Application of parametric function in construction of particle shape and discrete element simulation

In previous studies, particles, such as ellipsoids and super-ellipsoids, are mainly described by implicit functions. However, parametric functions can define more surfaces that can be used to represent a wider variety of particle shapes. In this study, parametric functions are used to construct particles, and the algorithm to determine the multi-point contact between concave particles is also given. Furthermore, the geometric parameters at the contact point, for instance, normal vectors, curvatures, and overlapping, are formulated by the parameters in the parametric function. To verify the proposed method, we use the discrete element method to simulate the systems of torus-shaped particles nested within each other, and analyze the momentum and kinetic energy changes with time. The equilibrium state with approximately zero energy is obtained, which means that the algorithm for multi-point contact of concave particles is suitable and stable. In addition to the torus-shaped particle, another two concave particles defined by parametric function are modeled. The simulation results indicate that the method is universal. Any plane curve can be used to construct surfaces by using the method provided in this work, and computational efficiency of simulations of particles defined by parametric function is higher because calculations of the inverse Jacobian matrix in the Newton-Raphson method are unnecessary. The parametric function method extends the scope of previous studies on particle shape.