Numerical Methods for Wave Equations in Time and Frequency Domain

波的一个固有特征是它们能够在不改变形状的情况下远距离传播。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。波还可以用来探测地球、人体或建筑物或桥梁等工程结构的内部。这种探测可以通过求解反问题的方法转化为内部图像，并在扩展时，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，首席研究人员将开发计算模拟工具，以提高我们利用波传播特性实现公共利益的能力。该项目开发的工具还可以用来设计具有自然界中找不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，如微型扬声器、5G组件和其他毫米波技术。这项研究将使用一种新的思想，使使用波动方程的时间域方法来设计频域Helmholtz型解算器。这种方法值得注意的是，基本的线性算子对应于一个对称的正定矩阵，允许解决强制问题，而不是不定的Helmholtz问题。由于所提出的亥姆霍兹解算器仅依赖于波动方程的演化，因此它们将是大规模并行的、可伸缩的和高精度的。这项研究的一个目标是在比目前可能的更多的核心上，以更高的频率在三维中求解亥姆霍兹方程。该研究还将寻求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来改进时间域不连续Galerkin方法的时间步长约束。这样的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。

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.