This paper presents a novel method to couple Finite Element Methods (FEM), typically employed for modeling Lagrangian solids such as flesh, cloth, hair, and rigid bodies, with Material Point Methods (MPM), which are well-suited for simulating materials undergoing substantial deformation and topology change, including Newtonian/non-Newtonian fluid, granular materials, and fracturing materials. The challenge of coupling these diverse methods arises from their contrasting computational needs: implicit FEM integration is often favored to enjoy stability and large timesteps, while explicit MPM integration benefits from its allowance for efficient GPU optimization and flexibility of applying different plasticity models, which only allows for moderate timesteps. To bridge this gap, a mixed implicit-explicit time integration (IMEX) approach is proposed, utilizing principles from time splitting for partial differential equations and optimization-based time integrators. This method adopts incremental potential contact (IPC) to define a variational frictional contact model between the two materials, serving as the primary coupling mechanism. Our method enables implicit FEM and explicit MPM to coexist with significantly different timestep sizes while preserving two-way coupling. Experimental results demonstrate the potential of our method as a strong foundation for future exploration and enhancement in the field of multi-material simulation. READ MORE