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

摘要: 波动模拟是工程结构抗震分析的重要手段之一。Chebyshev谱元法是一种高精度的微分方程求解方法，模拟波动问题时具有高精度和高效率的特点，因此受到广泛关注。Chebyshev谱元法求解波动问题时的质量矩阵是一致质量矩阵，即空间耦合矩阵，当和时域分析方法联用时，为时空耦合格式，每步计算需要联立求解线性方程组，计算效率受到了制约。本文在Chebyshev谱元法一致质量矩阵的基础上导出了两种集中质量矩阵：数学集中质量矩阵和物理集中质量矩阵，给出了两者的数学表达，并采用这两种集中质量矩阵下的Chebyshev谱元法结合时域中心差分法求解一维波动问题，此种模拟方案为时空解耦方法。数值分析表明，采用Chebyshev集中质量矩阵配合时域中心差分法模拟波动的方案具有较高的计算精度，并且每步计算不需要联立求解线性方程组，可以大幅度提高计算效率。其中，数学集中质量矩阵的计算精度要高于物理集中质量矩阵。

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.