1. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. SAND No. 2011-XXXXP Analysis of a Fluid-Structure Interaction Problem Decoupled by Optimal Control Paul Kuberry and Hyesuk Lee Clemson University / Sandia National Laboratories Applications Optimization-based Approach Introduction Conclusions We consider a fluid on domain in contact with a structure on domain over the time dependent interface. Fluid-Structure Interaction (FSI) occurs when non-negligible forces are exerted between a fluid and a structure, generally resulting in a deformation of their interface. Example: Photo: Nottingham Trent University Photo: 3ds.com Blood flow Stent placement Heart pump Aeroelasticity Airplane wing Wind turbine Industrial Processing Inkjet printer Microfluidics Offshore Oil Platforms Photo: DataCenterTactics.com At each time step, find a that minimizes the functional subject to to the flow and structure constraint equations. Use to replace with and with, i.e. the unknown traction force is used as a control. Allows for use of partitioned solvers Can easily be applied to a wide variety of physics Optimal convergence rates in space and time Few nonlinear solves per time step Gauss-Newton approach allows for matrix factorization reuse for inner optimization loops Scales well with mesh refinement  P.K., H.L., Convergence of a fluid-structure interaction problem decoupled by a Neumann control over a single time-step, submitted. Error estimates over a single time step Standard Stokes and linear elasticity error rates Convergence of the gradient method from any initial guess, using a sufficiently small time step Steepest descent over a single time step at t=0.8 s Δt = 1e-6 s ε tol = 1e-10 Finite Element Convergence Rates Computation over time steps from t=0.5 to t=1.0 s Outer Gauss-Newton ε tol = 1e-10, Inner CG ε tol = 1e-13 t=0.025 st=0.500 s Navier—Stokes fluid ν f = 0.035 poise, ρ f = 1 g/cm 3, Straight vessel of radius 0.5 cm and length 5 cm St. Venant—Kirchhoff structure (Nonlinear) ρ s = 1.2 g/cm 3, E = 3.0e+6 dynes/cm 2, ν= 0.3 Surrounding structure thickness of 0.1 cm Very weak dependence of Work Factor on DOFs Around 4-5 Gauss-Newton iterations per time step Problem Description Overpressure on inlet boundary of 1.3332e+4 dynes/cm 2 for 0≤ t ≤0.005 s Inlet and outlet boundaries are clamped Δt = 1e-4 s 3D Pulsatile Flow Through a Cylinder t=0.075 s t=0.100 s Gauss-Newton + CG over many time steps Photo: boem.gov Photo: Massachusetts General Hospital Manufactured Solution Computational Domain Work factor for a sequence of refined meshes Optimization-based Algorithm It is possible that. An alternate functional to minimize is since and are well defined on the interface. Proves that an optimal solution to this minimization problem exists Lagrange multipliers are proven to exist, allowing the PDE-constrained optimization problem to be written as an unconstrained minimization problem  P.K., H.L., Analysis of a fluid-structure interaction problem recast in an optimal control setting, submitted. Not sequentially staggered Parallel computation of state and adjoint solutions Compatible with black-box solvers Advantages