Distribution coefficient algorithm for small mass nodes in material point method

When using the time explicit material point method to simulate interaction of materials accompanied by large deformations and fragmentation, one often encounters a numerical instability caused by small node mass, because acceleration on a mesh node is obtained by dividing the total force on the node by the mass of the node. When the material points are in the far sides of the cells containing the node, typically happening near material interfaces, the node mass can be very small leading to artificially large acceleration and then numerical instability. For the case of small material deformations, this instability is typically avoided by placing the material points away from cell boundaries. For cases with large deformations, with the exception of initial conditions, there is no control on locations of the material points. The instability caused by small mass nodes is often encountered. To avoid this instability tiny time steps are usually required in a numerical calculation.In this work, we present a numerical algorithm to treat this instability. We show that this algorithm satisfies mass and momentum conservation laws. The error in energy conservation is proportional to the second order of the time step, consistent with the explicit material point method. Numerical implementation of the algorithm is described. Numerical examples show effectiveness of the algorithm.