Mixed material point method formulation, stabilization, and validation for a unified analysis of free-surface and seepage flow

TL;DR

This work develops a stabilized mixed material point method (MPM) to unify free-surface and seepage flow by coupling the Navier–Stokes equations with the Darcy–Brinkman–Forchheimer model in a single displacement–pressure framework. A variational multiscale (VMS) stabilization is integrated alongside equal-order interpolation and blurred porosity interfaces, enabling seamless transitions between non-porous, porous, and multi-porous regions. The method is verified against 1D analytical benchmarks and validated through 2D and 3D experiments, showing superior volume conservation, smoother pressure fields, and stability across challenging flows, including dam-break scenarios. The approach yields substantial computational advantages over fractional-step methods in low-permeability regimes and provides a versatile, scalable tool for geotechnical and hydraulic applications where free-surface and porous flows interact.$

Abstract

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.

Figures (30)

• Figure 1: (left) Geometrical settings of the problem domain and (right) a schematic illustration of their discretization in MPM.
• Figure 2: Example of a blurred porous interface with two different porosity values, $\theta=0.3$ and 0.5 with the shape of a sine function: (a) discretization into porous particles and background grid with a cell size of $h_{\mathrm{ref}}=0.2$. The reconstructed nodal porosity fields in different cell sizes are plotted in (b) $h=h_{\mathrm{ref}}$ and (c) $h=h_{\mathrm{ref}}/4$. The one-dimensional porosity profile and its gradient over sections A-A' are blown up in (d) and (e), respectively, for different cell sizes.
• Figure 3: One-dimensional porosity profile and its gradient generated from two different formulations: Eqs. \ref{['eq:nodal_porosity']} and \ref{['eq:nodal_porosity_NG']}, assuming linear and quadratic B-Spline basis functions.
• Figure 4: 1D verification suite: geometrical settings of six cases and boundary conditions.
• Figure 5: 1D verification suite - Case 1: evolution of pressure field at different time snapshots $t=0.1$, 0.5, 1.0, 2.0, and 5.0 s. The red arrow highlights volumetric and pressure errors obtained by the fractional-step method.

Theorems & Definitions (6)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6