Bernstein-Bezier Techniques for High Order Time-Domain Discontinuous Galerkin Methods

高阶时域间断伽辽金方法的 Bernstein-Bezier 技术

• 批准号: 1719818
• 负责人: Jesse Chan
• 金额: $ 20万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-01 至 2022-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1719818&HistoricalAwards=false
• 关键词: Bernstein; Bezier; Techniques; Order; Time

Simulations of wave propagation are the backbone of numerous applications in a diverse set of areas, such as medical imaging, predictive seismology, and engineering design. Many applications require repeated high resolution simulations of waves in domains with complex boundaries or features. For example, brain imaging methods require dealing with wave propagation through changing media within the human body, such as transitions between different types of tissue or transitions from tissue to bone. The target application areas require numerical simulations which are reliable, accurate, and efficient. However, incorporating accurate approximations of complex media and geometries typically sacrifices one or more of these properties, resulting in simulations which are accurate but slow, fast but inaccurate, or unstable outside of specific settings. The proposed project aims to develop reliable, provably stable methods for wave propagation in complex media and geometries which retain accuracy and efficiency. The project will study numerical wave propagation using high order discontinuous Galerkin (DG) methods, and consists of two main tasks. The first task will leverage high performance architectures such as Graphics Processing Units (GPUs), as well as the recently developed weight-adjusted and Bernstein-Bezier DG methods. Weight-adjusted DG methods achieve provable stability and high order accuracy in the presence of variable media, but have a high computational complexity with respect to the order of approximation. Bernstein-Bezier DG methods achieve an optimal computational complexity with respect to the order, but require low-resolution models of media and geometry in order to do so. The first task will combine these two approaches to construct methods for wave propagation in varying media which are stable, high order accurate, and have low computational complexity. The second task will be to improve the reliability of modeling complex boundaries using curvilinear unstructured meshes. Typical methods for producing curvilinear meshes can result in elements unsuitable for numerical simulation. The second task will seek robust methods for constructing high order approximations of geometry by leveraging Bernstein-Bezier representations of polynomials, which are closely related to shape properties of the underlying function.

波传播的模拟是医学成像、预测地震学和工程设计等不同领域中众多应用的基础。 许多应用需要在具有复杂边界或特征的域中重复进行高分辨率的波模拟。 例如，脑成像方法需要处理通过改变人体内的介质的波传播，例如不同类型的组织之间的过渡或从组织到骨骼的过渡。 目标应用领域需要可靠、准确、高效的数值模拟。 然而，结合复杂介质和几何形状的精确近似通常会牺牲一个或多个这些属性，导致模拟准确但缓慢，快速但不准确，或在特定设置之外不稳定。 该项目旨在为复杂介质和几何形状中的波传播开发可靠、可证明稳定的方法，同时保持准确性和效率。 该项目将研究使用高阶间断伽辽金（DG）方法的数值波传播，并包括两个主要任务。 第一个任务将利用高性能架构，如图形处理单元（GPU），以及最近开发的权重调整和Bernstein-Bezier DG方法。 权值调整DG方法在可变介质的情况下具有可证明的稳定性和高阶精度，但在近似阶数方面具有较高的计算复杂度。 Bernstein-Bezier DG方法实现了相对于阶数的最佳计算复杂度，但需要低分辨率的介质和几何模型才能实现。 第一个任务是将这两种方法联合收割机结合起来，构造出稳定的、高阶精度的、计算复杂度低的波动在变化介质中传播的方法。 第二个任务是提高使用曲线非结构网格模拟复杂边界的可靠性。 用于产生曲线网格的典型方法可能导致不适合于数值模拟的单元。第二个任务将寻求鲁棒的方法，通过利用多项式的Bernstein-Bezier表示来构建几何的高阶近似，这与底层函数的形状属性密切相关。

项目成果

Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: Explicit time-stepping and efficient mass matrix inversion

• DOI: 10.1016/j.cma.2018.01.022
• 发表时间: 2017-08
• 期刊: Computer Methods in Applied Mechanics and Engineering
• 影响因子: 7.2
• 作者: Jesse Chan;John A. Evans

Weight-adjusted discontinuous Galerkin methods: Matrix-valued weights and elastic wave propagation in heterogeneous media: Weight-adjusted discontinuous Galerkin methods: Matrix-valued weights and elastic wave propagation in heterogeneous media

权重调整间断伽辽金方法：异质介质中的矩阵定值权重和弹性波传播：权重调整间断伽辽金方法：异质介质中的矩阵定值权重和弹性波传播

• DOI: 10.1002/nme.5720
• 发表时间: 2018
• 期刊: International Journal for Numerical Methods in Engineering
• 影响因子: 2.9
• 作者: Chan, Jesse

EFFICIENT ENTROPY STABLE GAUSS COLLOCATION METHODS

• DOI: 10.1137/18m1209234
• 发表时间: 2019-01-01
• 期刊: SIAM JOURNAL ON SCIENTIFIC COMPUTING
• 影响因子: 3.1
• 作者: Chan, Jesse;Fernandez, David C. Del Rey;Carpenter, Mark H.
• 通讯作者: Carpenter, Mark H.

On discretely entropy stable weight-adjusted discontinuous Galerkin methods: curvilinear meshes

• DOI: 10.1016/j.jcp.2018.11.010
• 发表时间: 2019-02-01
• 期刊: JOURNAL OF COMPUTATIONAL PHYSICS
• 影响因子: 4.1
• 作者: Chan, Jesse;Wilcox, Lucas C.
• 通讯作者: Wilcox, Lucas C.

On discretely entropy conservative and entropy stable discontinuous Galerkin methods

• DOI: 10.1016/j.jcp.2018.02.033
• 发表时间: 2018-06-01
• 期刊: JOURNAL OF COMPUTATIONAL PHYSICS
• 影响因子: 4.1
• 作者: Chan, Jesse