An efficient, accurate and robust solution technique for variable coefficient elliptic partial differential equations in complex geometries

复杂几何中变系数椭圆偏微分方程的高效、准确和稳健的求解技术

• 批准号: 2110886
• 负责人: Adrianna Gillman
• 金额: $ 29.49万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2110886&HistoricalAwards=false
• 关键词: efficient; accurate; robust; solution; technique

Numerical simulations play a key role in scientific discovery and device development because they have the ability to reduce the cost of testing theories and ideas. These simulations often involve the solution of problems that are prescribed by physics models. In many cases, the geometry where the problem is posed is complex and/or the equation that is modeling the physical phenomena results in geometry complexities. Applications involving complex geometries include materials design, inverse scattering and fluid simulations. Due to the sophisticated nature of solutions, it is desirable to keep the computational cost as low a possible. The key to doing this is to use as few degrees of freedom as possible to capture the physics and to couple this with efficient solution techniques. A recent numerical technique called the hierarchical Poincare-Steklov (HPS) method is able to do this for many problems. This method has been demonstrated to be effective for high frequency scattering problems and has been integrated into inverse scattering simulations. However, the current version of the HPS method is not able to handle the complex geometries that arise in most applications. This research will address this shortfall allowing the method to be applied to complex geometries. This work will also connect the new version of the HPS method to existing software for complex geometries allowing it to be integrated into simulation packages.The numerical simulations under consideration in this project are linear elliptic partial differential equations with variable coefficients including high frequency Helmholtz problems. These equations arise in the modeling of physical phenomena such as scattering, electrostatics, and when using many time-stepping techniques for solving time dependent problems (i.e. in fluid simulations). The current version of the HPS method can only handle geometries that can be easily mapped from a square or cube. This is problematic as in most applications, there are geometric features that do not fall into either of these categories. This project will extend the HPS method to a general range of geometries and will integrate seamlessly with existing mesh generation software. Additionally, this project will make the HPS method efficient for three dimensional problems (which it currently is not) and develop the necessary analysis to support the numerical results that are observed in practice. Thanks to the robustness of the HPS method, it can be used in material design and inverse scattering with a known a priori computational cost. This means that practitioners can confidently use this technique for scattering applications knowing that the method is achieving the desired accuracy.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.

数值模拟在科学发现和设备开发中发挥着关键作用，因为它们能够降低测试理论和想法的成本。这些模拟通常涉及物理模型规定的问题的解决方案。在许多情况下，提出问题的几何形状是复杂的和/或对物理现象建模的方程导致几何形状复杂性。涉及复杂几何形状的应用包括材料设计、逆散射和流体模拟。 由于解决方案的复杂性，希望保持尽可能低的计算成本。做到这一点的关键是使用尽可能少的自由度来捕获物理学，并将其与有效的解决方案技术相结合。最近的数值技术称为分层庞加莱-斯捷克洛夫（HPS）方法是能够做到这一点的许多问题。该方法已被证明是有效的高频散射问题，并已被集成到逆散射模拟。然而，HPS方法的当前版本不能处理在大多数应用中出现的复杂几何形状。这项研究将解决这一不足，使该方法适用于复杂的几何形状。这项工作还将连接的HPS方法的新版本，以现有的软件，复杂的几何形状，允许它被集成到simulation packages.在这个项目中考虑的数值模拟是线性椭圆偏微分方程的变系数，包括高频亥姆霍兹问题。这些方程出现在诸如散射、静电等物理现象的建模中，以及当使用许多时间步进技术来解决时间相关问题时（即在流体模拟中）。HPS方法的当前版本只能处理可以轻松从正方形或立方体映射的几何形状。这是有问题的，因为在大多数应用中，存在不属于这些类别中的任一个的几何特征。该项目将HPS方法扩展到一般的几何形状范围，并将与现有的网格生成软件无缝集成。此外，该项目将使HPS方法对三维问题有效（目前还不是），并开发必要的分析来支持在实践中观察到的数值结果。由于HPS方法的鲁棒性，它可以用于材料设计和逆散射与已知的先验计算成本。这意味着从业者可以放心地使用这种技术来分散应用，知道该方法正在实现所需的精度。该奖项反映了NSF的法定使命，并已被认为是值得支持的，通过评估使用基金会的知识价值和更广泛的影响审查标准。