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Nonlinear Partial Differential Equations

Date: November 7, 2017

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Suncica Canic, University of Houston

  

Title: Nonlinear PDEs Seminar

Abstract:

Title: A mathematical framework for proving existence of weak solutions to a class of fluid-structure interaction problems

Abstract:
The focus of this talk will be on nonlinear moving-boundary problems involving incompressible, viscous fluids and elastic structures. The fluid and structure are coupled via two sets of coupling conditions, which are imposed on a deformed fluid-structure interface. The main difficulty in studying this class of problems from the analysis and numerical points of view comes from the strong geometric nonlinearity due to the nonlinear fluid-structure coupling. We have recently developed a robust framework for proving existence of weak solutions to this class of problems, allowing the treatment of various structures (Koiter shell, multi- layered composite structures, mesh-supported structures), and various coupling conditions (no-slip and Navier slip). The existence proofs are constructive: they are based on the time-discretization via Lie operator splitting, and on our generalization of the famous Lions-Aubin-Simon’s compactness lemma to moving boundary problems. The constructive proof strategy can be used in the design of a loosely-coupled partitioned scheme, in which the fluid and structure sub-problems are solved separately, with the cleverly designed boundary conditions to enforce the coupling in a way that approximates well the continuous energy of the coupled problem. This provides stability and uniform energy estimates, important for the convergence proof of the numerical scheme. Applications of this strategy to the simulations of real-life problems will be shown. They include the flow of blood in a multi-layered coronary artery treated with vascular devices called stents (with Dr. Paniagua (Texas Heart Institute) and Drs. Little and Barker, Methodist Hospital, Houston), and optimal design of micro- swimmers and bio-robots (with biomed. engineer Prof. Zorlutuna, Notre Dame).

Parts of the mathematical work are joint with B. Muha