用于结构动力时程分析的无条件稳定显式算法

发布时间：2018-04-24 12:33

本文选题：时程分析 + 显式算法　；参考：《华侨大学》2017年硕士论文

【摘要】：在结构动力时程分析中,直接积分法常常被用来求解结构运动方程。直接积分法可以分为显式算法和隐式算法。相比隐式算法,显式算法的计算效率较高,但较差的稳定性限制了其应用,尤其是结构进入非线性后,线性系统中无条件稳定的显式算法可能会退化为条件稳定。另一方面,部分算法拥有的数值阻尼可以将数值结果中虚假的高频震荡成分迅速剔除。但这些数值阻尼特性不能方便地引入到其他算法之中。针对这两个问题,本文开展了以下研究:(1)在状态空间下,将结构动力方程改写成了一阶常微分方程的形式,并利用积分因子法导出了该问题含有隐式积分项的精确解。利用Pade近似,对上述积分项进行近似处理,提出了一类显式单步算法,记为Pade-based算法。该算法对位移和速度都具有二阶精度,且在线性系统和非线性系统中均无条件稳定。(2)基于显式Adams法并利用Pade近似与高斯数值积分法,对上述隐式积分项进行近似处理,构造出了一类显式多步算法,记为Adams-based算法。该算法可以表达为具有任意高阶精度的一般形式,且在线性系统和非线性系统中都保持无条件稳定。通过控制Pade近似的形式,新算法的稳定性可以在A稳定和L稳定之间转换。(3)基于广义Pade近似,提出了一种用于构造可控数值阻尼的一般方法。该方法通过调整单一参数?,达到控制数值阻尼大小的目的。合适的数值阻尼可以使计算结果中虚假的高频震荡成分被剔除,同时保留真实的低频成分。
[Abstract]:In structural dynamic time history analysis, direct integration method is often used to solve structural equations of motion. Direct integration method can be divided into explicit algorithm and implicit algorithm. Compared with the implicit algorithm, the explicit algorithm is more efficient, but its application is limited by its poor stability. Especially when the structure is nonlinear, the unconditionally stable explicit algorithm in the linear system may degenerate into conditional stability. On the other hand, the partial numerical damping can quickly eliminate the false high frequency oscillation in the numerical results. However, these numerical damping characteristics can not be easily introduced into other algorithms. For these two problems, the following research is carried out: 1) in the state space, the structural dynamic equation is rewritten into the form of the first order ordinary differential equation, and the exact solution of the problem with implicit integral term is derived by using the integral factor method. By using Pade approximation, the above integral terms are approximated, and a class of explicit one-step algorithm is proposed, which is described as Pade-based algorithm. The algorithm has second-order accuracy for displacement and velocity, and is unconditionally stable in linear and nonlinear systems. Based on explicit Adams method and using Pade approximation and Gao Si numerical integration method, the implicit integral terms mentioned above are approximated. A class of explicit multistep algorithm is constructed, which is called Adams-based algorithm. The algorithm can be expressed as a general form with arbitrary higher order accuracy and is unconditionally stable in both linear and nonlinear systems. By controlling the form of Pade approximation, the stability of the new algorithm can be transformed between A stability and L stability. Based on the generalized Pade approximation, a general method for constructing controllable numerical damping is proposed. The numerical damping is controlled by adjusting a single parameter. With proper numerical damping, the false high frequency oscillation components can be eliminated and the true low frequency components can be retained.
【学位授予单位】：华侨大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：TU311.3

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