# Chebyshev谱元法模拟波动的两种集中质量矩阵Two Kinds of Lumped Mass Matrixes of Simulation of Wave Problem with Chebyshev Spectral Element Method

Abstract: Wave simulation is an important procedure for seismic analysis of engineering structure. Chebyshev Spectral Element Method is a kind of method to solve differential equations with high accuracy. Chebyshev Spectral Element Method has properties of high accuracy and high efficiency when it is used to simulate wave problem. The mass matrix of Chebyshev Spectral Element Method solving wave problem is consistent mass matrix which is space coupled. When it is used together with normal analysis procedure in time domain, it is time and space coupled. In each time step, it is needed to solve linear equations, and the efficiency is limited. In this paper, two kinds of lumped mass matrixes are derived based on Chebyshev Spectral Element Method consistent mass matrix. The math equations of the two mass matrixes are given. A procedure is given to solve one dimension wave problem. Chebyshev Spectral Element Method with the two matrixes is used in space domain, and central differential method is used in time domain. This procedure is decoupled in time and space domain. Numerical analysis shows that, this procedure has high accuracy, and it is not needed to solve linear equations in each time step, so compute efficiency can be largely enhanced. Within the two mass matrixes, math lumped mass matrix has higher accuracy than physical lumped mass matrix.

1. 前言

2. 波动模拟的Chebyshev谱元模型

$\rho \frac{{\partial }^{2}u}{\partial {t}^{2}}=\mu \frac{{\partial }^{2}u}{\partial {x}^{2}}$ (1)

$\underset{\Omega }{\int }\rho v\stackrel{¨}{u}\text{d}x+\underset{\Omega }{\int }\nabla v\mu \nabla u\text{d}x=0$ (2)

$\underset{\Omega }{\int }{\rho }^{e}v{\stackrel{¨}{u}}^{e}\text{d}x+\underset{\Omega }{\int }\nabla v{\mu }^{e}\nabla {u}^{e}\text{d}x=0$ (3)

${u}^{e}\left(x\right)=\underset{i=0}{\overset{N-1}{\sum }}{a}_{i}{A}_{i}\left(x\right)$ (4)

${\xi }_{i}=\mathrm{cos}\left(\frac{\pi i}{m}\right),\text{\hspace{0.17em}}i=0,1,\cdots ,m$ (5)

${u}^{e}\left(x\right)=\underset{i=0}{\overset{N-1}{\sum }}{u}^{e}\left({x}_{i}\right){l}_{i}\left(x\right)$ (6)

${M}^{e}{\stackrel{¨}{u}}^{e}\left(t\right)+{K}^{e}{u}^{e}\left(t\right)=0$ (7)

${M}_{ij}^{e}=\rho J\frac{4}{{N}^{2}{\stackrel{¯}{c}}_{i}{\stackrel{¯}{c}}_{j}}\underset{k,l=0}{\overset{N}{\sum }}\left(\frac{1}{{\stackrel{¯}{c}}_{k}{\stackrel{¯}{c}}_{l}}{T}_{k}\left({\epsilon }_{i}\right){T}_{l}\left({\epsilon }_{j}\right)\underset{-1}{\overset{1}{\int }}{T}_{k}\left(\epsilon \right){T}_{l}\left(\epsilon \right)\text{d}\epsilon \right)$ (8)

${K}_{ij}^{e}=J{\left({J}^{-1}\right)}^{2}\frac{4}{{N}^{2}{\stackrel{¯}{c}}_{i}{\stackrel{¯}{c}}_{j}}\underset{k,l=0}{\overset{N}{\sum }}\left(\frac{1}{{\stackrel{¯}{c}}_{k}{\stackrel{¯}{c}}_{l}}{T}_{k}\left({\epsilon }_{i}\right){T}_{l}\left({\epsilon }_{j}\right)\underset{-1}{\overset{1}{\int }}{{T}^{\prime }}_{k}\left(\epsilon \right){{T}^{\prime }}_{l}\left(\epsilon \right)\text{d}\epsilon \right)$ (9)

$J=\frac{\Delta x}{2}$${J}^{-1}=\frac{2}{\Delta x}$

$M\stackrel{¨}{u}\left(t\right)+Ku\left(t\right)=0$ (10)

3. 集中质量矩阵模型

3.1. 数学集中质量矩阵

$M=\left[\begin{array}{ccccc}0.0444& -0.0071& -0.0158& 0.0198& -0.0079\\ -0.0071& 0.2539& 0.0507& -0.0507& 0.0198\\ -0.0158& 0.0507& 0.3301& 0.0507& -0.0158\\ 0.0198& -0.0507& 0.0507& 0.2539& -0.0071\\ -0.0079& 0.0198& -0.0158& -0.0071& 0.0444\end{array}\right]$

${M}_{s}\left(i,i\right)=\underset{j=1}{\overset{N}{\sum }}M\left(i,j\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,N;j=1,2,\cdots ,N$ (11.a)

${M}_{s}\left(i,j\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,N;j=1,2,\cdots ,N;i\ne j$ (11.b)

3.2. 物理集中质量矩阵

${M}_{w}\left(i,i\right)=\frac{{x}_{i+1}-{x}_{i-1}}{2}\rho ,i=2,\cdots ,N-1$ (12.a)

${M}_{w}\left(i,i\right)=\frac{{x}_{2}-{x}_{1}}{2}\rho ,i=1$ (12.b)

${M}_{w}\left(i,i\right)=\frac{{x}_{N}-{x}_{N-1}}{2}\rho ,i=N$ (12.c)

${M}_{w}\left(i,j\right)=0,i=1,2,\cdots ,N;j=1,2,\cdots ,N;i\ne j$ (12.d)

${M}_{s}\stackrel{¨}{u}\left(t\right)+Ku\left(t\right)=0$ (13)

${M}_{w}\stackrel{¨}{u}\left(t\right)+Ku\left(t\right)=0$ (14)

4. 时空解耦模拟方案

${M}_{s}{\stackrel{¨}{u}}_{i}\left(t\right)+K{u}_{i}\left(t\right)=0$ (15)

${\stackrel{¨}{u}}_{i}=\frac{{u}_{i+1}-2{u}_{i}+{u}_{i-1}}{\Delta {t}^{2}}$ (16)

${M}_{s}\frac{{u}_{i+1}-2{u}_{i}+{u}_{i-1}}{\Delta {t}^{2}}+K{u}_{i}=0$ (17)

$\frac{1}{\Delta {t}^{2}}{M}_{s}{u}_{i+1}=-\left(K{u}_{i}-\frac{2}{\Delta {t}^{2}}{M}_{s}\right){u}_{i}-\frac{2}{\Delta {t}^{2}}{M}_{s}{u}_{i-1}$ (18)

5. 数值实验

$P\left(\tau \right)=16\left[G\left(\tau \right)-4G\left(\tau -\frac{1}{4}\right)+6G\left(\tau -\frac{1}{2}\right)-4G\left(\tau -\frac{3}{4}\right)+G\left(\tau -1\right)\right]$ (19)

$G\left(\tau \right)={\tau }^{3}H\left(\tau \right)$ (20)

Figure 1. Spatial waveform comparison

Figure 2. Comparison of displacement time history

Figure 3. Comparison of peak relative error

6. 结论

1) 两种集中质量矩阵都可以很好地模拟波动的传播过程，且具有较高的计算精度。

2) 数学集中质量矩阵的计算精度高于物理集中质量矩阵。在本文给定参数下，数学集中质量矩阵的相对误差约为物理集中质量矩阵的1/5~1/3。

3) 采用本文提出的两种集中质量矩阵的谱元法结合时间域中心差分法可以使得计算过程成为一个时空解耦的过程，不需每时步联立求解线性方程组，可以大幅度提高计算效率。

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