A New Computational Approach for Wave Propagation

波浪运动是许多科学领域感兴趣的关键机制，如力学、声学、地震学、海洋学、海岸和近海工程、电磁学等。尽管波浪现象的物理外观千奇万怪，但基于它们实现的数学模型和数值方法面临可靠性和准确性问题，特别是在高频和冲击载荷下。该奖项将支持对波传播具有最佳精度的新数值方法的研究。新的建模技术将能够为与国家健康和安全相关的广泛民用和军事应用提供准确的解决方案，其中波浪传播是一个问题。该项目将为研究生和本科生提供波浪传播理论和建模方面的教育和培训机会，重点关注代表性不足的学生。在这个项目下，还将为高中教师开设一个关于动态行为的讲习班。许多基于偏微分方程弱形式公式的现代数值技术不能提供离散方程的最佳精度。这导致现实世界的波传播问题的计算时间大得令人望而却步。在这个项目中，波在不规则区域的笛卡尔网格上传播的一种新的高阶数值方法将直接用未知系数的离散方程来表示。这些系数是通过最小化局部截断误差来计算的，并提供了最优的精度顺序。一些重要的应用将包括霍普金森棒中的高频波传播和裂纹尖端和缺陷附近的弹性波传播。由于固定的笛卡尔网格，不需要对移动裂缝的区域进行网格划分。在相同的笛卡尔网格上，用相应的高阶边界条件来处理运动裂纹。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。

Wave motion is the key mechanism of interest to many fields of science, such as mechanics, acoustics, seismology, oceanography, coastal and offshore engineering, electromagnetism, etc. Despite an extreme variety of physical appearances of wave phenomena, mathematical models and numerical methods implemented from them face reliability and accuracy concerns, especially at high frequencies and under impact loadings. This award will support research on a new numerical approach with optimal accuracy for wave propagation. The new modeling technique will be able to provide accurate solutions for a broad range of civil and military applications related to national health and security, respectively, where wave propagation is a concern. The project will provide opportunities to educate and train graduate and undergraduate students on the theoretical and modeling aspects of wave propagation, with a focus on underrepresented students. A workshop on dynamic behavior will also be developed for high school teachers under this project.Many modern numerical techniques based on weak-form formulations of partial differential equations do not provide optimal accuracy in the discrete equations. This leads to a prohibitively large computation time for real-world wave propagation problems. In this project, a novel high-order numerical approach for wave propagation on Cartesian meshes for irregular domains will be directly formulated in terms of discrete equations with unknown coefficients. These coefficients are calculated by the minimization of the local truncation error and provide the optimal order of accuracy. Some of the important applications will include high-frequency wave propagation in the Hopkinson Bar and elastic wave propagation in the vicinity of crack tips and defects. Due to stationary Cartesian meshes, there will be no need to remesh the domain for a moving crack. The moving crack will be treated by the corresponding high-order boundary conditions on the same Cartesian grid.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.