求解无限域动力刚度矩阵的双渐近算法

采用连分式算法可以有效地求解无限域动力刚度表示的比例边界有限元方程, 它具有收敛范围广、收敛速度快等优点. 该文在高频渐近连分式算法的基础上考虑了低频渐近, 发展了一种针对矢量波动方程的双渐近算法. 随着展开阶数的增加, 双渐近算法可以在全频域范围内快速逼近准确解. 引入了系数矩阵?X⁽ⁱ⁾来增强连分式算法的数值稳定性. 通过在高频极限、低频极限时满足动力刚度表示的比例边界有限元方程, 建立了递推关系以求得动力刚度矩阵. 通过二维半无限楔形体、三维均质弹性半空间数值算例表明, 双渐近算法比单渐近算法更稳定、优越.     

等横截面无限声学水域的连分式公式

基于比例边界有限元法连分式理论,提出了等横截面无限声学水域的连分式公式,推导了高频连分式公式与双渐近连分式公式,比较了连分式公式与动态质量矩阵模拟等截面无限水域的计算效率,发现前者效率优于后者。利用该公式分析了等截面无限声学水域在顺河向激励下的瞬态响应。数值模拟结果表明高频连分式公式的稳定性与收敛性有待改进,而双渐近连分式则具有更好的稳定性和收敛性,能正确模拟等截面无限水域。     

基于连分式与有限元法的坝库耦合瞬态分析方法

为高效模拟地震激励下坝库耦合瞬态响应,建立了无限水库的连分式与有限元法的耦合公式。结合坝体有限元公式,利用坝库耦合项,发展了坝库耦合瞬态分析迭代算法。利用该算法分析了水平向地震激励下重力坝的瞬态响应。比较了基于连分式法、动态刚度矩阵法、动态质量矩阵法模拟坝库耦合问题的计算效率。数值算例表明该耦合算法模拟坝库耦合瞬态响应的正确性及高效性。该方法继承了比例边界有限元法的精度高、离散单元少等特点,又避免了其卷积积分,提升其计算效率,为坝库耦合瞬态响应提供了一种高效分析方法。     

一种新的求解坝面动水压力的半解析方法

基于比例边界有限元理论,推导了综合考虑库水可压缩性和库底边界吸收的坝面动水压力方程,提出了一种新的求解坝面动水压力的半解析方法,并通过算例验证了这种方法的精度和效率。结果表明,与传统方法相比,这种方法具有精度高、计算工作量小的优点。     

基于连分式与有限元法的坝库耦合瞬态分析方法EI北大核心CSCD

为高效模拟地震激励下坝库耦合瞬态响应,建立了无限水库的连分式与有限元法的耦合公式。结合坝体有限元公式,利用坝库耦合项,发展了坝库耦合瞬态分析迭代算法。利用该算法分析了水平向地震激励下重力坝的瞬态响应。比较了基于连分式法、动态刚度矩阵法、动态质量矩阵法模拟坝库耦合问题的计算效率。数值算例表明该耦合算法模拟坝库耦合瞬态响应的正确性及高效性。该方法继承了比例边界有限元法的精度高、离散单元少等特点,又避免了其卷积积分,提升其计算效率,为坝库耦合瞬态响应提供了一种高效分析方法。     

结构动力方程一种新的级数形式的解析解

将结构的位移及速度响应作为状态变量,采用Lyapunov(李雅普诺夫)人工小参数法求解状态方程,导出状态方程的一个新的级数形式的解析解,该解析解还可以推广到非线性动力方程的计算。将秦九韶算法引入级数解的计算,提高了计算的效率和稳定性,同时给出了算法的计算格式和步骤。该算法无需对转换矩阵H求逆,仅使用矩阵向量相乘,计算稳定,精度仅由收敛项数控制,很容易达到任意精度要求,而且适合并行计算及压缩存储。最后通过算例进一步证实了该算法的精度和效率。     

动力失稳的拟静力刚度准则与能量判别准则

以可模拟非线性保守结构体系的一个计算模型为例,重点分析、对比了判别结构动力稳定性的拟静力刚度准则和能量判别准则。拟静力刚度准则依靠切线刚度非正定判定结构发生动力失稳可能导致误判。能量判别准则适用于具有屈曲后不稳定平衡路径的结构,利用屈曲后不稳定平衡路径上鞍点处的总势能作为动力失稳临界能量,结构总能量超越临界能量则判定为动力失稳。振动极值位移随荷载变化的曲线可以作为一种动力平衡路径,在接近临界荷载时,荷载的微小增量会导致结构振动极大位移显著增大,最终在临界点发生跃越失稳。     

拉压刚度不同桁架的动力参变量变分原理和保辛算法

对于一些展开结构,为达到其设计性能,必须采用特殊的索、膜结构,这些索、膜部件表现出不同的拉压性质。具有拉、压不同性质的材料或结构的力学分析,体现出较强的非线性特征,需要针对这类问题发展有效的求解算法。本文建立了由拉压刚度不同杆单元组成的桁架结构的动力学参变量变分原理,将拉压刚度不同桁架问题的非线性动力分析转换为线性互补问题求解。结合时间有限元方法构造了求解此问题的保辛数值积分方法,此方法不需要迭代和刚度矩阵更新,避免了迭代求解方法的收敛问题,计算过程稳定、高效。     

基于动力刚度法的体外预应力梁自振频率分析

用预应力梁-杆组合结构模拟体外预应力梁,以体外预应力简支梁为例,建立了预应力梁杆单元动力刚度矩阵。采用动力刚度法,进一步推导出预应力梁-杆组合结构的整体动力刚度矩阵,利用Williams-Wittrick算法求解频率。本文以理论推导为基础,引入了动力刚度法。最后通过算例,讨论了各种因素对梁横向的振动特性的影响,并与试验值比较。计算结果表明:动力刚度法能够精确有效的求解体外预应力混凝土梁的横向振动问题。     

可靠性优化的一种新的启发式算法

建立了可靠性冗余优化模型,分析了各种优化方法的优缺点。分析了几种常见的启发式算法,根据拉格朗日乘子法和K-T方程,提出了一种新的启发式算法,结果表明该方法比较有效。     

An improved continued‐fraction‐based high‐order transmitting boundary for time‐domain analyses in unbounded domains

A high‐order local transmitting boundary to model the propagation of acoustic or elastic, scalar or vector‐valued waves in unbounded domains of arbitrary geometry is proposed. It is based on an improved continued‐fraction solution of the dynamic stiffness matrix of an unbounded medium. The coefficient matrices of the continued‐fraction expansion are determined recursively from the scaled boundary finite element equation in dynamic stiffness. They are normalised using a matrix‐valued scaling factor, which is chosen such that the robustness of the numerical procedure is improved. The resulting continued‐fraction solution is suitable for systems with many DOFs. It converges over the whole frequency range with increasing order of expansion and leads to numerically more robust formulations in the frequency domain and time domain for arbitrarily high orders of approximation and large‐scale systems. Introducing auxiliary variables, the continued‐fraction solution is expressed as a system of linear equations in iω in the frequency domain. In the time domain, this corresponds to an equation of motion with symmetric, banded and frequency‐independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable in the frequency and time domains. Analytical and numerical examples demonstrate the superiority of the proposed method to an existing approach and its suitability for time‐domain simulations of large‐scale systems. Copyright © 2011 John Wiley & Sons, Ltd.     

Transient analysis of three-dimensional dynamic interaction between multilayered soil and rigid foundation

The study of dynamic soil-structure interaction is significant to civil engineering applications, such as machine foundation vibration, traffic-induced vibration, and seismic dynamic response. The scaled boundary finite element method (SBFEM) is a semi-analytical algorithm, which is used to solve the dynamic response of a three-dimensional infinite soil. It can automatically satisfy the radiation boundary condition at infinity. Based on the dynamic stiffness matrix equation obtained by the modified SBFEM, a continued fraction algorithm is proposed to solve the dynamic stiffness matrix of layered soil in the frequency-domain. Then, the SBFEM was coupled with the finite element method (FEM) at the interface to solve the dynamic stiffness matrices of the rigid surface/buried foundation. Finally, the mixed-variable algorithm was used to solve the three-dimensional transient dynamic response of the foundation in the time domain. Numerical examples were performed to verify the accuracy of the proposed algorithm in solving the dynamic stiffness matrix of the infinite domain in the frequency domain and the dynamic transient displacement response of the foundation in the time domain. Compared with the previous numerical integration technique, the dynamic stiffness matrix in the frequency domain calculated by using the proposed algorithm has higher accuracy and higher efficiency.     

A continued‐fraction‐based high‐order transmitting boundary for wave propagation in unbounded domains of arbitrary geometry

A high‐order local transmitting boundary is developed to model the propagation of elastic waves in unbounded domains. This transmitting boundary is applicable to scalar and vector waves, to unbounded domains of arbitrary geometry and to anisotropic materials. The formulation is based on a continued‐fraction solution of the dynamic‐stiffness matrix of an unbounded domain. The coefficient matrices of the continued fraction are determined recursively from the scaled boundary finite element equation in dynamic stiffness. The solution converges rapidly over the whole frequency range as the order of the continued fraction increases. Using the continued‐fraction solution and introducing auxiliary variables, a high‐order local transmitting boundary is formulated as an equation of motion with symmetric and frequency‐independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable for evaluating the response in the frequency and time domains. Analytical and numerical examples demonstrate the high rate of convergence and efficiency of this high‐order local transmitting boundary. Copyright © 2007 John Wiley & Sons, Ltd.     

A Frobenius solution to the scaled boundary finite element equations in frequency domain for bounded media

The scaled boundary finite element method (FEM) is a recently developed semi‐analytical numerical approach combining advantages of the FEM and the boundary element method. Although for elastostatics, the governing homogeneous differential equations in the radial co‐ordinate can be solved analytically without much effort, an analytical solution to the non‐homogeneous differential equations in frequency domain for elastodynamics has so far only been obtained by a rather tedious series‐expansion procedure. This paper develops a much simpler procedure to obtain such an analytical solution by increasing the number of power series in the solution until the required accuracy is achieved. The procedure is applied to an extensive study of the steady‐state frequency response of a square plate subjected to harmonic excitation. Comparison of the results with those obtained using ABAQUS shows that the new method is as accurate as a detailed finite element model in calculating steady‐state responses for a wide range of frequencies using only a fraction of the degrees of freedom required in the latter. Copyright © 2006 John Wiley & Sons, Ltd.     

A high‐order approach for modelling transient wave propagation problems using the scaled boundary finite element method

A high‐order time‐domain approach for wave propagation in bounded and unbounded domains is proposed. It is based on the scaled boundary FEM, which excels in modelling unbounded domains and singularities. The dynamic stiffness matrices of bounded and unbounded domains are expressed as continued‐fraction expansions, which leads to accurate results with only about three terms per wavelength. An improved continued‐fraction approach for bounded domains is proposed, which yields numerically more robust time‐domain formulations. The coefficient matrices of the corresponding continued‐fraction expansion are determined recursively. The resulting solution is suitable for systems with many DOFs as it converges over the whole frequency range, even for high orders of expansion. A scheme for coupling the proposed improved high‐order time‐domain formulation for bounded domains with a high‐order transmitting boundary suggested previously is also proposed. In the time‐domain, the coupled model corresponds to equations of motion with symmetric, banded and frequency‐independent coefficient matrices, which can be solved efficiently using standard time‐integration schemes. Numerical examples for modal and time‐domain analysis are presented to demonstrate the increased robustness, efficiency and accuracy of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.     

The scaled boundary finite element method in structural dynamics

The scaled boundary finite element method is extended to solve problems of structural dynamics. The dynamic stiffness matrix of a bounded (finite) domain is obtained as a continued fraction solution for the scaled boundary finite element equation. The inertial effect at high frequencies is modeled by high‐order terms of the continued fraction without introducing an internal mesh. By using this solution and introducing auxiliary variables, the equation of motion of the bounded domain is expressed in high‐order static stiffness and mass matrices. Standard procedures in structural dynamics can be applied to perform modal analyses and transient response analyses directly in the time domain. Numerical examples for modal and direct time‐domain analyses are presented. Rapid convergence is observed as the order of continued fraction increases. A guideline for selecting the order of continued fraction is proposed and validated. High computational efficiency is demonstrated for problems with stress singularity. Copyright © 2008 John Wiley & Sons, Ltd.     

An improved time-varying mesh stiffness algorithm and dynamic modeling of gear-rotor system with tooth root crack

When a tooth crack failure occurs, the vibration response characteristics caused by the change of time-varying mesh stiffness play an important role in crack fault diagnosis. In this paper, an improved time-varying mesh stiffness algorithm is presented. A coupled lateral and torsional vibration dynamic model is used to simulate the vibration response of gear-rotor system with tooth crack. The effects of geometric transmission error (GTE), bearing stiffness, and gear mesh stiffness on the dynamic model are analyzed. The simulation results show that the gear dynamic response is periodic impulses due to the periodic sudden change of time varying mesh stiffness. When the cracked tooth comes in contact, the impulse amplitude will increase as a result of reductions of mesh stiffness. Amplitude modulation phenomenon caused by GTE can be found in the simulation signal. The lateral–torsional coupling frequency increases greatly within certain limits and thereafter reaches a constant while the lateral natural frequency nearly remains constant as the gear mesh stiffness increases. Finally, an experiment was conducted on a test bench with 2 mm root crack fault. The results of experiment agree well with those obtained by simulation. The proposed method improves the accuracy of using potential energy method to calculate the time-varying mesh stiffness and expounds the vibration mechanism of gear-rotor system with tooth crack failure.     

A scaled boundary finite element formulation for dynamic elastoplastic analysis

This study presents the development of the scaled boundary finite element method (SBFEM) to simulate elastoplastic stress wave propagation problems subjected to transient dynamic loadings. Material nonlinearity is considered by first reformulating the SBFEM to obtain an explicit form of shape functions for polygons with an arbitrary number of sides. The material constitutive matrix and the residual stress fields are then determined as analytical polynomial functions in the scaled boundary coordinates through a local least squares fit to evaluate the elastoplastic stiffness matrix and the residual load vector semianalytically. The treatment of the inertial force within the solution of the nonlinear system of equations is also presented within the SBFEM framework. The nonlinear equation system is solved using the unconditionally stable Newmark time integration algorithm. The proposed formulation is validated using several benchmark numerical examples.     

A boundary condition in Padé series for frequency‐domain solution of wave propagation in unbounded domains

A boundary condition satisfying the radiation condition at infinity is frequently required in the numerical simulation of wave propagation in an unbounded domain. In a frequency domain analysis using finite elements, this boundary condition can be represented by the dynamic stiffness matrix of the unbounded domain defined on its boundary. A method for determining a Padé series of the dynamic stiffness matrix is proposed in this paper. This method starts from the scaled boundary finite‐element equation, which is a system of ordinary differential equations obtained by discretizing the boundary only. The coefficients of the Padé series are obtained directly from the ordinary differential equations, which are not actually solved for the dynamic stiffness matrix. The high rate of convergence of the Padé series with increasing order is demonstrated numerically. This technique is applicable to scalar waves and elastic vector waves propagating in anisotropic unbounded domains of irregular geometry. It can be combined seamlessly with standard finite elements. Copyright © 2006 John Wiley & Sons, Ltd.