PDE solvers: Frequency-domain, time-domain and hybrids---with applications to materials science and engineering

PDE 求解器：频域、时域和混合求解器——在材料科学和工程中的应用

• 批准号: 1411876
• 负责人: Oscar Bruno
• 金额: $ 47万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-15 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1411876&HistoricalAwards=false
• 关键词: PDE; solvers; Frequency; domain; time

This effort concerns development of mathematical tools which can be used to make accurate predictions about physical phenomena, with impact on areas of significant societal interest such as electrical engineering (optics, electronics, photonics) communications (antennas), atmospheric science, medicine (tomography, imaging, diagnostic and therapeutic ultrasound, targeted drug delivery), military and civilian remote sensing (radar, sonar, stealth), renewable energy production and energy policy (Doppler signature of wind farms) etc. The methodologies to be pursued represent a change in paradigm in the mathematical approach. Oversimplifying for the sake of clarity, the new methods can be visualized as using some sort of a computational version of the french curve tool rather than a straight ruler in such a way that much more accurate representations of physical reality as well as greatly reduced computing costs result -- to the point that previously unfeasible simulations become possible. While such "smooth-curve representations" (or, in mathematical nomenclature, high-order/spectral methods) have been available for many years, the novelty of the proposed work is that it enables utilization of such highly accurate methodologies at vastly reduced computing costs and for highly complex engineering problems--such as those mentioned above--including complex electronic components, full vehicles, etc.Technically, this project concerns development and analysis of high-performance, highly accurate numerical algorithms for solution of Partial Differential Equations (PDE), with application to a wide range of problems in materials science and engineering. The proposed algorithms emphasize accuracy, efficiency as well as generally applicability on the basis of spectral and high-order methodologies; the associated theoretical discussions, in turn, seek to provide necessary background and performance guarantees. This effort considers two broad application areas, namely, I. Frequency-domain, time-harmonic acoustics and electromagnetism, and II. Time-domain PDE in general three-dimensional domains, with applicability to acoustics, electromagnetism and elasticity as well as compressible and incompressible fluid-dynamics.

这项工作涉及数学工具的开发，可用于对物理现象进行准确预测，对具有重要社会意义的领域产生影响，例如电气工程（光学、电子、光子）通信（天线）、大气科学、医学（断层扫描、成像、诊断和治疗超声、靶向药物输送）、军事和民用遥感（雷达、声纳、隐形）、可再生能源生产和能源 政策（风电场的多普勒特征）等。所追求的方法代表了数学方法范式的变化。为了清楚起见，我们将新方法可视化为使用某种计算版本的法式曲线工具而不是直尺，从而可以更准确地表示物理现实，并大大降低计算成本，从而使以前不可行的模拟成为可能。虽然这种“平滑曲线表示”（或者，用数学术语来说，高阶/谱方法）已经存在很多年了，但所提出的工作的新颖之处在于，它能够以大大降低的计算成本利用这种高精度的方法，并解决高度复杂的工程问题（例如上面提到的那些问题），包括复杂的电子元件、整车等。从技术上讲，该项目涉及开发和分析 用于求解偏微分方程 (PDE) 的高性能、高精度数值算法，可应用于材料科学和工程中的各种问题。所提出的算法在谱和高阶方法的基础上强调准确性、效率以及普遍适用性；相关的理论讨论反过来又寻求提供必要的背景和绩效保证。这项工作考虑了两个广泛的应用领域，即 I. 频域、时谐声学和电磁学，以及 II. 频域、时谐声学和电磁学。一般三维域中的时域偏微分方程，适用于声学、电磁学和弹性以及可压缩和不可压缩流体动力学。