Variational formulation for the smooth particle hydrodynamics (SPH) simulation of fluid and solid problems Original Research Article

The paper describes the variational formulation of smooth particle hydrodynamics for both fluids and solids applications. The resulting equations treat the continuum as a Hamiltonian system of particles where the constitutive equation of the continuum is represented via an internal energy term. For solids this internal energy is derived from the deformation gradient of the mapping in terms of a hyperelastic strain energy function. In the case of fluids, the internal energy term is a function of the density. Once the internal energy terms are established the equations of motion are developed as equations of Lagrange, where the Lagrangian coordinates are the current positions of the particles. Since the energy terms are independent of rigid body rotations and translations, this formulation ensures the preservation of physical constants of the motion such as linear and angular momentum.