A Total-Lagrangian material point method for fast and stable hydromechanical modeling of porous media

Abstract

Modeling the incompressible fluid flow in porous media has long been a challenging task in the Material Point Method (MPM). Although widely used, conventional Updated Lagrangian MPM (ULMPM) often suffers from numerical stability and computational efficiency issues in the hydromechanical analysis of saturated porous media. To address these issues, we herein present a novel semi-implicit Total Lagrangian MPM (TLMPM). The proposed TLMPM leverages the fractional step method to decouple pore pressure from kinematic fields and employs the semi-implicit scheme to bypass the small time step constraint imposed by permeability and fluid compressibility. Unlike its UL counterpart, the TLMPM evaluates weighting functions and their gradients only once in the reference configuration, eliminating material point tracking and inherently resolving cell-crossing instabilities. Given the consistent set of active degrees of freedom throughout simulations, the proposed method greatly reduces computational costs associated with system matrix assembly for both kinematics and pore pressure and with free-surface node detection. Furthermore, this feature also facilitates the efficient Cholesky factorization, resulting in a substantial acceleration of the solver performance. The proposed approach has been validated against various benchmark tests, and our results have highlighted the remarkable performance of TLMPM, which can achieve up to 63 times speedup over conventional methods, scaling favorably with problem size, and retaining numerical stability even with low-order basis functions. These advancements position the TLMPM as a transformative tool for poroelastic analysis, with broader applicability to large-deformation problems in geomechanics, energy systems, and environmental engineering.