一种数值阻尼耗散可控的结构动力方程积分方法  

An Integration Method With Controllable Numerical Damping Dissipation for Structural Dynamic Equations

作　　者：刘伟 童小龙[1] 金蓉 LIU Wei;TONG Xiaolong;JIN Rong(College of Civil Engineering&Architecture,Hunan Institute of Science and Technology,Yueyang,Hunan 414006,P.R.China;Nanhu College,Hunan Institute of Science and Technology,Yueyang,Hunan 414006,P.R.China)

机构地区：[1]湖南理工学院土木建筑工程学院,湖南岳阳414006 [2]湖南理工学院南湖学院,湖南岳阳414006

出　　处：《应用数学和力学》2024年第7期922-935,共14页Applied Mathematics and Mechanics

基　　金：湖南省教育厅科学研究项目(22B0681);湖南省自然科学基金(2023JJ50281);2022年国家级大学生创新创业计划(S202212658006)。

摘　　要：数值耗散是数值积分方法的重要特性,直接影响数值仿真结果的准确性.对于含有虚假高频振动的动力系统,数值耗散能够改善数值仿真结果,但是对于具有真实高频振动的动力系统,数值耗散则会导致计算结果失真.该研究针对结构动力系统的求解,提出了一种数值耗散可控的两子步隐式数值积分方法.基于理论推导,详细介绍了新积分方法的谱半径、稳定性、振幅衰减和周期延长等数值特性.新隐式积分方法通过算法参数α能够对高频虚假振动数值耗散完全可控,相应的耗散比例为1-α,其中-1≤α≤1.通过单自由度动力系统、高频虚假振动系统和多自由度非线性弹簧质量系统三个典型算例,分别证明了新隐式积分方法在计算精确性、高频数值耗散和非线性求解能力方面的优越性.Numerical dissipation is an important characteristic of numerical integration methods,which directly affects the accuracy of numerical simulation results.Numerical dissipation can improve numerical simulation re-sults for dynamic systems with spurious high-frequency vibrations,but it can also cause distorted calculation re-sults for dynamic systems with real high-frequency vibrations.A 2-sub-step implicit numerical integration meth-od was proposed with controllable numerical damping dissipation to solve structural dynamic systems.Through theoretical derivations,the numerical properties of the new integration method,including the spectral radii,stability,amplitude decay,and period elongation,were introduced in detail.The new implicit integration meth-od can utilize algorithm parameterαto control the numerical dissipation of spurious high-frequency vibration,with a corresponding dissipation ratio of 1-α,where-1≤α≤1.The advantages of the new method in terms of the computational accuracy,the high-frequency numerical dissipation,and the nonlinear solving ability were demonstrated through 3 typical examples of a 1-DOF dynamic system,a high-frequency spurious vibration system,and a multi-DOF nonlinear spring-mass system.

关 键 词：结构动力方程 隐式积分方法 数值阻尼 放大矩阵 谱半径 

分 类 号：O39[理学—工程力学]