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We present an arbitrary updated Lagrangian Material Point Method (A-ULMPM) toalleviate issues, such as the cell-crossing instability and numerical fracture,that plague state of the art Eulerian formulations of MPM, while still allowingfor large deformations that arise in fluid simulations. Our proposed frameworkspans MPM discretizations from total Lagrangian formulations to Eulerianformulations. We design an easy-to-implement physics-based criterion thatallows A-ULMPM to update the reference configuration adaptively for measuringphysical states including stress, strain, interpolation kernels and theirderivatives. For better efficiency and conservation of angular momentum, wefurther integrate the APIC[Jiang et al. 2015] and MLS-MPM[Hu et al. 2018]formulations in A-ULMPM by augmenting the accuracy of velocity rasterizationusing both the local velocity and its first-order derivatives. Our theoreticalderivations use a nodal discretized Lagrangian, instead of the weak formdiscretization 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 Eulerianformulation. A-ULMPM does not require significant changes to traditionalEulerian formulations of MPM, and is computationally more efficient since itonly updates interpolation kernels and their derivatives when large topologychanges occur. We present end-to-end 3D simulations of stretching and twistinghyperelastic solids, splashing liquids, and multi-material interactions withlarge deformations to demonstrate the efficacy of our novel A-ULMPM framework.

A-ULMPM: An Arbitrary Updated Lagrangian Material Point Method for Efficient Simulation of Solids and Fluids