Material point methods implementable in unstructured meshes and their consistency and accuracy improvements

Material point method (MPM) uses Eulerian mesh and Lagrangian particles. The original
MPM can be easily used in unstructured mesh as the finite element method. Because of
the discontinuity of the gradients of the shape functions across cell boundaries, as
particles move across them, numerical noises on nodal forces are generated, leading to
the failure of a calculation. To address this issue, the generalized interpolation material
point (GIMP) and the convective particle domain integration (CPDI) methods are
developed. In these methods a particle is not a point but a finite domain, which can
occupy multiple cells. Nonlocal operations are required to map between the particle and
nodal quantities making the method difficult to implement with an unstructured mesh.
To address the cell-crossing noise, the dual domain material point (DDMP) method maps
part of the stress on particles to nodes. The nodal force is then calculated with two
contributions, from the nodal stress and from the remaining stress at particles. As a
particle approaches a cell boundary, the remaining stress is reduced to zero to eliminate
the discontinuity on the nodal force. In DDMP, particles remain as geometric points and
all mappings between particles and nodes are local. This method can be implemented
with unstructured meshes. Recently, DDMP is further improved with the local stress
difference (LSD) scheme, significantly increasing numerical accuracy and consistence
while reducing mesh dependency.