Nonlinear Geometric Models: Algorithms, Analysis, and Computation

非线性几何模型：算法、分析和计算

基本信息

批准号：
1908267
负责人：
Ricardo Nochetto
金额：
$ 108.36万
依托单位：
University of Maryland, College Park
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1908267&HistoricalAwards=false
关键词：
Nonlinear Geometric Models Algorithms Analysis

项目摘要

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.

对新材料和智能材料的制造和操纵，特别是在纳米技术和生物技术的战略领域，需要了解受几何局部微分方程（PDE）控制的非线性现象。在小尺度上，微型和纳米尺度，表面张力和弯曲效应主导了散装效应，从而使小型设备的致动和控制成为现实。这导致了科学有趣且技术上有用的配置和动态行为。生物医学科学（药物输送囊泡，细胞包裹设备和传感器）和工程（光伏设备，光学器件，能源储能，微型机器，微型机器，微植物，微毛管和可自适应可变形镜）中的例子比比皆是。但是，微加工是耗时，昂贵且通常不稳定的，这使得在工程和科学中具有至关重要的预测计算工具的发展。该项目涉及材料科学，生物物理学，等离子体物理学和机器人技术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，以令人兴奋的，数学和计算上的挑战，并且实际上相关的当代研究领域。以最简单和最冷漠的模型在科学和工程中以最简单和最粗略的模型来接近非线性现象的基本行为。这允许理解基本机制，用于模拟和控制设备的有效数值方法的设计和实施以及模型和算法的分析。现代研究的这些关键方面纳入了以下四个相互交织的项目中：具有约束的几何PDE（双层执行器和胶片的胶片，血浆限制的形状优化，以及完全非线性的PDE）；复杂流体的致动（由电场和温度引起的液晶以及由磁场作用的铁熔花）；非局部模型（用于线性和非线性分数扩散和随机控制的有效求解器）；后验误差分析和适应性（高阶方法，分数PDE和自由边界问题）。非线性几何PDE的数值处理是由于几何形状的动态变形，强烈的非线性的存在以及自渗透结构和拓扑变化的发展，这是一项巨大的科学挑战。有效的算法应优化和平衡计算工作，从而捕获小尺度而不会过度解决其他量表，从而导致准确的接口描述。该项目开发具有后验误差控制（自适应FEM）和多级求解器的结构保存有限元方法（FEM），这允许通过具有相对适度的计算资源的非常不同的时空量表解决问题。几何，非线性，非局部性和自适应近似的作用渗透到研究，从基本问题在非线性PDE的数值分析中，到国家利益的战略领域的应用。研究生和博士后学生参加了该项目的研究。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评论标准来评估值得支持的。