Material point methods applied to one-dimensional shock waves and dual domain material point method with sub-points

Although in the present paper, the numerical examples are one-dimensional, all derivations are for multidimensional problems. With the technique of approximately tracking particle domains of CPDI, the extension of this sub-point method to multidimensional problems is straightforward. This new method preserves the conservation properties of the DDMP method, which conserves mass and momentum exactly and conserves energy to the second order in both spatial and temporal discretizations.