Nonlinear Geometric Models: Algorithms, Analysis, and Computation

新型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，需要理解由几何偏微分方程（PDEs）控制的非线性现象。在小尺度上，比如微纳米尺度，表面张力和弯曲效应主导了体积效应，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。这样的例子在生物医学科学（药物传递囊泡、细胞封装装置和传感器）和工程（光伏装置、光学、储能、微电机、微钳、微阀和自适应变形镜）中比比皆是。然而，微加工耗时、昂贵，而且常常不稳定，这使得预测计算工具的发展在工程和科学中至关重要。本项目涉及材料科学、生物物理学、等离子体物理学和机器人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，并帮助教育学生和博士后在令人兴奋的，数学和计算的挑战，以及当代研究的实际相关领域。用最简单和最粗糙的模型捕捉非线性现象的基本行为是科学和工程的基础。这允许对基本机制的理解，设计和实现有效的数值方法模拟和设备控制，以及模型和算法的分析。现代研究的这些关键方面被纳入以下四个相互交织的项目：具有约束的几何pde（双层致动器和预紧膜，等离子体约束的形状优化，以及完全非线性pde）；复杂流体的驱动（由电场和温度驱动的液晶，以及由磁场驱动的铁磁流体）；非局部模型（线性和非线性分数扩散和随机控制的有效解算器）；后验误差分析和自适应（高阶方法，分数偏微分方程和自由边界问题）。由于几何结构的动态变形、强非线性、自穿透结构的发展和拓扑结构的变化，非线性偏微分方程的数值处理是一项艰巨的科学挑战。有效的算法应该优化和平衡计算工作量，从而捕获小尺度而不过度解析其他，从而导致准确的接口描述。该项目开发具有后判误差控制（自适应fem）和多级求解器的结构保留有限元方法（fem），允许用相对适度的计算资源解决非常不同的时空尺度的问题。从非线性偏微分方程数值分析中的基本问题到在国家利益战略领域的应用，几何、非线性、非定域性和自适应近似的作用渗透到研究中。研究生和博士后参与项目研究。该奖项反映了美国国家科学基金会的法定使命，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。

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.