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 准确预测动力系统的响应是其设计和最终控制的必要的第一步。 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、评估系统性能和安全性的充分性以及建议纠正措施的方法。 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，并且完整且准确地确定这些响应通常是一件非常重要的事情。 通过适当构建模型，可以促进许多此类问题的解决，使得响应过程是马尔可夫的，因此完全由转移概率密度函数表征，通常通过求解前向柯尔莫哥洛夫或福克-普朗克方程来获得。 该项目的目标是开发有效的算法来求解受到加法和乘法（即参数）激励的线性和非线性系统的多维福克-普朗克方程，并将这些算法引入工程实践。 将研究几类求解方法，包括与直接（特别是显式）求解器相结合的有限元方法。 这些消除了对离子高维相空间中出现的运算矩阵进行上三角化的需要。 该解不仅会产生响应过程的一阶概率密度函数，而且在软件开发后，还会产生响应的边际密度、响应矩以及上交叉和峰值统计，从而完整地表征随机响应过程的基本性质。 随着时间的推移，解决方案的可视化允许分析人员观察动态系统的丰富行为。 因此，将花费大量精力来确定在低维空间中查看高维问题的解决方案的最佳方法，以便保留最大量的重要信息。 在许多应用中，福克-普朗克方程仅对其中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着的优势。 例如，算子分裂方法试图将具有极大内存和计算要求的多维问题简化为一系列更小的、更简单的问题。 在目前情况下，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在每个时间步长的一部分上交替进行数值求解，并且解从网格的一侧逐列传播到另一侧。 这些方法有时被称为交替方向方法，并且将详细检查它们对当前类别问题的适用性。 预计边界元法等其他计算方法也将得到评估。 此外，将合作检查上述问题的可视化方面。