Analyis & Visualization of Evolutionary Response Characteristics for Stochastic Dynamical Eng Systems: Solution of the Multidimensional Fokker-Planck Equation

9411589 Spencer准确预测动态系统的响应是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、评估系统在性能和安全方面的充分性以及建议纠正措施的方法。 最近在南加州发生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，这些响应的完整和准确的确定通常是一个不平凡的事情。 许多这些问题的解决方案是方便的适当建设的模型，使响应过程是马尔可夫的，因此，完全由一个过渡概率密度函数，通常通过求解前向柯尔莫哥洛夫或福克-普朗克方程。 该项目的目标是开发有效的算法来求解线性和非线性系统的多维福克-普朗克方程，这些系统都受到加法和乘法（即，参数）激励，并将这些算法引入工程实践。 几类解决方案的方法将被检查，包括有限元方法结合直接，特别是显式，求解器。 这些消除了需要上三角化的操作矩阵，出现离子高维相空间。 该解决方案将不仅产生响应过程的一阶概率密度函数，而且在软件开发之后，还产生响应的边缘密度、响应矩、上交和峰值统计，从而完全表征随机响应过程的基本性质。 可视化的解决方案，因为它演变的时间允许分析观察丰富的行为的动力系统。 因此，显着的努力将花费，以确定最佳的方法，查看高维问题的解决方案，在低维空间，以保持最大量的重要信息。 在许多应用中，福克-普朗克方程只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提供了显著的优点。 例如，算子分裂方法寻求将多维问题（其具有令人望而却步的大存储器和计算需求）简化为一系列小的、更简单的问题。 在目前的情况下，微分算子可以分裂成一个近似序列的一维问题。 这些一维问题中的每一个在每个时间步长的一部分上交替地数值求解，并且解从网格的一侧逐列地传播到另一侧。 这些方法有时被称为交替方向法，它们对当前问题的适用性将被详细研究。 预计还将评估其他计算方法，如边界元法。 此外，上述问题的可视化方面将进行合作研究。

9411589 Spencer 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 analys t 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.