Long ellipsoid particle and wall surface collision efficient simulation method based on discrete elements

Abstract

The invention discloses an efficient simulation method for collision between long ellipsoid particles and a wall surface based on discrete elements, and belongs to the technical field of gas-solid two-phase flow numerical simulation. The method comprises the steps that a universal long ellipsoid particle-wall surface collision elastic force model is established, and the model represents a universal function relation between elastic force and particle geometry, material attributes and kinematics state parameters; based on the elastic force model and the momentum theorem, constructing a particle-wall surface collision dynamic model; based on the kinetic equation, a discrete time method and a numerical integration algorithm are adopted, and force, speed and energy parameters, changing along with time, of the long ellipsoid particles and the wall face in the whole collision period are solved. According to the method, the analytic model is embedded into a discrete element simulation framework to replace calculation-intensive finite element solution, so that second-level efficient solution is realized while the calculation precision is ensured, and the calculation efficiency is remarkably improved.

Description

Technical Field

[0001] This invention relates to the field of numerical simulation technology for gas-solid two-phase flow, and in particular to an efficient simulation method for collisions between long ellipsoidal particles and walls based on discrete element method. Background Technology

[0002] In gas-solid two-phase flow systems in industries such as energy, chemicals, and pharmaceuticals, particle-wall collisions are a key process determining the system's dynamic behavior, energy dissipation, and equipment wear. Accurate simulation of this process is crucial for optimizing the design of industrial equipment. The discrete element method (DEM) is the core numerical approach for studying such problems, and its accuracy is highly dependent on the particle-wall contact model.

[0003] Currently, commercial software generally lacks contact models for non-spherical particles, forcing industrial practice and research to approximate non-spherical particles with spherical models. However, actual industrial particles (such as pharmaceutical and agricultural particles) are mostly non-spherical, with elongated ellipsoids being particularly common. Their collision dynamics differ significantly from those of spheres, leading to substantial deviations in the application of spherical models.

[0004] Specifically, the discrete element method faces two major challenges in accurately simulating the motion of particles on a long ellipsoid: First, there is a lack of a general elastic force model for long ellipsoids. Existing models are unable to accurately construct the constitutive relationship between their anisotropic geometry and contact forces, and the calculation accuracy of normal elastic forces is insufficient. Second, there is a lack of a collision dynamics model that can couple translation and rotation and is applicable to arbitrary incident conditions, which cannot realistically reproduce key behaviors such as velocity, angle and energy transfer after a collision.

[0005] In pursuit of accuracy, the finite element method (FEM) is often used as an alternative, but its computational cost is extremely high: a detailed simulation of a single collision can take several hours, and a large number of collisions in engineering will lead to a dramatic increase in computation time, which is completely unable to meet the efficiency requirements of large-scale design optimization. This contradiction between accuracy and efficiency severely restricts the industrial application of realistic particle shape simulation technology. Summary of the Invention

[0006] The technical problem solved by this invention is to overcome the shortcomings of existing technologies and provide an efficient simulation method for collisions between elongated ellipsoidal particles and walls based on discrete element methods. Addressing the issues of insufficient accuracy in existing discrete element simulations due to the lack of dedicated models and the forced use of spherical approximations, and the high accuracy but prohibitive computational cost of the finite element method, this invention proposes an analytical-numerical hybrid framework integrating a universal elastic force model and a complete collision dynamics model. This framework achieves high-precision and high-efficiency simulation of the collision process between elongated ellipsoidal particles and walls, providing a reliable tool for accurate prediction of the behavior of non-spherical particles and equipment optimization in industrial gas-solid systems.

[0007] To address the aforementioned technical problems, this invention discloses an efficient simulation method for collisions between elongated ellipsoidal particles and walls based on the discrete element method, comprising:

[0008] A universal elastic force model for particle-wall collision on a long ellipsoid is established, wherein the elastic force model characterizes the general functional relationship between elastic force and particle geometry, material properties and kinematic state parameters;

[0009] Based on the elastic force model and momentum theorem, a particle-wall collision dynamics model is constructed.

[0010] Based on the aforementioned dynamic equations, the discrete-time method and numerical integration algorithm are used to solve for the force, velocity, and energy parameters of the elongated ellipsoidal particle and the wall as a function of time throughout the entire collision cycle, providing a basis for the design of dynamic systems containing elongated ellipsoidal particles and the optimization of particle parameters.

[0011] Furthermore, the elastic force model for the collision between the elongated ellipsoidal particle and the wall is expressed as follows:

[0012]

[0013] Where F e E represents the elastic force experienced by a particle during the collision with a rigid wall. * R represents the equivalent Young's modulus, y represents the depth of compression, and R represents the equivalent Young's modulus. s ξ(λ,β) represents the equivalent sphere's volume radius; ξ(λ,β) represents the geometric parameters of the ellipsoidal particle's aspect ratio λ and incident angle β; h represents the maximum compression depth that the ellipsoidal particle can achieve during the collision. s This represents the maximum compression depth of the equivalent sphere.

[0014] Furthermore, the equivalent sphere's volume radius R s The aspect ratio of the ellipsoidal particles satisfies:

[0015]

[0016] Where a represents the major axis radius of the ellipsoidal particle, and b represents the minor axis radius of the ellipsoidal particle; the equivalent Young's modulus E * satisfy:

[0017]

[0018] Where E1 and ν1 are the elastic modulus and Poisson's ratio of the elongated ellipsoidal particles, and E2 and ν2 are the elastic modulus and Poisson's ratio of the wall, respectively.

[0019] Furthermore, the geometric shape parameters ξ(λ,β) satisfy a polynomial fitting relationship:

[0020] ξ(λ,β)=k0+k1β+k2β 2 +k3β 3 +k4β 4 +k5β 5 +k6β6

[0021] Where k0, k1, ..., k6 are the fitting coefficients related to the aspect ratio λ. When λ = 1.5, 2, 3, the fitting coefficients are as follows:

[0022]

[0023] β is the incident angle of the ellipsoidal particle.

[0024] Furthermore, the maximum compression depth h of the equivalent sphere s as follows:

[0025]

[0026] Where m is the particle mass. The initial normal velocity before particle collision.

[0027] Furthermore, the maximum compression depth h of the ellipsoidal particle is as follows:

[0028]

[0029] Among them, F e,max This represents the maximum elastic force during the collision.

[0030] Furthermore, the maximum elastic force F during the collision process e,max as follows:

[0031]

[0032] Where η(λ,β) represents a dimensionless parameter related to the aspect ratio and the angle of incidence, and ρ represents the density of the particles.

[0033] Furthermore, the dimensionless parameter η(λ,β) satisfies a polynomial fitting relationship:

[0034] η(λ,β)=c0+c1β+c2β 2 +c3β 3 +c4β 4 +c5β 5 +c6β 6

[0035] Where c0, c1, ..., c6 are the fitting coefficients related to the aspect ratio λ. For example, when the aspect ratio λ is 1.5, 2, or 3, the coefficient matrix satisfies:

[0036]

[0037] Furthermore, the particle-wall collision dynamics model is expressed as follows:

[0038] The equation of translational dynamics is:

[0039]

[0040] The rotational dynamics equation is:

[0041]

[0042] Where m is the mass of the elongated ellipsoidal particle, U is the particle translational velocity vector (including normal and tangential components), t is the collision time, and F... e For the elastic force (normal action) as described in claim 2, F f ω is the frictional force (tangential force) between the particle and the wall, I is the moment of inertia of the particle, a and b are the radii of the major and minor axes of the ellipsoidal particle as described in claim 3, ω is the angular velocity vector of the particle, and r is the position vector from the contact point to the center of mass of the particle.

[0043] Compared with the prior art, the beneficial effects of the present invention are:

[0044] 1. This invention constructs a universal analytical model of elastic force between elongated ellipsoidal particles and their walls, systematically coupling particle aspect ratio, incident angle, material properties, and kinematic state (compression depth, initial velocity). By introducing an equivalent spherical volume radius and geometric parameters, this model overcomes the inherent dependence of traditional contact mechanics on the spherical assumption, fundamentally solving the simulation distortion problem caused by the forced use of spherical approximation due to the lack of a dedicated model, and achieving accurate characterization of the anisotropic contact behavior of elongated ellipsoidal particles.

[0045] 2. This invention establishes a collision dynamics model that couples translation and rotation. This model can simultaneously solve the evolution of linear momentum and angular momentum of particles, accurately capturing key physical phenomena such as rotational effects and tangential rebound. At the same time, based on the solution process of discrete time method and numerical integration algorithm, it successfully breaks through the bottleneck of high computational cost of high-precision finite element method. Under the premise of ensuring equivalent accuracy, it shortens the simulation time of a single collision from several hours to seconds or milliseconds, providing a practical and feasible technical path for large-scale, high-precision simulation of industrial particle systems. Attached Figure Description

[0046] Figure 1 This is an overall flowchart of the method of the present invention;

[0047] Figure 2 This is a schematic diagram of the collision between the elongated ellipsoidal particle and the wall in this invention;

[0048] Figure 3 This is the curve showing the variation of the geometric shape parameter ξ(λ,β) with the incident angle in this invention;

[0049] Figure 4Comparison of the dynamic response of the model of this invention and FEM;

[0050] Figure 5 This is a comparison diagram of the dynamic response of the model of this invention and the spherical equivalent model;

[0051] Figure 6 This is a comparison chart of the time consumption of the method of this invention and finite element simulation. Detailed Implementation

[0052] To make the objectives, technical solutions, and advantages of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings.

[0053] Reference Figure 1 In this embodiment, the efficient simulation method for collisions between long ellipsoidal particles and walls based on discrete element method includes the following steps:

[0054] Step 1: Establish a universal elastic force model for collisions between long ellipsoidal particles and walls.

[0055] This step forms the foundation for building a high-precision simulation method. Its goal is to establish an analytical model that accurately describes the normal contact force when a long ellipsoidal particle collides with a wall. The core of this model lies in overcoming the limitations of traditional spherical particle models by systematically coupling the geometry, material properties, and motion state of the long ellipsoidal particle. The specific implementation is as follows:

[0056] First, define the particle geometry and equivalent parameters, referring to... Figure 2 The diagram showing the collision between a long ellipsoidal particle and a wall illustrates how to calculate the aspect ratio λ = a / b and the equivalent sphere's volume radius based on the major axis radius 'a' and minor axis radius 'b' of the long ellipsoidal particle. Based on the material properties of the particles and the wall (elastic modulus (E1, E2) and Poisson's ratio (ν)) 1, ν2)) Calculate the equivalent Young's modulus

[0057] Then, the analytical expression for elastic force is constructed, and the elastic force model formula is expressed as follows:

[0058]

[0059] Where F e E represents the elastic force experienced by a particle during the collision with a rigid wall. * R represents the equivalent Young's modulus, y represents the depth of compression, and R represents the equivalent Young's modulus. s Let λ represent the equivalent sphere's volume radius; ξ(λ,β) represent the geometric parameters of the ellipsoidal particle's aspect ratio λ and incident angle β, such as... Figure 3 As shown; h represents the maximum compression depth that the ellipsoidal particle can reach during the collision process, h sThis represents the maximum compression depth of the equivalent sphere.

[0060] The geometric shape parameter ξ(λ,β) satisfies a polynomial fitting relationship:

[0061] ξ(λ,β)=k0+k1β+k2β 2 +k3β 3 +k4β 4 +k5β 5 +k6β 6

[0062] Where k0, k1, ..., k6 are the fitting coefficients related to the aspect ratio λ. When λ = 1.5, 2, 3, the fitting coefficients are as follows:

[0063]

[0064] β is the incident angle of the ellipsoidal particle.

[0065] The maximum compression depth h of the equivalent sphere s as follows:

[0066]

[0067] Where m is the particle mass. The initial normal velocity before particle collision.

[0068] The maximum compression depth h of the ellipsoidal particle is as follows:

[0069]

[0070] Among them, F e,max This represents the maximum elastic force during the collision.

[0071] Furthermore, the maximum elastic force F during the collision process e,max as follows:

[0072]

[0073] Where η(λ,β) represents a dimensionless parameter related to the aspect ratio and the angle of incidence, and ρ represents the density of the particles.

[0074] Furthermore, the dimensionless parameter η(λ,β) satisfies a polynomial fitting relationship:

[0075] η(λ,β)=c0+c1β+c2β 2 +c3β 3 +c4β 4 +c5β 5 +c6β 6

[0076] Where c0, c1, ..., c6 are the fitting coefficients related to the aspect ratio λ. When the aspect ratio λ is 1.5, 2, or 3, the coefficient matrix satisfies:

[0077]

[0078] Step 2: Establish a particle-wall collision dynamics model.

[0079] Based on the universal elastic force model established in step 1, and combined with the momentum theorem, a complete model describing the dynamic behavior of an elongated ellipsoidal particle throughout the entire collision cycle is constructed. The key feature of this model is that it simultaneously couples the translational and rotational motions of the particle, thereby accurately describing the dynamic phenomena unique to non-spherical particles, such as rotational effects and tangential rebound.

[0080] The particle-wall collision dynamics model is represented as follows:

[0081] The equation of translational dynamics is:

[0082]

[0083] The rotational dynamics equation is:

[0084]

[0085] Where m is the mass of the elongated ellipsoidal particle, U is the particle translational velocity vector (including normal and tangential components), t is the collision time, and F... e For elastic force (normal force), F f The frictional force between the particles and the wall (F) f =μF e μ is the coefficient of friction, and I is the moment of inertia of the particle. ω is the particle angular velocity vector, and r is the position vector from the contact point to the particle's center of mass.

[0086] Step 3: Solve the particle motion state using the discrete-time method and numerical integration algorithm.

[0087] Based on the elastic force model and particle-wall collision dynamics model established in steps 1 and 2, this step uses numerical calculation methods to efficiently and stably solve the coupled translational and rotational dynamics equations, thereby obtaining complete time history information of the particle motion state throughout the entire collision cycle.

[0088] The specific implementation of the solution process is as follows:

[0089] First, a solution framework is constructed by simultaneously establishing the translational and rotational equations. This is combined with the relationships between elastic force and compression depth, the dynamic relationship between the contact point normal velocity and compression depth, and the geometric mapping between the contact point velocity and the center point velocity, forming a closed system of ordinary differential equations. The goal of solving this system is to obtain the variation of the particle's velocity vector U(t), angular velocity vector ω(t), and compression depth y(t) with time t.

[0090] Next, a discrete-time method is used to discretize the continuous collision process in the time domain. A reasonable time step Δt is set (in this embodiment, it is 5e-). 13 (Note: Here, micron-sized ellipsoidal particles are used, so the time step needs to be very short.) The total collision time t is divided into n tiny time steps, i.e., t0, t1, ..., tn. n , where t k+1 =t k +Δt.

[0091] Then at each time step [t] k ,t k+1 Within the [section / area], the velocity and compression depth of the particles are calculated until the particles separate from the wall, at which point the collision event ends. Finally, the force, velocity, angular velocity, position, and energy parameters at all time steps are output, forming a complete collision dynamics response time history curve. The comprehensive performance of this invention is verified through system comparisons in the following three dimensions:

[0092] For accuracy verification, typical working conditions were adopted (particulate material: E = 63 GPa, ν = 0.24, ρ = 2500 kg / m³). 3 λ=1.5, β=30°, R s =1.225μm,U n (0) =0.1m / s; comparative analysis of rigid wall surfaces (e.g., wall surface with a speed of 0.1m / s). Figure 4 As shown, the present invention and the finite element method exhibit a high degree of consistency in key dynamic parameters (both normalized). The elastic force, compression depth, normal velocity, and angular velocity curves of the two methods have extremely high overlap, with a maximum relative error of less than 3%, which confirms that the model of the present invention has a computational accuracy comparable to that of the high-precision finite element method.

[0093] In terms of model necessity verification, the importance of establishing a dedicated non-spherical particle model is highlighted by comparing it with the spherical equivalent model. For example... Figure 5 As shown, under the same operating conditions (β=45°, U n (0)=1m / s, other parameters are the same as in the above case). The spherical model shows systematic deviations in angular velocity prediction and tangential rebound characteristics, especially in characterizing particle rotation effects with orders of magnitude errors. This fully demonstrates the inherent limitations of the traditional spherical assumption in dealing with the collision problem of long ellipsoidal particles.

[0094] In terms of efficiency verification, system testing across the entire incident angle range (β = 0°-90°) demonstrated the breakthrough in computational efficiency of this invention. For example... Figure 6 As shown, this method reduces the simulation time for a single collision from hours to milliseconds in the finite element method, achieving an efficiency improvement of 4-6 orders of magnitude, while maintaining good computational stability.

[0095] In this embodiment, the overall implementation process of the efficient simulation method for the motion of long ellipsoidal particles based on discrete element method is as follows: First, based on the geometric parameters (major axis radius a, minor axis radius b, aspect ratio λ) and material properties (elastic modulus E1, Poisson's ratio ν1, density ρ) of the long ellipsoidal particles, as well as the wall material properties (elastic modulus E2, Poisson's ratio ν2), the equivalent sphere volume radius R is calculated. s (The major axis radius a and minor axis radius b of the long ellipsoid are calculated according to the principle of volume equivalence) and the equivalent Young's modulus E * The elastic modulus and Poisson's ratio of the particles and the wall are calculated using the equivalent formulas of contact mechanics. This is combined with the given initial motion conditions (initial normal velocity U). n (0) Using the incident angle β, and substituting it into the elastic force model, calculate the normal elastic force F during the particle-wall collision process at different compression depths. e Secondly, the calculated normal elastic force F e The tangential frictional force F calculated according to the Coulomb friction model f The particles are substituted into the particle-wall collision dynamics model, including translational and rotational dynamics equations, forming a closed set of dynamic equations. Finally, based on the established complete dynamics model, the discrete-time method and numerical integration algorithm are used to iteratively solve the problem within a sufficiently small time step to obtain the transient changes in key parameters such as normal velocity, tangential velocity, angular velocity, compression depth, and collision force of the particles throughout the entire collision cycle.

[0096] This invention addresses two major bottlenecks in existing technologies: first, existing particle simulation software lacks a dedicated contact model for elongated ellipsoidal particles, forcing the use of spherical approximations that lead to accuracy distortion; second, while the finite element method achieves the required accuracy, its high computational cost makes it unsuitable for large-scale industrial simulations. This invention proposes an efficient simulation method for the motion of elongated ellipsoidal particles based on the discrete element method. The core innovation of this invention is the construction of a hybrid framework that integrates analytical modeling and numerical solution, combining a universal elastic force model with a complete collision dynamics model. Specifically, the breakthroughs are reflected in two aspects: first, by establishing an analytical elastic force model that systematically couples the particle's aspect ratio, incident angle, material properties, and motion state, it overcomes the limitations of the traditional "spherical assumption" in contact mechanics, accurately depicting the anisotropic contact behavior of elongated ellipsoidal particles; second, by establishing a translational-rotational coupled collision dynamics model, combined with the discrete-time method and numerical integration for efficient solution, while maintaining equivalent accuracy to the finite element method, it reduces the simulation time for a single collision from hours to milliseconds, achieving an order-of-magnitude improvement in computational efficiency. This invention provides a reliable technical tool with both high precision and high efficiency for predicting the movement of non-spherical particles and optimizing equipment in industrial gas-solid systems (such as pharmaceutical particle conveying, chemical raw material separation, and powder processing equipment).

[0097] Although the technical solutions of this invention have been described in detail through the foregoing embodiments, this does not constitute a limitation on the scope of protection of this invention. Any person skilled in the art, within the scope of the technical principles disclosed in this invention, may make reasonable modifications or equivalent substitutions to the technical features of the described embodiments, and all such modifications or substitutions will fall within the scope of protection of this invention.

Claims

1. A high-efficiency simulation method for collision between prolate spheroid particles and wall surface based on discrete elements, characterized in that, The method comprises: establishing a universal long-ellipsoid particle-wall collision elastic force model, wherein the elastic force model represents a general function relationship between elastic force and particle geometry, material properties and kinematic state parameters; based on the elastic force model and the momentum theorem, a particle-wall collision dynamics model is constructed; based on the dynamics equation, a discrete time method and a numerical integration algorithm are used to solve the force, velocity and energy parameters of the long-ellipsoid particle and the wall surface varying with time in the entire collision period, thereby providing a basis for the design of a dynamic system containing long-ellipsoid particles and the optimization of particle parameters.

2. The discrete element-based long prolate spheroid particle and wall collision efficient simulation method according to claim 1, characterized in that, The long-ellipsoid particle-wall collision elastic force model is represented as follows: where F e represents the elastic force experienced by the particle during the end of the collision process with the rigid wall, E * represents the equivalent Young's modulus, y represents the compression depth, R s represents the equivalent sphere volume radius; ξ(λ,β) represents a geometric shape parameter of the long-ellipsoid particle aspect ratio λ and the incident angle β; h represents the maximum compression depth that the ellipsoidal particle can reach during the collision, h s represents the maximum compression depth of the equivalent sphere.

3. The method according to claim 2, wherein, The equivalent sphere volume radius R s and the aspect ratio of the ellipsoidal particles satisfies: wherein a represents the long-axis radius of the ellipsoid particle, and b represents the short-axis radius of the ellipsoid particle; the equivalent Young's modulus E * satisfies: wherein E1 and v1 are the elastic modulus and Poisson's ratio of the long-ellipsoid particle, and E2 and v2 are the elastic modulus and Poisson's ratio of the wall surface, respectively.

4. The discrete element-based long prolate spheroid particle and wall collision efficient simulation method according to claim 2, characterized in that, The geometric shape parameter ξ(λ,β) satisfies a polynomial fitting relationship as follows: ξ(λ, β) = k0+ k1β+ k2β 2 + k3β 3 + k4β 4 + k5β 5 + k6β 6 ; wherein k0, k1, …, k6 are fitting coefficients related to the aspect ratio λ, and when λ = 1.5, 2 and 3, the fitting coefficients are respectively: β is the incident angle of the ellipsoid particle.

5. The method according to claim 2, wherein, The maximum compression depth h of the equivalent sphere s is represented as follows: where m is the mass of the particle, is the initial normal velocity of the particle before the collision.

6. The discrete element-based long prolate spheroid particle and wall collision efficient simulation method according to claim 2, characterized in that, The maximum compression depth h of the ellipsoid particle is represented as follows: where F e,max represents the maximum elastic force during the collision.

7. The discrete element-based long prolate spheroid particle and wall collision efficient simulation method according to claim 6, characterized in that, The maximum elastic force F during the collision of the ellipsoidal particles e,max is represented as follows: wherein η(λ,β) represents a dimensionless parameter related to the aspect ratio and the incident angle, and ρ represents the density of the particle.

8. The discrete element-based long prolate spheroid particle and wall collision efficient simulation method according to claim 7, characterized in that, The dimensionless parameter η(λ,β) satisfies a polynomial fitting relationship as follows: η(λ, β) = c0+ c1β+ c2β 2 + c3β 3 + c4β 4 + c5β 5 + c6β 6 ; wherein c0, c1, …, c6 are fitting coefficients related to the aspect ratio λ, and when the aspect ratio λ is 1.5, 2 and 3, the coefficient matrix satisfies: 9.The method according to claim 1, wherein, The particle-wall collision dynamics model is represented as follows: The translational dynamics equation is: The rotational dynamics equation is: where m is the mass of the prolate spheroid particle, U is the translational velocity vector of the particle, t is the collision time, F e is the elastic force, F f is the friction force between the particle and the wall, I is the moment of inertia of the particle, a and b are the radii of the long and short axes of the spheroid particle, ω is the angular velocity vector of the particle, and r is the position vector from the contact point to the center of mass of the particle.

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