For decades, uncertainty quantification techniques based on the spectral approach have been demonstrated to be computationally more efficient than the Monte Carlo method for a wide variety of problems, particularly when the dimensionality of the probability space is relatively low. The time-dependent generalized polynomial chaos (TD-gPC) is one such technique that uses an evolving orthogonal basis to better represent the stochastic part of the solution space in time. In this paper, we present a new numerical method that uses the concept of enriched stochastic flow maps to track the evolution of the stochastic part of the solution space in time. The computational cost of this proposed flow-driven stochastic chaos (FSC) method is an order of magnitude lower than TD-gPC for comparable solution accuracy. This gain in computational cost is realized because, unlike most existing methods, the number of basis vectors required to track the stochastic part of the solution space does not depend upon the dimensionality of the probability space. Four representative numerical examples are presented to demonstrate the performance of the FSC method for long-time integration of second-order stochastic dynamical systems in the context of stochastic dynamics of structures.

几十年来，基于谱方法的不确定性量化技术已被证明在计算上比蒙特卡罗方法更有效，可以解决各种问题，尤其是当概率空间的维数相对较低时。瞬态广义多项式混沌 (TD-gPC) 就是这样一种技术，它使用不断发展的正交基来更好地及时表示解空间的随机部分。在本文中，我们提出了一种新的数值方法，该方法使用了丰富的随机流图的概念及时跟踪解空间随机部分的演化。这种提出的流驱动随机混沌 (FSC) 方法的计算成本比 TD-gPC 低一个数量级，以获得可比的解决方案精度。实现计算成本的这种增益是因为与大多数现有方法不同，跟踪解空间的随机部分所需的基向量数量不依赖于概率空间的维数。四个有代表性的数值例子展示了 FSC 方法在结构随机动力学背景下二阶随机动力系统长期积分的性能。