4.4. Force Field

The force field encompasses a broader set of operators and mechanisms responsible for computing forces acting on particles within the particle environment. It includes various types of forces such as gravitational forces, contact forces, and other external influences that affect particle dynamics.

4.4.1. Contact Force Laws

In the Discrete Element Method (DEM), the equations of motion (translations and rotations) are discretized in time. Only the rigid-body displacements are considered. Small overlaps between the particles are allowed and used as strain variables. The total contact force between particle \(i\) and particle \(j\) is given by

\[\textbf{f}_{ij} = f_n \textbf{n} + \textbf{f}_t\]

Twhere \(f_n\) the normal component of the contact force and \(\textbf{f}_t\) is the tangential force vector. These forces are expressed in the local contact frame \((\textbf{n},\textbf{t},\textbf{s})\) as a function of the overlaps and tangential displacements. They are calculated from force laws that generally describe frictional contact interactions. An important feature of DEM is to allow the particles to overlap. This overlap \(\delta_n\) represents a normal strain localized in the vicinity of the contact point. A simple linear relation is assumed between normal contact force and \(\delta_n\) . This is consistent with the fact that the overlaps allow for a penalty-based explicit formulation of particle motions, i.e., the elastic repulsion force is mobilized to prevent penalizing the overlap. The condition of particle undeformability implies that the overlaps must stay small compared to particle size. In this linear approximation, the normal component of the contact force is given by

\[f_n = - k_n \delta_n + \nu_n v_n\]

where \(k_n\) is the normal stiffness coefficient, \(v_n\) the normal component of the relative velocity (between particle \(i\) and particle \(j\)), and \(\nu_n\) is the viscous damping coefficient. \(\nu_n\) is related to the restitution coefficient \(e_n\) by

\[ \begin{align}\begin{aligned}\nu_n = \alpha_n \sqrt{2 m_{\text{eff}} v_n}\\\alpha_n = \frac{- \ln{e_n}}{\sqrt{\ln^2{e_n} + \pi^2}}\end{aligned}\end{align} \]

where:

\(\nu_n\) is the viscous damping rate during the collision
\(\alpha_n\) is the damping parameter, calculated from the restitution coefficient \(e_n\)
\(m_{\text{eff}}\) is the effective mass of the two colliding particles, defined by \(m_{\text{eff}} = \frac{m_i m_j}{m_i + m_j}\)

Note

The formulas are identical to those used in Rockable (see Rockable Force Laws) but are implemented differently to align with the exaDEM data structure.

4.4.2. Tangential Component

The tangential force represents the frictional resistance between particles when they slide against each other. This force is calculated based on the relative tangential velocity (\(v_t\)) and a tangential stiffness parameter (\(k_t\), keyword ktContact).

The Coulomb friction model is used to limit the tangential force, ensuring that it does not exceed the product of the friction coefficient \(\mu\) and the normal force (\(f_n\)):

\[f_t \leq \mu \cdot f_n\]

The tangential force \(f_t\) is incrementally updated at each time step according to the following relation, starting from the onset of contact:

\[f_t = f_t + k_t \cdot v_t \cdot \Delta t\]

where: - \(f_t\) is the tangential force, - \(k_t\) is the tangential stiffness, - \(v_t\) is the relative tangential velocity, and - \(\delta t\) is the time step.

The friction force is reset to zero as soon as contact is lost.

4.4.3. Contact’s Law Operators

Contact's Law in the context of the Discrete Element Method (DEM) refers to the principle used to calculate forces between particles based on their relative displacements. In DEM simulations, Contact's Law is applied to model interactions between particles, enabling the simulation of elastic deformation and linear force behaviors within particle-based systems.

Variables:

Variable	Description
\(cp\)	The contact position
\(r_i\)	The position of the particle i
\(r_j\)	The position of the particle j
\(v_i\)	The velocity of the particle i
\(v_j\)	The velocity of the particle j
\(vrot_i\)	The angular velocity of the particle i
\(vrot_j\)	The angular velocity of the particle j
\(m_i\)	The mass of the particle i
\(m_j\)	The mass of the particle j
\(d_n\)	The interpenetration / particle overlap

And constants:

Constant	Description
\(\Delta_t\)	The timestep increment
\(\alpha_n\)	The damping rate
\(k_n\)	The normal stiffness coefficient
\(k_t\)	The tangential stiffness coefficient
\(v_t\)	The relative tangential velocity
\(k_r\)	The rotational stiffness coefficient
\(mu\)	The coefficient of friction
\(fc\)	The cohesive coefficient
\(dncut\)	X

Three possibilities depending on the value of the interpenetration between \(d_n\) two particles:

\( d_n < -dncut \)
\( -dncut < d_n < 0 \)
\( 0 < d_n < dncut \)

Warning

Cohesive forces (dncut) are only applied if you use the operators: contact_sphere_with_cohesion or contact_polyhedron_with_cohesion. Otherwise we only consider the case ::math`d_n < 0.0`.

Formula between particle i and particle j if \( d_n < -dncut \) :

\[\textbf{f}_{ij} = -k_n . d_n + \alpha_n \sqrt{2.m_{eff}} v_n + k_t . v_t . \Delta_t\]

with the effective mass:

\[m_{eff} = \frac{m_i.m_j}{m_i+m_j}\]

and the relative velocity norm:

\[v_n = (v_i - (cp - r_i) \wedge vrot_i) - (v_j - (cp - r_j) \wedge vrot_j)\]

Formula between particle i and particle j if \( -dncut < d_n < 0 \) :

\[\textbf{f}_{ij} = f_n + k_t . v_t . \Delta_t\]

with:

\[\begin{split}f_n =\left \{ \begin{array}{lcl} -k_n . d_n + \alpha_n \sqrt{2.m_{eff}} v_n & if & -k_n . d_n + \alpha_n \sqrt{2.m_{eff}} v_n >= -fc \\ & & \\ -fc & if & -k_n . d_n + \alpha_n \sqrt{2.m_{eff}} v_n < -fc \end{array} \right.\end{split}\]

Formula between particle i and particle j if \( 0 < d_n < dncut \) :

\[\textbf{f}_{ij} = (\frac{fc}{dncut} . d_n - fc) . \textbf{n}\]

with n normalized vector from particle i to particle j

Name: contact_sphere, contact_polyehdron, contact_sphere_with_cohesion, or contact_polyehdron_with_cohesion
Description: These operators compute forces between particles and particles/drivers using the contact’s law.
Parameter:

symetric	Activate or disable symmetric updates (do not disable it with polyhedron).
config	Data structure that contains contact force parameters (dncut, kn, kt, kr, fc, mu, damp_rate). Type = exaDEM::ContactParams. No default parameter.
config_driver	Data structure that contains contact force parameters (dncut, kn, kt, kr, fc, mu, damp_rate). Type = exaDEM::ContactParams. This parameter is optional.
save_interactions	Store interactions into the classifier data structure. Default is false.

Here are 4 examples with YAML:

- contact_sphere:
   symetric: true
   config: { kn: 100000, kt: 100000, kr: 0.1, mu: 0.9, damp_rate: 0.9}

- contact_polyhedron:
   config: { kn: 10000, kt: 10000, kr: 0.1, mu: 0.1, damp_rate: 0.9}
   config_driver: { kn: 10000, kt: 10000, kr: 0.1, mu: 0.3, damp_rate: 0.9}

- contact_sphere_with_cohesion:
   symetric: true
   config: { dncut: 0.1 m, kn: 100000, kt: 100000, kr: 0.1, fc: 0.05, mu: 0.9, damp_rate: 0.9}

- contact_polyhedron_with_cohesion:
   config: { dncut: 0.1 m, kn: 10000, kt: 10000, kr: 0.1, fc: 0.05, mu: 0.1, damp_rate: 0.9}
   config_driver: { dncut: 0.1 m, kn: 10000, kt: 10000, kr: 0.1, fc: 0.05, mu: 0.3, damp_rate: 0.9}

Note

It is important to check that interaction lists have been built with this option enabled. By default, exaDEM always builds interaction lists using the symmetry option to limit the number of calculations.

Note

Contact Force With Cohesion operator includes a cohesion force from rcut to rcut+dncut with the cohesion force parameter fc.

Note

The contact_sphere and contact_sphere_with_cohesion operators are designed to process interactions built in nbh_sphere (please, include the config_spheres.msp file).
The contact_polyhedron and contact_polyhedron_with_cohesion operators are designed to process interactions built in nbh_polyhedron (please, include the config_polyhedra.msp file).

4.4.4. External Forces

External forces are additional influences acting on particles within the simulation environment, originating from sources outside the particle system itself. These forces can include environmental factors like wind, fluid flow, or magnetic fields, as well as user-defined forces applied to specific particles or regions.

4.4.4.1. Gravity Operator

Formula:

(1)\[\textbf{f} = m.\textbf{g}\]

With f the forces, m the particle mass, and g the gravity constant.

Operator Name: gravity_force
Description: This operator computes forces related to the gravity.
Parameter:

gravity

Define the gravity constant in function of the gravity axis, default value are x axis = 0, y axis = 0 and z axis = -9.807

YAML example:

- gravity_force:
   gravity: [0,0,-0.009807]

4.4.4.2. Quadratic Drag Force

Formula:

\[\textbf{f} = -\mu.cx.\|v\|.\textbf{v}\]

With f the particle forces, cx the aerodynamic coefficient, and \(\mu\) the drag coefficient, ||v|| the norm of the particle velocity, and v the particle velocity.

Operator Name: quadratic_force
Description: External forces that model air or fluid, f = - mu * cx * norm(v) * vector(v).
Parameter:

cx	aerodynamic coefficient, default value is for air = 0.38.
mu	drag coefficient. default value is for air = 0.000015.

YAML example: see example quadratic-force-test/QuadraticForceInput.msp

- quadratic_force:
   cx: 0.38
   mu: 0.0000015

4.4.4.3. Fluid Grid Force

Operator Name: sphere_fluid_friction
Description: External forces that model a fluid computed from a grid such as: f = ||fv - pv||

\[dv = fv - pv\]

\[f = cx . dv . ||dv|| . \pi . r . r.\]

With fv the fluid velocity, pv the particle velocity, r the particle radius, and cx a coefficient set to 1 by default.

Note

The fluid velocity fv for each point of the grid has been defined by the operator set_cell_values (pure exaNBody operator).