CDS&E: Stochastic Isogeometric Analysis by Hierarchical B-Spline Sparse Grids

CDS

• 批准号: 1607398
• 负责人: Sharif Rahman
• 金额: $ 39.99万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-15 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1607398&HistoricalAwards=false
• 关键词: CDS& Stochastic; Isogeometric; Analysis; Hierarchical

A great many complex materials and engineered structures are plagued by variability, that is, uncertainty, due to imprecise knowledge of operating environment, insufficient information about material properties, and intrinsic randomness in manufacturing processes. Conventional modeling and simulation procedures rely on heuristically derived safety factors and do not quantitatively account for the statistical variation of a system response. Consequently, the resulting predictions are either too conservative and overcompensate for uncertainty, or unknowingly risky due to unresolved ambiguities. This project will conduct fundamental research on modeling and simulation of complex systems subject to uncertainty; in other words, new computational methods will be developed to quantify the effect of uncertainty on system response. By addressing uncertainty head-on, effective measures to manage and mitigate uncertainty can be devised. Potential engineering applications include microstructure-property relationship in advanced materials, fatigue and fracture of engineering structures, and design of nanoelectromechanical systems, among others. Beyond engineering, the results from this research will benefit the U.S. economy and society through application in areas where uncertainty quantification plays a vital role, such as energy sciences--nuclear energy, carbon sequestration; statistical physics--medicine, quantum mechanics; geosciences--seismology, reservoir modeling; and bioinformatics--drug delivery, agriculture. This research is multi-disciplinary, involving engineering, applied mathematics, and statistics, and will help broaden participation of underrepresented groups in research as well as positively impact engineering education.The objective of this project is to advance the theory of isogeometric analysis, accompanied by robust numerical algorithms, for uncertainty quantification of a high-dimensional response from complex materials and structures. The effort will involve: (1) new randomized non-uniform rational B-splines (NURBS) for the stochastic matrix equation and NURBS-based random field discretization for a material body; (2) new stochastic isogeometric methods entailing the hierarchical B-spline sparse grids for high-dimensional function interpolation; and (3) new formulae and scalable algorithms for predicting the statistical moments and probability density functions of a complex structural response. The research will bridge geometric modeling, stress analysis, and stochastic simulation by interacting natively upon the same mathematical building blocks, forming a seamless uncertainty quantification pipeline of the future. Due to innovative formulation of the sparse grid interpolation, the resulting stochastic method will be efficiently implemented regardless of the size of an uncertainty quantification problem. New computational algorithms will be generated for efficiently estimating the statistical moments and probability density function of a structural response, including error estimates that will result in a rigorous assessment of the sparse grid approximation. The overall effort will effectively integrate research, education, training, and outreach.

许多复杂的材料和工程结构都受到可变性的困扰，即由于对操作环境的不精确，有关材料属性的信息不足以及制造过程中的固有随机性而引起的不确定性。 常规的建模和仿真程序依赖于启发性的安全因子，并且不能定量解释系统响应的统计变化。 因此，由此产生的预测过于保守，并且由于未解决的歧义而导致的不确定性过于补偿，或者在不知不觉中冒险。 该项目将对受到不确定性的复杂系统建模和模拟进行基础研究；换句话说，将开发新的计算方法来量化不确定性对系统响应的影响。通过正面解决不确定性，可以设计有效管理和减轻不确定性的有效措施。 潜在的工程应用包括高级材料中的微观结构 - 特制关系，工程结构的疲劳和断裂，以及纳米机电系统的设计。 除工程外，这项研究的结果还将通过在不确定性量化起着至关重要作用的领域（例如能源科学） - 核能源，碳固存；统计物理学 - 医学，量子力学；地球科学 - 学科，储层建模；和生物信息学 - 药物分娩，农业。 这项研究是多学科的，涉及工程学，应用数学和统计数据，将有助于扩大代表性不足的群体参与研究的参与，并对工程教育产生积极影响。该项目的目的是提高等符号分析理论，并伴随着鲁棒的数值算法响应高度质量的构成量化，并伴随着高度的质量量化。 这项工作将涉及：（1）用于随机矩阵方程和基于NURBS的材料机构的新的随机非均匀理性B型（NURB）； （2）新的随机等几何方法需要用于高维函数插值的层次B-Spline稀疏网格； （3）用于预测复杂结构响应的统计矩和概率密度函数的新公式和可扩展算法。 这项研究将通过在相同的数学构建块上本地相互作用，形成未来的无缝不确定性定量管道，从而桥接几何建模，应力分析和随机模拟。 由于稀疏的网格插值的创新配方，无论不确定性定量问题的大小如何，所得随机方法都将有效地实现。 将生成新的计算算法，以有效估计结构响应的统计矩和概率密度函数，包括错误估计值，这些估计将导致对稀疏网格近似的严格评估。 总体努力将有效地整合研究，教育，培训和外展。