Nonlinear Geometric Models: Algorithms, Analysis, and Computation

新型和智能材料的制造和操作，特别是在纳米技术和生物技术的战略领域，需要了解由几何偏微分方程组(PDE)控制的非线性现象。在小尺度上，比如微米和纳米尺度，表面张力和弯曲效应主导着整体效应，从而使小型设备的驱动和控制成为现实。这导致了科学上有趣和技术上有用的配置和动态行为。在生物医学科学(药物输送囊泡、细胞封装设备和传感器)和工程(光伏设备、光学、能量存储、微马达、微抓手、微阀和自适应变形反射镜)中，例子比比皆是。然而，微制造是耗时、昂贵的，而且往往是不稳定的，这使得预测计算工具的开发在工程和科学中至关重要。这个项目涉及材料科学、生物物理学、等离子体物理学和机器人学中感兴趣的几何问题的建模、分析和计算。它增强了建模和预测能力，并帮助学生和博士后在激动人心的、在数学和计算方面具有挑战性的、与当代研究相关的领域进行教育。用最简单和最原始的模型捕捉非线性现象的基本行为是科学和工程的基础。这有助于理解基本机制，设计和实现用于模拟和控制设备的有效数值方法，以及分析模型和算法。现代研究的这些关键方面被合并到以下四个相互交织的项目中：带约束的几何偏微分方程组(双层致动器和预应变薄膜，等离子体约束的形状优化，以及完全非线性偏微分方程组)；复杂流体的驱动(电场和温度驱动的液晶，以及磁场驱动的磁流体)；非局部模型(线性和非线性分数阶扩散和随机控制的有效求解器)；后验误差分析和自适应(高阶方法，分数阶偏微分方程组，和自由边界问题)。由于几何的动态变形、强非线性的存在以及自穿透结构和拓扑变化的发展，非线性几何偏微分方程组的数值处理是一项艰巨的科学挑战。高效的算法应该优化和平衡计算努力，从而在不过度解析其他尺度的情况下捕获小尺度，从而导致准确的接口描述。该项目开发了具有后验误差控制(自适应FEMS)和多级解算器的结构保持有限元方法(FEMS)，允许以相对较少的计算资源来解决具有非常不同的时空尺度的问题。几何、非线性、非局部性和自适应近似的作用渗透到研究中，从非线性偏微分方程组的数值分析的基本问题到国家利益的战略领域的应用。研究生和博士后学生参与该项目的研究。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。

Fabrication and manipulation of new and smart materials, particularly in the strategic areas of nanotechnology and biotechnology, require understanding of nonlinear phenomena governed by geometric partial differential equations (PDEs). At small scales, say micro and nano scales, surface tension and bending effects dominate bulk effects, thereby making the actuation and control of small devices a reality. This leads to scientifically interesting and technologically useful configurations and dynamic behavior. Examples abound in biomedical sciences (drug delivery vesicles, cell encapsulation devices, and sensors) and engineering (photovoltaic devices, optics, energy storage, micromotors, microgrippers, microvalves, and adaptive deformable mirrors). However, microfabrication is time-consuming, expensive, and often erratic, which makes the development of predictive computational tools of paramount importance in engineering and science. This project deals with modeling, analysis, and computation of geometric problems of interest in materials science, biophysics, plasma physics, and robotics. It enhances modeling and prediction capabilities and helps educate students and postdocs in exciting, mathematically and computationally challenging, and practically relevant areas of contemporary research.Capturing the essential behavior of nonlinear phenomena with the simplest and crudest models is fundamental in science and engineering. This allows for understanding of basic mechanisms, the design and implementation of efficient numerical methods for simulation and control of devices, and the analysis of both models and algorithms. These crucial aspects of modern research are incorporated into the following four intertwined projects: geometric PDEs with constraints (bilayer actuators and prestrained films, shape optimization for plasma confinement, and fully nonlinear PDEs); actuation of complex fluids (liquid crystals actuated by electric fields and temperature, and ferrofluids actuated by magnetic fields); nonlocal models (efficient solvers for linear and nonlinear fractional diffusion and stochastic control); a posteriori error analysis and adaptivity (high-order methods, fractional PDEs, and free boundary problems). Numerical treatment of nonlinear geometric PDEs is a formidable scientific challenge due to the dynamic deformation of geometries, the presence of strong nonlinearities, and the development of self-penetrating structures and topological changes. Efficient algorithms should optimize and balance the computational effort and thus capture small scales without over-resolving others, thereby leading to accurate interface description. This project develops structure-preserving finite element methods (FEMs) with a posteriori error control (adaptive FEMs) and multilevel solvers, which allow for the resolution of problems with very disparate space-time scales with relatively modest computational resources. The roles of geometry, nonlinearity, nonlocality, and adaptive approximation permeate the research, from basic questions in numerical analysis of nonlinear PDEs to applications in strategic areas of national interest. Graduate and postdoctoral students participate in the research of the project.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.