内域介质波动数值模拟的若干研究
发布时间：2018-11-28 20:42
【摘要】：内域介质波动数值模拟是大型结构-基础相互作用分析中的一个重要方面。本文基于有限元-有限差分法及ABAQUS/Explicit积分格式,研究离散网格中波动的传播规律,提出计算精度的分析方法及改善数值频散的方案。主要工作有:1.对于ABAQUS/Explicit积分格式假定的正确性进行验证;建立了有阻尼体系显式积分格式计算精度的一种新方法,并应用于ABAQUS/Explicit积分格式的精度分析。推得单自由度体系质点振动的周期延长率、峰值递减率随频率、阻尼比及计算时间步长变化的解析式。取工程常用的阻尼比,对积分格式的计算精度进行了定量分析。2.基于ABAQUS/Explicit积分格式应用于显式有限元-有限差分法时的递推形式,推导出传递函数矩阵,给出离散有限元网格中波动的传播规律。无阻尼情况下存在截止频率,大于截止频率的波动成分无法传播;有阻尼情况时,小于Nyquist频率的波动成分可以在离散网格中传播,接近以及大于截止频率ωu的波动成分的幅值会迅速衰减。时间积分步长At取值越小或者空间步距Ax取值越大,截止频率ωu越小,波动中的高频成分衰减越快,积分格式的能耗越大。3.通过与理论解进行对比来评价算法的计算精度,结果表明:低频区时数值计算结果和理论解很接近;阻尼系数β取值比较小时,数值解的幅值衰减项与理论解拟合程度不高,随着β取值增大,计算结果精确性提高。4.通过计算相速度和群速度分析了一维有阻尼体系数值模拟的频散效应。采用集中质量矩阵时,在满足稳定性的条件下,时间步长取值越小,对波动中高频成分的抑制越明显,但波动中的频散也会增大;采用一致质量矩阵时,时间步长At取值越小,数值模拟的结果精度越高,波动的频散越小。采用不同的质量矩阵形式时时间步长的选择很重要。一致质量矩阵时,阻尼系数β越大,数值模拟的波动频散越大,这与理论上波动频散规律相同,但是β对集中质量矩阵的频散影响相反。5.将集中质量矩阵与一致质量矩阵进行线性组合,这种方法能提高计算结果的精度并且有效地压制频散,最优的线性组合系数为0.5。
[Abstract]:Numerical simulation of internal medium wave is an important aspect in the analysis of large-scale structure-foundation interaction. Based on the finite element finite difference method and ABAQUS/Explicit integral scheme, the propagation law of wave in discrete mesh is studied, and the analytical method of calculation accuracy and the scheme of improving numerical dispersion are put forward in this paper. The main work is: 1. The correctness of the assumption of ABAQUS/Explicit integral scheme is verified and a new method for calculating the accuracy of explicit integral scheme of damped system is established and applied to the accuracy analysis of ABAQUS/Explicit integral scheme. An analytical formula for the variation of the period prolongation rate of particle vibration and the peak decline rate with frequency damping ratio and calculating time step of single degree of freedom system is derived. Taking the damping ratio commonly used in engineering, the calculation accuracy of integral scheme is quantitatively analyzed. 2. Based on the recursive form of ABAQUS/Explicit integral scheme applied to explicit finite-difference finite element method, the transfer function matrix is derived, and the propagation law of wave in discrete finite element mesh is given. In the case of undamped, the fluctuation component larger than the cutoff frequency can not propagate because of the existence of cutoff frequency. In the case of damping, the wave components smaller than the Nyquist frequency can propagate in the discrete grid, and the amplitude of the wave components close to and larger than the cutoff frequency 蠅 u will decay rapidly. The smaller the At value of the time integral step or the greater the Ax value of the spatial step, the smaller the cutoff frequency 蠅 u, the faster the attenuation of the high frequency component in the fluctuation, and the greater the energy consumption of the integral scheme. The accuracy of the algorithm is evaluated by comparing with the theoretical solution. The results show that the numerical results in the low frequency region are very close to the theoretical solution. When the damping coefficient 尾 is small, the amplitude attenuation term of the numerical solution is not well fitted with the theoretical solution. With the increase of 尾 value, the accuracy of the calculation results is improved. 4. The dispersion effect of numerical simulation of one-dimensional damped system is analyzed by calculating phase velocity and group velocity. When the lumped mass matrix is adopted, the smaller the time step is, the more obvious the suppression of the high frequency component in the wave is, but the frequency dispersion in the fluctuation will increase. When the uniform mass matrix is adopted, the smaller the time step At is, the higher the accuracy of the numerical simulation results is and the smaller the dispersion of the fluctuation is. The choice of time step is very important when adopting different mass matrix forms. In the case of uniform mass matrix, the larger the damping coefficient 尾, the larger the wave dispersion of numerical simulation, which is the same as the theoretical wave dispersion law, but the effect of 尾 on the dispersion of lumped mass matrix is opposite. 5. The linear combination of lumped mass matrix and uniform mass matrix can improve the accuracy of the calculation results and suppress dispersion effectively. The optimal linear combination coefficient is 0.5.
【学位授予单位】：中国水利水电科学研究院
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O302
[Abstract]:Numerical simulation of internal medium wave is an important aspect in the analysis of large-scale structure-foundation interaction. Based on the finite element finite difference method and ABAQUS/Explicit integral scheme, the propagation law of wave in discrete mesh is studied, and the analytical method of calculation accuracy and the scheme of improving numerical dispersion are put forward in this paper. The main work is: 1. The correctness of the assumption of ABAQUS/Explicit integral scheme is verified and a new method for calculating the accuracy of explicit integral scheme of damped system is established and applied to the accuracy analysis of ABAQUS/Explicit integral scheme. An analytical formula for the variation of the period prolongation rate of particle vibration and the peak decline rate with frequency damping ratio and calculating time step of single degree of freedom system is derived. Taking the damping ratio commonly used in engineering, the calculation accuracy of integral scheme is quantitatively analyzed. 2. Based on the recursive form of ABAQUS/Explicit integral scheme applied to explicit finite-difference finite element method, the transfer function matrix is derived, and the propagation law of wave in discrete finite element mesh is given. In the case of undamped, the fluctuation component larger than the cutoff frequency can not propagate because of the existence of cutoff frequency. In the case of damping, the wave components smaller than the Nyquist frequency can propagate in the discrete grid, and the amplitude of the wave components close to and larger than the cutoff frequency 蠅 u will decay rapidly. The smaller the At value of the time integral step or the greater the Ax value of the spatial step, the smaller the cutoff frequency 蠅 u, the faster the attenuation of the high frequency component in the fluctuation, and the greater the energy consumption of the integral scheme. The accuracy of the algorithm is evaluated by comparing with the theoretical solution. The results show that the numerical results in the low frequency region are very close to the theoretical solution. When the damping coefficient 尾 is small, the amplitude attenuation term of the numerical solution is not well fitted with the theoretical solution. With the increase of 尾 value, the accuracy of the calculation results is improved. 4. The dispersion effect of numerical simulation of one-dimensional damped system is analyzed by calculating phase velocity and group velocity. When the lumped mass matrix is adopted, the smaller the time step is, the more obvious the suppression of the high frequency component in the wave is, but the frequency dispersion in the fluctuation will increase. When the uniform mass matrix is adopted, the smaller the time step At is, the higher the accuracy of the numerical simulation results is and the smaller the dispersion of the fluctuation is. The choice of time step is very important when adopting different mass matrix forms. In the case of uniform mass matrix, the larger the damping coefficient 尾, the larger the wave dispersion of numerical simulation, which is the same as the theoretical wave dispersion law, but the effect of 尾 on the dispersion of lumped mass matrix is opposite. 5. The linear combination of lumped mass matrix and uniform mass matrix can improve the accuracy of the calculation results and suppress dispersion effectively. The optimal linear combination coefficient is 0.5.
【学位授予单位】：中国水利水电科学研究院
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O302
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