Efficient spectrally accurate global basis methods for high frequency wave scattering, chaotic eigenmodes, and photonics

适用于高频波散射、混沌本征模和光子学的高效光谱精确全局基础方法

• 批准号: 0811005
• 负责人: Alexander Barnett
• 金额: $ 31.05万
• 依托单位: Dartmouth College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-15 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0811005&HistoricalAwards=false
• 关键词: Efficient; spectrally; accurate; global; basis

Accurate and rapid numerical solution of the Helmholtz and related partial differential equations in complex geometries is key to future progress in device design, in imaging, and in basic science. However, at high frequencies (many wavelengths across the system) this becomes prohibitively challenging using direct discretization, due to the multiscale nature of the problem. The investigator seeks to build upon boundary-based methods which have been uniquely successful (up to a thousand times faster than the competition) in solving eigenmode problems hundreds of wavelength in size with spectral accuracy in two dimensions, and to extend them to the scattering problem, to more general media and periodic boundary conditions, and to three dimensions. These methods are global approximation by particular solution basis sets, and the scaling method for Dirichlet eigenmodes.Proposed extensions include: 1) use of fundamental solutions basis sets, and their analysis via the role of singularities in the analytic continuation of the wave field, 2) exploiting a little-known analytic formula for the fundamental solution in linear graded-index materials, enabling non-piecewise-constant media to be solved on the boundary, 3) error analysis of a reformulation of the scaling method via the Dirichlet-to-Neumann map for the domain, 4) application of such methods to the spectrally accurate solution of dielectric photonic crystal band structure, and to `quantum chaos' (the wave and spectral properties of cavities with ergodic ray dynamics).The impact of our technology such as radar, microwave communication (eg cellphones), optics and lasers, acoustics, medical ultrasound imaging, and miniaturized quantum devices has been, and will continue to be, profound and far-reaching. To design all such devices, one must calculate how they will reflect, guide and trap waves, and this is a time-intensive, difficult and sometimes unreliable computation.The computer algorithms proposed by the investigator will make such calculations faster and more accurate, particularly when the objects are large or complicated in shape. This is expected to lead to improvements in the design of, for example, optical signal-processing devices (which rely on microscopic periodic structures the size of the wavelength of light), promising candidates for the next generation of fast (post-silicon) computers. A deeper grasp of quantum chaos (the behavior of waves trapped in cavities which cause chaotic bouncing ofrays) would impact nanoscale quantum wave devices such as quantum dots, super-fast quantum computers, as well as areas of pure mathematics and physics theory. The proposal also provides training in applied and computational mathematics at both graduate and undergraduate levels, and a course on the ``Mathematics of Music and Sound'' introducing non-majors to waves, modes, and resonance.

复杂几何形状中Helmholtz和相关偏微分方程的准确数值解是设备设计，成像和基础科学中未来进步的关键。 但是，由于问题的多尺度性质，在高频（整个系统上的许多波长）下，这种直接离散化变得极为挑战。 研究者试图基于基于边界的方法，这些方法在解决本本本征局问题方面取得了独特成功（比竞争快一千倍），在两个维度的频谱准确性方面求解了数百个波长，并将其扩展到散射问题，并将其扩展到更一般的媒体和周期性的边界条件，并将其扩展到三个维度。 这些方法是通过特定溶液基集的全球近似值，以及dirichlet eigenmodes的缩放方法。已提供的扩展包括：1）使用基本解决方案基集，以及它们通过奇点在波领域的分析中的作用来进行分析，2）在波浪场中的作用；在边界上解决的解决方案，3）通过Dirichlet到Neumann映射的重新分析域的​​重新分析，4）将这种方法应用于介电光子晶体带结构的光谱准确的解决方案，以“量子和光谱”（Ergodic Ray Dynamics）的范围（Ergodic Ray Dynamics）的范围（Ergodic Ray Dynamics）的波动（量子和光谱）。激光，声学，医学超声成像和微型量子设备已经并且将继续是深刻而深远的。 要设计所有此类设备，必须计算它们的反射，引导和捕获波，这是一个时间密集型，困难，有时是不可靠的计算。研究人员提出的计算机算法将更快，更准确地进行计算，尤其是当对象大型或复杂的形状时。 预计这将导致改进的设计，例如，光学信号处理设备（依赖于微观周期性结构的光的波长的大小），有望成为下一代快速（后 - 硅）计算机的候选者。 对量子混乱的更深入的掌握（在腔体中捕获混乱的弹跳的波浪的行为）会影响纳米级量子波器设备，例如量子点，超快速量子计算机以及纯数学和物理学理论的领域。 该提案还提供了研究生和本科级别的应用和计算数学的培训，以及``音乐和声音的数学''的课程，向波浪，模式和共鸣介绍了非缩影。