Computational Modeling of Complex Interfacial Structures with Nonlinear and Nonlocal Interactions

具有非线性和非局部相互作用的复杂界面结构的计算建模

• 批准号: 2142500
• 负责人: Yanxiang Zhao
• 金额: $ 16.05万
• 依托单位: George Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2142500&HistoricalAwards=false
• 关键词: Computational; Modeling; Complex; Interfacial; Structures

Modeling interfacial structures and dynamics is of great importance in many applications such as biology, physics, and materials science. Many existing computational models can sometimes predict unphysical structures and also suffer from inefficiency in large scale computations, especially when the system of interest involves nonlinear and nonlocal interactions. On the other hand, some powerful numerical methods have been designed to approximate the interfacial structures in a stable and efficient manner. This can only happen if numerical methods are designed to preserve the underlying physical structures of the system of interest. The purpose of this project is to bring together researchers with complementary backgrounds to come up with a unified computational model to investigate the interfacial structure and dynamics with nonlocal and nonlinear interactions. The model will largely improve the efficiency of computations for interfacial structures, as well as correctly describe key quantities of interest in equilibria and dynamic structures of an underlying interfacial system. In addition, the mathematical modeling techniques and computational methods of this project will address key scientific challenges in applied mathematics, and meet basic research needs and provide necessary modeling tools for the applications to other systems involving interface problems. Besides, the computational model can provide theoretical guidance on producing nanostructured materials, which will ultimately promote a wide range of contemporary engineering applications such as materials synthesis, nanomedicine, and nanotechnology. Further important impacts will be research oriented curriculum development, mentoring undergraduate/graduate students to take part in the project, and the engagement of various outreach activities.This project focuses on developing a unified computational phase field model to investigate the complex interfacial structures with nonlinear and nonlocal interactions. Though some existing phase field approaches can be applied to model simple periodic structures such as lamellar, spherical, bicontinuous syroidsin block copolymers, some other interesting patterns are overlooked and have not been well studied theoretically. Therefore one needs to examine the variational problem in its full generality from a mathematically more sophisticated point of view, one which in particular allows for a fuller analysis of the competition between different terms in the system of interest. The inclusion of the general nonlocal and nonlinear interactions in this research project can characterize a broader class of features of microphase separation and pattern formation for interfacial structures, and provide more insights on theoretical studies of these subjects. The PIs will develop efficient, stable and accurate numerical methods for the system of interest. More specifically, asymptotically compatible, maximum principle preserving and energy stable schemes will be explored to preserve specific physical structures of the system of interest at the level of numerical approximations. Additionally, the designed numerical solvers will be used for the systematic study of materials science applications such as block copolymer melts and the bubble assemblies of the block copolymer system.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.

界面结构和动力学模型在生物学、物理学和材料科学等许多应用中具有重要意义。许多现有的计算模型有时可以预测非物理结构，但在大规模计算中也存在效率低下的问题，特别是当系统涉及非线性和非局部相互作用时。另一方面，一些强大的数值方法已经被设计来近似的界面结构，在一个稳定和有效的方式。只有当数值方法被设计成保留感兴趣系统的底层物理结构时，才能发生这种情况。该项目的目的是将具有互补背景的研究人员聚集在一起，提出一个统一的计算模型，以研究具有非局部和非线性相互作用的界面结构和动力学。该模型将在很大程度上提高界面结构的计算效率，以及正确地描述感兴趣的关键量的平衡和动态结构的一个潜在的界面系统。此外，本项目的数学建模技术和计算方法将解决应用数学中的关键科学挑战，满足基础研究需求，并为涉及接口问题的其他系统的应用提供必要的建模工具。此外，计算模型可以提供理论指导，生产纳米结构的材料，这将最终促进广泛的当代工程应用，如材料合成，纳米医学和纳米技术。进一步的重要影响将是以研究为导向的课程开发，指导本科生/研究生参加项目，并参与各种推广活动。本项目的重点是开发一个统一的计算相场模型，以研究具有非线性和非局部相互作用的复杂界面结构。虽然一些现有的相场方法可以应用于模拟简单的周期性结构，如层状，球形，双连续syroids嵌段共聚物，一些其他有趣的模式被忽视，并没有得到很好的理论研究。因此，人们需要从数学上更复杂的观点来考察变分问题的全部一般性，这种观点特别允许对利息系统中不同项之间的竞争进行更充分的分析。在本研究项目中纳入一般的非局部和非线性相互作用可以表征更广泛的一类界面结构的微相分离和图案形成的特征，并为这些主题的理论研究提供更多的见解。PI将为感兴趣的系统开发高效，稳定和准确的数值方法。更具体地说，渐近兼容，最大值原理保持和能量稳定的计划将被探索，以保持特定的物理结构的系统的利益在数值近似的水平。此外，设计的数值求解器将用于材料科学应用的系统研究，如嵌段共聚物熔体和嵌段共聚物系统的气泡组装。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Second-order stabilized semi-implicit energy stable schemes for bubble assemblies in binary and ternary systems

• DOI: 10.3934/dcdsb.2021246
• 发表时间: 2021
• 期刊: Discrete & Continuous Dynamical Systems - B
• 作者: H. Choi;Yanxiang Zhao

Supervised Optimal Transport

监督最优运输

• DOI: 10.1137/22m1469171
• 发表时间: 2022
• 期刊: SIAM Journal on Applied Mathematics
• 影响因子: 1.9
• 作者: Cang, Zixuan;Nie, Qing;Zhao, Yanxiang
• 通讯作者: Zhao, Yanxiang