Application of Multidemsional Fokker-Planek Equation to Engineering Systems

[224828]伯格曼对动力系统响应的准确预测是设计和最终控制动力系统所必需的第一步。结构系统和激励过程的精确模型的制定提供了确定系统响应的手段，评估系统在其性能和安全性方面的充分性，并提出纠正措施。南加州最近的地震活动提醒我们，环境负荷在本质上是随机的。此外，几乎每个物理系统的性质都存在一定程度的不确定性。因此，许多工程系统的响应将是随机过程，而这些响应的完整和准确的确定通常是一件很重要的事情。通过适当的模型构建，使得响应过程是马尔可夫的，从而完全由一个转移概率密度函数来表征，通常通过求解正向Kolmogorov或Fokker-Planck方程来获得。该项目的目标将是开发有效的算法来解决线性和非线性系统在加性和乘法（即参数）激励下的多维福克-普朗克方程，并将这些算法引入工程实践。几类的解决方法将被检查，包括有限单元法结合直接，特别是显式，求解。这就消除了对高维相空间中的运算矩阵进行上三角化的需要。该解决方案不仅将产生响应过程的一阶概率密度函数，而且在软件开发之后，还将产生响应的边际密度、响应矩以及响应的上交叉和峰值统计，从而完全表征随机响应过程的基本性质。随着时间的推移，解决方案的可视化使分析人员能够观察到动力系统的丰富行为。因此，为了保留最大量的重要信息，将花费大量的精力来确定在低维空间中查看高维问题解的最佳方法。在许多应用中，福克-普朗克方程仅对一个自变量具有二阶导数。利用这种特殊结构的方法具有显著的优势。例如，操作符分割方法试图将具有令人望而却步的大内存和计算需求的多维问题简化为一系列较小的、更简单的问题。在这种情况下，微分算子可以被分解成一维问题的近似序列。这些一维问题中的每一个都在每个时间步的一部分上交替地进行数值解决，并且解决方案从网格的一边一列一列地传播到另一边。这些方法有时被称为交替方向方法，它们对当前这类问题的适用性将被详细研究。预计其他计算方法，如边界元方法也将被评估。此外，如上所定义的问题的可视化方面将被合作研究。

9224828 Bergman The accurate prediction of the response of a dynamical system is a necessary first step toward its design and eventual control. Formulation of accurate models of the structural system and excitation processes provides the means to determine system response, assess the adequacy of the system in terms of its performance and safety, and suggest corrective actions. Recent seismic activity in southern California serves to remind us that environmental loads are random in nature. Furthermore, a degree of uncertainty exists in the properties of virtually every physical system. Thus, the responses of many engineering systems will be stochastic processes, and the complete and accurate determination of these responses is generally a nontrivial matter. The solution of many of these problems is facilitated by the appropriate construction of the model such that the response process is Markovian and is, thus, completely characterized by a transition probability density function, usually obtained by solving a forward Kolmogorov or Fokker-Planck equation. The object of this project will be to develop efficient algorithms to solve the multidimensional Fokker-Planck equation for linear and nonlinear systems subjected to both additive and multiplicative (i.e., parametric) excitations and to introduce these algorithms into engineering practice. Several classes of solution methods will be examined, including finite element methods combined with direct, particularly explicit, solvers. These eliminate the need to upper triangularize the operational matrix that occur ion high dimensional phase spaces. The solution will yield not only the first order probability density function of the response process but also, after software development, the marginal densities, response moments, and upcrossing and peak Statistics of the response, thus completely characterizing the fundamental nature of the stochastic response process. Visualization of the solution as it evolves in time permits the analyst to observe the rich behavior of the dynamical system. Thus, significant effort will be expended to determine optimal methods of viewing the solutions of higher dimensional problems in low dimensional spaces in order to preserve the maximum amount of important information. In many applications in the Fokker-Planck equation possesses a second derivative for only one of the independent variables. Methods that take advantage of this special structure offer significant advantages. For example, operator splitting methods seek to reduce a multidimensional problem, with its prohibitively large memory and computational requirements, to a sequence of small, simpler problems. In the present situation, the differential operator can be split into an approximating sequence of one dimensional problems. Each of these one dimensional problems is alternately solved numerically over a portion of each time step, and the solution is propagated from one side of the mesh to the other, column-by-column. These methods are sometimes referred to as alternating direction methods, and their applicability to the current class of problems will be examined in great detail. It is anticipated that other computational approaches such as boundary element methods will also be evaluated. Furthermore, the visualization aspects of the problem as defined above will be examined cooperatively.