Bodhinanda (Nanda) Chandra

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.

This study focuses on solving the numerical challenges of imposing absorbing boundary conditions for dynamic simulations in the material point method (MPM). To attenuate elastic waves leaving the computational domain, the current work integrates the Perfectly Matched Layer (PML) theory into the implicit MPM framework. The proposed approach introduces absorbing particles surrounding the computational domain that efficiently absorb outgoing waves and reduce reflections, allowing for accurate modeling of wave propagation and its further impact on geotechnical slope stability analysis. The study also includes several benchmark tests to validate the effectiveness of the proposed method, such as several types of impulse loading and symmetric and asymmetric base shaking. The conducted numerical tests also demonstrate the ability to handle large deformation problems, including the failure of elasto-plastic soils under gravity and dynamic excitations. The findings extend the capability of MPM in simulating continuous analysis of earthquake-induced landslides, from shaking to failure.

This paper presents a novel stabilized mixed material point method (MPM) designed for the unified modeling of free-surface and seepage flow. The unified formulation integrates the Navier-Stokes equation with the Darcy-Brinkman-Forchheimer equation, effectively capturing flows in both non-porous and porous domains. In contrast to the conventional Eulerian computational fluid dynamics (CFD) solver, which solves the velocity and pressure fields as unknown variables, the proposed method employs a monolithic displacement-pressure formulation adopted from the mixed-form updated-Lagrangian finite element method (FEM). To satisfy the discrete inf-sup stability condition, a stabilization strategy based on the variational multiscale method (VMS) is derived and integrated into the proposed formulation. Another distinctive feature is the implementation of blurred interfaces, which facilitate a seamless and stable transition of flows between free and porous domains, as well as across two distinct porous media. The efficacy of the proposed formulation is verified and validated through several benchmark cases in 1D, 2D, and 3D scenarios. Conducted numerical examples demonstrate enhanced accuracy and stability compared to analytical, experimental, and other numerical solutions.

This paper proposes novel and robust stabilization strategies for accurately modeling incom- pressible fluid flow problems in the material point method (MPM). To address the modeling of Newtonian fluids with incompressibility constraints, a new mixed implicit MPM formulation is proposed. Here, instead of solving the velocity and pressure fields as the unknown variables like the typical Eulerian computational fluid dynamics (CFD) solver, the proposed approach adopts a monolithic displacement–pressure formulation inspired by the mixed-form updated- Lagrangian Finite Element Method (FEM). To satisfy the inf–sup stability condition, two stabilization strategies are integrated into the formulation: the variational multiscale method (VMS) and the pressure-stabilization Petrov–Galerkin method (PSPG). By concurrently solving the displacement and pressure fields, the developed monolithic solver obviates the need for free-surface detection as well as Dirichlet and Neumann pressure imposition, in contrast to the fractional-step method. This attribute mitigates spurious pressure and velocity oscillations in simulating dynamic and transient flow problems. This study also addresses other prevalent challenges in MPM simulations, such as the pressure oscillations triggered by cell-crossing errors, particle-distribution-induced quadrature errors, and particle-grid information transfer. To resolve these issues, the quadratic B-Spline basis function, the delta-correction method, and the Taylor particle-in-cell method are incorporated into the proposed mixed MPM formulation, thereby enhancing numerical stability. The efficacy of the proposed stabilized incompressible MPM framework is validated through several benchmark cases, comparing the obtained results with other numerical methods and analytical solutions. Furthermore, the method’s capability in simulating real-world problems involving violent free-surface fluid motion is demonstrated through comparisons with experimental results of water sloshing and dam break scenarios.