A Hamiltonian particle method for non-linear elastodynamics

Particle methods are meshless simulation techniques in which motion of continua is approximated by discrete dynamics of a finite number of particles. They have a great degree of flexibility, for instance, in dealing with complex large deformations or the fragmentation of solids. In this paper, a particle method for non-linear elastodynamics of compressible and incompressible materials is developed based on a discretization of the Lagrangian from which the governing equations of elastodynamics are derived using the principle of least action. The discretized Lagrangian leads to a finite-dimensional Hamiltonian system via the Legendre transformation. If the material is incompressible, the Hamiltonian system is accompanied by holonomic constraints. Depending on whether the material is compressible or incompressible, the symplectic scheme adopted for numerical time integration is either the Störmer/ Verlet scheme or the RATTLE method, respectively. The resulting particle method inherits the symplectic structure possessed by the governing equations of elastodynamics. In the case of incompressible materials, incompressibility is strictly enforced at each time step. Some numerical tests indicate the excellence of the method for conservation of mechanical energy besides that of linear and angular momenta.