Numerical Methods for Wave Equations in Time and Frequency Domain

波的一个内在特征是它们能够传播很长的距离而不改变它们的形状。这种能力允许波携带信息，无论是通过语音还是电子数据传输。波也可以用来探测地球内部，人体或建筑物或桥梁等工程结构。这种探测可以通过求解逆问题的方式转化为内部图像，并在扩展中，通过准确预测地震引起的地面运动来减轻地震灾害。在这个项目中，主要研究者将开发计算模拟工具，以提高我们利用波传播特性的能力。该项目开发的工具也可用于设计具有自然界中无法找到的奇异特性的现代材料。这种超材料可以实现更好的传感技术和更快的声学和电磁电路组件，如小型化扬声器，5G组件和其他毫米波技术。这项研究将使用一种新的想法，使使用时域方法的波动方程设计频域亥姆霍兹型求解器。该方法是显着的，因为基本的线性算子对应于一个对称正定矩阵，允许强制问题的解决方案，而不是一个不定的亥姆霍兹问题。由于所提出的亥姆霍兹解算器仅依赖于演化波动方程，因此它们将是大规模并行的、可扩展的和高阶精度的。该研究的一个目标是以更高的频率在三维空间中求解亥姆霍兹方程，并且比目前可能的核心数量更多。研究还将寻求改善时域不连续Galerkin方法的时间步长约束，通过利用建立在等距节点数据的离散周期扩展上的近似空间。这些改进将导致更快的模拟时间和更准确的预测。将与学术机构和国家实验室的研究人员合作，将这些方法应用于微极材料的建模和地震波的模拟。该奖项反映了NSF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。

An intrinsic feature of waves is their ability to propagate over large distances without changing their shape. This ability allows waves to carry information, be it through speech or electronic transmission of data. Waves can also be used to probe the interior of the earth, the human body or engineered structures like buildings or bridges. This probing can be turned into images of the interior by the means of solving inverse problems, and in the extension, mitigate seismic hazards by accurate predictions of ground motion caused by earthquakes. In this project the principal investigator will develop computational simulation tools that increases our ability to exploit the properties of wave propagation for the common good. The tools developed in the project can also be used to design modern materials with exotic properties that cannot be found in nature. Such metamaterials can enable better sensing technologies and faster acoustic and electromagnetic circuit components such as miniaturized speakers, 5G components and other millimeter wave technologies. The research will use a new idea that enables the use of time domain methods for wave equations to design frequency domain Helmholtz type solvers. The approach is remarkable in that the underlying linear operator corresponds to a symmetric positive definite matrix allowing the solution of a coercive problem rather than an indefinite Helmholtz problem. As the proposed Helmholtz solvers rely solely on evolving the wave equation they will be massively parallel, scalable and high order accurate. A goal of the research is to solve the Helmholtz equation in three dimensions at higher frequencies, and on a larger number of cores than is currently possible. The research will also seek to improve the time-step constraints of time domain discontinuous Galerkin methods by exploiting approximation spaces built on discrete periodic extensions from equidistant node data. Such improvements will result in faster simulation times and more accurate predictions. Applications of the methods to modeling of micropolar materials and to simulation of seismic waves will be carried out in collaboration with researchers from academic institutions and national laboratories.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.