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

了解系统如何应对突发变化对科学和工程应用至关重要。典型的例子包括密闭区域的爆炸，活塞腔内压缩的气体，或水库大坝的破裂。重要的应用包括减震器的设计，这对于保护民用基础设施和汽车事故中的乘客至关重要。它们在航天器的部署和着陆中也起着至关重要的作用。本项目研究一类由非线性晶格建模的系统，即以非线性方式相互作用的粒子链的数学模型。基于折纸的材料模型将是本项目中主要考虑的非线性晶格，因为这种材料在应用中表现出理想的特性。例如，折纸晶格受到的撞击越重，波穿过它的速度就越慢。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晶格中波传播的理解。研究结果有望帮助设计减震装置。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经历。在某些系统遭受状态的突然变化，形成振荡波连接不同幅度的（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。在非线性格的情况下，威瑟姆调制方程过于复杂。在这个项目中，将使用数据驱动的方法寻求一个精确描述构成晶格中DSW的波的低维微分方程。这个低维微分方程将被用来获得DSW的简单解析描述。一个准连续统的方法也将被用来分析识别潜在的低维微分方程。对二维DSWs也将进行系统的研究。数值模拟，小幅度近似，调制理论和实验将被采用。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。

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.