the boundary conditions and additional condition are derived based on Lagrange’s multiplier method to achieve the prime convergence. Tikhonov regularization is introduced to improve the stability for the ill-posed system resulting from the noise, and the L-curve criterion is employed to select the optimum regularization parameter. No iteration scheme is required during the direct collocation computation which promotes the accuracy and stability for the solutions, while Galerkin-based methods demand the iteration procedure to deal with the inverse problems. High accuracy and good stability of the solution at very high noise level make this method a superior scheme for solving inverse problems.     

Regularized and Positive-constrained Inverse Methods in the Problem of Object Restoration

The restoration of incoherently illuminated objects, imaged by a perfect optical instrument, is a typical example of a linear ill-posed inverse problem where positive solutions are required. The purpose of this paper is two-fold: first, to discuss the limitations of regularization methods where the solution is not constrained to be positive; and second, to introduce a positive-constrained restoring method consisting in the solution of a linear programming problem. It is found that regularization methods are quite efficient in the restoration of smooth objects, while the solutions of the linear programming problem are considerably better in the restoration of objects with edges and sharp peaks. In order to justify the numerical results, the effect of positivity on numerical stability is carefully analysed. The extension of the results to other inverse problems is briefly discussed.