Analysis, Computation and Control of Coupled Partial Differential Equation Systems

耦合偏微分方程组的分析、计算与控制

• 批准号: 0908476
• 负责人: George Avalos
• 金额: $ 18.29万
• 依托单位: University of Nebraska-Lincoln
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2013-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0908476&HistoricalAwards=false
• 关键词: Analysis; Computation; Control; Coupled; Partial

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).In this project, the investigator embarks upon a nonlinear and numerical analysis study for systems of partial differential equations (PDE's) which constitute a coupling of two or more distinct PDE dynamics. The project is particularly focused on interactions between fluid and structural bodies--so-called fluid-structure interactions--which are omnipresent in nature. For these fluid-structure dynamics, as well as for other physically relevant coupled PDE models, a nonlinear theory is generated which will culminate in: (i) The derivation of control laws for these interactive models which can be successfully invoked so as to stabilize or steer both the fluid and structural components; (ii) numerical algorithms so as to approximate the profiles of the fluid-structure variables, for either controlled or uncontrolled regimes. In this part of the project, it is anticipated that a major role will be played by nonstandard implementations of the Babuska-Aziz and Babuska-Brezzi "inf-sup" theories. By means of such variational formulations, the coupling between fluid and structure components on the boundary interface will be resolved; resolution of this coupling is at the very heart of fluid-structure analysis. The qualitative and quantitative information gleaned from this project will provide a better understanding of the various physical phenomena which can described by interactive PDE models. For example, a fluid-structure PDE can be invoked to model the immersion of red blood cells within the plasma component of blood. These continuous and numerical approximation studies for modeling PDE dynamics would render it practicable to more accurately predict and simulate such blood flow dynamics. In particular, the project will culminate in the derivation of numerical algorithms which will be based upon the aforesaid Babuska-Brezzi variational formulations. As such, these algorithms would presumably have a higher degree of rigor than current available numerical methods. Moreover, the control laws we intend to consider in the project may lend insight into possible control engineering methodologies for the physical interactions governed by systems of coupled PDE's.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。在这个项目中，研究人员开始对偏微分方程（PDE）系统进行非线性和数值分析研究，该系统构成两个或多个不同PDE动力学的耦合。 该项目特别关注流体和结构体之间的相互作用-所谓的流体-结构相互作用-这在自然界中无处不在。对于这些流体-结构动力学，以及对于其他物理相关的耦合PDE模型，产生非线性理论，其将最终导致：（i）这些交互模型的控制律的推导，其可以被成功地调用以便稳定或操纵流体和结构部件;（ii）数值算法，以便近似流体-结构变量的轮廓，用于受控或非受控状态。在项目的这一部分，预计Babuska-Aziz和Babuska-Brezzi“inf-sup”理论的非标准实现将发挥重要作用。通过这样的变分公式，流体和结构组件之间的耦合的边界界面将被解决，这种耦合的分辨率是在流体-结构分析的核心。从这个项目中收集的定性和定量信息将提供一个更好的理解，可以描述的各种物理现象的交互式偏微分方程模型。例如，可以调用流体结构PDE来模拟红细胞在血液的血浆成分内的浸入。 这些连续的和数值近似的研究建模PDE动力学将使其切实可行的更准确地预测和模拟这样的血流动力学。特别是，该项目将最终在数值算法的推导，这将是基于上述巴布斯卡-Brezzi变分制剂。因此，这些算法可能比当前可用的数值方法具有更高的严格程度。此外，我们打算在该项目中考虑的控制律可能有助于深入了解耦合PDE系统所管理的物理相互作用的可能的控制工程方法。