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3D harmonic oscillator with stable numerical solution[J].计算力学学报,2020,37(1):119~130

3D harmonic oscillator with stable numerical solution
3D harmonic oscillator with stable numerical solution

DOI：10.7511/jslx20180912001

 作者 单位 E-mail 肖国峰 中国科学院 武汉岩土力学研究所 岩土力学与工程国家重点试验室, 武汉 430071 gfxiao@whrsm.ac.cn

谐振子广泛应用于物理系统的描述和物理现象的数值模拟。由于二维或三维谐振子对于系统参数、初始条件和边界条件的高度敏感性，很多物理过程的动力学模拟都会出现数值解不稳定的现象。近年来发展的无网格法、物质点法和近场动力学法等数值模拟方法均绕开了对固体材料固有构形的量化描述。本文引入了定常耗散项和弹簧耗散项，考虑随机微扰效应，提出了一种三维耗散谐振子，构建了基于蛙跳法和边界松弛技术的数值积分算法。应用三维谐振子构建了耗散型弹簧摆、简化弦和简化梁三个模型，设定了13个定解问题进行动力学模拟。数值试验结果表明，三维谐振子是稳定的。基于简化弦模型，模拟了拉弦、放弦和重弦三个有界弦振动问题；其中，拉弦和放弦问题成功模拟了有界弦的三维振形；重弦问题模拟再现了悬链线在水平向的微幅振荡现象。基于简化梁模型，模拟了三维梁的拉伸、剪切和扭转行为，验证了三维谐振子对于非线性大变形问题动力学模拟的描述能力，及其对外部作用的高速响应能力。本文方法可以为弦振动问题和材料力学非线性大变形问题的动力学模拟提供一条可行的实现途径。

The harmonic oscillator is widely used in the description of physical system and numerical simulation of physical phenomena.However,due to the high sensitivity of the harmonic oscillator to the system parameters,initial conditions and boundary conditions,many physical processes cannot be numerically solved by dynamic simulations.In recent years,the meshless method,the material point method and the peridynamics method have all bypassed the quantitative description of the natural structure of solid materials.In this paper,a 3D dissipative harmonic oscillator is proposed by using the constant dissipative term and the spring dissipation term,considering the random perturbation effect,and a numerical integration algorithm based on the leapfrog method and the boundary relaxation technique is constructed.Three models,dissipative spring pendulum,simplified string and simplified beam,are constructed by using 3D harmonic oscillator,and 13 definite solution problems are set up to dynamical simulation.The results of numerical experiments show that the 3D harmonic oscillator is stable.Based on the simplified string model,three problems of the bounded string vibrations such as plucking,release and heavy chord are simulated,in which the problem of heavy string simulates the phenomenon of micro-amplitude oscillation in horizontal direction of catenary.Based on the simplified beam model,the tensile,shearing and torsional behavior of a 3D beam is simulated,and the description ability of the 3D harmonic oscillator for the nonlinear large deformation problem is validated,and its high speed response to external action is verified.This work can provide a feasible way to simulate the string vibration problem and the nonlinear large deformation problem of material mechanics.