A New Decomposition Method for Solving Stochastic Eigenvalue Problems in Computational Dynamics

该提案的目标是通过开发一种创新方法（称为维度分解方法）来解决随机动力系统建模和仿真中的一般随机特征值问题，从而在爱荷华大学的计算力学领域获得核心竞争力。提出的工作将基于：(1)随机特征值问题的一般复值特征解的低维近似的新分解方法；(2)特征解概率特征的新的多点分解和单项预条件；(3)随机特征解的概率测度解析梯度的新设计灵敏度公式。拟议的研究是雄心勃勃的和新颖的，从根本上不同于大多数先前的研究在这一领域。待开发的方法将解决高度非线性的输入-输出转换，无限数量的随机变量或字段，以及随机输入的任意大的不确定性。由于解析导出的随机设计灵敏度的创新表述，动态系统的后续优化可以采用任何标准的基于梯度的算法进行。分解方法将有助于解决工程和科学中的大规模、多学科、随机特征值问题。这项研究将对民用、汽车和航空航天基础设施等众多商业和工业应用产生重大影响。潜在的工程应用包括土木结构的分析和设计；地面车辆系统的噪声-振动粗糙度；航空航天结构的疲劳耐久性；以及微电子和微机电系统的可靠性。除工程外，潜在的应用还包括核物理学、数论、计算生物学和计算金融学等。因此，这里提出的研究将对一些具有国家意义的领域产生积极影响。该项目创造的知识的转移和传播将通过与行业的持续合作、在ASME会议上组织专题讨论会、期刊出版物、在主要会议和机构上的演讲和出版物以及学生教育来实现。与两个政府和工业实验室的伙伴关系将使本项目开发的基本方法得以实施，以解决几个大规模的工业问题。教育目标包括从代表性不足的少数民族或女性群体中招募一名博士生，在爱荷华大学主要工程项目的升级课程中实施该项目的软件工具，并撰写一篇研究专著。

The objective of the proposal is to achieve a core competency in the field of computational mechanics at The University of Iowa through the development of an innovative method, referred to as the dimensional decomposition method, for solving a general random eigenvalue problem in modeling and simulation of stochastic dynamic systems. The proposed effort will be based on: (1) new decomposition method for lower-dimensional approximations of general complex-valued eigensolutions of random eigenvalue problems; (2) new multipoint decomposition and monomial preconditioner for probabilistic characteristics of eigensolutions; and (3) new design sensitivity formulation for analytic gradients of probabilistic measures of random eigensolutions. The proposed research is ambitious and novel, differing in fundamental ways from most prior research in this area. The methods to be developed will address highly nonlinear input-output transformations, an unlimited number of random variables or fields, and arbitrarily large uncertainty of random input. Due to innovative formulation of the analytically derived stochastic design sensitivities, subsequent optimization of dynamic systems can be conducted employing any standard gradient-based algorithm. The decomposition method will aid in solving large-scale, multidisciplinary, stochastic eigenvalue problems in engineering and science.The proposed research will be of significant benefit to numerous commercial and industrial applications, such as civil, automotive, and aerospace infrastructure. Potential engineering applications include analysis and design of civil structures; noise-vibration-harshness of ground vehicle systems; fatigue durability of aerospace structures; and reliability of microelectronics and micro-electro-mechanical systems. Beyond engineering, potential applications include nuclear physics, number theory, computational biology, and computational finance, among others. Therefore, the research proposed here will positively impact a number of areas of national significance. The transfer and dissemination of knowledge created by this project will take place through continued collaboration with industries, organization of symposia in ASME conferences, journal publications, presentations and publications at major conferences and institutions, and student education. Partnerships with two government and industrial laboratories will enable implementation of the basic methods developed in this project to resolve several large-scale industrial problems. The educational goals comprise recruitment of a Ph. D. student from underrepresented minority or women groups, implementation of software tools from this project in upgrading courses in The University of Iowa's principal engineering programs, and authoring a research monograph.