Application of Multidemsional Fokker-Planek Equation to Engineering Systems

多维福克-普朗克方程在工程系统中的应用

• 批准号: 9224828
• 负责人: Lawrence Bergman
• 金额: $ 15.43万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-05-15 至 1998-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9224828&HistoricalAwards=false
• 关键词: Application; Multidemsional; Fokker; Planek; Equation

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.

9224828 Bergman的精确预测动态系统的响应是迈向其设计和最终控制的必要第一步。 制定结构系统和激发过程的准确模型提供了确定系统响应，根据其性能和安全性评估适当性的方法，并提出纠正措施。 南加州最近的地震活动提醒我们，环境负荷本质上是随机的。 此外，几乎每个物理系统的属性中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，并且对这些响应的完整和准确确定通常是一个非平凡的问题。 许多这些问题的解决方案通过适当的模型构建促进了响应过程是马尔可夫人，因此完全以过渡概率密度函数为特征，通常通过求解前向kolmogorov或fokker-planck方程获得。 该项目的对象将是开发有效的算法来解决具有添加剂和乘法（即参数）激发的线性和非线性系统的多维Fokker-Planck方程，并将这些算法引入工程实践。 将检查几类解决方案方法，包括有限元方法以及直接的，尤其是显式求解器。 这些消除了上三角形的需要，将发生离子高尺寸相位空间的操作矩阵。 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，响应的边际密度，响应矩，上跨和峰值统计数据，从而完全表征了随机响应过程的基本性质。 随着溶液的及时演变，可视化允许分析师观察动力学系统的丰富行为。 因此，将花费大量精力来确定在低维空间中查看更高维问题的解决方案的最佳方法，以保留最大的重要信息量。 在Fokker-Planck方程中的许多应用中，仅对一个自变量之一具有第二个导数。 利用这种特殊结构的方法具有很大的优势。 例如，操作员拆分方法试图减少多维问题，并具有较大的记忆和计算要求，以使一系列小，简单的问题序列。 在当前情况下，差分运算符可以分为一个维度问题的近似序列。 这些一个维问题中的每一个都在每个时间步的一部分上交替求解，并且该解决方案从网格的一侧传播到另一侧，逐列。 这些方法有时被称为交替方向方法，它们对当前问题类别的适用性将进行详细检查。 预计还将评估其他计算方法，例如边界元素方法。 此外，将合作检查上述问题的可视化方面。