A convex cone programming based implicit material point method

For conventional Material Point Method (MPM), both explicit-based and implicit-based MPM have shortcomings: explicit MPM has high requirements on time steps, and implicit MPM has high requirements on convergence. To circumvent these limitations, this paper innovatively proposes a convex cone programming-based implicit MPM (CP-MPM) algorithm, which ensures excellent convergence of solving complex problems involving large deformation, regardless of the chosen time step. In the proposed CP-MPM, the governing equations are initially transformed into a stationary point of a multivariable functional, leveraging the generalized Hellinger-Reissner (HR) variational principle. This stationary point problem is subsequently reformulated as a min-max convex cone optimization problem, with constraints originating from elastoplastic constitutive equations. In addition, a novel particle-based adhesive-frictional contact algorithm is proposed to effectively tackle the interaction between MPM domain and rigid bodies. The contact inequality between material points and rigid bodies is transformed into convex cone constraints, which rigorously prevent material point penetration and facilitate the imposition of irregular boundary conditions. Both elastic and elastoplastic problems involving contact under static or dynamic loading are ultimately represented as standard second-order cone programming (SOCP) problems, which is effectively solved by employing the Primal-Dual Interior Point (PDIP) method. The robustness, accuracy and convergence of the proposed method are validated through a series of elastic and elastoplastic benchmark problems. All results demonstrate the CP-MPM is a very promising method for implicitly solving complex practices.