An unconditionally stable, energy–momentum consistent implementation of the material-point method

Conservation of linear and angular momenta, and conservation of energy are examined for the material-point method (MPM). It is shown that MPM can be formulated with implicit energy and momentum conserving mesh dynamics for hyperelastic materials. With a consistent mass matrix the resulting overall numerical method preserves the conservative properties of the mesh solution. Energy dissipation and angular momentum errors are also quantified for a lumped mass formulation. Properties of the method are illustrated in numerical examples.