Current Research project:

• Fluid-structure interaction problems
  
• Optimal control for flows with defective boundary conditions
  
• Optimization-based domain decomposition for multiphysic problems
  

Research Highlights:

• Fluid-Structure Interaction(FSI)

  The interaction of fluid flows with an elastic medium is of great interest for both industrial and biological uses. Examples of such systems include bronchial air ways, blood flow in arteries, micro-fluidic devices and resonant vibrations of aircraft components. A fluid-elastic system is typically highly linked through a strong coupling of governing equations, making its mathematical and numerical study very challenging. The mathematical description of the fluid-elastic problem is a system of time-dependent, coupled, nonlinear partial differential equations with moving boundaries. Studies on FSI have been focused on Newtonian flows so far, however, many industrial and biological fluids of considerable interest do not behave as Newtonian fluids. Motivation for this project stems from recent advances in computing viscoelastic and quasi-Newtonian fluid flows, including methods designed for fluid motion within deformable elastic structures
  
  • We investigated the numerical approximation of a nonlinear, time-dependent quasi-Newtonian flow problem formulated in the framework of Arbitrary Lagrangian Eulerian (ALE) method. We showed some stability results and convergence analysis of finite element solutions for semidiscrete and fully discretized problems. To our knowledge, this is the first work that shows rigorous numerical analysis of non-Newtonian flows in a moving domain, based on ALE.
    
  • We performed analytical numerical studies on the stability of numerical solutions of a viscoelastic flow in a movable domain. A numerical approximation scheme was developed based on the Arbitrary Lagrangian- Eulerian (ALE) formulation of the flow equations. First and second order timestepping schemes satisfying the geometric conservation law (GCL) were derived and analyzed, and numerical examples supported the theoretical results.
    
  • We are currently investigating coupled systems of (i) 1D nonlinear structure and 2D viscoelastic fluid, and (ii) 2D linear structure and 2D quasi-Newtonian fluid.
    

• Flows with Defective Boundary Conditions

  Many flow problems in real applications are modeled in an unbounded domain. Blood flows in the vascular system is one of the important applicative fields relative to such a situation. In simulating blood flow in a portion of vessel, an artificial boundary is introduced for upstream and downstream sections, and a Dirichlet or Neumann type boundary condition is imposed on the boundary. However, in practice, those boundary data are not usually available, and sometimes recirculation of flow can occur near or on a boundary section. For this reason, in numerical simulation of such flows, it is more realistic to implement measurable data on the boundary such as pressure and flow rates. The goal of this project is to study numerical solutions for non-Newtonian flows with defective boundary conditions.
  
  • We studied the numerical approximation of a quasi-Newtonian Stokes flow problem where only the flow rates are specified at the inflow and outflow boundaries. A variational formulation of the problem was developed using Lagrange multipliers to enforce the stated flow rates. The existence and the uniqueness to the continuous and discrete variational formulations of the solution were proved and an error estimate for the numerical approximation was also derived.
    
  • We investigated a generalized-Newtonian fluid with defective boundary conditions where flow rates or mean pressures were prescribed on parts of the boundary. The defect boundary condition problem was formulated as an optimal control problem in which a Neumann or Dirichlet boundary control was used for matching given flow rates or mean pressures. For the constrained optimization problem an optimality system was derived from which a solution of the problem was obtained.
    
  • We investigated numerical algorithms for viscoelastic fluid flows with defective boundary conditions. The defective boundary condition problem was formulated as a minimization problem, where we found boundary conditions of the flow equations which yielded an optimal functional value. Two different approaches were considered for computational algorithms, and robustness of the algorithms was verified by numerical tests.
    
  • We studied non-Newtonian fluid flows governed by the Cross model with flow rate boundary conditions. A constrained optimal control problem was formulated for the defective boundary conditions using a Neuman control. The control problem was analyzed for an existence result and the Lagrange multiplier rule.
    
  

• Optimization based domain decomposition

  • We developed and analyzed a domain decomposition algorithm for the linear/nonlinear Stokes-Darcy coupled problem based on optimization. The coupled system was formulated as a constrained optimal control problem, where a flow balance was forced across the interface, inflow, and outflow boundaries by minimizing a suitably defined functional. The functional was carefully designed so that its definition was consistent with the chosen function spaces for the Stokes and Darcy velocities, respectively. A computational algorithm was developed using a least squares approach, and accuracy and efficiency of the algorithm were verified by numerical experiments.
    
  • We are currently investigating optimization based domain decomposition methods for various multiphysics problems such as fluid-elastic structure interaction and fluid-poroelastic media interaction problems.