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徐亚洲, 田锐. 广义概率密度演化方程的Chebyshev拟谱法. 力学学报, 待出版. DOI: 10.6052/0459-1879-23-633
引用本文: 徐亚洲, 田锐. 广义概率密度演化方程的Chebyshev拟谱法. 力学学报, 待出版. DOI: 10.6052/0459-1879-23-633
Xu Yazhou, Tian Rui. A pseudo-spectral method for the generalized density evolution equation. Chinese Journal of Theoretical and Applied Mechanics, in press. DOI: 10.6052/0459-1879-23-633
Citation: Xu Yazhou, Tian Rui. A pseudo-spectral method for the generalized density evolution equation. Chinese Journal of Theoretical and Applied Mechanics, in press. DOI: 10.6052/0459-1879-23-633

广义概率密度演化方程的Chebyshev拟谱法

A PSEUDO-SPECTRAL METHOD FOR THE GENERALIZED DENSITY EVOLUTION EQUATION

  • 摘要: 概率密度演化方法(probability density evolution equation, PDEM)为非线性随机结构的动力响应分析提供了新的途径. 通过PDEM获得结构响应概率密度函数(probability density function, PDF)的关键步骤是求解广义概率密度演化方程(generalized probability density evolution equation, GDEE). 对于GDEE的求解通常采用有限差分法, 然而, 由于GDEE是初始条件间断的变系数一阶双曲偏微分方程, 通过有限差分法求解GDEE是可能会面临网格敏感性问题、数值色散和数值耗散现象. 文章在全局逼近的角度出发, 基于Chebyshev拟谱法为GDEE构造了全局插值格式, 解决了数值色散、数值耗散以及网格敏感性问题. 考虑GDEE的系数在每个时间步长均为常数, 推导了GDEE在每一个时间步长内时域上的序列矩阵指数解. 由于序列矩阵指数解形式上是解析的, 从而很好地克服了数值稳定性问题. 两个数值算例表明, 通过Chebyshev拟谱法结合时域的序列矩阵指数解求解GDEE得到的结果与精确解以及Monte Carlo模拟的结果非常吻合, 且数值耗散和数值色散现象几乎可以忽略. 此外, 拟谱法具有高效的收敛性且序列矩阵指数解不受(courant-friedrichs-lewy, CFL)条件的限制, 因此该方法具有良好的数值稳定性和计算效率.

     

    Abstract: Probability density evolution method (PDEM) provides a new approach for the dynamic response analysis of nonlinear stochastic structures. One of the key steps in obtaining the probability density function (PDF) of the dynamic response of nonlinear stochastic structures through PDEM is to solve the generalized probability density evolution equation (GDEE). The finite difference method is usually used to solve GDEE. Unfortunately, since GDEE is a first order hyperbolic partial differential equation with variable coefficients and discontinuous initial conditions, solving GDEE by finite difference method may encounter grid sensitivity problems, numerical dispersion and numerical dissipation. In this work, from the perspective of global approximation, a global interpolation scheme was constructed for GDEE based on the Chebyshev pseudo-spectral method, which overcame the problems of numerical dispersion, numerical dissipation, and grid sensitivity. Considering that the coefficients of GDEE remain constant at each time step, a sequential matrix exponential solution for GDEE in the time domain was derived. Since the sequent matrix exponential solution of GDEE is analytically formulated, this method can overcome numerical stability issues. The numerical results from two examples demonstrate that GDEE can be effectively solved using a combination of the sequential matrix exponential method for time integration and the Chebyshev pseudo-spectral method for spatial discretization. The numerical solution exhibited excellent agreement with both the exact solution and the results from Monte Carlo simulations, with negligible numerical dissipation and numerical dispersion. In addition, due to the high convergence of pseudo-spectral method and the fact that the sequential matrix exponential method is not restricted by CFL (courant-friedrichs-lewy) condition, the proposed method has excellent numerical stability and computational efficiency.

     

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