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

了解系统对突然变化的反应对于科学和工程应用至关重要。经典的例子包括密闭区域的爆炸，活塞室中压缩的气体，或者水库大坝的决口。重要的应用包括设计减震器，这在保护民用基础设施和汽车事故中的乘客方面非常重要。它们在航天器的部署和着陆中也发挥着至关重要的作用。这个项目研究了一类由非线性晶格建模的系统，即以非线性方式相互作用的粒子链的数学模型。折纸材料的模型将是本项目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，折纸晶格被击中得越重，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波在非线性格子中传播的理解。预计研究结果将有助于减震装置的设计。属于在科学领域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招募他们参加暑期研究体验。在一些受到状态突变影响的系统中，形成一个振荡波，它连接不同幅度的(局部)状态。这种振荡波称为色散激波(DSWs)。分析DSW的标准方法是基于Whitham调制理论。在非线性格子的情况下，Whitham调制方程复杂得令人望而却步。在这个项目中，将使用数据驱动的方法寻找一个低维的微分方程，它可以准确地描述组成晶格中的DSW的波。我们将利用这个低维的微分方程式来获得对DSW的简单解析描述。还将使用准连续方法来分析识别基本的低维微分方程式。还将对二维DSWs进行系统研究。将采用数值模拟、小幅度近似、调制理论和实验。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。

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.