# A Novel Contact Algorithm Based on a Distance Potential Function for the 3D Discrete-Element Method

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## Abstract

The combined finite–discrete-element method (FDEM) has made a groundbreaking progress in the computation of contact interaction. However, FDEM has a strict requirement on the element type, and the simulation result may be inconsistent due to a deficiency of physical meaning of the potential function. To address this problem, a new 3D discrete-element method based on a distance potential is proposed for a system consisting of a large number of arbitrary convex polyhedral elements. In this approach, a well-defined distance potential is proposed as a function of the penetration between the contact pairs. It exhibits a clear physical meaning and a precise measurement of the embedding between the elements in contact. The newly presented method provides a holonomic and accurate contact interaction without being influenced by the element shape. Therefore, the restraint of the element type in FDEM is removed and the proposed method can be used for arbitrary convex polyhedrons. In addition, an improved contact detection algorithm for non-uniform block discrete elements is provided to overcome the constraint of elements with the same size in the Munjiza-No Binary Search contact detection method. The new approach retains the merits of the FDEM and avoids its deficiencies. It is validated with well-known benchmark examples including an impact simulation, a friction experiment, a joint structure of a sliding rock mass, pillar impact, block accumulation, and analysis for the failure process of wedge slope. The results of this proposed method are in excellent agreement with the existing experimental measurements and analytical solutions.

## Keywords

Discrete-element modelling Arbitrary convex polyhedral element Distance potential function Tangential contact interaction Contact detection algorithm## Abbreviations

- CP
Common plane

- DEM
Discrete-element method

- DDA
Discontinuous deformation analysis

- FDEM
The combined finite–discrete-element method

- MMR
Multi-step Munjiza–Rougier algorithm

- NBS
No binary search

- NMM
Numerical manifold method

- LWSP
Left structural weak surfaces

- RWSP
Right structural weak surfaces

## List of symbols

- \(\varphi\)
Potential function

- \(k\)
Penalty parameter

- \(V\)
Volume

- \({{\varvec{f}}_{\text{n}}}\)
Normal contact force

- \({V_{{\text{t}} \cap {\text{c}}}}\)
Overlapping volume between the discrete elements \({\beta _{\text{t}}}\) and \({\beta _{\text{c}}}\)

- \({\varphi _{\text{c}}}\)
Potential function in \({V_{{\text{t}} \cap {\text{c}}}}\) belonging to the elements \({\beta _{\text{c}}}\)

- \({\varphi _{\text{t}}}\)
Potential function in \({V_{{\text{t}} \cap {\text{c}}}}\) belonging to the elements \({\beta _{\text{t}}}\)

- \({S_{{\text{t}} \cap {\text{c}}}}\)
Boundary surface of \({V_{{\text{t}} \cap {\text{c}}}}\)

- \({\varvec{n}}\)
Outward unit vector of the boundary surface \({S_{{\text{t}} \cap {\text{c}}}}\)

- \({\varphi _{\text{d}}}\)
Distance potential function

- \({h_{\rm I}}\)
Distance from the point

*p*to the base \({\alpha _{\rm I}}\) of the sub-polyhedron- \(r\)
Radius of the maximum inscribed sphere of a polyhedral element

- \(S\)
Intersection surface among the plane of the base of \({\beta _{\text{c}}}\) and the target sub-polyhedron of \({\beta _{\text{t}}}\)

- \({S_1},{S_2} \ldots ,{S_n}\)
Nodes of the intersection surface \(S\)

- \(B\)
Intersection polygon defined by the surface \(S\) and the base \(\alpha\) of \({\beta _{\text{c}}}\)

- \({B_1},{B_2} \ldots ,{B_n}\)
Nodes of the polygon \(B\)

- \({x_i},{y_i}\)
Local coordinates of the point on the polygonal surface \(B\)

- \({A_1},{A_2} \ldots ,{A_n}\)
Parameters of the formulation of distance potential function in local coordinate system

- \(({x_1},{y_1})\)
Local coordinates of \({B_1}\)

- \(({x_2},{y_2})\)
Local coordinates of \({B_2}\)

- \(({x_3},{y_3})\)
Local coordinates of \({B_3}\)

- \({{\varvec{f}}_{{\text{n}},B}}\)
Normal contact force over the polygonal surface \(B\)

- \({{\varvec{n}}_B}\)
Outward unit vector of the polygonal surface \(B\)

- \({k_{\text{n}}}\)
Normal contact stiffness

- \({{\varvec{M}}_{x,B}},{{\varvec{M}}_{y,B}}\)
Moments contributed by the contact normal force \({{\varvec{f}}_{{\text{n}},B}}\) in the local coordinate system \((x,y)\)

- \({N_i}(\eta ,\zeta )\)
Shape function

- \(m\)
Number of the divided triangular surfaces of the polygonal surface \(B\)

- \(\left| J \right|\)
Jacobi determinant of coordinate transformation

- \({C_1},{C_2} \ldots ,{C_n}\)
Parameters of the formulations of the normal contact force and moments in the natural coordinate system

- \({{\varvec{M}}_{\eta ,B}},{{\varvec{M}}_{\zeta ,B}}\)
Moments contributed by the contact normal force \({{\varvec{f}}_{{\text{n}},B}}\) in the natural coordinate system \((\eta ,\zeta )\)

- \({\eta _{\text{n}}},{\zeta _{\text{n}}}\)
Coordinates of the action position of the normal contact force in the natural coordinate system

- \({n_{\text{s}}}\)
Number of the boundary surfaces

- \({\varvec{f}}_{{\text{s}}}^{i}\)
Tangential contact force at step

*i*- \(\Delta \varvec{\delta}_{{{\text{s,t}}}}^{i}\)
Tangential increment displacement of each surface at step

*i*- \({k_{\text{s}}}\)
Tangential contact stiffness

- \({\varvec{v}^i}\)
Relative velocity of the contact element \({\beta _{\text{c}}}\) with respect to the target element \({\beta _{\text{t}}}\)

- \(\varvec{v}_{c}^{i}\)
Translational velocity of block \({\beta _{\text{c}}}\) at step

*i*- \(\varvec{v}_{t}^{i}\)
Translational velocity of block \({\beta _{\text{t}}}\) at step

*i*- \(\varvec{\omega}_{{\text{c}}}^{i}\)
Angular velocity of block \({\beta _{\text{c}}}\) at step

*i*- \(\varvec{\omega}_{{\text{t}}}^{i}\)
Angular velocity of block \({\beta _{\text{t}}}\) at step

*i*- \(\Delta \varvec{\delta}_{{\text{s}}}^{i}\)
Incremental tangential displacement at step

*i*- \(\Delta {{\varvec{s}}^i}\)
Incremental displacement between \({\beta _{\text{c}}}\) and \({\beta _{\text{t}}}\)

- \({{\varvec{n}}_{\text{n}}}\)
Unit direction vector of total normal contact force

- \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{{\varvec{r}}}\)
Rotation matrix that rotates the normal vector from step

*i*− 1 to the normal vector at the current step*i*- \(\Delta {\varvec{f}}_{{\text{s}}}^{i}\)
Incremental tangential contact force

- \({\varphi _\mu }\)
Maximum static friction angle

- \(C\)
Cohesion force

- \({({f_{\text{s}}})_{\rm{max} }}\)
Maximum possible value of the magnitude of the tangential force

- \({\varvec{M}}_{{\text{s}}}^{i}\)
Tangential contact moment at step

*i*- \({\varvec{\gamma}_{\text{s}}}\)
Vector from the force \({\varvec{f}}_{{\text{s}}}^{i}\) load position to the centroid of the element

- \({d_i}\)
Maximum circumradius for each group block

- \(D\)
Maximum circumradius among all the blocks

- \({x_{{\text{cent}}}},{y_{{\text{cent}}}},{z_{{\text{cent}}}}\)
Current coordinates of the centroids in the global coordinate system

- \({x_k},{y_k},{z_k}\)
Coordinates of the centroids when the elements are mapped into the cell

- \(l\)
Length of the cell

- \({l_{\text{c}}}\)
Distance between the centroids of the contact pairs

- \({l_{\rm{max} }}\)
Contact distance

- \({l_{1\rm{max} }},{l_{2\rm{max} }}\)
The longest distances between the vertexes and centroids of the two contact elements

- \({{\varvec{u}}_0}\)
Initial velocity of the sliding block

- \({{\varvec{s}}_0}\)
Initial displacement of the sliding block

- \(g\)
Gravity acceleration

- \(\theta\)
Angle of the inclined plane

- \(\mu\)
Friction coefficient

- \(t\)
Time

- \({V_{\text{w}}}\)
Wave velocity

- \(E\)
Elastic modulus

- \(\rho\)
Density of the block

- \({\varvec{F}}\)
Normal contact force between the blocks in impact simulation of a pillar

- \(\Delta \varvec{\delta}\)
Relative displacement between the contact blocks in impact simulation of a pillar

- \(\xi\)
Damping radio

## Notes

### Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant no. 51279050), the 15th Fok Ying-Tong Education Foundation for Young Teachers in the Higher Education Institutions of China (Grant no. 151073), the National Key R&D Program of China (Grant no. 2016YFC0401601), the project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (Grant YS11001), and the 111 Project.

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