We present an efficient algorithm for simulation of deformable bodies interacting with incompressible flows. The temporal and spatial discretizations of the Navier--Stokes equations in vorticity-stream function formulation are based on classical fourth-order Runge--Kutta method and compact finite differences, respectively. Using a uniform Cartesian grid we benefit from the advantage of a new fourth-order direct solver for the Poisson equation to ensure the incompressibility constraint down to machine zero over an optimal grid. For introducing a deformable body in fluid flow, the volume penalization method is used. A Lagrangian structured grid with prescribed motion covers the deformable body which is interacting with the surrounding fluid due to the hydrodynamic forces and the torque calculated on the Eulerian reference grid. An efficient law for controlling the curvature of an anguilliform fish, swimming toward a prescribed goal, is proposed which is based on the geometrically exact theory of nonlinear beams and quaternions. Furthermore to reduce the computational effort, better resolving the boundary layer and the vortical structures, adaptation of grid is performed by using multiresolution analysis. The method is based on Harten's point value representation, which through nonlinear filtering of the wavelet coefficients reduces the number of active grid points significantly. Finally an extension to three dimensional swimming is performed by adding the implicit volume penalization method to the Incompact3d open access code, to be able to take into account the deformable bodies interaction with incompressible flows. Validation of the developed method shows the efficiency and expected accuracy of the algorithm for fish-like swimming and also for a variety of fluid/solid interaction problems.
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