Lagrangian methods in SPH for complex systems

The application of Lagrangian mechanics in combination with the definition of the dissipative function concept, first introduced by Rayleigh, to the construction of Smoothed Particle Hydrodynamics (SPH) equations for complex systems and situations is reviewed in this article. To illustrate the method, we have addressed the non-trivial problem of a micropolar fluid for which we have derived the most general expression for the friction forces related to particle velocities coupled to particle spin, for translational as well as for the rotational motion of the SPH particles. For the latter we have also discussed the scaling of the moment of inertia in SPH with the range of the weight function, and also with regards to the microscopic moment of inertia. The comparison of the SPH results with known solutions of the dynamics of micropolar fluids yields an excellent agreement.