  We present an arbitrary updated Lagrangian Material Point Method (A-ULMPM) to
alleviate issues, such as the cell-crossing instability and numerical fracture,
that plague state of the art Eulerian formulations of MPM, while still allowing
for large deformations that arise in fluid simulations. Our proposed framework
spans MPM discretizations from total Lagrangian formulations to Eulerian
formulations. We design an easy-to-implement physics-based criterion that
allows A-ULMPM to update the reference configuration adaptively for measuring
physical states including stress, strain, interpolation kernels and their
derivatives. For better efficiency and conservation of angular momentum, we
further integrate the APIC[Jiang et al. 2015] and MLS-MPM[Hu et al. 2018]
formulations in A-ULMPM by augmenting the accuracy of velocity rasterization
using both the local velocity and its first-order derivatives. Our theoretical
derivations use a nodal discretized Lagrangian, instead of the weak form
discretization in MLS-MPM[Hu et al. 2018], and naturally lead to a "modified"
MLS-MPM in A-ULMPM, which can recover MLS-MPM using a completely Eulerian
formulation. A-ULMPM does not require significant changes to traditional
Eulerian formulations of MPM, and is computationally more efficient since it
only updates interpolation kernels and their derivatives when large topology
changes occur. We present end-to-end 3D simulations of stretching and twisting
hyperelastic solids, splashing liquids, and multi-material interactions with
large deformations to demonstrate the efficacy of our novel A-ULMPM framework.
