Early short time transient response of finite L-shaped Mindlin plate

The early short time transient response of an L-shaped plate is investigated by the method of reverberation ray matrix (MRRM) and the generalized ray method (GRM). The MRRM and GRM have been successfully used to study the transient elastic wave transmitting in the beams, planar trusses, space frames and infinite layered media. In the present paper, these two methods are extended to investigate the flexural wave propagation in a finite L-shaped Mindlin plate. The reverberation-ray matrix representing the multi-reflected and scattered waves in the L-shaped plate is presented. The rays in the GRM propagating in the L-shaped plate are determined by the shock source and the boundary conditions. From the numerical results, it indicates that the early short time transient responses of L-shaped plate calculated by the MRRM are the same as those by the GRM. The early short time transient acceleration responses are very large, while the early short time transient displacement responses are very small for the L-shaped plate subjected to the unit impulse. Furthermore, the effects of the different shock signals and the plate thickness on the early short time transient responses of the finite L-shaped Mindlin plates are also analyzed.     

基于绝对节点坐标法的弹性线方法研究

相比于浮动坐标系法, 绝对节点坐标法(absolute nodal coordinateformulation, ANCF)在处理柔性体非线性大变形问题上具有显著优势,ANCF将单元节点坐标定义在全局坐标系下,采用斜率矢量代替节点转角坐标, 具有常数质量阵,不存在科氏离心力等优点, 然而弹性力阵为非线性项,其求解将比较耗时且占用资源. 据此, 在弹性力求解方法中,引入弹性线方法(elastic line method, ELM),该方法将格林--拉格朗日应变张量定义在中心线上,采用曲率公式来定义弯曲应变, 转角公式来定义扭转应变.同时采用有限元法对三维柔性梁位移场进行离散,求解梁单元常数质量阵、广义刚度阵、广义力阵,进而得到单元的动力学方程, 通过转换矩阵得到三维梁的动力学方程.接着从理论上指出连续介质力学方法(continuum mechanics method,CMM)和弹性线方法在求解弹性力上的不同点, 并编制动力学仿真软件.最后分别采用连续介质力学方法和弹性线方法对柔性单摆以及履带式车辆的动力学问题进行仿真分析,结果表明:弹性线方法能在保证精度的前提下有效提高计算效率.      

求解多层弹性半空间轴对称问题的精确刚度矩阵法

本文首先从弹性力学的基本方程出发，利用Hankel积分变换等数学手段，推导出了单层弹性半空问轴对称问题的刚度矩阵，然后按传统的有限元方法组成总体刚度矩阵。通过求解由总体刚度矩阵所构成的代数方程和Hankel积分逆变换就可解出静荷载作用下多层弹性半空间轴对称问题的精确解。由于刚度矩阵的元素中不含有正指数项，计算时不会出现溢出的现象，从而克服了传递矩阵法的缺点。由于在推导过程中摒弃了应力函数的选择，使得问题的求解更加理论化和合理化，同时也为进一步研究这类问题如温度场，动力学等方向奠定了理论基础。最后，文中还给出了计算实例来证明推导结果的准确性。     

Fractal patterns from the dynamics of combined polynomial root finding methods

A solid tetrahedral finite element employing the absolute nodal coordinate formulation (ANCF) is presented. In the ANCF, the mass matrix and vector of the generalized gravity forces used in the equations of motion are constant, whereas the vector of the elastic forces is highly nonlinear. The proposed solid element uses translations of nodes as sets of nodal coordinates. The tetrahedral shape of the element makes it suitable for modeling structures with complex shapes, and the small number of the degrees of freedom enables good performance and versatile application to problems of structural dynamics. The accuracy and convergence of the element were investigated using statics and dynamics benchmarks and a practical industry application.     

On generalized Kelvin solutions in a multilayered elastic medium

This paper presents fundamental singular solutions for the generalized Kelvin problems of a multilayered elastic medium of infinite extent subjected to concentrated body force vectors. Classical integral transforms and a backward transfer matrix method are utilized in the analytical formulation of solutions in both Cartesian and cylindrical coordinates. The solution in the transform domain has no functions of exponential growth and is invariant with respect to the applied forces. The convergence of the solutions in the physical domain is rigorously and analytically verified. The solutions satisfy all required constraints including the basic equations and the interfacial conditions as well as the boundary conditions. In particular, singular terms of the generalized Kelvin solutions associated with the point and ring types of concentrated body force vectors are obtained in exact closed-forms via an asymptotic analysis. Numerical results presented in the paper illustrate that numerical evaluation of the solutions can be easily achieved with very high accuracy and efficiency and that the layering material inhomogeneity has a significant effect on the elastic field.The Canadian Government right to retain a non-exclusive, royalty-free licence in and to any copyright is acknowledged.     

层状弹性半空间非轴对称动力问题的奇异解

在柱坐标系下，利用关于方位角的Ｆｏｕｒｉｅｒ变换及关于径向的Ｈａｎｋｅｌ变换，将弹性力学基本方程组转化为非齐次的一阶常微分方程组的标准形式．采用求解微分方程组的矩阵法，建立了介质层的传递矩阵．由层间完全接触条件，导出了在任意埋藏源作用下层状弹性半空间频域奇异解，时域奇异解可通过关于频率的Ｆｏｕｒｉｅｒ积分得到．该方法可应用到固体、流体层的情况     

层状横观各向同性地基上异性矩形薄板的弯曲解析解

选用更具广泛性的层状横观各向同性弹性地基模型,来分析四边自由各向异性矩形地基板的弯曲解析解。先基于直角坐标下横观各向同性体的静力胡海昌通解,借助双重傅里叶变换及矩阵传递法,获得层状横观各向同性地基的静力位移场和应力场;然后将异性薄板的弯曲控制方程,与基于层状横观各向同性弹性地基的位移解建立的板与地基变形协调方程相结合,先按对称性分解,再用三角级数法,得出层状横观各向同性弹性地基上四边自由各向异性矩形薄板的弯曲解析解,包括地基反力、板的挠度及板的内力的解析表达式。克服了数值法的弊端,取消了对地基反力的假设,且避免了矩阵指数函数的计算;同时考虑了地基的层状性及板和地基的各向异性,从而得到板的内力及地基反力更切实际的分布规律。算例结果与文献的有限元结果吻合良好,证明本文方法是切实可行的。     

Theory of scattering from multilayered bodies of arbitrary shape

Green 's functions and boundary integral equation methods are used to derive a matrix set of equations for scattering from a multilayered homogeneous elastic body embedded in an infinite elastic material. The surfaces separating the layers have arbitrary shape. The formalism for the three-layer material is derived in detail and generalized to N-layers. A matrix factorization method (MFM) is shown to considerably simplify the computational problem. The relation to the problems of acoustic waves in fluids and electromagnetic waves in a dielectric material is briefly indicated.     

THE NEW CALCULATION METHOD FOR CONTACT FORCE IN DRILLSTRING MECHANICS

钻柱在井底的接触力计算是一个复杂的非线性问题. 选取较为精确的三维钻柱力学模型（简称ABIS）作为研究对象,力图从复杂的求解过程中探索出更直观的方法. 从钻柱某个节点的接触力与位移关系,推广到该节点接触力与钻柱其他所有节点的位移均存在同样的关系,进而建立了整根钻柱位移与作用在钻柱上的所有接触力的更直观的柔度关系式. 据此提出新的接触力计算方法,算例结果表明该方法显著提高迭代速度.     

轴对称横观各向同性层状弹性半空间问题受力分析

本文从柱坐标系弹性力学基本方程出发,将位移场和应力场在径向进行Ｈａｎｋｅｌ变换,利用常微分主程求解原理,直接得出在轴对称荷载作用下横观各向同性半无限弹性空间的位移场,利用此结果推导出单层元的刚度矩阵。     

精化杂交压电固体单元的研究

基于力、电耦合问题的三类交量广义交分原理，提出了广义杂交压电单元列式。为了进一步改进单元的性能和保证单元能够通过分片检验，通过引入非协调模式、放松电学方程约束条件和单元间的弱连续性条件，建立了新的、修正的广义交分原理，在此基础上成功地引入了应力、应交的正交化插值模式，从而建立了精化杂交压电单元法，它继承了常规精化杂交单元的全部优点。文中所推导的八节点精化杂交压电固体单元列式完全避免了矩阵求逆运算，较广义杂交压电单元和杂交应力压电单元均显著提高了计算效率。数值算例表明，与同类型其他单元相比，该单元明显具有更好的对歪斜网格的适应性。     

流固耦合地震波动问题的显式谱元模拟方法

流固耦合地震波动问题主要研究由流体和固体构成的复杂系统中地震波传播特性及其规律. 传统模拟方法中一般以声波方程、弹性波方程的数值解分别描述理想流体和弹性固体中的波动, 并实时地处理两种不同性质介质之间的相互耦合作用, 数值格式复杂且限制数值模拟精度与计算效率. 本文采用谱元法结合多次透射公式人工边界条件实现了一种流固耦合地震波动问题的高阶显式数值计算方法. 该方法利用了流固耦合问题统一计算框架,可将饱和多孔介质的Biot波动方程分别退化为理想流体的声波方程和弹性固体的弹性波方程. 通过P波垂直入射的水平成层理想流体-饱和多孔介质-弹性固体场地模型、P波斜入射的不规则层状界面以及任意形状界面的理想流体-饱和多孔介质-弹性固体场地模型等三个算例, 与传递函数法解析解以及集中质量有限元法计算结果进行对比分析, 证明了本文方法的正确性与有效性. 数值模拟结果表明, 本文方法相较传统有限元法可以少得多的节点数量获得更高的数值精度, 并且在较宽的频率范围内都能可靠地模拟出流固耦合系统的动力响应, 充分体现出本文方法兼顾高精度、计算效率和复杂场地建模灵活的特点.      

Numerical solution of three-dimensional static problems of elasticity for a body with a noncanonical inclusion

A boundary-element scheme is proposed for the numerical determination of the stress-strain state of a three-dimensional composite body, which is an elastic inclusion of arbitrary shape perfectly bonded to an infinite elastic matrix. The scheme involves the reduction of the original problem to six boundary integral equations for the components of interfacial displacements and forces and the boundary-element parametrization and discretization of these equations using generalized Gaussian integrals and topological maps with regularizing Jacobians. Numerical results are obtained for a cylindrical inclusion with rounded ends in a matrix subject at infinity to constant forces acting along this fiber. The influence of the length-to-radius ratio of the fiber and the ratio of the elastic moduli of the matrix and fiber on the stresses is examined __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 27–35, April 2007.     

分形煤层气藏数值模拟研究——Douglas-Jones预估-校正法的应用

传统的煤层气动力学模型多建立在欧几里得基础上,难以描述煤层气孔隙结构的复杂性和形状的不规则性。为此,以分形理论为基础,通过引入煤层基质和裂缝的分形维数来刻画煤层气孔隙结构的复杂性和吸附特性,建立了双重分形介质渗流模型,采用Douglas-Jones预估-校正法对非线性方程组进行离散,获得了无限大地层和有限地层定产量生产时拟稳态吸附模型的差分方程,求得数值解。结果表明,Douglas-Jones预估-校正法可以有效解决这类非线性模型的求解问题,获得无限大地层定产量生产时变形双重分形介质模型的数值解;分析各种分形参数下的煤层压力动态,做出了典型压力曲线图。对无限大地层,初期分形维数对压力影响不大,后期分形维数越小,压力越高。对有限地层,初期分形维数的影响明显,且分形维数越大,压力越低。压力随分形指数的减小量呈现先增大后减小的趋势,在末期压力平稳趋向同一值。     

Transient impact responses of laminated composite cylindrical shells

The generalized ray method (GRM) has been successfully used to study the transient elastic wave transmitting in the beams, planar trusses, space frames and infinite layered media. In this letter, the GRM is extended to investigate the early short time transient responses of laminated composite cylindrical shells under impact load. By using the Laplace transformation and referring to the boundary conditions, the ray groups transmitting in the finite laminated cylindrical shells under the shock load are obtained and the transient response related to each ray group can be derived via FFT algorithm. From the numerical results, it is shown that the early short time transient accelerations of the laminated composite cylindrical shell under impact loads are very large. But the short time transient shear strain and displacement are very small.     

基于人工弹簧模型的周期结构带隙计算方法研究

能量法具有将求解微分方程边值问题转化为泛函极值问题的优点,故而在结构动力学分析中被广泛使用, 近年来也被引入到周期结构带隙计算中. 然而,由于周期结构边界条件相对复杂,采用传统能量法(如Rayleigh-Ritz法)分析时位移函数构造难度大;且由于位移函数中包含波数项,扫描波数计算带隙的过程中质量、刚度矩阵需不断重算, 导致计算量较大. 鉴于此,本文对传统能量法进行改进,通过引入人工弹簧来模拟包含周期边界在内的各类边界条件,可将边界约束转化为人工弹簧的弹性势能,故而各能量分部中仅有周期边界弹性势能包含波数项,扫描波数时仅需重新计算与其对应的刚度矩阵,其余的质量、刚度矩阵只需要计算一次, 继而显著降低了计算量. 研究结果表明,本文方法准确、可靠, 且相较于传统能量法, 本文方法的计算效率更高,随着结构质量、刚度矩阵的维度增大, 或者扫描波数点数的增多,本文方法计算效率优势更加明显. 此外, 人工弹簧模型使用灵活、便捷,可进一步地拓展到更为复杂的周期性组合结构带隙分析中.      

空间刚架结构的广义逆力法

在结构计算中，根据算法中所采用的基本未知量的不同．结构分析方法可以分为力法、位移法和混合法。其中位移法由于适宜计算机处理而在结构计算领域得到了广泛的应用．经典力法相比之下应用就远不如位移法普遍，虽然力法本身在力学上有其独特的优势。广义逆矩阵做为一种较新的数学工具，自二十世纪五十年代诞生以来正日益表现出越来越旺盛的生命力。广义逆力法就是一种基于力法和广义逆矩阵理论的新的迭代算法。这种算法是一种完全适合计算机处理的力法方法。该算法的思路以及对于求解线弹性空间刚架结构问题的具体公式均在文中给出并给出了算例。从算例计算结果可以看到广义逆力法有着较好的计算效率和计算精度。该算法的提出为力法在计算机计算领域的应用开拓了新的发展空间。该算法在材料非线性问题和结构并行计算方面也有着较好的应用前景。     

A SEMI－ANALYTICAL AND SEMI－NUMERICAL METHOD FOR SOLVING 2－D SOUND－STRUCTURE INTERACTION PROBLEMS

Based on the transfer matrix method and the virtual source simulation technique,this paper proposes a novel semi-analytical and semi-numerical method for solving 2-D sound-structure interaction problems under a harmonic excitation. Within any integration segment,as long as its length is small enough, along the circumferential curvilinear coordinate, the nonhomogeneous matrix differential equation of an elastic ring of complex geometrical shape can be rewritten in terms of the homogeneous one by the method of extended homogeneous capacity proposed in this paper. For the exterior fluid domain, the multi-circular virtual source simulation technique is adopted. The source density distributed on each virtual circular curve may be ex-panded as the Fourier‘‘s series. Combining with the inverse fast Fourier transformation, a higher accuracy and efficiency method for solving 2-D exterior Helmholtz‘s problems is presented in this paper. In the aspect of solution to the coupling equations, the state vectors of elastic ring induced by the given harmonic excitation and generalized forces of coefficients of the Fourier series can be obtained respectively by using a high precision integration scheme combined with the method of extended homogeneous capacity put forward in this paper. According to the superposition principle and compatibility conditions at the interface between the elastic ring and fluid, the algebraic equation of system can be directly constructed by using the least square approximation. Examples of acoustic radiation from two typical fluid-loaded elastic rings under a harmonic concentrated force are presented. Numerical results show that the method proposed is more efficient than the mixed FE-BE method in common use.