Analysis, Computation and Control of Coupled Partial Differential Equation Systems

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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.