CAREER: A Stochastic Framework for Uncertainty Quantification on Complex Geometries: Application to Additive Manufacturing

该教师早期职业发展（CAREER）拨款将支持基础研究，重点是复杂几何形状在预测随机计算模型中的整合。最近在增材制造和组织工程等领域的技术突破彻底改变了材料和结构的加工、制造和制造方式。通过在多个长度尺度上生产具有前所未有的材料和几何复杂性的零件，这些突破也极大地增加了计算建模和实验测试的挑战。其中之一是复杂几何形状的零件响应不确定性的量化。该职业项目旨在开发一个随机建模框架，该框架将使复杂的几何特征自动且稳健地集成到高维、预测计算设置中。这种方法将为必须在现实世界几何形状上量化行为不确定性的领域的理论发展和虚拟测试范例铺平道路。作为该项目的一部分，还计划了广泛的教育和推广计划。该部分主要包括：(1) 为本科生和研究生提供实践研究机会，(2) 吸引和教育广大受众了解具有影响力的应用的基础科学概念的活动，以及 (3) 提高 K-12 学生和代表性不足群体对计算力学、材料科学和 STEM 的参与的活动。这项研究旨在弥合几何复杂性和不确定性量化方法之间的差距。尽管在计算物理学中考虑多种不确定性来源的概率框架的开发方面取得了相当大的进展，但将复杂（例如非凸）几何描述正确集成到随机方法中的方法仍然大多尚未探索。在这种情况下，几何特征的特征和材料不确定性的内在属性通过加工条件交织在一起，这对随机建模和不确定性量化的最先进技术提出了独特的挑战。为了推进新的知识和工具，该项目的目标包括：(1) 为跨（空间）尺度的广泛随机本构模型开发适当的概率表示，(2) 构建用于在复杂的大规模域上采样的高效生成器，以及 (3) 开发用于模型识别、传播和验证的稳健概率方法。为了解决这些问题，该研究将结合约束状态空间随机建模的理论推导、通过分数偏微分方程随机生成的计算发展、不确定统计逆问题的贝叶斯推理，以及增材制造的骨状钛支架的实验表征。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

This Faculty Early Career Development (CAREER) grant will support fundamental research focusing on the integration of complex geometries in predictive stochastic computational modeling. Recent technological breakthroughs in, e.g., additive manufacturing and tissue engineering, have revolutionized the way materials and structures are processed, fabricated, and manufactured. By enabling the production of parts with unprecedented levels of material and geometric complexities over multiple length scales, these breakthroughs have also greatly enhanced the challenges in computational modeling and experimental testing. One of them is the quantification of part response uncertainties over complex geometries. This CAREER project aims to develop a stochastic modeling framework that will enable the automatic and robust integration of complex geometrical features into high-dimensional, predictive computational settings. This approach will pave the way for theoretical developments and virtual testing paradigms in fields where uncertainty in behavior must be quantified on real-world geometries. As part of the project, an extensive educational and outreach plan is also planned. This component notably includes: (1) hands-on research opportunities for undergraduate and graduate students, (2) activities to engage and educate a broad audience on basic science concepts with impactful applications, and (3) activities to increase the participation of K-12 students and underrepresented groups in computational mechanics, materials science, and STEM at large. This research seeks to bridge the gap between geometrical complexity and uncertainty quantification methodologies. While there has been considerable progress in the development of probabilistic frameworks accounting for multiple sources of uncertainties in computational physics, the proper integration of complex (e.g., nonconvex) geometrical descriptions into stochastic approaches remains mostly unexplored. In this case, the characteristics of the geometrical features and the intrinsic properties of material uncertainties are intertwined through processing conditions, which uniquely challenges the state-of-the-art in stochastic modeling and uncertainty quantification. To advance new knowledge and tools, the objectives of this project include: (1) the development of appropriate probabilistic representations for a broad class of stochastic constitutive models across (spatial) scales, (2) the construction of efficient generators for sampling on complex large-scale domains, and (3) the development of robust probabilistic methodologies for model identification, propagation, and validation. To address these issues, the research will combine theoretical derivations for stochastic modeling on constrained state spaces, computational developments for random generation through fractional partial differential equations, Bayesian inference for underdetermined statistical inverse problems, and experimental characterization on additively-manufactured bone-like titanium scaffolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.