6 DoF Dynamics : Descrete Element Method (DEM) Simulation Dataset for Learning GNN Surrogate Model

Descrete Element Method (DEM) Dataset Description — Dynami-CAL GraphNet Vinay Sharma1and Olga Fink1 1Intelligent Maintenance and Operations Systems, EPFL, Lausanne, Switzerland This dataset accompanies the paper: “Dynami-CAL GraphNet: A Physics-Informed Graph Neural Network Conserving Linear and Angular Momentum for Dynamical Systems” V. Sharma & O. Fink, 2025 — arXiv: 2501.07373 It provides all Discrete Element Method (DEM) simulation data used for training,validation, and testing of the Dynami-CAL GraphNet model. Each archive contains time-resolved particle trajectories with explicit boundary conditions and contact parameters. The simulations were performed using the MFiX DEM framework [1]. 1 Homogeneous Granular Collisions Archive: RawData_ 60Spheres_ Homogeneous_ Interaction_ Inside_ Cuboidal_ Enclosure. zip Linked paper section: Section 2.2 — Confined Granular Collisions (Training Dataset) This archive forms the primary training dataset for Dynami-CAL GraphNet. It consists of five simulated trajectories of 60 identical spheres confined inside a stationary cuboidal enclosure, initialized with random velocities. The dataset evaluates the model’s ability to: •Generate stable long-horizon rollouts over 1500 simulation steps, •Capture the physically consistent evolution of system-level kinetic energy, linear momentum, and angular momentum, •Model open, dissipative systems in which energy and momentum are absorbed by stationary walls. Each trajectory corresponds to a distinct random initialization of particle velocities. Performance is evaluated under both within-distribution (interpolation) and out-of-distribution (extrapolation) conditions, as described in Supplementary Information §1.1 of the paper. Simulation Framework and Temporal Resolution The dataset was generated using the MFiX Discrete Element Method (DEM) solver. Each trajectory comprises 1500 time steps sampled at ∆t= 1 ×10−4s. Internally, the MFiX solver advances dynamics at δt = 1 ×10−7s to accurately resolve particle–particle collisions and wall contacts. 1 Simulation Parameters (No Gravity) Parameter Value Sphere Diameter (dp) 0.005 m Sphere Density (ρp) 4000 kg/m3 Coulomb Friction Coefficient (µs) 0.1 (sphere–sphere and sphere–wall) Normal Spring Constant (kn) 1000 N/m (sphere–sphere and sphere–wall) Tangential/Normal Stiffness Ratio (kt/kn) 2/7 (sphere–sphere and sphere–wall) Damping Tangential/Normal Ratio (ηt/ηn) 0.5 (sphere–sphere and sphere–wall) Coefficient of Restitution (enml) 0.9 (sphere–sphere and sphere–wall) Gravity (g) Not applied All simulations are conducted within a stationary cuboidal enclosure defined by the spatial bounds xmin = (0,0,0) and xmax = (0.03,0.03,0.03) m. Folder and File Structure The archive contains a top-level directory DATA/ with seven subfolders, each corresponding to a simulation case: •CASE01–CASE05: Training trajectories with unique random velocity magnitudes and directions. •CASE06: Validation case — velocity magnitude within training range but unseen during training. •CASE07: Extrapolation case — initial kinetic energy ≈3×the largest training case. Each folder (e.g., CASE01/) contains 1500 CSV files, one per recorded time step, named sequentially as: data_at_timestep_000.csv,data_at_timestep_001.csv, . . . , data_at_timestep_1499.csv CSV Schema and Units Each CSV file contains 60 rows (one per particle) and the following columns: [ Diameter, Velocity :0 , Velocity :1 , Velocity :2 , Angular_velocity:0, Angular_velocity:1, Angular_velocity:2, Orientation:0, Orientation:1, Orientation:2, Density , Particle_ID , coordinates:0, coordinates:1, coordinates:2 ] Units: •Diameter: meters (m) •Velocity,Angular_velocity: m/s and rad/s •Orientation: unitless quaternion-like representation •Density: kg/m3 •Coordinates: meters (m) The field Particle_ID is an integer from 1 to 60. Each CSV file thus represents the full instantaneous system state at that time step. Note: The Orientation fields (Orientation:0–2) are included in the raw solver output but are not used in any analysis or modeling. Their computation method within the MFiX 2 DEM solver was undocumented in the available release, and their physical meaning may not correspond to a standard quaternion or Euler representation. Users can safely ignore these fields for all downstream tasks. Example Rows (First Six Particles at a Single Time Step) Geom./ID Velocity Angular Vel. Orientation Density PID Coordinates D(m) vxvyvzωxωyωzqxqyqz(kg/m3)x y z 0.005 1 0 0 0 0 0 0 0 1 4000 1 0.005283 0.0042866 0.0060622 0.005 1 0 0 0 0 0 0 0 1 4000 2 0.010497 0.0042866 0.0060622 0.005 1 0 0 0 0 0 0 0 1 4000 3 0.015745 0.0042866 0.0060622 0.005 1 0 0 0 0 0 0 0 1 4000 4 0.0079341 0.0042866 0.0106090 0.005 1 0 0 0 0 0 0 0 1 4000 5 0.0131580 0.0042866 0.0106090 0.005 1 0 0 0 0 0 0 0 1 4000 6 0.0183600 0.0042866 0.0106090 (Shown values are illustrative; each file contains 60 rows for all particles.) 1.1 Test Benchmark: Oblique Collision Archive: RawData_ Benchmark_ 2Spheres_ Oblique_ Collision. zip Linked paper section: Section 2.1.2 — Oblique Collision: Conservation of Linear and Angular Momentum This dataset provides a controlled two-particle oblique collision benchmark designed to assess the model’s ability to conserve both linear and angular momentum in the presence of tangential interactions. Two identical spheres are initialized with offset positions and non-collinear velocities, causing a glancing collision under identical physical parameters as the head-on case. This configuration induces both translational and rotational motion, producing non-zero torque and angular momentum exchange during impact. The dataset evaluates whether the model—trained on the homogeneous 60-sphere dataset (Section 1)—can: •Correctly reproduce coupled translational–rotational dynamics, •Conserve total linear and angular momentum in a closed system, •Capture realistic post-collision trajectories and angular velocity evolution. Simulation Framework and Temporal Resolution The simulation spans 100 time steps, sampled at ∆t= 1 ×10−4s, and uses DEM solver parameters as detailed in Table 1. No external forces or wall boundaries are applied, ensuring an isolated system. During impact, tangential contact forces induce a small rotational response in both spheres, leading to angular momentum redistribution without energy loss beyond the restitution-based damping. Folder and File Structure The dataset archive consists of 100 CSV files, one per time step: data_at_timestep_000.csv,data_at_timestep_001.csv, . . . , data_at_timestep_099.csv Each file follows the same schema as defined in Section 1, with two rows (one per sphere): 3 [ Diameter, Velocity :0 , Velocity :1 , Velocity :2 , Angular_velocity:0, Angular_velocity:1, Angular_velocity:2, Orientation:0, Orientation:1, Orientation:2, Density , Particle_ID , coordinates:0, coordinates:1, coordinates:2 ] The velocity and angular velocity vectors can be used to compute total linear and angular momentum at each step for conservation analysis. 1.2 Test Benchmark: Angled Wall Collisions Archive: RawData_ Benchmark_ 1Sphere_ Multiple_ Wall_ Collision. zip Linked paper section: Supplementary Information §1.1.7 — Benchmarking the Accuracy of Learned Interactions in Dynami-CAL GraphNet This dataset serves as a test-only benchmark to evaluate the predictive accuracy of the trained models. The experiment follows the standard oblique impact protocol commonly used in DEM benchmark studies [2] and examines the model’s ability to reproduce collision outcomes at various impact angles. The setup consists of a single sphere initialized above a horizontal wall (located at z= 0) with a prescribed initial velocity vector such that it impacts the wall at specific oblique angles ranging from 10◦to 90◦. This task isolates the model’s learned contact dynamics with the wall — specifically restitution, damping, and frictional effects — in a single-sphere environment, enabling precise quantitative comparison of simulated rebound trajectories and energy dissipation. Simulation Framework and Temporal Resolution The simulation spans 200 time steps, sampled at ∆t= 1 ×10−4s, and uses DEM solver parameters as detailed in Table 1. It consists of following setup : •The sphere starts from an initial position r0= (0,0,0.005) m. •The wall coincides with the z= 0 plane (the xy-plane). •No Gravity is applied. •The initial velocity determines the angle of impact. Each trajectory captures the complete preand post-impact motion of the sphere, including the bounce height and tangential sliding. Folder and File Structure The dataset archive contains five subfolders corresponding to impact angles (in degrees): 10/,30/,45/,60/,90/ Each subfolder contains 200 CSV files, one per recorded time step: data_at_timestep_000.csv,data_at_timestep_001.csv, . . . , data_at_timestep_199.csv Each CSV file represents the instantaneous physical state of the sphere at that time step and follows the same schema as defined in Section 1: 4 [ Diameter, Velocity :0 , Velocity :1 , Velocity :2 , Angular_velocity:0, Angular_velocity:1, Angular_velocity:2, Orientation:0, Orientation:1, Orientation:2, Density , Particle_ID , coordinates:0, coordinates:1, coordinates:2 ] Each file contains a single row corresponding to the sphere. 2 Heterogeneous Granular Collisions in Gravity Archive: RawData_ 60Spheres_ Gravity_ Inside_ Cuboidal_ Enclosure. zip Linked paper section: Section 2.1.3 — Extrapolation to Moving Boundaries: Rotating Cylindrical Hopper and Section 1.2 — External Gravitational Force with Heterogeneous Interactions Dataset: GravityDriven Confined Collisions (Training Dataset) This dataset consists of 60 spheres initialized near the center of a stationary cuboidal box (see Section 1), with gravity and heterogeneous contact parameters applied to simulate dissipative granular motion. The enclosure is defined by the spatial limits xmin = (0,0,0) and xmax = (0.03,0.03,0.03) m. Particles are initialized with random velocities drawn from a uniform distribution, ensuring a diverse set of collision dynamics across trajectories. Each sphere experiences both inter-particle and wall interactions with distinct friction and damping properties, resulting in a heterogeneous, dissipative system characterized by gravitational settling and energy dissipation through inelastic collisions. The dataset is designed to: •Train models to capture gravity-driven dynamics and heterogeneous particle–wall interactions, •Evaluate robustness of learned physics under external force fields and mixed boundary effects. Simulation Framework and Temporal Resolution All simulations were conducted using the MFiX Discrete Element Method (DEM) solver. Each trajectory comprises 1500 recorded time steps, sampled at ∆t= 1×10−4s. Internally, the solver advances particle states at δt = 1 ×10−7s to maintain numerical stability during contact events and gravitational acceleration. Simulation Parameters (With Gravity) Parameter Particle–Particle Particle–Wall Particle Diameter (dp) 0.005 m 0.005 m Density (ρp) 4000 kg/m3— Coulomb Friction Coefficient (µs) 0.1 0.3 Normal Spring Constant (kn) 10000 N/m 10000 N/m Tangential/Normal Stiffness Ratio (kt/kn) 2/7 2/7 Damping Tangential/Normal Ratio (ηt/ηn) 0.3 0.5 Coefficient of Restitution (enml) 0.95 0.9 Gravity (g) 9.81 m/s2 All simulations are performed within a stationary cuboidal enclosure spanning coordinates (x, y, z)∈ [0,0.03]3m, with rigid boundary walls. 5 Folder and File Structure The archive contains a top-level directory DATA/ with seven subfolders: •CASE01–CASE05: Training trajectories with random initial velocities under gravity. •CASE06: Validation trajectory sampled from the same initialization distribution but unseen during training. •CASE07: Extrapolation trajectory with higher initial kinetic energy. Each folder (e.g., CASE01/) contains 1500 CSV files, one per recorded time step: data_at_timestep_000.csv,data_at_timestep_001.csv, . . . , data_at_timestep_1499.csv Each CSV file includes 60 rows (one per particle) with the following schema: [ Diameter, Velocity :0 , Velocity :1 , Velocity :2 , Angular_velocity:0, Angular_velocity:1, Angular_velocity:2, Orientation:0, Orientation:1, Orientation:2, Density , Particle_ID , coordinates:0, coordinates:1, coordinates:2 ] Note: The fields Orientation:0–2 are included in the raw solver output but are not used in any subsequent modeling or analysis. 2.1 Granular Mixing in Rotating Cylinder Archive: RawData_ Extrapolation_ 2073Spheres_ Gravity_ Inside_ Rotating_ Cylinder. zip Linked paper section: Section 2.1.3 — Extrapolation to Moving Boundaries: Rotating Cylindrical Hopper This dataset represents the final extrapolation benchmark for evaluating the generalization capability of the Dynami-CAL GraphNet to large-scale, real-world granular mixing systems. It simulates a rotating cylindrical hopper containing 2073 spheres subjected to gravity and a time-varying rotational acceleration profile. The configuration replicates industrial-scale rotating drum or mixer dynamics, involving coupled effects of shear, gravity, and wall-driven motion. All interaction parameters are identical to those defined in Section 2 (Heterogeneous Granular Collisions in Gravity), ensuring a consistent physical basis for direct extrapolation analysis. Simulation Setup The simulation domain consists of a vertically oriented cylindrical enclosure bounded by: (x, y, z)∈[−0.05,0.05] ×[−0.048,0.052] ×[0.0,0.1] m, yielding a cylinder of radius r= 0.05 m and height h= 0.1m. The cylinder’s central axis is aligned with the global z-axis, and its geometric center is located at (0.0,0.002,0.05) m. The curved wall acts as a rotating rigid boundary, while the top and bottom surfaces remain fixed, forming a closed cylindrical volume. A total of 2073 spherical particles are initially distributed uniformly within the enclosure and allowed to settle under gravity prior to the onset of rotation. The cylinder then undergoes controlled angular motion, imposing shear along the curved wall and inducing granular mixing. 6 Rotational Boundary Condition The cylinder rotates about the z-axis following a smooth, time-varying angular velocity function: ω(t) =        4t, 0≤t < 0.5, 4−4t, 0.5≤t≤1.5, 0, t>1.5, where ω(t)is expressed in normalized units and scaled by 2πto yield radians per second. This rotation profile produces an acceleration phase, followed by a symmetric deceleration phase, after which the system comes to rest. The direction of rotation remains constant about the cylinder’s longitudinal axis. Simulation Framework and Temporal Resolution All simulations were performed using the MFiX Discrete Element Method (DEM) framework. The trajectory spans 2000 recorded time steps sampled at ∆trecord = 1 ×10−3s. Each DEM integration step internally advances with a much finer temporal increment δt = 1 ×10−7s to ensure numerical stability under dense collisions and rotational acceleration. The Dynami-CAL GraphNet model, trained on the simpler cuboidal 60-sphere configuration, learns to generalize and simulate the dynamics of this complex rotating system. It operates at a finer temporal resolution of ∆tmodel = 1 ×10−4s. Therefore, reproducing the full trajectory from this dataset requires a20 000-step rollout to synchronize model predictions with the 2000 recorded snapshots. Folder and File Structure The archive contains a single subfolder within the top-level directory DATA/: •CASE08: Extrapolation trajectory comprising 2073 particles within the rotating cylindrical domain. The folder (CASE08/) contains 2000 CSV files, one per recorded time step: data_at_timestep_000.csv,data_at_timestep_001.csv, . . . , data_at_timestep_1999.csv Each CSV file contains 2073 rows, representing individual particles, and follows the same schema defined in Section 1: [ Diameter, Velocity :0 , Velocity :1 , Velocity :2 , Angular_velocity:0, Angular_velocity:1, Angular_velocity:2, Orientation:0, Orientation:1, Orientation:2, Density , Particle_ID , coordinates:0, coordinates:1, coordinates:2 ] Note: The Orientation fields are present in the raw solver output but are not utilized in subsequent modeling or evaluation due to non-documented computation within the MFiX release. 7 References [1] R. Garg, J. Galvin, T. Li, and S. Pannala, “Documentation of open-source mfix–dem software for gas-solids flows,” From URL https://mfix. netl. doe. gov/download/mfix/mfix_current_documentation/dem_doc_2012-1. pdf, 2012. [2] A. Di Renzo and F. P. Di Maio, “Comparison of contact-force models for the simulation of collisions in dem-based granular flow codes,” Chemical engineering science, vol. 59, no. 3, pp. 525–541, 2004. 8