小波方法及其非线性力学问题应用分析

小波分析是近几十年来发展起来的重要数学分支,被誉为“数学显微镜”,其独具的多分辨分析和大量可供选择的,可兼具正交性、紧支性、对称性、低通滤波、线性相位及插值性等优良数学品质的小波基函数为强非线性微分方程的数值求解带来了新的契机。自上世纪90年代以来,诸如小波伽辽金法、小波配点法、小波有限单元法和小波边界单元法等数值方法被先后构建出来并成功应用于各类力学问题的定量研究之中。本文从小波提出的历史背景及作为其理论基础的多分辨分析出发,对现有基于小波理论的各类数值方法进行梳理,总结各自的优点、缺点和下一步可能的发展方向,为未来基于小波理论的定量分析方法的发展及其在复杂非线性力学问题中的应用研究提供参考。     

不规则区域平面弹性问题的正则区域重心插值配点法

将不规则区域嵌入到规则的矩形区域,在矩形区域上将弹性平面问题的控制方程采用重心Lagrange插值离散,得到控制方程矩阵形式的离散表达式。在边界节点上利用重心插值离散边界条件,规则区域采用置换法施加边界条件,不规则区域采用附加法施加边界条件,得到求解平面弹性问题的过约束线性代数方程组,采用最小二乘法进行求解,得到整个规则区域上的位移数值解。利用重心插值计算得到不规则区域内任意节点的位移值,计算精度可到10^-14以上。数值算例验证了所建立方法的有效性和计算精度。     

非线性MEMS微梁的重心有理插值迭代配点法分析

文章利用重心有理插值迭代配点法分析计算非线性MEMS微梁问题。通过处理MEMS微梁的几何通过假设初始函数,将微梁非线性控制方程转换为线性化微分方程,建立逼近非线性微分方程的线性化迭代格式。采用重心有理插值配点法求解线性化微分方程,提出了数值分析MEMS微梁非线性弯曲问题的重心插值迭代配点法。给出了非线性微分方程的直接线性化和Newton线性化计算公式,详细讨论了非线性积分项的计算方法和公式。利用重心有理插值微分矩阵,建立了矩阵-向量化的重心插值迭代配点法的计算公式。数值算例结果表明,重心插值迭代配点法求解微梁非线性弯曲问题,具有计算公式简单、程序实施方便和计算精度高的特点。     

A WAVELET METHOD FOR BENDING OF CIRCULAR PLATE WITH LARGE DEFLECTION

A wavelet method for solving strongly nonlinear boundary value problems is described, which has been demonstrated early to have a convergence rate of order 4, almost independent of the nonlinear intensity of the equations. By using such a method, we study the bending problem of a circular plate with arbitrary large deflection. As the deflection increases,the bending behavior usually exhibits a so-called plate-to-membrane transition. Capturing such a transition has ever frustrated researchers for decades. However, without introducing any additional treatment, we show in this study that the proposed wavelet solutions can naturally cover the plate-membrane transition region as the plate deflection increases. In addition, the high accuracy and efficiency of the wavelet method in solving strongly nonlinear problems is numerically confirmed, and applicable scopes for the linear, the membrane and the von Karman plate theories are identified with respect to the large deformation bending of circular plates.     

应用分形有限元方法于外域声场计算

 应用二级分形有限元方法计算了外域声场. 用一人工边界把外域声场分为两 部分，人工边界以内使用常规有限元方法，人工边界以外的无限大区域使用分形有限元方法. 使用分形有限元方法的优点是：一方面形成几何自相似网格使得相邻层之间的单元刚度矩阵 和质量矩阵具有非常简单的关系；另一方面引用自动满足无限远辐射条件的全域插值函数把 节点自由度变换为一组广义坐标，因而计算量可以大大减少. 数值算例表明：该方法对于计 算无限大外域声场是有效的.     

A simultaneous space-time wavelet method for nonlinear initial boundary value problems

A high-precision and space-time fully decoupled numerical method is developed for a class of nonlinear initial boundary value problems. It is established based on a proposed Coiflet-based approximation scheme with an adjustable high order for the functions over a bounded interval, which allows the expansion coefficients to be explicitly expressed by the function values at a series of single points. When the solution method is used, the nonlinear initial boundary value problems are first spatially discretized into a series of nonlinear initial value problems by combining the proposed wavelet approximation and the conventional Galerkin method, and a novel high-order step-by-step time integrating approach is then developed for the resulting nonlinear initial value problems with the same function approximation scheme based on the wavelet theory. The solution method is shown to have the N th-order accuracy, as long as the Coiflet with [0, 3 N-1]compact support is adopted, where N can be any positive even number. Typical examples in mechanics are considered to justify the accuracy and efficiency of the method.     

算子自定义小波弹性板单元构造及自适应分析

提出一类用于分析弹性板问题的算子自定义小波弹性板单元构造方法。该方法的优点在于根据工程问题的求解需要灵活构造具有解耦特性的算子自定义小波基,使得系统多尺度刚阵具有沿对角线的强稀疏性,从而实现了该算法在每个尺度上独立、快速求解,系统方程的求解效率得到较大提高。建立多分辨Lagrange有限元空间和多尺度计算理论,提出基于稳定完备法的算子自定义小波弹性板单元构造方法及解耦条件。依据两尺度相对误差估计,提出自适应算子自定义小波有限元算法。数值算例证明,算子自定义小波弹性板单元具有求解精度与计算效率高等特点。     

一种适用于强非线性结构力学问题数值求解的修正小波伽辽金方法

论文通过对有限区间上的任一连续函数在边界处采用基于泰勒展开的延拓处理,构造了一种与任意边界条件相协调的改进小波尺度基函数及在此基础上建立了小波逼近格式,由此可有效避免小波逼近在求解微分方程时在边界处的跳跃或抖动问题.在此基础上,结合论文后两位作者提出的广义小波高斯积分法,关于未知函数的任意非线性项的小波展开可以显式地用...     

A finite wavelet domain method for wave propagation analysis in thick laminated composite and sandwich plates

An efficient numerical method is developed for the simulation of three dimensional transient dynamic response in thick laminated composite and sandwich plate structures involving very high frequencies and wave numbers. The proposed method incorporates Daubechies wavelet scaling functions for the interpolation of the in-plane displacements with a Galerkin formulation. It further explores the orthonormality and compact support of wavelet scaling functions to produce near diagonal consistent mass matrices and banded stiffness matrices. Hence, an uncoupled equivalent discrete spatial dynamic system is formulated, synthesized and rapidly solved in the wavelet domain using an explicit time integration scheme. The in-plane wavelet interpolation is further combined with an efficient high order layerwise laminate plate theory, that implements Hermite cubic splines for the through-the-thickness approximation of displacement fields. Numerical results are presented on the prediction of guided waves in laminated and thick sandwich composite plates and compared with respective solutions obtained by analytical, semi-analytical and time domain spectral element models. The method yielded higher convergence rates and substantial reductions in computational effort compared to respective time domain spectral finite elements.     

A mixed Newton-Tikhonov method for nonlinear ill-posed problems

Newton type methods are one kind of the efficient methods to solve nonlinear ill-posed problems, which have attracted extensive attention. However, computational cost of Newton type methods is high because practical problems are complicated. We propose a mixed Newton-Tikhonov method, i.e., one step Newton-Tikhonov method with several other steps of simplified Newton-Tikhonov method. Convergence and stability of this method are proved under some conditions. Numerical experiments show that the proposed method has obvious advantages over the classical Newton method in terms of computational costs.     

三维位势问题的梯度边界积分方程的新解法

三维位势问题的边界元分析中,关于坐标变量的边界位势梯度的计算是一个困难的问题. 已有一些方法着手解决这个问题,然而,这些方法需要复杂的理论推导和大量的数值计算. 本文提出求解一般边界位势梯度边界积分方程的辅助边值问题法. 该方法构造了与原边界值问题具有相同解域的辅助边值问题,该辅助边值问题具有已知解,因此通过求解此辅助边值问题,可获得梯度边界积分方程对应的系统矩阵,然后将此系统矩阵应用于求解原边值问题,求解过程非常简单,只需求解一个线性系统即可获得原边值问题的解. 值得注意的是,在求解原边值问题时,不再需要重新计算系统矩阵,因此辅助边值问题法的效率并不很差. 辅助边值问题法避免了强奇异积分的计算,具有数学理论简单、程序设计容易、计算精度高等优点,为坐标变量梯度边界积分方程的求解提供了一个新的途径. 3个标准的数值算例验证了方法的有效性.      

大规模边界元模态分析的高效数值方法

随着大规模快速边界元计算技术的发展,在复杂结构的动态设计、振动与噪声分析中愈来愈多地采用边界元法,因此求解大规模边界元特征值问题、进行复杂结构和声场模态分析,成为工程应用中一个十分重要,但却极具挑战性的课题,目前国际上还没有十分有效的数值方法.本文针对边界元法中典型的非线性特征值问题,提出了一种通用、高效的数值解法,称为基于预解矩阵采样的Rayleigh-Ritz投影法,记为RSRR.首先,通过求解一系列频域边界元问题来构造特征向量搜索空间,进而可以采用Rayleigh-Ritz投影,将原问题转化为一个可以采用现有方法求解的小规模缩减特征值问题;其次,为了降低Rayleigh-Ritz投影过程的计算量,基于解析函数的Cauchy积分公式,构造了边界元系数矩阵的插值近似方法,以及缩减特征值问题系数矩阵的快速计算方法,给出了插值项数的估计策略;最后,将RSRR与声学快速边界元法结合,应用于大规模吸声结构的复模态分析.数值算例表明,RSRR方法能够可靠地求出给定频段内的全部特征值和特征向量,具有计算效率高、精度高、通用等优点.     

A boundary-field equation method for a nonlinear exterior elasticity problem in the plane

This paper presents a new method for solving a nonlinear exterior boundary value problem arising in two-dimensional elasto-plasticity. The procedure is based on the introduction of a sufficiently large circle that divides the exterior domain into a bounded region and an unbounded one. This allows us to consider the Dirichlet-Neumann mapping on the circle, which provides an explicit formula for the stress in terms of the displacement by using an appropriate infinite Fourier series. In this way we can reduce the original problem to an equivalent nonlinear boundary value problem on the bounded domain with a natural boundary condition on the circle. Hence, the resulting weak formulation includes boundary and field terms, which yields the so called boundary-field equation method. Next, we employ the finite Fourier series to obtain a sequence of approximating nonlinear problems from which the actual Galerkin schemes are derived. Finally, we apply some tools from monotone operators to prove existence, uniqueness and approximation results, including Cea type error estimates for the corresponding discrete solutions.     

An optimization method based on the generalized polynomials for nonlinear variable-order time fractional diffusion-wave equation

In this paper, an optimization method based on a new class of basis functions, namely generalized polynomials (GPs), is proposed for nonlinear variable-order time fractional diffusion-wave equation. Variable-order time fractional derivative is expressed in the Caputo sense. In the proposed method, solution of the problem under consideration is expanded in terms of GPs with unknown free coefficients and control parameters. In this way, some new operational matrices of the ordinary and fractional derivatives are derived for these basis functions. The residual function and its 2-norm are employed for converting the problem under study to an optimization one and then choosing the unknown free coefficients and control parameters optimally. As a useful result, the necessary conditions of optimality are derived as a system of nonlinear algebraic equations with unknown free coefficients and control parameters. The validity and effectiveness of the method are demonstrated by solving some numerical examples. The results demonstrate that the proposed method is a powerful algorithm with good accuracy for solving such kind of problems.     

求解Helmholtz方程基于核重构思想的最小二乘配点法

基于核重构思想构造近似函数，将配点法和最小二乘原理相结合对微分方程进行离散， 建立了Helmholtz方程的最小二乘配点格式，并分别研究了Helmholtz方程的波传播问题和 边界层问题. 通过数值算例可以发现，给出的数值计算结果非常接近于精确解，计算精度明显高于SPH 法的数值结果，且随着节点数目的增加，其精确度越来越高，具有良好的收敛性.     

Dynamic system for reference signal transmission and reconstruction in distributed MIMO radars

The renormalization method based on the Taylor expansion for asymptotic analysis of differential equations is generalized to difference equations. The proposed renormalization method is based on the Newton–Maclaurin expansion. Several basic theorems on the renormalization method are proven. Some interesting applications are given, including asymptotic solutions of quantum anharmonic oscillator and discrete boundary layer, the reductions and invariant manifolds of some discrete dynamics systems. Furthermore, the homotopy renormalization method based on the Newton–Maclaurin expansion is proposed and applied to those difference equations including no a small parameter. In addition, some subtle problems on the renormalization method are discussed.     

两点边值问题的小波配点法

根据多分辨分析,提出用任意连续的尺度函数构造区间上的插值基函数,形成以尺度函数为基础的求解两点边值问题的小波配点法.该方法中,尺度函数不受紧支撑、插值等性质的限制,计算复杂度小,数值解收敛性由多分辨分析理论保证.同时,给出边值条件的积分处理方法,能够方便地处理任意边界条件,当尺度函数不具有高阶导数时,该方法也能有效使用.数值算例表明,该方法是一个高效、高精度的算法.     

动态断裂力学的无限相似边界元法

对弹性动力学的相似边界元法进行了进一步研究,推导了相应的计算公式,并在此基础上提出了动态断裂力学的无限相似边界元法.与传统的边界元法相比,相似边界元法由于只需在少数单元上进行数值积分,大大减少了计算量.对动态断裂力学问题,无限相似边界元法由于在裂纹尖端的边界上设置了逼近于裂纹尖端的无限个相似边界单元,可直接得到裂纹尖端具有奇异性的应力,而不需要设置奇异单元,从而突破了奇异单元对应力奇异性阶次的局限.另外,还讨论了无限相似边界元法得到的无限阶的线性代数方程组的求解方法.     

A high‐order backward forward sweep interpolating algorithm for semi‐Lagrangian method

Conventional semi‐Lagrangian methods often suffer from poor accuracy and imbalance problems of advected properties because of low‐order interpolation schemes used and/or inability to reduce both dissipation and dispersion errors even with high‐order schemes. In the current work, we propose a fourth‐order semi‐Lagrangian method to solve the advection terms at a computing cost of third‐order interpolation scheme by applying backward and forward interpolations in an alternating sweep manner. The method was demonstrated for solving 1‐D and 2‐D advection problems, and 2‐D and 3‐D lid‐driven cavity flows with a multi‐level V‐cycle multigrid solver. It shows that the proposed method can reduce both dissipation and dispersion errors in all regions, especially near sharp gradients, at a same accuracy as but less computing cost than the typical fourth‐order interpolation because of fewer grids used. The proposed method is also shown able to achieve more accurate results on coarser grids than conventional linear and other high‐order interpolation schemes in the literature. Copyright © 2017 John Wiley & Sons, Ltd.     

A meshfree-based local Galerkin method with condensation of degree of freedom for elastic dynamic analysis简

Condensation technique of degree of freedom is first proposed to improve the computational efficiency of meshfree method with Galerkin weak form for elastic dy- namic analysis. In the present method, scattered nodes with- out connectivity are divided into several subsets by cells with arbitrary shape. Local discrete equation is established over each cell by using moving Kriging interpolation, in which the nodes that located in the cell are used for approxima- tion. Then local discrete equations can be simplified by con- densation of degree of freedom, which transfers equations of inner nodes to equations of boundary nodes based on cells. The global dynamic system equations are obtained by as- sembling all local discrete equations and are solved by using the standard implicit Newmark＇s time integration scheme. In the scheme of present method, the calculation of each cell is carried out by meshfree method, and local search is imple- mented in interpolation. Numerical examples show that the present method has high computational efficiency and good accuracy in solving elastic dynamic problems.