In the stochastic dynamic analysis, the probability density evolution method (PDEM) provides an optional way to capture the complete probability distribution of the stochastic response of general nonlinear systems. In the PDEM, the key point is to solve the generalized probability density evolution equation (GDEE), which governs the evolution of the joint probability density function (PDF) of the response and the randomness. In this paper, a new numerical method based on the reproducing kernel particle method (RKPM) is proposed. The GDEE can be approximated through the RKPM. By some particles in the response domain, the instantaneous PDF and its partial derivative with respect to response are smoothly expressed. Then, the approximated GDEE can be discretized directly at the collocation points in the response domain. At the same time, discretization in the time domain is achieved by the difference scheme. Therefore, the RKPM-based formulation to obtain the numerical solution of GDEE is formed. The implementation procedure of the proposed method is given in detail. The accuracy and efficiency of this method are illustrated with some numerical examples. Some details of parameter analysis are also discussed.

在随机动态分析中，概率密度演化方法 (PDEM) 提供了一种可选的方法来捕获一般非线性系统随机响应的完整概率分布。在 PDEM 中，关键点是求解广义概率密度演化方程 (GDEE)，该方程控制响应和随机性的联合概率密度函数 (PDF) 的演化。本文提出了一种新的基于再生核粒子法(RKPM)的数值方法。GDEE 可以通过 RKPM 来近似。通过响应域中的一些粒子，可以平滑地表达瞬时 PDF 及其相对于响应的偏导数。然后，近似的 GDEE 可以直接在响应域中的搭配点进行离散化。同时，时域的离散化是通过差分方案实现的。因此，形成了用于获得 GDEE 数值解的基于 RKPM 的公式。详细给出了所提出方法的实现过程。通过一些数值例子说明了该方法的准确性和效率。还讨论了参数分析的一些细节。