Domain-Element Model for Adaptive DEM

• Domain-Element Model is a framework that extends DEM by decomposing the simulation domain into subdomains with different resolutions and explicit inter-domain coupling.
• It employs a single-parameter spring–dashpot rolling friction model, where the critical rolling angle determines stiffness, damping, and torque capacity to mimic non-spherical effects.
• Adaptive fine–coarse coupling enables efficient large-scale simulations while preserving particle-scale behaviors in regions requiring high resolution.

Searching arXiv for the specified papers and closely related DEM coarse-graining work to ground the article. In the cited literature, DEM denotes the Discrete Element Method, while one source explicitly interprets its adaptive multi-resolution methodology as a template for a broader Domain-Element Model in which the computational domain is decomposed into subdomains with different levels of resolution and coupled through volumetric passing of boundary conditions (Queteschiner et al., 2017). Within this framework, granular materials are represented by particles advanced through Newton’s laws, and recent work extends coarse-grained DEM by introducing a single-parameter spring-dashpot rolling friction model in which the critical rolling angle $\theta_0$ determines rolling stiffness, damping, and torque capacity, thereby providing a physically interpretable route for representing non-spherical particle effects with spherical particles in industrial-scale simulations (Widartiningsih et al., 6 Feb 2026).

1. Conceptual scope and terminology

In the standard sense of the cited work, DEM is the Discrete Element Method for particulate flows: each particle is tracked individually, contact forces are resolved explicitly, and translational and rotational motion are integrated in time (Queteschiner et al., 2017). The expression Domain-Element Model appears as a broader interpretation of adaptive coarse-grain DEM, where the domain is decomposed into elements understood as subdomains with different effective particle sizes, different averaging meshes, and explicit inter-subdomain coupling rules.

This interpretation is most directly associated with adaptive, hybrid fine–coarse DEM. A master DEM simulation uses a coarse-grain representation, while one or more fine-scale subdomains are embedded where particle-size-dependent phenomena must be resolved accurately. Coupling is established through cell-based exchange of mass flow, velocity, volume fraction, and stress. The same framework can be extended recursively to multiple levels of coarse-graining, so that fine, medium, and coarse subdomains coexist within one hierarchy (Queteschiner et al., 2017).

A plausible implication is that “Domain-Element Model” is best understood not as a distinct particle mechanics law, but as a domain-decomposition interpretation of DEM in which resolution, scaling laws, and feedback controllers become integral modeling elements. Under that reading, the rolling-friction formulation proposed for coarse-grained DEM is a constitutive enhancement that enlarges the range of particle-shape effects such a framework can preserve (Widartiningsih et al., 6 Feb 2026).

2. Governing mechanics and contact laws

Both cited works formulate DEM through Newtonian particle dynamics. In the adaptive coarse-graining study, each particle $i$ satisfies

$$m_i \ddot{\mathbf{x}}_i = \mathbf{f}_i, \qquad I_i \dot{\bm{\omega}}_i = \mathbf{t}_i,$$

with total force composed of body forces and contact forces, and torque generated primarily by tangential contact forces acting at the contact point (Queteschiner et al., 2017). In the rolling-friction study, translational motion is likewise written in terms of contact force, fluid drag, pressure-gradient force, and gravity, while rotational motion is augmented by a rolling resistance moment (Widartiningsih et al., 6 Feb 2026).

The contact mechanics differ slightly between the two formulations. One uses a non-linear damped Hertzian spring–dashpot model with normal and tangential components,

$$\mathbf{f}_{n,ij} = k_n \,\bm{\delta}_{n,ij} - \gamma_n \,\dot{\bm{\delta}}_{n,ij}, \qquad \mathbf{f}_{t,ij} = k_t \,\bm{\delta}_{t,ij} - \gamma_t \,\dot{\bm{\delta}}_{t,ij},$$

together with the Coulomb condition $f_{t,ij} \le \mu f_{n,ij}$ (Queteschiner et al., 2017). The other uses a Voigt spring–dashpot with Coulomb friction for the normal and tangential contact force and then introduces rolling resistance as an additional torque law (Widartiningsih et al., 6 Feb 2026).

The mechanical motivation for rolling resistance is that industrial DEM commonly adopts spheres because contact detection is easy and time integration is robust, whereas real particles often have flat faces and edges that interlock and resist rolling. Explicit non-spherical DEM with ellipsoids, superquadrics, polyhedra, or multi-spheres is much more expensive. Rolling resistance therefore functions as an approximation that allows spherical particles to reproduce macroscopic consequences of particle shape, including higher shear strength, larger angle of repose, and stronger arches (Widartiningsih et al., 6 Feb 2026).

3. Single-parameter spring–dashpot rolling friction model

The spring–dashpot rolling friction model is formulated as a rotational analogue of the tangential contact law. For each particle, rotational motion is written as

$$I \frac{d\boldsymbol{\omega}}{dt} = \mathbf{T}_{\text{ext}} + \mathbf{M},$$

where $\mathbf{M}$ is the rolling resistance moment summed over contacts (Widartiningsih et al., 6 Feb 2026). At a contact, the rolling torque follows a spring–dashpot law with a yield cap:

$\mathbf{M}= \begin{cases} - k_\theta \boldsymbol{\theta}_{\text{roll}} - n_\theta \boldsymbol{\omega}_{\text{roll}}, & |\mathbf{M}| \le |M_{\text{ref}}|,\[4pt] - M_{\text{ref}} \dfrac{\mathbf{M}}{|\mathbf{M}|}, & |\mathbf{M}| > |M_{\text{ref}}|. \end{cases}$

Here $\boldsymbol{\theta}_{\text{roll}}$ is the relative rolling angular displacement, $\boldsymbol{\omega}_{\text{roll}}$ is the relative rolling angular velocity, $i$0 is the angular stiffness, $i$1 is the angular viscous damping, and $i$2 is the maximum rolling resistance moment (Widartiningsih et al., 6 Feb 2026).

The rolling spring variable is updated incrementally as

$i$3

which makes $i$4 the rotational analogue of tangential displacement in standard DEM. The model then derives rolling stiffness and damping from an idealized circular contact area of radius $i$5, using spring and dashpot coefficients per unit area. This produces the proportionalities

$i$6

so rolling stiffness and rolling damping are not independent input parameters but are tied to the normal contact parameters and contact geometry (Widartiningsih et al., 6 Feb 2026).

The decisive simplification is the introduction of the critical rolling angle $i$7, also called the critical slope angle. It is defined as the slope angle at which a particle resting on an inclined plane transitions from static equilibrium to rolling motion. Force and torque balance give

$i$8

and therefore

$i$9

while the torque cap becomes

$$m_i \ddot{\mathbf{x}}_i = \mathbf{f}_i, \qquad I_i \dot{\bm{\omega}}_i = \mathbf{t}_i,$$0

Given $$m_i \ddot{\mathbf{x}}_i = \mathbf{f}_i, \qquad I_i \dot{\bm{\omega}}_i = \mathbf{t}_i,$$1, $$m_i \ddot{\mathbf{x}}_i = \mathbf{f}_i, \qquad I_i \dot{\bm{\omega}}_i = \mathbf{t}_i,$$2, particle radius $$m_i \ddot{\mathbf{x}}_i = \mathbf{f}_i, \qquad I_i \dot{\bm{\omega}}_i = \mathbf{t}_i,$$3, and a single additional physical parameter $$m_i \ddot{\mathbf{x}}_i = \mathbf{f}_i, \qquad I_i \dot{\bm{\omega}}_i = \mathbf{t}_i,$$4, all three rolling quantities are determined: stiffness, damping, and torque limit (Widartiningsih et al., 6 Feb 2026).

This construction also separates static rolling from kinetic rolling. Below $$m_i \ddot{\mathbf{x}}_i = \mathbf{f}_i, \qquad I_i \dot{\bm{\omega}}_i = \mathbf{t}_i,$$5, the spring–dashpot law governs the contact and can balance small external moments. Once $$m_i \ddot{\mathbf{x}}_i = \mathbf{f}_i, \qquad I_i \dot{\bm{\omega}}_i = \mathbf{t}_i,$$6, torque is capped and additional rolling tendency is accommodated by rotation. No separate rolling friction coefficient is needed (Widartiningsih et al., 6 Feb 2026).

4. Comparison with conventional rolling models and equilibrium properties

The principal comparison is with the Directional Constant Torque (DCT) model, in which the rolling torque is

$$m_i \ddot{\mathbf{x}}_i = \mathbf{f}_i, \qquad I_i \dot{\bm{\omega}}_i = \mathbf{t}_i,$$7

Its torque magnitude is constant for a given contact and independent of rolling angle or angular velocity. The cited study notes that this single-parameter model is empirically calibrated and lacks a clear, easily measurable physical interpretation. Near equilibrium, a constant torque can be too large, causing particles to overshoot and oscillate (Widartiningsih et al., 6 Feb 2026).

The same study contrasts the proposed model with classical S–D rolling formulations associated with Iwashita and Oda, Luding, and Jiang et al. Those models typically introduce multiple independent rolling parameters—rolling stiffness, rolling damping, rolling friction coefficient or maximum rolling moment, and sometimes twisting stiffness, twisting damping, and a twisting friction coefficient. Because these parameters are treated as independent tunables and their effects on bulk behavior are interdependent, calibration can become non-unique and experimentally burdensome (Widartiningsih et al., 6 Feb 2026).

The single-parameter S–D model retains the dissipative structure of spring–dashpot dynamics while reducing calibration to the critical rolling angle. This matters for both numerics and physical interpretation. The time-step restriction must satisfy translational and rotational stability simultaneously,

$$m_i \ddot{\mathbf{x}}_i = \mathbf{f}_i, \qquad I_i \dot{\bm{\omega}}_i = \mathbf{t}_i,$$8

so once $$m_i \ddot{\mathbf{x}}_i = \mathbf{f}_i, \qquad I_i \dot{\bm{\omega}}_i = \mathbf{t}_i,$$9 is chosen, the rotational constraint follows directly from $$\mathbf{f}_{n,ij} = k_n \,\bm{\delta}_{n,ij} - \gamma_n \,\dot{\bm{\delta}}_{n,ij}, \qquad \mathbf{f}_{t,ij} = k_t \,\bm{\delta}_{t,ij} - \gamma_t \,\dot{\bm{\delta}}_{t,ij},$$0 (Widartiningsih et al., 6 Feb 2026).

Equilibrium behavior distinguishes the model most sharply. In particle assembly collapse tests, particle positions became static by about $$\mathbf{f}_{n,ij} = k_n \,\bm{\delta}_{n,ij} - \gamma_n \,\dot{\bm{\delta}}_{n,ij}, \qquad \mathbf{f}_{t,ij} = k_t \,\bm{\delta}_{t,ij} - \gamma_t \,\dot{\bm{\delta}}_{t,ij},$$1 in all cases, but the rotational states differed. With the proposed S–D rolling model, angular-velocity maps at $$\mathbf{f}_{n,ij} = k_n \,\bm{\delta}_{n,ij} - \gamma_n \,\dot{\bm{\delta}}_{n,ij}, \qquad \mathbf{f}_{t,ij} = k_t \,\bm{\delta}_{t,ij} - \gamma_t \,\dot{\bm{\delta}}_{t,ij},$$2 and $$\mathbf{f}_{n,ij} = k_n \,\bm{\delta}_{n,ij} - \gamma_n \,\dot{\bm{\delta}}_{n,ij}, \qquad \mathbf{f}_{t,ij} = k_t \,\bm{\delta}_{t,ij} - \gamma_t \,\dot{\bm{\delta}}_{t,ij},$$3 were nearly identical and mostly very low, and the mean absolute angular velocity decayed to nearly zero and stayed there. With the DCT model, particle positions also became static, yet many particles retained finite angular velocities and the mean absolute angular velocity continued to fluctuate significantly after the pile was static. A common misconception is that any one-parameter rolling model automatically yields physically consistent static behavior; these results indicate that a constant-torque formulation can sustain non-physical oscillatory rolling, whereas the spring–dashpot formulation damps motion into rotational equilibrium (Widartiningsih et al., 6 Feb 2026).

5. Coarse-graining, adaptive coupling, and domain decomposition

Coarse-graining is central to the domain-element interpretation because it assigns different effective particle sizes to different regions. In the rolling-friction study, a coarse-grain ratio $$\mathbf{f}_{n,ij} = k_n \,\bm{\delta}_{n,ij} - \gamma_n \,\dot{\bm{\delta}}_{n,ij}, \qquad \mathbf{f}_{t,ij} = k_t \,\bm{\delta}_{t,ij} - \gamma_t \,\dot{\bm{\delta}}_{t,ij},$$4 is defined by

$$\mathbf{f}_{n,ij} = k_n \,\bm{\delta}_{n,ij} - \gamma_n \,\dot{\bm{\delta}}_{n,ij}, \qquad \mathbf{f}_{t,ij} = k_t \,\bm{\delta}_{t,ij} - \gamma_t \,\dot{\bm{\delta}}_{t,ij},$$5

while the critical rolling angle remains unchanged under coarse-graining. From equality of angular kinetic energy and rolling strain energy, the model derives

$$\mathbf{f}_{n,ij} = k_n \,\bm{\delta}_{n,ij} - \gamma_n \,\dot{\bm{\delta}}_{n,ij}, \qquad \mathbf{f}_{t,ij} = k_t \,\bm{\delta}_{t,ij} - \gamma_t \,\dot{\bm{\delta}}_{t,ij},$$6

and rewrites the rolling torque scaling as

$$\mathbf{f}_{n,ij} = k_n \,\bm{\delta}_{n,ij} - \gamma_n \,\dot{\bm{\delta}}_{n,ij}, \qquad \mathbf{f}_{t,ij} = k_t \,\bm{\delta}_{t,ij} - \gamma_t \,\dot{\bm{\delta}}_{t,ij},$$7

This scaling is intended to preserve rolling behavior at the macro-scale when larger representative particles are used (Widartiningsih et al., 6 Feb 2026).

In the adaptive coarse-grain study, the analogous coarse-grain ratio is $$\mathbf{f}_{n,ij} = k_n \,\bm{\delta}_{n,ij} - \gamma_n \,\dot{\bm{\delta}}_{n,ij}, \qquad \mathbf{f}_{t,ij} = k_t \,\bm{\delta}_{t,ij} - \gamma_t \,\dot{\bm{\delta}}_{t,ij},$$8, with

$$\mathbf{f}_{n,ij} = k_n \,\bm{\delta}_{n,ij} - \gamma_n \,\dot{\bm{\delta}}_{n,ij}, \qquad \mathbf{f}_{t,ij} = k_t \,\bm{\delta}_{t,ij} - \gamma_t \,\dot{\bm{\delta}}_{t,ij},$$9

while density, Young’s modulus, coefficient of restitution, and coefficient of friction are kept unchanged. Contact stiffness scales as $f_{t,ij} \le \mu f_{n,ij}$0, damping as $f_{t,ij} \le \mu f_{n,ij}$1, and the allowable time step scales approximately as $f_{t,ij} \le \mu f_{n,ij}$2 (Queteschiner et al., 2017).

Quantity	Coarse-grained DEM with rolling resistance	Adaptive coarse-grain DEM
Size scaling	$f_{t,ij} \le \mu f_{n,ij}$3	$f_{t,ij} \le \mu f_{n,ij}$4
Mass scaling	$f_{t,ij} \le \mu f_{n,ij}$5	$f_{t,ij} \le \mu f_{n,ij}$6
Additional scaling	$f_{t,ij} \le \mu f_{n,ij}$7, $f_{t,ij} \le \mu f_{n,ij}$8, $f_{t,ij} \le \mu f_{n,ij}$9	$$I \frac{d\boldsymbol{\omega}}{dt} = \mathbf{T}_{\text{ext}} + \mathbf{M},$$0, $$I \frac{d\boldsymbol{\omega}}{dt} = \mathbf{T}_{\text{ext}} + \mathbf{M},$$1, $$I \frac{d\boldsymbol{\omega}}{dt} = \mathbf{T}_{\text{ext}} + \mathbf{M},$$2

The adaptive framework operationalizes this scaling through explicit subdomain coupling. Coarse-to-fine transfer uses boundary surface cells of about 3–10 coarse-grain diameters, within which mass flow rate, particle velocity, and particle size distribution are averaged over a coupling interval. Fine particles are then inserted using a Sequential Simple Inhibition (SSI) algorithm to avoid overlap and artificial clustering. Inside the fine region, a boundary layer of hexahedral cells computes Eulerian averages such as stress tensor, volume fraction, and particle velocities; a discrete PI controller uses coarse-grain stress as setpoint and applies correcting forces to fine particles. The controller is updated every 15 time steps, its force is limited so that no fine particle exceeds the maximum coarse-grain velocity, and it is faded out within a distance of $$I \frac{d\boldsymbol{\omega}}{dt} = \mathbf{T}_{\text{ext}} + \mathbf{M},$$3 fine-particle diameters from the boundary (Queteschiner et al., 2017).

Fine-to-coarse transfer is handled differently. Rather than inserting coarse particles when fine particles exit the refined region, the coarse particles are preserved and adjusted through a proportional controller in an overlapping hexahedral grid. In regions where matching mass flow is essential, the coarse target velocity is obtained by multiplying the fine-scale velocity by the volume-fraction mismatch; elsewhere the correction can focus on velocity alone. This two-way, cell-based coupling is the clearest basis for describing the framework as a Domain-Element Model: each subdomain has its own particle scale and averaging mesh, while consistency is maintained through feedback control and volumetric exchange of boundary information (Queteschiner et al., 2017).

The same study extends this to multiple levels with $$I \frac{d\boldsymbol{\omega}}{dt} = \mathbf{T}_{\text{ext}} + \mathbf{M},$$4. In a hypothetical large system, a fully resolved simulation of about $$I \frac{d\boldsymbol{\omega}}{dt} = \mathbf{T}_{\text{ext}} + \mathbf{M},$$5 fine particles is replaced by nested subdomains whose combined particle count is about 46.5 million, with an estimated speedup $$I \frac{d\boldsymbol{\omega}}{dt} = \mathbf{T}_{\text{ext}} + \mathbf{M},$$6 when time-step scaling is also used (Queteschiner et al., 2017).

6. Validation, industrial relevance, and limitations

Validation in a DEM–CFD incinerator system uses the FELMI framework and a vertical rectangular chamber of height 200 mm, width 40 mm, and depth 40 mm, with a circular control plate of diameter 23 mm located 140 mm above the bottom, and upward air flow at 1.5 m/s. The original system contains 100 $$I \frac{d\boldsymbol{\omega}}{dt} = \mathbf{T}_{\text{ext}} + \mathbf{M},$$7m glass beads and 2.56 million particles. Coarse-grain ratios of $$I \frac{d\boldsymbol{\omega}}{dt} = \mathbf{T}_{\text{ext}} + \mathbf{M},$$8 and $$I \frac{d\boldsymbol{\omega}}{dt} = \mathbf{T}_{\text{ext}} + \mathbf{M},$$9 are compared both without rolling resistance and with the proposed rolling resistance using $\mathbf{M}$0 (Widartiningsih et al., 6 Feb 2026).

Without rolling resistance, coarse-grained DEM–CFD reproduces particle and gas flow patterns qualitatively, but the angle of repose on the control plate decreases as coarse-graining increases: Case 2-1 gives $\mathbf{M}$1, Case 2-2 gives $\mathbf{M}$2, and Case 2-3 gives $\mathbf{M}$3. With the proposed rolling resistance, coarse-grained cases preserve particle accumulation and maintain the angle of repose much more closely: Case 3-1 gives $\mathbf{M}$4, Case 3-2 gives $\mathbf{M}$5, and Case 3-3 gives $\mathbf{M}$6. The study states that all deviations are within the adopted 10% tolerance considered acceptable in the DEM literature. Simple enlarged-particle simulations without proper coarse-grain scaling perform poorly because drag scaling is wrong, and particles hardly reach the region above the plate (Widartiningsih et al., 6 Feb 2026).

Validation of adaptive domain decomposition is provided by a silo filling and discharge problem. The fine reference simulation contains 187,504 particles of diameter 2.8 mm; a pure coarse-grain simulation with $\mathbf{M}$7 uses 23,438 particles of diameter 5.6 mm; and the two-level model combines the same coarse region with a fine subdomain in the lower quarter containing about 44,000 particles. Relative runtimes show a speedup of about $\mathbf{M}$8 for the coarse-grain-only simulation and about $\mathbf{M}$9 for the two-level simulation (Queteschiner et al., 2017).

Accuracy results reveal the strengths and limits of the framework. During filling, the average granular pressure in the transition cells drops and stabilizes at a lower value of about $\mathbf{M}= \begin{cases} - k_\theta \boldsymbol{\theta}_{\text{roll}} - n_\theta \boldsymbol{\omega}_{\text{roll}}, & |\mathbf{M}| \le |M_{\text{ref}}|,\[4pt] - M_{\text{ref}} \dfrac{\mathbf{M}}{|\mathbf{M}|}, & |\mathbf{M}| > |M_{\text{ref}}|. \end{cases}$0 without stress correction because the fine region lacks the weight of particles above it; with the PI controller, the fine-region pressure follows the reference closely. During discharge, the pure coarse-grain model underpredicts the mass flow rate by more than 20%, with $\mathbf{M}= \begin{cases} - k_\theta \boldsymbol{\theta}_{\text{roll}} - n_\theta \boldsymbol{\omega}_{\text{roll}}, & |\mathbf{M}| \le |M_{\text{ref}}|,\[4pt] - M_{\text{ref}} \dfrac{\mathbf{M}}{|\mathbf{M}|}, & |\mathbf{M}| > |M_{\text{ref}}|. \end{cases}$1 instead of the reference $\mathbf{M}= \begin{cases} - k_\theta \boldsymbol{\theta}_{\text{roll}} - n_\theta \boldsymbol{\omega}_{\text{roll}}, & |\mathbf{M}| \le |M_{\text{ref}}|,\[4pt] - M_{\text{ref}} \dfrac{\mathbf{M}}{|\mathbf{M}|}, & |\mathbf{M}| > |M_{\text{ref}}|. \end{cases}$2, while the two-level coupled model gives $\mathbf{M}= \begin{cases} - k_\theta \boldsymbol{\theta}_{\text{roll}} - n_\theta \boldsymbol{\omega}_{\text{roll}}, & |\mathbf{M}| \le |M_{\text{ref}}|,\[4pt] - M_{\text{ref}} \dfrac{\mathbf{M}}{|\mathbf{M}|}, & |\mathbf{M}| > |M_{\text{ref}}|. \end{cases}$3, corresponding to $\mathbf{M}= \begin{cases} - k_\theta \boldsymbol{\theta}_{\text{roll}} - n_\theta \boldsymbol{\omega}_{\text{roll}}, & |\mathbf{M}| \le |M_{\text{ref}}|,\[4pt] - M_{\text{ref}} \dfrac{\mathbf{M}}{|\mathbf{M}|}, & |\mathbf{M}| > |M_{\text{ref}}|. \end{cases}$4 deviation from reference (Queteschiner et al., 2017).

These studies delimit the model class clearly. Pure coarse-graining fails when behavior depends intrinsically on absolute particle size, as in Beverloo-type discharge or deposition shape. Correct coarse-grain scaling plus rolling friction can recover shape-sensitive macroscopic observables in coarse-grained DEM–CFD, while adaptive fine–coarse coupling can restore size-dependent behavior in localized regions. At the same time, the adaptive framework requires tuning of coarse-grain ratio, subdomain placement, controller gains, and update frequency, and mass and momentum conservation at interfaces are approximated through controllers and cell-based insertion rather than being enforced at the level of individual grains (Queteschiner et al., 2017). A plausible implication is that the Domain-Element interpretation is most effective when particle-size-dependent effects are spatially confined, whereas systems dominated by such effects throughout the full domain still require fine resolution over most or all of the geometry.