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.

拟议工作的主要思想是使用熵平衡方程的离散形式作为控制有限元（FE）可压缩气体模拟中人工耗散量的理由。 可以根据所谓的修改熵函数重新解释熵控制作为非线性稳定性估计。 初步结果包括对拟议思想的实际验证，使用Taylor-Galerkin离散化在时间方法中与Hughes和Johnson提出的人工粘度模型相结合，所有这些都在H-自适应线性有限元的背景下。获得的数值结果证实，熵控制确实可以为FE离散化的稳定性和高阶分辨率之间的谨慎平衡提供基础。 拟议的研究包括：对局部熵控制的研究（到目前为止获得的结果，仅确保了全球稳定性），对其他时间离散化方法和人工耗散模型的研究，以及扩展到更高阶空间大约。 拟议的工作涉及以可压缩的navier-stokes和Euler方程为模型的超音速粘性和无粘性流。 可能的应用程序包括对完整飞机或部分飞机的流量进行建模，以及主要与燃烧问题（发动机）相关的内部流动。 所描述的研究应有助于更好地理解熵功能在可压缩气体模拟的全球和局部稳定性中的基本作用。基于熵控制，应将控制稳定性的内部机制内置在现有的FE代码中，从而导致新的，可靠和强大的近似类别。