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

该学院早期职业发展（CAREER）资助将支持基础研究，重点是预测随机计算建模中复杂几何的整合。最近的技术突破，例如，增材制造和组织工程已经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，这些突破也大大提高了计算建模和实验测试的挑战。其中之一是量化复杂几何形状的零件响应不确定性。这个CAREER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能够自动和鲁棒地集成到高维预测计算环境中。这种方法将铺平道路的理论发展和虚拟测试范例的行为的不确定性，必须量化现实世界的几何形状的领域。作为该项目的一部分，还计划了一项广泛的教育和外联计划。这一部分主要包括：（1）为本科生和研究生提供动手研究的机会，（2）通过有效的应用吸引和教育广大受众了解基础科学概念的活动，以及（3）增加K-12学生和代表性不足的群体参与计算力学，材料科学和STEM的活动。本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。虽然在计算物理学中解释多个不确定性来源的概率框架的发展方面已经取得了相当大的进展，但复杂的（例如，非凸）的几何描述到随机方法仍然主要是未开发的。在这种情况下，几何特征的特性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新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）为广泛的随机本构模型跨（空间）尺度的适当概率表示的发展，（2）在复杂的大规模域采样的有效发电机的建设，和（3）模型识别，传播和验证的鲁棒概率方法的发展。为了解决这些问题，研究将结合联合收割机理论推导的随机建模的约束状态空间，计算发展的随机生成通过分数偏微分方程，贝叶斯推断欠定统计逆问题，以及增材制造骨骼的实验表征-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进行评估，被认为值得支持审查标准。

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.