Hamiltonian formulation of the variable-h SPH equations

In this paper a variational formulation of smooth particle hydrodynamics for dynamic problems is presented. The resulting equations treat the continuum as a Hamiltonian system of particles where the constitutive equation of the material is represented via an internal energy term. In the case of fluids the internal energy term is a function of density. The new formulation introduces a variable smoothing length for the evaluation of density and incorporates a consistent approach for the treatment of rigid boundaries. The method overcomes some problems faced by standard SPH approaches that use constant smoothing lengths and provides a variational context for a variable smoothing length formulation. A numerical example shows the capabilities of this novel formulation.