A new Galerkin ®nite element method for the solution of the Navier±Stokes equations in enclosures containing internal parts which may be moving is presented. Dubbed the virtual ®nite element method, it is based upon optimization techniques and belongs to the class of ®ctitious domain methods. Only one volumetric mesh representing the enclosure without its internal parts needs to be generated. These are rather discretized using control points on which kinematic constraints are enforced and introduced into the mathematical formulation by means of Lagrange multipliers. Consequently, the meshing of the computational domain is much easier than with classical ®nite element approaches.
First, the methodology will be presented in detail. It will then be validated in the case of the two-dimensional Couette cylinder problem for which an analytical solution is available. Finally, the three-dimensional ¯uid ¯ow inside a mechanically agitated vessel will be investigated. The accuracy of the numerical results will be assessed through a comparison with experimental data and results obtained with a standard ®nite element method.