Geometric Potential Research Articles

A revisit of common normal method for discrete modelling of non-spherical particles

Discrete element method (DEM) is prominent for studying granular materials at particle scale. However, how to model non-spherical particles in DEM is still challenging. In light of the present contact detection algorithms in the literature, common normal (CN) and geometric potential (GP) are two methods used for particles with smooth surfaces. Yet it has been long believed that CN gives erroneous results while GP is more preferable since they were firstly proposed for ellipsoidal particles decades ago. A revisit of CN in this work identifies two problems in the original CN, and then a new CN is proposed which can overcome these problems. Based on the comparison to sub-particle scale finite element analyses, the new CN has been further shown to be able to predict the contact plane more accurately than the original CN and GP. Such an advantage is found for the modelling of ellipsoidal and superquadric particles. The study not only proposes an improved CN algorithm but also demonstrates that CN should receive more attentions in DEM, though GP is now much more widely used.

Evaluation of contact force models for discrete modelling of ellipsoidal particles

Discrete element method (DEM) has been widely used to study granular materials. However, how to model non-spherical particles is still challenging. Ellipsoidal particles are a typical kind of non-spherical particles in DEM simulations. There are three common methods to calculate the overlap and contact force between two ellipsoidal particles, namely, geometric potential (GP), common normal (CN) and overlap region (OR) methods. These methods are based on different physical concepts and hence will give different results. However, the comprehensive evaluation of these methods is still lacking, leaving DEM users no solid reference for selecting algorithms. In this paper, we conduct detailed comparisons on the penetration depth, contact plane and contact point predicted by the three methods. Particularly, using the orientation discretization method, the results are compared in all orientations quantitatively. It is shown that the difference between GP and CN is the largest whereas OR is always in between. The difference in contact point is relatively small when overlap ratio is small, whereas the difference in contact plane can always be large. Further, the results are directly compared to those obtained from sub-particle scale finite element analyses, which reveals that CN can always accurately predict contact plane and most times contact point, while GP are relatively better in predicting force magnitude. This study not only gives a more clear and comprehensive evaluation of different contact force models for ellipsoidal particles but also establishes an effective framework for comparing and verifying contact force models for general non-spherical particles.

Curvature-induced bound states and coherent electron transport on the surface of a truncated cone

We study the curvature-induced bound states and the coherent transport properties for a particle constrained to move on a truncated cone-like surface. With longitudinal hard wall boundary condition, the probability densities and spectra energy shifts are calculated, and are found to be obviously affected by the surface curvature. The bound-state energy levels and energy differences decrease as increasing the vertex angle or the ratio of axial length to bottom radius of the truncated cone. In a two-dimensional (2D) GaAs substrate with this geometric structure, an estimation of the ground-state energy shift of ballistic transport electrons induced by the geometric potential (GP) is addressed, which shows that the fraction of the ground-state energy shift resulting from the surface curvature is unnegligible under some region of geometric parameters. Furthermore, we model a truncated cone-like junction joining two cylinders with different radii, and investigate the effect of the GP on the transmission properties by numerically solving the open-boundary 2D Schrödinger equation with GP on the junction surface. It is shown that the oscillatory behavior of the transmission coefficient as a function of the injection energy is more pronounced when steeper GP wells appear at the two ends of the junction. Moreover, at specific injection energy, the transmission coefficient is oscillating with the ratio of the cylinder radii at incoming and outgoing sides.