Computational Modeling of Complex Interfacial Structures with Nonlinear and Nonlocal Interactions

界面结构和动力学模型在生物、物理、材料科学等领域具有重要的应用价值。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，特别是当感兴趣的系统涉及非线性和非局部相互作用时。另一方面，已经设计了一些强大的数值方法来稳定和高效地逼近界面结构。只有当数值方法被设计成保留感兴趣系统的基本物理结构时，这种情况才会发生。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来研究具有非局域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，并正确地描述对底层界面系统的平衡和动态结构感兴趣的关键量。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，该计算模型可以为制备纳米结构材料提供理论指导，最终将促进材料合成、纳米医学和纳米技术等广泛的当代工程应用。更重要的影响将是以研究为导向的课程开发，指导本科生/研究生参与该项目，以及各种外展活动的参与。该项目侧重于开发一个统一的计算相场模型来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用来模拟嵌段共聚物中的片状、球形、双连续等简单的周期结构，但其他一些有趣的模式被忽视了，理论上还没有得到很好的研究。因此，人们需要从数学上更复杂的观点来研究变分问题的全部一般性，这尤其允许对感兴趣系统中不同项之间的竞争进行更全面的分析。将一般的非局域和非线性相互作用纳入这一研究项目，可以更广泛地描述界面结构的微相分离和图案形成的特征，并为这些课题的理论研究提供更多的见解。PIS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，将探索渐近相容、保持最大值原理和能量稳定的方案，以在数值近似的水平上保持感兴趣系统的特定物理结构。此外，设计的数值求解器将用于材料科学应用的系统研究，如嵌段共聚熔体和嵌段共聚体系的气泡组件。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.