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

许多复杂的材料和工程结构都受到可变性的困扰，即不确定性，这是由于对操作环境的不精确了解，对材料特性的信息不足，以及制造过程中的内在随机性。传统的建模和模拟程序依赖于启发式推导的安全系数，不能定量地解释系统响应的统计变化。因此，最终的预测要么过于保守，过度补偿不确定性，要么由于未解决的模糊性而不知不觉地存在风险。本项目将对不确定性复杂系统的建模与仿真进行基础研究；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。通过正面处理不确定性，可以设计出有效的措施来管理和减轻不确定性。潜在的工程应用包括先进材料的微结构-性能关系，工程结构的疲劳和断裂，以及纳米机电系统的设计等。除了工程学，这项研究的结果还将通过在不确定性量化发挥重要作用的领域的应用，使美国经济和社会受益，例如能源科学——核能、碳封存；统计物理——医学、量子力学；地球科学——地震学、储层建模；生物信息学——药物输送，农业。这项研究是多学科的，涉及工程、应用数学和统计学，将有助于扩大代表性不足的群体在研究中的参与，并对工程教育产生积极影响。该项目的目标是推进等几何分析理论，伴随着鲁棒的数值算法，对复杂材料和结构的高维响应进行不确定性量化。该工作将涉及：(1)随机矩阵方程的新随机非均匀有理b样条（NURBS）和基于NURBS的材料体随机场离散化；(2)基于分层b样条稀疏网格的高维函数插值随机等几何方法；(3)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。该研究将在几何建模、应力分析和随机模拟之间架起桥梁，通过在相同的数学构建块上进行交互，形成未来无缝的不确定性量化管道。由于稀疏网格插值的创新公式，无论不确定性量化问题的大小，所得到的随机方法都将有效地实现。新的计算算法将用于有效地估计结构响应的统计矩和概率密度函数，包括误差估计，这将导致对稀疏网格近似的严格评估。整体努力将有效地整合研究、教育、培训和推广。

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.