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.

本研究的目标是解决基本问题，在强大的和有效的计算方法模拟波浪的发展。由波传播控制的问题跨越了我们所经历的许多物理现象，并在通信和成像的工程系统以及地球环境的自然发生方面发挥着重要作用。具体例子包括：与地震和海啸等自然灾害有关的波动现象;机场和城市附近的声污染等环境刺激物;对国防和民用应用具有重要意义的电磁现象，如雷达成像;以及医学应用，如高频超声和组织的相互作用。该项目将开发用于模拟波浪的改进工具，并将设计相关的通用开放源码软件，这些软件可能在一系列重要应用领域产生重大影响。随着计算能力的惊人增长，我们预计将模拟更困难和更全面的物理现象模型。对于在时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，涉及与复杂几何特征的相互作用。为了有效地处理这些问题，需要使用高阶离散化方法，以尽量减少色散和耗散的影响。这项工作将集中在基本的数学问题所需的进一步发展的强大，高阶波解算器。这些措施包括：混合结构-非结构网格上能量稳定的高阶/高分辨率离散化方法的发展和分析。具体来说，我们将研究耦合高阶迎风不连续Galerkin方法的非结构网格附近的复杂边界和材料接口更有效的结构网格方法，如新的谱元素方法的基础上Hermite-Birkhoff插值（也称为喷射计划）或迎风差分方法构造分段多项式或带限插值函数。一阶和二阶双曲方程组都将被考虑。二.这些方法的hp适应策略的开发、分析和实施。三.与开源辐射边界条件库（预计2014年底发布）耦合，该库包含完整辐射边界条件（CRBC）的各种实现。这些允许先验确定的边界条件参数，以保证在远场的各向同性，均匀模型的任何所需的精度。 四.利用CRBC对任何弗里德里希系统都是稳定的这一事实，将其适用性扩展到更复杂的物理模型，包括各向异性。