The Method of Fundamental Solutions for Inverse and Moving Rigid Body Problems

The method of fundamental solutions (MFS) is proposed to deal with the inverse and moving rigid body problems. Firstly, the MFS with condition number analysis is carried out for the inverse problems in 2D Laplace equation, Cauchy problems, problem with missing boundary condition and internal data, problem with scattered data, and shape identification problem. Then, the MFS based on the steady Stokeslets has been employed to solve the inverse Stokes problems with over- and under-specified boundary segments. The coefficients of the Stokeslets can be obtained from any two field variables among the u, v velocity, pressure, vorticity or stream function. The numerical results are almost identical with the analytical solutions and other numerical results.
Furthermore, the unsteady Stokes flow in semi-infinite domain can be handled according to the experiences of dealing with the inverse problems. The MFS with unsteady Stokeslets can directly solve the semi-infinite domain problem without any iteration or regularization. In the next, the unsteady Stokes flow with various driven boundaries, square cavity and circular cavity will be solved. The variations also clearly demonstrate the phenomena of flow system with one moveable piece, two rotating belts and eccentric rotating cylinder. Finally, the Eulerian-Lagrangian method of fundamental solutions (ELMFS), which is a combination of the MFS and the Eulerian-Lagrangian method (ELM), is applied to solve the Navier-Stokes equations with moving rigid body. Further, the benchmark Navier-Stokes flow in lid-driven cavity is validated by the ELMFS based on the unsteady Stokelslets with Re=10 and Re=50. Finally, the phenomena of Navier-Stokes flow with a moving cylinder will be obtained and simulated. The ELMFS can be used clearly and directly to describe the moving rigid body phenomena in the fluid. The numerical results show good agreements with immersed-boundary finite element method (FEM).