Energy Conservation Error in the Material Point Method for Solid Mechanics

The material point method (MPM) for solid mechanics conserves mass and momentum by construction, but energy conservation is not explicitly enforced. Material constitutive response and internal energy are carried on discrete points (material points), while the governing equations are solved on an overlying grid. The constitutive response (and internal energy) may be updated at the beginning or end of a numerical time step without affecting mass or momentum conservation properties. Both versions of the algorithm have been applied in the literature. Here energy conservation on the material points is investigated and found to depend strongly on the version of the algorithm used. The energy error is found and partitioned into two terms, one of which is of definite sign (and dissipative). The other term is indefinite, and of opposite sign for the two algorithmic variations. It is shown analytically for a single-material-point free-vibration example that one version of the algorithm is strictly dissipative. For the other version the error terms cancel each other out and energy is conserved. The same trends are borne out in numerical solutions of free axial vibration of continuum bars as the wavelength of the vibrational mode begins to approach the computational cell size. The dissipative algorithm may be described as tending to damp out unresolved modes. For resolved modes, both algorithms give identical results, with no perceptible energy error or dissipation. It is suggested that the dissipative algorithm is a better choice in general, as the damping is consistent with the accuracy of the solution.