摘要
对于大型结构及复杂场地地震反应特别是非线性地震反应数值模拟计算,显式方法相对于隐式方法在计算量上有明显的优势。李小军等人给出的有阻尼体系动力方程求解的显式积分格式不仅具有与中心差分法相当的二阶计算精度,而且它对于任意阻尼体系的动力问题均能实现显式格式求解。另外,利用这一积分格式与局部透射边界相结合进行无限介质波动的数值分析时,积分格式所具有的数值计算能耗特性可以起到控制高频失稳的作用。所以,这种积分格式在处理大型结构及复杂场地地震反应问题方面,不失为一种值得利用的积分格式。
本文本着进一步探讨这一显式积分格式特性的目的,通过理论推导论证及数值试验计算结果分析,着重开展了以下几个方面的研究工作,并初步得到了一些有意义的结果:
1.显式积分格式能耗特性的理论分析
以单自由度动力方程求解为例,基于理论分析推导了这种显式积分格式的数值计算能耗特性以及其随频率和物理阻尼的变化规律,并进行数值计算,展示了这种数值计算能耗特性在不同情况下的特征的理论推证结果。研究表明:①该积分格式引入的人工阻尼随频率的增大而逐渐增大,并且这种趋势随物理阻尼的增加而更加明显;②在求解单自由度动力方程时,在保证该积分格式稳定的情况下,存在一个频率值ω_0,当频率小于该值时,数值解为振动解;当频率大于该值时,数值解为振荡(非振动)衰减解,且振荡的形式随频率与物理阻尼的不同而不同。
2.显式积分格式在一维波动有限元模拟中的应用分析
本文以该显式积分格式与局部透射边界相结合应用于一维波动的有限元模拟为例,分析了积分格式对波动在离散网格中的传播特性的影响,并且提出了用循环系数来分析积分格式的数值计算能耗特性在抑制局部透射人工边界引入的高频失稳方面的作用,然后分别使用代表波源问题和散射问题的两种模型,进行数值试验计算,验证了理论分析的结果。得到的结论包括:
①该显式积分格式应用于波动的数值模拟时也存在截止频率ω_u,并且这个截止频率ω_u即为空间离散后波动在网格中传播的截止频率ω_c;②理论分析
For numerical computation of earthquake responses of large-scale structure and complex site, especially non-linear response, the use of explicit algorithms is generally preferred over implicit algorithms in view of the amount of computational time. An explicit integration formula used to solve dynamic equation of damped structure, suggested by Li Xiaojun and others, has not only two-order calculating accuracy as that of the central differential integration formula, but also general applicability. In one hand, the integration formula can be applied to solve dynamic problem of system with any damping, on the other hand, when the integration formula is applied to the numerical analysis of wave motion in infinite space, the numerical dissipation of the formula can depress or eliminate the high-frequency instability induced by Local Transmitting Boundary. So the explicit formula is an ideal integration formula for solving the earthquake responses of large-scale structure and complex site.The objective of this dissertation is to further study characteristics of the explicit integration formula in order to make good use of the formula. Trough theoretical studies and numerical experiments, the following problems are analyzed, and some applicable results are obtained.1. The numerical dissipation of this explicit integration formulaThe relationship between frequency, damping and algorithmic dissipation is studied by using this explicit integration formula in solving dynamic equation of single-degree of freedom. Besides, the influences of the algorithmic dissipation on numerical computation are showed by numerical experiments. Research results show that: a) the numerical dissipation of this explicit integration formula is stronger in higher modes, the larger the real damping is, this phenomenon is moreobvious; b) There is a cut-off frequency ω0 when this integration formula is applied to solveSDF dynamic problems under computation stability condition, if frequency is larger than ω0, thesolution become oscillation, and the kind of oscillation is different to diverse damping and frequency.2. The application of this explicit integration formula in wave motion simulation of a one-dimensional discrete finite element modelIn this dissertation, the effect of this integration formula on wave propagation in discrete finite element model is analyzed by studying one-dimensional finite element simulation of wave motion
引文
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