A Variational Formulation for Deformable Particle Simulations and its Level Set Discrete Element Method Implementation

TL;DR

A deformable Discrete Element Method is presented that extends the classical rigid-particle formulation through a reduced-order description of elastic grain-scale deformation, while enabling physically grounded grain-scale deformability at a computational cost of the same order of magnitude as rigid DEM.

Abstract

We present a deformable Discrete Element Method (DEM) that extends the classical rigid-particle formulation through a reduced-order description of elastic grain-scale deformation. The method hinges on two developments. First, an energetic variational formulation based on the Lagrange--d'Alembert principle extends classical rigid-body dynamics to incorporate particle deformability by embedding translational, rotational, and deformation degrees of freedom within a unified energetic description. Second, particle deformation is realized within the Level Set DEM formalism through evolving level sets. The framework applies broadly to general particle geometries and topologies, and supports arbitrary deformation modes. The resulting deformable DEM retains the robustness, geometric and physical clarity, and scalability of classical DEM, while enabling physically grounded grain-scale deformability at a computational cost of the same order of magnitude as rigid DEM. Comparisons with full finite-element simulations demonstrate excellent agreement at both particle and system scales, establishing a general and extensible variational framework for modeling deformation in particulate systems.

Figures (6)

• Figure 1: Representative particulate systems across length scales and physical contexts, illustrating the breadth of particulate matter considered in this work: material systems composed of discrete entities whose collective mechanical behavior arises from inter-particle interactions and particle deformation. Examples include geomaterials (e.g., rocks), biological lamellar composites (e.g., nacre), architected assemblies of bonded or frictionally interlocking discrete units (e.g., topologically interlocked structures), cellular aggregates in living matter (e.g., cancer tumors), and industrial granular systems (e.g., pharmaceutical powders). The figure highlights the unifying discrete mechanics perspective motivating the deformable DEM framework developed herein.
• Figure 2: Schematic illustration of the deformable LS--DEM framework. At time $t$, each grain is characterized by its position $X_i^{t}$, orientation $\Theta_i^{t}$, and modal deformation coordinates $\varepsilon_{\alpha}^{t}$, which determine the level set field $\phi^{t}(\boldsymbol{x})$. Interaction forces $f_k^{\,t}$ are computed on the grain surface and drive the evolution of the state variables to $X_i^{t+\Delta t}$, $\Theta_i^{t+\Delta t}$, and $\varepsilon_{\alpha}^{t+\Delta t}$ at time $t+\Delta t$. The right panel shows the updated grain geometry and level set field after deformation. The bottom row highlights the distinction between rigid DEM, where only $X_i$ and $\Theta_i$ evolve, and deformable DEM, where the additional internal deformation coordinates $\varepsilon_{\alpha}$ allow grains to undergo shape changes.
• Figure 3: Illustration of several possible approaches for updating the level set field under particle deformation (shown in 2D for ease of illustration). (a) Direct recomputation of the level set field from the updated particle surface at time $t+\Delta t$. (b) Advection of the existing level set field by solving a transport equation using nodal surface velocities. (c) Semi-Lagrangian advection by mapping from the current configuration to the reference configuration, and computing, for each current position, its corresponding location in the undeformed configuration and evaluating the reference level set field there. (d) Interpolation from precomputed level sets parameterized by deformation coordinates.
• Figure 4: Three-point bending benchmark test for an analytical deformation mode. (a) Undeformed and deformed configuration of the analytical first bending mode, shown together with the grain obtained from the deformable LS-DEM simulation. The analytical mode is scaled to the same modal amplitude for qualitative comparison of the deformation pattern, as defined by the displacement field in Eq. \ref{['eq:bending_mode_shape']}. Color indicates the magnitude of the displacement field. (b) Quantitative comparison between the numerical normalized force–displacement response and the analytical stiffness reference derived from the generalized stiffness.
• Figure 5: Comparison between the results obtained from the high-fidelity FEM indentation simulation and the results obtained from the deformable LS-DEM framework under the same indentation setup, using a deformation mode extracted from the FEM simulation. (a) Deformed shapes obtained in FEM and in deformable LS-DEM for the same value of the generalized modal displacement used to scale the extracted deformation mode. Color indicates the magnitude of the displacement field; the same color scale is used for both representations. (b) Quantitative comparison between the normalized force–displacement response from the deformable LS-DEM simulation and the FEM force–displacement curve.