Analysis, Computation and Control of Coupled Partial Differential Equation Systems

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，研究者开始了一个非线性和数值分析研究系统的偏微分方程（PDE的），它构成了两个或多个不同的偏微分方程动力学的耦合。该项目特别关注流体和结构体之间的相互作用，即所谓的流体-结构相互作用，这在自然界中无处不在。对于这些流体-结构动力学以及其他与物理相关的耦合偏微分方程模型，产生了一个非线性理论，其最终结果是：(i)推导出这些相互作用模型的控制律，这些模型可以成功地调用，以稳定或引导流体和结构组件；（ii）数值算法，以便对受控或非受控状态近似流体结构变量的轮廓。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主要作用。通过这种变分公式，可以解决边界界面上流体与结构部件之间的耦合问题；这种耦合的解析是流体结构分析的核心。从这个项目中收集到的定性和定量信息将有助于更好地理解可由相互作用的偏微分方程模型描述的各种物理现象。例如，流体结构PDE可以用来模拟红细胞在血液的血浆成分中的浸泡。这些对PDE动力学建模的连续和数值近似研究将使更准确地预测和模拟这种血流动力学变得可行。具体地说，该项目最终将以上述Babuska-Brezzi变分公式为基础推导出数值算法。因此，这些算法可能比目前可用的数值方法具有更高的严谨性。此外，我们打算在项目中考虑的控制律可以为耦合PDE系统所控制的物理相互作用的可能的控制工程方法提供见解。

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.