Application of Multidemsional Fokker-Planek Equation to Engineering Systems

9224828伯格曼对动力系统响应的准确预测是其设计和最终控制的必要的第一步。建立结构系统和激励过程的准确模型提供了确定系统响应、评估系统在性能和安全性方面的充分性以及建议纠正措施的手段。最近南加州的地震活动提醒我们，环境负荷本质上是随机的。此外，几乎每个物理系统的属性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，而完整和准确地确定这些响应通常是一件不平凡的事情。通过适当地构造模型，使得响应过程是马尔可夫过程，从而完全由转移概率密度函数来描述，通常通过解正向Kolmogorov或Fokker-Planck方程来获得，从而促进了许多问题的解决。本项目的目标是开发有效的算法来求解线性和非线性系统在加性和乘性(即参数)激励下的多维Fokker-Planck方程，并将这些算法引入工程实践。将考察几类求解方法，包括结合直接求解器、特别是显式求解器的有限元方法。这消除了对出现高维相空间的运算矩阵进行上三角化的需要。该解不仅给出了响应过程的一阶概率密度函数，而且在软件开发后还得到了响应的边际密度、响应矩、响应的上交叉统计量和峰值统计量，从而完整地刻画了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员能够观察到动力系统的丰富行为。因此，将花费大量的努力来确定查看低维空间中的高维问题的解的最佳方法，以便保存最大量的重要信息。在福克-普朗克方程的许多应用中，只有一个自变量具有二阶导数。利用这种特殊结构的方法提供了显著的优势。例如，算子分裂方法寻求将一个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小的、更简单的问题。在目前的情况下，微分算子可以分解为一维问题的近似序列。这些一维问题中的每一个都交替地在每个时间步长的一部分上进行数值求解，并且解逐列地从网格的一侧传播到另一侧。这些方法有时被称为交替方向法，我们将非常详细地研究它们对当前问题类别的适用性。预计还将对边界单元法等其他计算方法进行评价。此外，上面定义的问题的可视化方面将被合作检查。

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.