RUI: Dispersive Shock Waves in Nonlinear Lattices: Theory to Application

了解系统如何应对突然变化对科学和工程应用至关重要。经典的例子包括在封闭区域内的爆炸，在活塞室中压缩的气体，或水库大坝的破坏。重要的应用包括冲击吸收器的设计，这对于在汽车事故中保护民用基础设施和乘客非常重要。它们在航天器的部署和着陆方面也发挥着至关重要的作用。本项目研究一类由非线性晶格建模的系统，非线性晶格是以非线性方式相互作用的粒子链的数学模型。折纸为基础的材料的模型将在这个项目中考虑的主要非线性晶格，因为这种材料表现出理想的应用性能。例如，折纸格子受到的撞击越大，波通过它的速度就越慢。该项目旨在通过数学建模、计算机模拟和实验来提高对非线性格子中波传播的理解。预计结果将有助于设计的冲击缓解装置。属于科学代表性不足的群体的学生将接受培训，并通过勤工俭学计划参加夏季研究体验。在某些系统中，当系统的状态发生突变时，会形成一个振荡波，连接不同振幅的（局部）状态。这种振荡波被称为色散冲击波（DSW）。分析DSW的标准方法是基于Whitham调制理论。在非线性晶格的情况下，Whitham调制方程非常复杂。在这个项目中，一个低维微分方程，准确地描述了波，使DSW在一个晶格将寻求使用数据驱动的方法。这个低维微分方程将被利用，以获得一个简单的分析描述DSW。一个准连续的方法也将被用来分析识别底层的低维微分方程。还将对二维DSW进行系统研究。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Understanding how systems respond to sudden changes is critical to science and engineering applications. Classic examples include explosions in confined areas, gases compressed in a piston chamber, or breaking of water reservoir dams. Important applications include design of impact absorbers, which are important in protecting civil infrastructure and passengers in automobile accidents. They also play a crucial role in the deployment and landing of spacecraft. This project studies a class of systems modeled by nonlinear lattices, mathematical models of chains of particles that interact with each other in a nonlinear fashion. Models of origami-based materials will be the primary nonlinear lattices considered in this project since such materials exhibit desirable properties for applications. For example, the harder an origami lattice is hit, the slower a wave will travel through it. The project aims to improve understanding of wave propagation in nonlinear lattices through mathematical modeling, computer simulation, and experimentation. Results are expected to aid in the design of impact-mitigating devices. Students who belong to groups underrepresented in the sciences will be trained and recruited to participate in summer research experiences via a work-study program. In some systems subjected to a sudden change in state, an oscillating wave is formed that connects (local) states of different amplitude. Such oscillating waves are called dispersive shock waves (DSWs). The standard approach to analyze a DSW is based on Whitham modulation theory. In the case of nonlinear lattices, the Whitham modulation equations are prohibitively complex. In this project, a low-dimensional differential equation that accurately describes the waves that make up a DSW in a lattice will be sought using data-driven methodologies. This low-dimensional differential equation will be exploited to obtain a simple analytical description of a DSW. A quasi-continuum approach will also be employed to analytically identify the underlying low-dimensional differential equation. A systematic study of two-dimensional DSWs will also be conducted. Numerical simulations, small-amplitude approximations, modulation theory, and experimentation will be employed.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.