Mathematical Sciences: Entropy-Controlled Adaptive Finite Element Simulations of Compressible Gas Flow

数学科学：可压缩气体流动的熵控制自适应有限元模拟

• 批准号: 9414480
• 负责人: Leszek Demkowicz
• 金额: $ 7.5万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-05-01 至 1998-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9414480&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Entropy; Controlled; Adaptive

The principal idea of the proposed work consists in using a discrete form of the entropy balance equation as a rationale for controlling the optimal amount of artificial dissipation in Finite Element (FE) compressible gas simulations. The entropy control can be reinterpreted as a nonlinear stability estimate in terms of the so-called modified entropy function. Preliminary results include a practical verification of the proposed ideas, using the Taylor-Galerkin discretization in time method combined with an artificial viscosity model proposed by Hughes and Johnson, all in the context of h-adaptive linear finite elements. The obtained numerical results confirm that the entropy control indeed may provide a basis for the careful balance between stability and higher-order resolution in FE discretizations. The proposed investigations include: a study of a local entropy control (in the results obtained so far, only a global stability was ensured), a study of other time discretization methods and artificial dissipation models, and extensions to higher order spatial approxima tions. The proposed work deals with the supersonic viscous and inviscid flows, modeled by compressible Navier-Stokes and Euler equations. Possible applications include modeling of flows around a complete aircraft or part of it, and internal flows related mostly to combustion problems (engines). The described research should contribute to a better understanding of the fundamental role of the entropy function in global and local stability of compressible gas simulations. Based on the entropy control, an internal mechanism for controlling stability should be built into the existing FE codes, resulting in a new, reliable and powerful class of approximations.

这项工作的主要思想是在有限元(FE)可压缩气体模拟中使用离散形式的熵平衡方程作为控制最优人工耗散量的理论基础。该熵控制可以被重新解释为根据所谓的修正的熵函数的非线性稳定性估计。初步结果包括使用Taylor-Galerkin时间离散化方法结合Hughes和Johnson提出的人工粘性模型对所提出的思想进行了实际验证，所有这些都是在h-自适应线性有限元的背景下进行的。得到的数值结果证实，熵控制确实可以为有限元离散中稳定性和高阶分辨率之间的谨慎平衡提供基础。所提出的研究包括：局部熵控制的研究(在迄今的结果中，仅确保了全局稳定性)，其他时间离散化方法和人工耗散模型的研究，以及对高阶空间近似的扩展。该工作涉及超音速粘性和无粘性流动，由可压缩的Navier-Stokes方程和Euler方程模拟。可能的应用包括对整个飞机或其部分周围的流动进行建模，以及主要与燃烧问题(发动机)有关的内部流动。所描述的研究应该有助于更好地理解熵函数在可压缩气体模拟的全局和局部稳定性中的基本作用。在熵控制的基础上，应在现有的有限元程序中建立控制稳定性的内部机制，从而产生一种新的、可靠的、功能强大的逼近类。