岩土力学参数概率分布的切比雪夫多项式推断

提出了较大样本岩土力学参数概率分布的切比雪夫多项式逼近法。基于数值逼近原理,直接根据试验样本矩,运用切比雪夫多项式推断岩土力学参数的概率密度函数,并用精度较高的K—S检验法,从理论上证明所求密度函数的正确性和实用性。该方法直接根据试验样本信息和统计方法推断,而不是事先假定成经典的理论概率分布,因此数学和物理意义更加充分。通过对各种经典分布曲线（正态分布、指数分布、对数正态分布等）数值检验,结果表明所得到的逼近表达式有很好的拟合性能。根据样本数据得出的某岩石抗压强度概率密度函数,与实际统计所得分布频率非常接近,可以满足岩土工程可靠性分析的要求。     

基于信号分解和切比雪夫多项式的多体系统区间不确定性分析方法

参数的不确定性会对多体系统动力学响应产生显著影响, 区间分析方法只需根据不确定性参数的边界信息, 便可实现在多体系统动力学分析中考虑参数的不确定性. 考虑区间参数不确定性, 采用切比雪夫区间方法(Chebyshev interval method, CIM)在分析多体系统动力学响应时, 随着时间的增大, 响应边界精度会越来越低. 为解决CIM的这一问题, 本文将信号分解技术与切比雪夫多项式结合, 采用切比雪夫多项式分别对HHT变换(Hilbert-Huang transform, HHT)和局域均值分解(local mean decomposition, LMD)得到的瞬时幅值和瞬时相位近似, 提出CIM-HHT方法和CIM-LMD方法, 以获得含区间参数的长周期动力学响应边界. HHT和LMD分解能够将多体系统的多分量响应分解为多个单分量和一个趋势分量(残余分量)之和, CIM-HHT和CIM-LMD对每个分量的瞬时幅值和瞬时相位、和趋势分量采用切比雪夫多项式近似, 进而建立系统响应的耦合模型, 可以得到系统的动力学响应边界. 最后, 考虑单摆和曲柄滑块机构中的参数不确定性, 验证了CIM-HHT和CIM-LMD方法的有效性. 结果表明, 相比CIM, 在长周期区间动力学响应分析中CIM-HHT和CIM-LMD能够获得较准确的结果. 此外, 相比CIM-HHT, CIM-LMD具有更弱的末端效应, 计算精度更高.      

求非线性动力系统周期解的切比雪夫多项式法

周期运动是一种在客观世界中普遍存在的运动形式,它与混沌运动之间存在十分密切的关系,因而具有很重要的研究价值。利用切比雪夫多项式的若干良好性质,对自治非线性动力系统进行分析,将状态矢量在主周期上展开为切比雪夫多项式的形式,从而将原问题转变为非线性代数方程组的求解问题,得出一种可以方便、迅速地获得周期轨道近似多项式表达式的方法。此方法不依赖于小参数假设,可以用于分析强非线性问题,而且对参数激励系统同样有效。在计算机条件允许时,对高维系统也能迅速、精确地得到其周期轨道的近似多项式表达式。以三维Rossler系统和五维非线性磁浮转子系统周期轨道的计算为例,通过与四阶Runge-Kutta数值积分结果比较,说明此方法的精确、高效性。     

用切比雪夫多项式研究短梁的弯曲计算

考虑剪切效应,利用切比雪夫多项式构造严格满足表面切应力边界条件的轴向位移表达式,建立了短梁弯曲问题的新理论．利用奇异函数把作用在短梁上的复杂外载荷表示为分布载荷,推导出了短梁弯曲时的截面正应力公式及挠曲线表达式．把采用切比雪夫多项式推导出短梁的弯曲计算公式计算结果与弹性理论计算结果进行比较,可知该方法的计算精度较高．研究结果表明：在复杂外载荷作用下,当长高比小于等于6时,剪切变形对梁的弯曲挠度影响较大,而当长高比小于3时,剪切变形对梁的弯曲应力影响较大;因此建议采用切比雪夫多项式方法给出的挠度表达式、弯曲应力进行计算,因为切比雪夫多项式方法不但给出了复杂外载荷作用下梁截面挠度、弯曲应力的计算通式,而且该方法具有计算过程简便、精度高的优点．     

基于切比雪夫多项式的旋变轴角快速解调算法

双通道旋转变压器在定点汇编层实现轴角解调时,传统方法运算量大、占用存储空间多。文中根据粗（精）测角所对应的正余弦值大小及其符号,依据反正切函数的性质将求角的定义域从[-∞,+∞]转化到[0,1],设计了在[0,1]区间上基于切比雪夫多项式快速逼近arctan( x)的低阶分段多项式,用来解决其解调问题;提出了一种通过粗测角,在其附近寻找最佳粗精组合角值的轴角组合及纠错方法;最后在某型号导引头系统的内场试验中进行了测试。试验结果表明,应用本文方法比调用反正切函数法的计算时间减少了50%,比应用查表法的计算精度提高了103倍;该方法具有较好的解码速度和精度,能够用于某些既需要综合考虑功能、体积、重量等要求,又需要快速在定点汇编层实现反正切求角解调的导航系统。     

考虑道砟侵切土质路基的有砟轨道离散元-有限元耦合方法

论文考虑有砟轨道道床结构中底层道砟颗粒会逐渐侵切土质路基上表层的实际情况,引入基于双曲线函数的土阻力公式,发展了离散元颗粒与有限元网格体内耦合的接触力算法,建立了考虑道砟颗粒侵切路基表层的有砟轨道离散元-有限元耦合模型.其中道床中的非规则形状道砟采用离散元镶嵌单元构造,路基采用有限元20节点六面体实体单元构造.分析了道砟侵切路基下列车荷载及路基软硬程度对道床整体沉降的影响,绘制了有砟道床-路基的横截面应力及位移云图.研究表明提出的耦合模型不仅能反映道床自身的弹性变形和累积塑性变形,还能真实反映底层道砟侵切软土路基表层所导致的侵切沉降,为深入了解有砟道床的沉降劣化机理提供重要借鉴作用.     

有限散射信号下二维缺陷形状识别的罚函数方法

研究在有限照射角度和频带宽度下二维缺陷的形状识别问题。首先,通过引进介质参数扰动函数,建立介质参数扰动函数和弹性波散射场之间的非线性关系,并将所关心的缺陷的形状识别问题转化为关于扰动函数的反演;然后,利用变分技术和优化方法求解,为了弥补散射数据的不足,在总的目标函数中,采用附加度量函数作为罚函数;最后,对后场散射远场测量时有限照射角度和频带宽度下几种典型缺陷进行了模拟识别,表明了;表明了罚函数法的有效性。     

基于模态振型和平稳小波变换的悬臂梁微小缺陷识别研究

结构损伤识别方法有很多种,通常结构振动模态振型对缺陷的损伤识别较为敏感,振型体现结构的固有属性及结构局部的特征,可以用于检测缺陷的存在及其位置。但是缺陷比较微小时,仅仅通过振型难以进行损伤识别。因此本文通过对振型进行平稳小波变换处理来检测悬臂梁的微小缺陷。通过有限元模态分析获得含不同缺陷深度、不同缺陷宽度、不同缺陷位置的悬臂梁模型的振型并利用平稳小波变换进行分析处理。结果表明:该方法可以准确判断缺陷的存在及其位置,并且平稳小波细节系数突变峰值随着缺陷深度增大而增大,随着缺陷宽度增大而增大;另外,该方法受振型节点影响,在工程实际应用时应综合前几阶次振型进行缺陷识别。     

考虑道砟侵切路基的有砟铁路离散元-有限元耦合方法

本文考虑有砟铁路道床结构中底层道砟颗粒会逐渐侵切路基上表层的实际情况,引入基于双曲线函数的土阻力公式,发展了离散元颗粒与有限元网格体内耦合的接触力算法,建立了考虑道砟颗粒侵切路基表层的有砟铁路离散元-有限元耦合模型。其中道床中的非规则形状道砟采用离散元镶嵌单元构造,路基采用有限元20节点六面体实体单元构造。分析了道砟侵切路基下列车荷载及路基软硬程度对道床整体沉降的影响,绘制了有砟道床-路基的横截面应力及位移云图。研究表明提出的耦合模型不仅能反映道床自身的弹性变形和累积塑性变形,还能真实反映底层道砟侵切软土路基表层所导致的侵切沉降,为深入了解高速重载下有砟道床的沉降劣化机理提供重要借鉴作用。     

基于混合编码遗传算法和有限元分析的压电结构载荷识别

与传统的优化算法相比,遗传算法不需要计算目标函数的导数信息,便于迭代,可实现全局寻优.因此,本文提出一种采用混合编码的遗传算法与有限元分析相结合,对复合材料层合板、壳进行载荷识别的新方法.在遗传算法求解过程中,设计变量的编码方法选择是其重要环节,二进制编码容易产生连续函数离散化时的映射误差,且其求解精度与染色体的编码长度紧密相关,过长的染色体描述虽可提高精度,但会显著降低算法的求解效率.为此,本文提出采用混合编码的方法进行载荷识别,即用二进制编码表征载荷作用位置,浮点数编码表示载荷的大小.这一方法大大降低了染色体的长度,并显著提高了计算效率和精度.     

Discrete non‐local absorbing boundary condition for exterior problems governed by Helmholtz equation

The finite element method is employed to approximate the solutions of the Helmholtz equation for water wave radiation and scattering in an unbounded domain. A discrete, non‐local and non‐reflecting boundary condition is specified at an artificial external boundary by the DNL method, yielding an equivalent problem that is solved in a bounded domain. This procedure formulates a boundary value problem in a bounded region by imposing a relation in the discrete medium between the nodal values at the two last layers. For plane geometry, this relation can be found by straightforward eigenvalue decomposition. For circular geometry, the plane condition is applied at the external layer and this condition is condensed through a structured annular region, resulting in a condition at an inner radius. Exterior problems with a bounded internal physical obstacle are considered. It is well‐known that these kind of problems are well‐posed, and have a unique solution. Numerical studies based on standard Galerkin methodology examine the dependence of the DNL condition with respect to the circular annular region width. The DNL condition is compared with local boundary conditions of several orders. Numerical examples confirm the important improvement in accuracy obtained by the DNL method over standard conditions. Copyright © 1999 John Wiley & Sons, Ltd.     

Absorbing boundary conditions for elastic waves in transversely isotropic media

In the numerical simulation of elastic wave propagation in the solid, it is essential to introduce absorbing boundary conditions to limit the large or unbounded domain of computation. In this paper, the absorbing boundaries for transversely isotropic media are composed of simple first-order partial differential operators, and each of the operators can perfectly absorb a plane wave outgoing at a certain angle. To test the absorbing ability, the reflection coefficient formulas for the quasi-P and quasi-S wave on the absorbing boundary are derived based on the potential functions theory of the elastic wave. Numerical examples show that the absorbing effect is good. The boundary conditions given here have a practical meaning.Supported by National Natural Science Foundation of China.     

Perfectly matched layer boundary condition for two-dimensional Euler equations in generalized coordinate system

In the past, perfectly matched layer (PML) equations have been constructed in Cartesian and spherical coordinates. In this article, the focus is on the development of a PML absorbing technique for treating numerical boundaries, especially those with unbounded domains, in a generalized coordinate system for a flow in an arbitrary direction. The PML equations for two-dimensional Euler equations are developed in split form through a space–time transformation involving a complex variable transformation with the application of a pseudo-mean-flow in the PML domain. A numerical solver is developed using conventional numerical schemes without employing any form of filtering or artificial dissipation to solve the governing PML equations for two-dimensional Euler equations in a generalized coordinate system. Physical domains of arbitrary shapes are considered and numerical simulations are carried out to validate and demonstrate the effectiveness of the PML as an absorbing boundary condition in generalized coordinates.     

FEM-based reconstruction of sound pressure field damped by partially absorbing boundary conditions

This paper presents an inverse formulation of the acoustic boundary value problem featuring arbitrary admittance boundary conditions. The problem is discretised by using finite elements to reconstruct the sound pressure field of a cavity based on a preferably small number of measurements. For that, a modal approach is investigated to handle the generally under-determined and ill-conditioned system of equations. The viability of the algorithm is tested to reconstruct sound pressure field inside a two-dimensional sedan passenger compartment.     

NUMERICAL SOLUTION OF THE SINGULARLY PERTURBED PROBLEM WITH NONLOCAL BOUNDARY CONDITION

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弹性空腔膨胀中边界条件的转换方法

以弹性空腔膨胀为研究对象,利用速度和应力2种边界条件下运动场势函数相等的原理,运用Laplace变换及其卷积定理,得到了两种边界条件的相互转换关系,建立了球面波运动场中速度场与应力场的转换关系。以双指数应力时程和正弦指数衰减速度时程为例,研究了球面波运动场转换的特点。结果表明,球面波上的应力和速度之间不是简单的线性关系,与平面波相比,球面波上的质点速度较小。而影响这种差异大小的主要因素是波的传播距离和介质中波的传播速度,波传播距离越近,传播速度越快,这种差异越大。     

Interface damage modeled by spring boundary conditions for in-plane elastic waves

In-plane elastic wave propagation in the presence of a damaged interface is investigated. The damage is modeled as a distribution of small cracks and this is transformed into a spring boundary condition. First the scattering by a single interface crack is determined explicitly in the low frequency limit for the case of a plane wave normally incident to the interface. The transmission at an interface with a random distribution of small cracks is then determined and is compared to periodically distributed cracks. The cracked interface is then described by a distributed spring boundary condition. As an illustration the dispersion relation of the first modes in a thick plate with a damaged interface in the middle is given.     

Finite element method for shallow water equation including open boundary condition

A method to deal with an open boundary condition in the analysis of water surface waves, the tide, etc. by means of the finite element method is proposed in this paper. The present method has two important features relating to the treatment of the open boundary condition. The first feature is to consider the non-reflective virtual boundary condition which has been developed in the numerical wave analysis method. The incident wave conditions without spurious reflected waves can be imposed at the open boundary. The second feature is to identify the amplitude of the components of incident waves in terms of observed water elevations in the field of standing waves. This can be done as a parameter identification based on an optimization problem by applying the conjugate gradient method. The applicability of this method to wave propagation problems is verified by several numerical computations.     

A numerical method for coupling three dimensional nonlinear finite elements with elastic boundary elements

This paper systematically deals with the following three problems: (1) Some numerical schemes in coupling FEM- and BEM: including condensation of the boundary integral equation, symmetrization of equivalent stiffness matrix and treatment of traction discontinuity, (2) Coupling of elastoplastic finite elements to elastic boundary elements, (3) Coupling of elasto-viscoplastic finite elements to elastic boundary elements and numerical stability condition.     

PML absorbing boundary conditions for the linearized and nonlinear Euler equations in the case of oblique mean flow

For the case of uniform mean flow in an arbitrary direction, perfectly matched layer (PML) absorbing boundary conditions are presented for both the linearized and nonlinear Euler equations. Although linear perfectly matched side layers with an oblique mean flow have been studied in previous works, we propose in the present paper a construction of corner layer equations that are dynamically stable. Stability issues are investigated by examining the dispersion relations of linear waves supported by the corner layer equations. For increased efficiency, a pseudo mean flow is included in the derivation of the PML equations for the nonlinear case. Numerical examples are given to support the validity of the proposed equations. Specifically, the linear PML formulation is tested for the case of acoustic, vorticity, and entropy waves traveling with an oblique mean flow. The nonlinear formulation is tested with an isentropic vortex moving diagonally with a constant velocity. Copyright © 2008 John Wiley & Sons, Ltd.