移动简谐荷载作用下桥梁响应的高效计算

在计算移动荷载过桥问题中广泛使用的Newmark方法必须在每一时间步内限制荷载的大小和作用位置都不能改变。精细积分法虽然允许荷载的大小在每一时间步长内发生变化,但是仍假定其作用位置是不变的,未能采取措施以描述荷载沿着桥面的连续移动性。本文提出三种精细积分格式,在每一时间步内不但允许移动荷载的大小按简谐规律连续变化,而且模拟了简谐荷载在空间域的连续移动。通过与Newmark方法和简单问题的解析解进行数值比较,表明用本文提出的方法可以用较粗的结构单元和较大的时间步长而获得很高的计算精度。在精度相同的前提下,计算效率比Newmark方法可提高1~2个数量级。     

增维精细积分法的适用范围研究

对线性定常结构动力系统提出的增维精细积分法,能够将非齐次动力方程转化为齐次动力方程,不用对状态矩阵求逆就能方便高效地求解出结构的动力响应。本文在仔细分析增维精细积分法性质的基础上,提出了其适用条件,进一步拓宽了其应用范围,并给出了将荷载项展开成傅里叶级数时,相应增维精细积分法的表达式。同时,在一个时间步长内,通过对非齐次项作线性化假设,成功地将增维精细积分法应用到了非线性动力分析领域。本文方法计算格式统一,易于编程,具有很高的计算效率。数值算例证明了本文方法的有效性。     

精细积分法在迷宫密封转子系统非线性动力学特性计算中的应用

基于Muszynska密封力模型，建立了迷宫密封转子系统的非线性动力学模型，将精细积分法推广应用于非线性情况，计算了迷宫密封不平衡转子系统的动力学特性，依据Floquet理论讨论其分岔特性。研究表明：在2^N类算法计算指数矩阵基础上提出的精细积分法和传统的数值计算方法相比，其精度高，在分析中通过取不同步长计算对比，表明该方法在某些情况下可以采取较大时间步长，有效提高了计算速度。     

非线性结构动力学方程的迭代时程积分方法

本文把文（１）提出的精细时程积分法推广到非线性动力学方程的响应分析中。时间步预估一迭代的方法减少了非线性迭代的步数，改善了其收敛性。计算的结果表明，该方法具有较高的精度。     

饱和两相介质动力固结问题时域求解的精细时程积分方法

针对u-p形式的饱和两相介质波动方程,采用精细时程积分方法计算固相位移u,采用向后差分算法求解流体压力p,建立了饱和两相介质动力固结问题时域求解的精细时程积分方法。针对标准算例,对该方法的计算精度进行了校核。开展了该方法相关算法特性的研究,对采用不同数值积分方法计算非齐次波动方程特解项计算精度的差异进行了对比研究,并对采用不同积分点数目的高斯积分法计算特解项条件下计算精度的差异进行了对比研究。研究结果表明,(1)该方法具有良好的计算精度。(2)计算非齐次波动方程特解项的数值积分方法中,梯形积分法的计算精度最差,高斯积分法、辛普生积分法和科茨积分法都具有较好的计算精度。(3)增加高斯积分点数目对于提高计算精度的作用并不显著。     

精细积分法应用于地震碰撞力反应谱计算研究

相邻结构的碰撞时程分析是计算地震碰撞力反应谱的基础。结构碰撞时程分析要求采用稳定性好、精度高及计算效率高的数值分析方法。精细积分法将二阶动力微分方程通过增元降阶的方式转换成Hamilton对偶变量体系,得到了动力微分方程的精确解。基于此,本文将精细积分法引入结构碰撞时程分析及地震碰撞力反应谱计算中。在推导精细积分法公式的基础上,在MATLAB环境下编制了结构碰撞时程分析程序和碰撞力反应谱计算程序,并实现了碰撞力反应谱程序的并行化。经算例验证,精细积分法应用于结构碰撞时程分析及地震碰撞力反应谱计算是可行的,程序计算结果准确。     

波动方程的精细逐步积分法

应用现有的波动方程求解方法解决工程实际问题尚存在一定的局限性。本文在结构动力方程精细逐步积分的基础上,提出了波动方程初边值问题的精细逐步积分法,并分别给出了不同边界条件下的精细逐步积分格式。此数值方法虽然是显式积分方法,却是无条件稳定的。分别用精细逐步积分法和其它已有的方法对两个算例进行了计算,一个是有解析解的例子,该例验证了此方法的准确性,另一个例子是求解由波动方程及初始条件和边界条件组成的有杆抽油系统预测模型,此例验证了精细逐步积分法的高效性。     

基于精细积分技术的非线性动力学方程的同伦摄动法

将精细积分技术（PIM）和同伦摄动方法（HPM）相结合，给出了一种求解非线性动力学方程的新的渐近数值方法。采用精细积分法求解非线性问题时，需要将非线性项对时间参数按Taylor级数展开，在展开项少时，计算精度对时间步长敏感；随着展开项的增加，计算格式会变得越来越复杂。采用同伦摄动法，则具有相对筒单的计算格式，但计算精度较差，应用范围也限于低维非线性微分方程。将这两种方法相结合得到的新的渐近数值方法则同时具备了两者的优点，既使同伦摄动方法的应用范围推广到高维非线性动力学方程的求解，又使精细积分方法在求解非线性问题时具有较简单的计算格式。数值算例表明，该方法具有较高的数值精度和计算效率。     

精细时程积分法及其数值衍生格式应用评估

旋翼气动弹性耦合动力学方程本质上是一组刚性比较大的非线性偏微分方程。在有限元结构离散后,可改写为非齐次微分方程组,其中非齐次项是桨叶运动量(位移与速度)和气动载荷的函数。针对这类方程,本文尝试引入精细积分法及其衍生格式,借助数值方法计算Duhamel积分项。从积分精度与数值稳定性方面比较研究具有代表性的精细库塔法和高精度直接积分法。结合隐式积分算法,评估精细积分法应用于旋翼动力学方程的可行性。算例表明,精细积分法对矩形直桨叶动力学方程具有足够的求解精度。     

精细时程积分法及其数值衍生格式应用评估

旋翼气动弹性耦合动力学方程本质上是一组刚性比较大的非线性偏微分方程。在有限元结构离散后，可改写为非齐次微分方程组，其中非齐次项是桨叶运动量（位移与速度）和气动载荷的函数。针对这类方程，本文尝试引入精细积分法及其衍生格式，借助数值方法计算Duhamel积分项。从积分精度与数值稳定性方面比较研究具有代表性的精细库塔法和高精度直接积分法。结合隐式积分算法，评估精细积分法应用于旋翼动力学方程的可行性。算例表明，精细积分法对矩形直桨叶动力学方程具有足够的求解精度。     

精细积分方法的发展与扩展应用

钟万勰院士于1991年首先提出计算矩阵指数的精细积分方法,其要点是2^N类算法和增量存储。精细积分方法可给出矩阵指数在计算机意义上的精确解,为常微分方程的数值计算提供了高精度、高稳定性的算法,现已成功应用于结构动力响应、随机振动、热传导以及最优控制等众多领域。本文首先介绍矩阵指数精细积分方法的提出、基本思想和发展;然后依次介绍在时不变/时变线性微分方程、非线性微分方程以及大规模问题求解中发展起来的各种精细积分方法,分析了其优缺点和适用范围;最后介绍了精细积分方法的基本思想在两点边值问题、椭圆函数和病态代数方程等问题的扩展应用,进一步展示了该思想的特色。     

分层介质中Love波的扩展W-W算法

应用精细积分法(PIM)和扩展Wittrick-Williams(W-W)算法求解横观各向同性分层半空间中的Love波问题.Love波对应于波数-频率域线性常微分方程的本征值问题.精细积分法是求解线性常微分方程两端边值问题和初值问题的高精度算法.利用本征值计数技术,扩展W-W算法可以不遗漏地找到所有本征值.因此,文中使用的方法可以得到计算机精度意义下的精确解.     

A coupled immersed boundary‐lattice Boltzmann method with smoothed point interpolation method for fluid‐structure interaction problems

The immersed boundary‐lattice Boltzmann method has been verified to be an effective tool for fluid‐structure interaction simulation associated with thin and flexible bodies. The newly developed smoothed point interpolation method (S‐PIM) can handle the largely deformable solids owing to its softened model stiffness and insensitivity to mesh distortion. In this work, a novel coupled method has been proposed by combining the immersed boundary‐lattice Boltzmann method with the S‐PIM for fluid‐structure interaction problems with large‐displacement solids. The proposed method preserves the simplicity of the lattice Boltzmann method for fluid solvers, utilizes the S‐PIM to establish the realistic constitutive laws for nonlinear solids, and avoids mesh regeneration based on the frame of the immersed boundary method. Both two‐ and three‐dimensional numerical examples have been carried out to validate the accuracy, convergence, and stability of the proposed method in consideration of comparative results with referenced solutions.     

Nonlinear aeroelastic analysis of an airfoil-store system with a freeplay by precise integration method

The aeroelastic system of an airfoil-store configuration with a pitch freeplay is investigated using the precise integration method (PIM). According to the piecewise feature, the system is divided into three linear sub-systems. The sub-systems are separated by switching points related to the freeplay nonlinearity. The PIM is then employed to solve the sub-systems one by one. During the solution procedures, one challenge arises when determining the vibration state passing the switching points. A predictor-corrector algorithm is proposed based on the PIM to tackle this computational obstacle. Compared with exact solutions, the PIM can provide solutions to the precision in the order of magnitude of 10⁻¹². Given the same step length, the PIM results are much more accurate than those of the Runge–Kutta (RK) method. Moreover, the RK method might falsely track limit cycle oscillations (LCOs), bifurcation charts or chaotic attractors; even the step length is chosen much smaller than that for the PIM. Bifurcations and LCOs are obtained and analyzed by the PIM in detail. Interestingly, it is found that multiple LCOs and chaotic attractors can exist simultaneously. With this magnitude of precision and efficiency, the PIM could become a solution technique with excellent potential for piecewise nonlinear aeroelastic systems.     

Numerical analysis of Biot's consolidation process by radial point interpolation method

An algorithm is proposed to solve Biot's consolidation problem using meshless method called a radial point interpolation method (radial PIM). The radial PIM is advantageous over the meshless methods based on moving least-square (MLS) method in implementation of essential boundary condition and over the original PIM with polynomial basis in avoiding singularity when shape functions are constructed. Two variables in Biot's consolidation theory, displacement and excess pore water pressure, are spatially approximated by the same shape functions through the radial PIM technique. Fully implicit integration scheme is proposed in time domain to avoid spurious ripple effect. Some examples with structured and unstructured nodes are studied and compared with closed-form solution or finite element method solutions.     

多层地基条带基础动力刚度矩阵的精细积分算法

提出应用精细积分算法计算多层地基的动力刚度问题. 精细积分是计算层状介质中波传播的高效而精确的数值方法. 利用傅里叶积分变换将层状地基的波动方程转换为频率-波数域内的两点边值问题的常微分方程组, 运用精细积分方法求解格林函数, 最后再将得到的频率-波数域内地基表面的动力刚度矩阵转换到频率-空间域内, 进而得到刚性条带基础频率域的动力柔度或刚度矩阵. 所建议的精细积分算法, 可以避免一般传递矩阵计算中的指数溢出问题, 对各种情况有广泛的适应性, 计算稳定, 在高频段可以保障收敛性, 并能达到较高的计算精度.     

A PRECISE METHOD FOR SOLVING WAVE PROPAGATION IN HOLLOW SANDWICH CYLINDERS WITH PRISMATIC CORES

Wave propagation in infinitely long hollow sandwich cylinders with prismatic cores is analyzed by the extended Wittrick-Williams(W-W) algorithm and the precise integration method(PIM). The effective elastic constants of prismatic cellular materials are obtained by the homogenization method. By applying the variational principle and introducing the dual variables, the canonical equations of Hamiltonian system are constructed. Thereafter, the wave propagation problem is converted to an eigenvalue problem. In numerical examples, the effects of the prismatic cellular topology, the relative density, and the boundary conditions on dispersion relations,respectively, are investigated.     

Improved precise integration method for differential Riccati equation

An improved precise integration method(IPIM) for solving the differential Riccati equation(DRE) is presented.The solution to the DRE is connected with the exponential of a Hamiltonian matrix,and the precise integration method(PIM) for solving the DRE is connected with the scaling and squaring method for computing the exponential of a matrix.The error analysis of the scaling and squaring method for the exponential of a matrix is applied to the PIM of the DRE.Based on the error analysis,the criterion for choosing two parameters of the PIM is given.Three kinds of IPIMs for solving the DRE are proposed.The numerical examples show that the IPIM is stable and gives the machine accuracy solutions.     

Dynamic analysis of an aeroelastic airfoil with freeplay nonlinearity by precise integration method based on Padé approximation

Nonlinear Dynamics - An improved precise integration method (PIM) incorporated with Padé approximation (PadéPIM) is proposed, and the aeroelastic behavior of an aeroelastic airfoil with...