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年的《美国回收与再投资法》（公共法111-5）进行资助的。在该项目中，研究人员开始了针对偏微分方程（PDE）系统的非线性和数值分析研究，该研究构成了两种或多种不同不同的PDE动力学的耦合。 该项目尤其集中在流体和结构体之间的相互作用（如此被称为流体结构的相互作用）本质上是无处不在的。对于这些流体结构动力学以及其他物理相关的耦合PDE模型，生成了非线性理论，该理论将最终导致：（i）可以成功调用这些交互式模型的控制定律的推导，以稳定或引导流体和结构组件； （ii）数值算法，以便为受控或不受控制的态度近似流体结构变量的曲线。在该项目的这一部分中，预计Babuska-Aziz和Babuska-Brezzi“ Inf-Sup”理论的非标准实施将扮演重要角色。通过这种变异配方，将解决边界界面上流体和结构组件之间的耦合。该耦合的分辨率是流体结构分析的核心。该项目收集的定性和定量信息将更好地理解各种物理现象，这些现象可以通过交互式PDE模型描述。例如，可以调用流体结构PDE，以建模浸入血浆血浆中的红细胞。 这些用于建模PDE动力学的连续和数值近似研究将使更准确地预测和模拟此类血流动力学的可行。特别是，该项目将在数值算法的推导下达到顶点，该算法将基于上述Babuska-Brezzi变化表述。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，我们打算在项目中考虑的控制法可能会深入了解耦合PDE系统控制的物理相互作用的可能控制工程方法。