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

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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.