On Efficient Numerical Methods for Simulating the Dynamics of Fluid-Structure Interaction Problems(2/2)

The study of fluid-structure interaction problems is of great importance in many applicationsof sciences and engineering, and it usually involves complex structure geometries.The body-fitted approach is a conventional method that is frequently used to simulate flowwith a complex boundary. In that approach, the incompressible Navier-Stokes equations arespatially discretized on a curvilinear or unstructured grid that conforms to the immersedstructure boundaries. Therefore, the internal boundary conditions can be imposed easily.However, the body-fitted discretizations have to re-mesh the spatial domain at every timestep correspondingly when the body deforms or moves in the fluid. Since the grid generationcan become a large computational overhead, it would be desirable to avoid the needof re-meshing at each time step. Thus, one should seek the help of the Cartesian grid basednon-boundary conforming methods, such as the so-called immersed boundary (IB) method,to address the complex fluid-structure interaction problems.The main purpose of this three-year project is to develop efficient numerical methodsfor simulating the dynamics of fluid-structure interaction problems. We will focus on thefollowing three topics:1. The direct-forcing IB projection method for the fluid-solid interaction, where the immersedsolid object is moving in the fluid governed by the equations of motion.2. The direct-forcing IB projection method for the fluid-elastic body interaction, wherethe elastic structure is quasi-static, isotropic and homogeneous which undergoes smalldeformations.In the above two direct-forcing IB projection methods, the solid/elastic body domainis treated like a fluid with an additional virtual force field applied to it so that it wouldact like a solid/elastic body. Actually, this virtual force is added to the momentumequations to accommodate the interaction between the solid/elastic body and fluidsuch that the boundary condition at the immersed boundary is exactly satisfied.3. We will also study the IB method combined with the artificial compressibility methodfor solving fluid-structure interaction problems, in which the incompressibility constraintr? u = 0 is replaced by the artificial compressibility equation, #¶tp +r? u = 0,and # > 0 is a small parameter depending on the time step length. The development ofefficient solvers for the linear systems associated with the fully discrete vector-valuedproblems to be solved will be the focus of this penalty approach.