CAREER: Randomized Multiscale Methods for Heterogeneous Nonlinear Partial Differential Equations

职业：异质非线性偏微分方程的随机多尺度方法

基本信息

批准号：
2145364
负责人：
Kathrin Smetana
金额：
$ 46.2万
依托单位：
Stevens Institute of Technology
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-06-01 至 2027-05-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2145364&HistoricalAwards=false
关键词：
CAREER Randomized Multiscale Methods Heterogeneous

项目摘要

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Heterogeneous systems with salient features at multiple scales are ubiquitous in science and engineering. A direct numerical simulation that aims at capturing relevant phenomena at all scales requires an often prohibitively large amount of computation time. To simulate such systems, multiscale methods include the local behavior of a numerical solution in the approximation process, thus taking into account the various scales. For example, in modeling a wind turbine made from composites, deformations during operation can be simulated for portions of the wind turbine blade. The multiscale approximation for the deformation of the whole wind turbine is then built from these local solutions. Multiscale methods that can guarantee that the error between the multiscale approximation and the global solution is below a given tolerance are of particular interest. The goal of this project is to design and analyze such multiscale methods for the numerical solution of nonlinear partial differential equations that are used in simulating deformations in (realistic) wind turbines. It is anticipated that the new methods will be crucial in building digital twins, that is, mathematical models of physical objects that can be employed in real time to assess, for example, the structural health of a system. Application of the results in digital twins for wind turbines will support the generation of renewable energy for society. The project includes a closely integrated educational plan to increase participation and retention of students from groups underrepresented in STEM by (i) designing and leading courses for high school students, helping them discover via creative and project-based learning techniques how the concepts of mathematics they are learning have important applications; and (ii) establishing a mentoring program for undergraduate mathematics students from underrepresented groups.To develop the desired multiscale methods, in this project, the local ansatz functions will be constructed to (quasi-)optimally approximate the nonlinear set of local solutions of the partial differential equation (PDE). To approximate the latter, randomized versions of model order reduction methods will be developed. While deterministic model reduction algorithms construct provably the optimal space to approximate a set of solutions of a PDE dependent on a parameter (here arbitrary Dirichlet boundary data), they suffer from the curse of dimensionality for high-dimensional parameter sets. Randomizing these methods is expected to break the curse of dimensionality and allow analysis of the error in novel ways suitable for nonlinear systems. The three research objectives of the project are: development and analysis of randomized multiscale methods for (i) elliptic and (ii) parabolic nonlinear PDEs, where the local ansatz functions can be constructed parallel in time, and (iii) application to the simulation of the deformation of wind turbines.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.

该奖项是根据2021年《美国救援计划法》（公法117-2）全部或部分资助的。具有多个尺度的显着特征的异质系统在科学和工程中无处不在。旨在捕获各个规模的相关现象的直接数值模拟需要大量的计算时间。为了模拟此类系统，多尺度方法包括在近似过程中数值解决方案的局部行为，从而考虑了各种量表。例如，在对由复合材料制成的风力涡轮机进行建模时，可以为风力涡轮刀片的一部分模拟操作过程中的变形。然后，由这些本地解决方案构建整个风力涡轮机变形的多尺度近似。可以保证多尺度近似和全局解决方案之间的误差低于给定公差的多尺度方法特别感兴趣。该项目的目的是设计和分析此类多尺度方法，用于用于模拟（现实）风力涡轮机中的变形的非线性部分微分方程的数值解。预计新方法对于建立数字双胞胎至关重要，即可以实时使用的物理对象的数学模型来评估系统的结构健康。将结果在数字双胞胎中应用于风力涡轮机，将支持为社会的可再生能源产生。该项目包括一项紧密融合的教育计划，以通过（i）为高中生设计和领导课程来增加seat中代表性不足的学生的参与和保留，从而通过创造性和项目的学习技术帮助他们发现他们所学习的数学概念如何具有重要的应用；（ii）为来自代表性不足的群体的本科数学学生建立指导计划。为了开发所需的多尺度方法，在该项目中，本地的ANSATZ函数将构建为（Quasi-）最佳近似于偏微分方程（PDE）的局部端口局部统计的非线性集合（PDE）。为了近似后者，将开发模型订购降低方法的随机版本。尽管确定模型还原算法构建了近似于参数的PDE解决方案的最佳空间（此处是任意的dirichlet边界数据），但它们遭受了高维参数集的维度诅咒。预计这些方法将破坏维度的诅咒，并允许以适合非线性系统的方式分析误差。该项目的三个研究目标是：（i）椭圆形和（ii）抛物面非线性PDE的随机多尺度方法的开发和分析，可以及时构建本地ANSATZ函数，以及（iii）应用于风力涡轮机的变形。该奖项反映了NSF的Infort of Noctiral of Suttrunial taustrucation。影响审查标准。