Robust High-Order Methods for Wave Equations in the Time Domain

时域波动方程的鲁棒高阶方法

• 批准号: 1418871
• 负责人: Thomas Hagstrom
• 金额: $ 39.9万
• 依托单位: Southern Methodist University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1418871&HistoricalAwards=false
• 关键词: Robust; Order; Methods; Wave; Equations

The goal of this research is to address basic issues in the development of robust and efficient computational methods for simulating waves. Problems governed by wave propagation span much of the physical phenomena we experience and play a fundamental role both in engineered systems for communication and imaging as well as naturally-occurring aspects of earth's environment. Specific examples include: wave phenomena involved with natural disasters such as earthquakes and tsunamis; environmental irritants such as acoustic pollution near airports and in cities; electromagnetic phenomena of importance to defense and civilian applications, such as radar imaging; and applications in medicine such as the interaction of high-frequency ultrasound and tissue. This project will develop improved tools for simulating waves and will design associated general-purpose open-source software with the potential for significant impact in a range of important application areas.With the staggering increases in computational power that have been and continue to be achieved, we expect to simulate more difficult and comprehensive models of physical phenomena. For wave propagation problems posed in the time domain, this means problems with many wavelengths within the computational domain involving interactions with complex geometrical features. To treat such problems efficiently requires the use of high-order discretization methods to minimize the effects of dispersion and dissipation. This work will be focused on fundamental mathematical issues required for the further development of robust, high-order wave solvers. These include: i. Development and analysis of energy-stable high-order/high-resolution discretization methods on hybrid structured-unstructured grids. Specifically we will investigate coupling high-order upwind discontinuous Galerkin methods on unstructured grids near complex boundaries and material interfaces with more efficient structured grid methods such as novel spectral element methods based on Hermite-Birkhoff interpolation (also known as jet schemes) or upwind difference methods constructed from piecewise polynomial or band-limited interpolation functions. Both first-order and second-order hyperbolic systems will be considered. ii. Development, analysis, and implementation of hp-adaptive strategies for these methods. iii. Coupling with an open-source radiation boundary condition library (expected release late 2014) containing various implementations of complete radiation boundary conditions (CRBC). These allow a priori determination of the boundary condition parameters to guarantee any desired accuracy for isotropic, homogeneous models in the far field. iv. Leveraging the fact that CRBCs are stable for any Friedrichs system, extend their applicability to more complex physical models including anisotropy.

这项研究的目的是解决用于模拟波的鲁棒和有效计算方法开发的基本问题。由波传播控制的问题涵盖了我们经历的许多物理现象，并且在工程系统中的通信和成像以及地球环境的自然疾病方面都发挥了基本作用。具体例子包括：与地震和海啸等自然灾害有关的波浪现象；环境刺激物，例如机场和城市附近的声污染；对防御和平民应用重要性的电磁现象，例如雷达成像； 以及医学中的应用，例如高频超声和组织的相互作用。该项目将开发改进的模拟波动的工具，并将设计相关的通用开源软件，并有可能在一系列重要的应用领域中产生重大影响。随着计算能力的惊人增长一直并继续实现，我们希望模拟更加困难，更全面的物理现象模型。对于时间域中提出的波传播问题，这意味着在计算域内具有许多波长的问题，涉及与复杂的几何特征相互作用。为了有效地处理此类问题，需要使用高级离散方法来最大程度地减少分散和耗散的影响。这项工作将集中在进一步发展强大的高阶波解决器所需的基本数学问题上。这些包括：i。在混合结构化的无结构网格上的高阶/高分辨率离散方法的开发和分析。具体而言，我们将研究在复杂边界和材料界面附近的非结构性网格上耦合的耦合，并具有更有效的结构化网格方法，例如基于Hermite-birkhoff插值（也称为JET架构）或从碎裂的多个跨度界面式构建的型号的差异方法，具有更有效的结构化网格方法，例如新型的光谱元素方法。将考虑一阶和二阶双曲系统。 ii。针对这些方法的HP自适应策略的开发，分析和实施。 iii。与包含完整辐射边界条件（CRBC）的各种实现的开源辐射边界条件库（预期发布）耦合。这些允许对边界条件参数的先验确定可以保证远场中各向同性，同质模型的任何预期准确性。 iv。利用CRBC在任何Friedrichs系统中都是稳定的事实，将其适用性扩展到包括各向异性在内的更复杂的物理模型。