2020 Volume 61 Issue 6 Pages 1158-1163
This paper describes mesh simplification of 3D-scanned objects, as used in discrete-element modelling (DEM). The mesh simplification algorithm parameter (Rp) is conventionally determined by trial-and-error based on the users’ judgment. We proposed a new criterion to quantify a balance of shape approximation accuracy and a quantity of meshes. A satisfactory Rp value was uniquely determined through minimization of the criterion. The proposed method was applied to mesh simplification in 3D-scanned waste electrical and electronic equipment, and successfully determined a satisfactory Rp value. To evaluate the determined Rp value, we performed a series of angle-of-repose DEM simulations using particles with different Rp values. The simulated repose angle converged to a constant value when the Rp was greater than a specific value. Its precise shape approximation reproduced the rotational resistance due to interparticle moment transmission that occurred upon contact of nonspherical particles. The Rp value was sufficient to obtain a constant value of repose angle, which indicated that the proposed method is valid for decision-making with respect to the selection of the Rp value used.
The discrete-element method (DEM)1) is a simulation method for frictional rigid particle systems, and is widely used in various industries. For example in material processing in mining engineering, DEM has been used to model angle of repose,2) particle comminution3,4) and flotation process.5) The mining industry conventionally targets natural ores. Given the rapid increase in the generation of waste electrical and electronic equipment (WEEE), WEEE has been widely recognized as a novel raw material because of its potential high resource availability and the importance of sustainability.6) Accordingly, DEM simulations of printed circuit boards7) and electronic cables8) have been conducted.
The raw material used in the processing of minerals, including natural ores and WEEE, is composed of a particle assembly whose particle characteristics such as three-dimensional (3D) shape, size distribution and metal content vary. In the present study, we focused on their 3D shape. Cundall and Strack1) used polygonal particles when they firstly proposed the DEM; thereafter, spheres (or circles) were conventionally used in numerous studies. However, particle shape modelling is absolutely necessary for quantitatively simulating the granular bulk behavior. Accordingly, DEM simulations using nonspherical particles were conducted in previous studies. One popular DEM approach is modelling a clump that approximates the 3D particle shape using multiple spheres connected in a rigid manner.9–15) The clump particle can be used in commonly used DEM software such as PFC and EDEM and has been used in previous DEM simulations of angle of repose,2,16) dynamic behavior of railway ballast17) and soil mechanical laboratory tests.18–20) Conventional DEM studies for mineral processing have been targeted at natural granular materials such as sand and rock fragments because their particle shapes form angular and equidimensional shape. Contrastingly, smartphone — a rapidly increasing WEEEs — form thin-rectangular shape. Such thin-shaped objects can hardly be found in the nature. The authors cannot find the recycling equipment for the thin-shaped object in the market, and hence aim to develop the new equipment suitable for WEEE recycling. The authors believe that DEM simulation using the nonspherical particles of WEEE is essential to create unprecedented equipment effectively and rapidly. Hence, the authors describe the DEM simulations using non-spherical particles obtained from 3D scan of WEEEs.
The sphere assembly in clump is appropriate to approximate the rounded shape. A lot of spheres should be used to fill such thin-shaped object. Computation time of DEM simulation depends on number of spheres (particles) used. In other words, DEM simulation using clump of WEEE requires huge computational load.
The other particle shape modelling in DEM is to use polyhedral particles.21–23) The polyhedral particle information obtained from 3D scanning can be directly used in DEM simulations.24,25) In the present study, we target shape modelling with polyhedral particles. Recent 3D high-resolution scans indicate that the 3D shape data of scanned objects include a huge number of meshes Np representing the object surface. We refer to the 3D shape data of the scanned objects as the ‘3D data’. The DEM simulations run directly using the 3D data require huge computation times; hence, the value of Np in 3D data should be reduced to increase computational efficiency.
Various mesh simplification algorithms (MSAs) have been proposed in the 3D computer graphics (CG) field and are available in many computer-aided design software packages. In the most of such MSAs, users can determine the value of Np by controlling the value of parameters that influences the reduction. A small Np value enhances the computational efficiency of DEM simulations; however, its shape can differ from indicated in the original 3D data. The parameter value of the mesh simplification is conventionally determined by a visual comparison of the shapes before and after a MSA has been applied. In the 3D CG, the priority is to display 3D animation smoothly compared with the precise shape approximation of an object. The shape approximation accuracy directly influences the granular bulk response obtained by the DEM simulation, including angle of repose.16) As far as we know, there is no method to quantitatively determine the parameter of mesh simplification.
In the present study, a new method to determine the parameter of mesh simplification is proposed. From the viewpoint of industrial importance, 3D shapes of WEEE were scanned, and the DEM simulation was applied. The parameter value of mesh simplification was systematically varied, and then a MSA was applied to the 3D data of scanned WEEE. The influence of the parameter of mesh simplification on the surface area and volume were evaluated quantitatively. We proposed a new criterion to quantify a balance between shape approximation accuracy and the value of Np and developed a method to determine the optimum value of the parameter of mesh simplification. Moreover, we conducted a series of DEM simulation of angle of repose and discussed the parameter value determined by the proposed method.
The material of interest was WEEE. We used nine samples of WEEE, consisting of three feature phones (FP), three digital cameras (DC) and three smartphones (SP). We refer to DC sample No. 1 as DC-1 in the present study.
A structured light scanning system (David SLS-3 stereo 3D scanner) was used to capture the 3D shape of the WEEE samples. The resolution of a mesh of the 3D data was as high as 0.05% of the scan size (up to 0.05 mm). Figure 1 shows a photo taken during a 3D scan. The structured light projects a striped pattern onto the object positioned on a rotating stage. A single scan provides the 3D surface profile on the basis of the change in the striped patterns in the absence and presence of the object. The 3D shape is obtained from 15 scans performed every 24° of rotations. This procedure cannot measure the bottom area of the object; hence, the WEEE was inclined by 90° and 180° and re-scanned. Note that the 3D scanner tends to produce erroneous surface profiles for black, shiny surfaces. To avoid such erroneous measurements, the object was covered with baby powder before scanning.
Photograph of the 3D scanning process of WEEE.
Figure 2 shows the measured 3D shape of the nine WEEE samples. The 3D-shape file format used in the present study was stereolithography (STL). Each STL file describes an assembly of triangular faces, i.e., meshes. The value of Np and the mesh resolution of the measured 3D shape of FP-1 are 762,008 and 0.27 mm, respectively, which indicates that the 3D scanner precisely captured the 3D shape of the WEEE samples.
The 3D data of the nine WEEE samples, as obtained by the 3D scanner.
The MSA used was quadratic edge collapse decimation (QECD) algorithm implemented in MeshLab, which is commonly used software for manipulating STL files. The user is required to set a parameter called the percentage reduction (Rp) to apply the QECD algorithm. The range of Rp values is 0 < Rp < 1. The Np is automatically reduced to the target value depending on the Rp; e.g., the Np becomes one-tenth of that of the original STL file when Rp = 0.1.
2.1.3 Influence of percentage reduction on the 3D shape propertiesFigures 3(a), (b), (c), (d), (e) and (f) show snapshots of the STL objects obtained from 3D scanning, i.e., the STL objects the QECD algorithm with Rp = 10−1, 10−2, 10−3, 10−4 and 10−5 was applied, respectively. The sharp corners and edges gradually round with decreasing Rp values. The overall shape for Rp < 10−4 is obviously different from that of the original 3D data; by contrast, for Rp > 10−3, a difference in shape is barely noticeable.
Snapshots of (a) the 3D shape of an FP measured by the 3D scanner and the 3D shapes after the QECD algorithm was applied for percentage reduction (Rp) of (b) 10−1, (c) 10−2, (d) 10−3, (e) 10−4 and (f) 10−5, respectively.
We varied the Rp value between 10−5 and 10−1 and generated STL files for the nine WEEE samples. The surface area S of the samples described by the STL files was obtained using the ‘Compute Geometric Measures’ function implemented in MeshLab. Figure 4(a) shows dependence of Rp on the surface error εs, which is determined as
| \begin{equation} \varepsilon_{\text{s}} = \frac{|S - S_{\text{m}}|}{S_{\text{m}}} \end{equation} | (1) |
Dependence of (a) the surface error εs and (b) the new criterion Ip on the percentage reduction Rp.
The value of εs monotonically decreases with increasing Rp, which implies that the Rp value used is based on the user’s intuitions. The origin of this concern is the error index is used to determine the Rp value. In the present study, we propose a new criterion, Ip, as follows:
| \begin{equation} I_{\text{p}} = \varepsilon_{\text{s}} + \frac{N_{\text{p}}^{\text{r}}}{N_{\text{p}}^{\text{m}}} \end{equation} | (2) |
Figure 4(b) shows the relationship between Rp and Ip. The value of Ip decreases for Rp > 10−3; thereafter, it increases. The values of Ip vary considerably for 10−5 < Rp < 5 × 10−3. Some WEEE samples, including the SPs, form simple convex shapes; such simple shapes can be approximated even though the number of meshes is small. In such cases, the value of Ip tends to be lower than in the case of concave, complex-shaped WEEE samples, including the DCs and FPs; consequently, the value of Ip fluctuates for 10−5 < Rp < 5 × 10−3. For Rp > 5 × 10−3, the fluctuation of Ip fades and the Ip values monotonically increase because the penalty becomes dominant compared with the reduction of εs due to the increasing number of meshes. In other words, the quantity of Np is too large for representing the 3D shapes of the WEEE samples. The values of εs decreases with increasing Rp, implying that users make a subjective, empirical judgement to determine the Rp value used. By contrast, the users can uniquely determine the Rp value by using the criterion Ip. The proposed method is not the only solution available for determining the Rp value; however, its quantitative capability, objectivity and automation potential of the proposed method are advantages over conventional trial-and-error processing.
2.3 Discrete-element modelling of angle of reposeHere, the commercial DEM software Rocky, which can directly import the user-defined polyhedral particles, was used. We used a contact model consisting of linear springs, dashpots and a frictional slider. The Young’s moduli of particles and boundary walls were 108 and 1011 N/m2, respectively. The Poisson’s ratio of both the particles and the boundary walls was 0.3. The damping coefficient of the dashpot correlates with the restitution coefficient of particle (eb), where, eb = 0.3. The particle-particle and particle-wall friction coefficients were both set to 0.5.
The model geometry was composed of a bottom disc and a hollow cylinder with diameters of 0.4 and 3 m, respectively. The 1197 particles, which were duplicated from the original set of the nine WEEE samples, were randomly generated from the top of the cylinder and packed under gravity for 6 s; thereafter, the particles were stabilized for 3 s. Then, the cylinder was displaced upward at a speed of 0.1 m/s for 11 s to form a steady-state repose angle. The repose angle — an inclination angle of the pile composed of the particle assembly — was calculated using a built-in macro-function implemented in the software Rocky. The initial configuration and orientation of particles may influence the repose angle; hence, ten simulations were conducted using different initial configurations and orientations of the particles. The preliminary simulation for Rp > 5 × 10−3 required long computation time; hence, the Rp value varied between 10−5 and 2 × 10−3.
Figure 5 shows how the Rp influences the repose angle. The data points and error bars in the figure represent the average and standard deviation of the ten simulations. The value of the repose angle increases with increasing Rp at relatively low Rp values. For Rp > 10−4, the repose angle converges to a constant value. The particle shape influences the repose angle: the repose angle of spherical particles is smaller than that of nonspherical particles.16)
Relationship between the repose angle and Rp.
Using the standard deviation, we calculated the coefficient of variation (CV) for the repose angle. Figure 5 shows influence of Rp on CV. The trend of CV value is decreasing, which means that the reproducibility of simulation enhances as Rp increases.
We also conducted angle of repose simulations using equivalent-volume diameter spheres of the WEEE samples. Snapshots at the final state of the simulations using the spheres and the particles with Rp = 10−5 and Rp = 10−3 are shown in Fig. 6(a), (b) and (c), respectively. The spherical particles could not form a repose angle because the particles on the bottom plate continued to displace with particle rotation.
Snapshots at the final state of (a) spheres, and particles with (b) Rp = 10−5 and (c) Rp = 10−3.
Figure 7 shows time evolutions of the average value of the particles’ absolute rotational velocities for Rp = 10−5, 10−4 and 10−3 and the equivalent-volume-diameter spheres. The plots in Fig. 7 are the average of the time evolutions of rotational velocity for the ten simulations. The constraint of rotations for spherical particles is due to friction forces at particle-particle or particle-wall contact points. By contrast, not only the contact frictions but also the moment transmissions contribute to the constraint of rotation of nonspherical particles. The strong rotational resistance of nonspherical particles causes them to have a relatively low rotational velocity compared with that of the spherical particles.
Time evolutions of particles’ average rotational velocities.
The nonspherical particles exhibit shape irregularities in various scales, such as overall shape irregularity, medium shape irregularities (e.g., edges and projections of the prticles) and the small shape irregularities (e.g., surface roughness). The difference in shape between spherical and nonspherical particles (Rp = 10−5, 10−4 and 10−3) is characterized by the overall shape irregularity, which is responsible for the substantial difference in rotational velocity between the spherical and nonspherical particles. Medium shape irregularities are found in certain components of the WEEE samples, such as the antenna unit of the FPs and the lens unit in the DCs; these irregularities contribute to the difference in shapes between the particles with Rp = 10−5, 10−4 and 10−3. The moment transmission due to the medium shape irregularity is added to the rotational resistance due to the overall shape irregularity. Such additional moment transmission is spontaneously reproduced for the high Rp value. The Rp of 10−5 is insufficient for 3D shape approximation of the WEEE samples because the rotational velocity of Rp = 10−5 particles is higher than those of the Rp = 10−4 and 10−3 particles. The plots of Rp = 10−3 and 10−4 hardly differ because the STL objects for Rp > 10−4 can approximate the medium shape irregularity of the WEEE samples. Accordingly, the additional moment transmission due to medium shape irregularities arises; as a result, the repose angles become a constant value for Rp > 10−5.
We used the εs in the proposed method to determine the Rp value to precisely approximate the 3D shape of the WEEE samples. The repose angle can be simulated by the additional moment transmission due to the shape approximation. The Rp value 10−3, as determined by the proposed method, is sufficiently high to model the repose angle. This result justifies the use of the Rp value determined by the proposed method in the DEM simulation.
The relationship between Rp and computation time is shown in Fig. 8. We used a computer with an NVIDIA GPU: Quadro GP100. The computation time in Fig. 8 is the time required to finish the angle of repose simulation using the GPU. The curve represents a fourth-order polynomial fitting, which means that the computation time increases with the order of O(Rp4). The computation time for Rp = 10−3 was successfully reduced to 58% of that for Rp = 2 × 10−3. A recent computational engineering design repeats a series of numerical process, i.e., the design change using CAD software and its performance evaluation by computer-aided engineering (CAE) analysis such as CFD26) and also granular dynamics (DEM).27,28) We refer to this computational design process as an optimum design. The CFD is based on partial differential equation (PDE); hence the results of simulations hardly vary. Even though the same initial and boundary conditions, the results of DEM simulations will vary due to various factors including initial particle configuration. To obtain statistical average response, we need to conduct multiple simulations under the same initial conditions. Accordingly, the computation time of the optimum design analysis using DEM tends to be longer than that using the PDE. The proposed method easily reduces the computation time of the DEM simulation without worsening of simulation accuracy.
Influence of Rp on the computation time.
We proposed a quantitative method to reduce the Np of 3D scanned objects. The proposed method was applied to determine the value of Rp in the QECD algorithm for WEEE samples; we obtained Rp = 10−3. To evaluate the determined Rp value, we conducted a series of angle of repose simulations using the polyhedral particles for a wide range of Rp (10−5 < Rp ≤ 2 × 10−3). The results show that the repose angle converged for Rp > 10−4. A particle with Rp > 10−4 can approximate not only the overall 3D shape but also medium shape irregularities such as angular projections and sharp edges. The interparticle moment transmissions due to the precise shape approximation enhanced the simulated repose angle for Rp > 10−4.
Here, we focused on the shape of housing body of WEEE. The housing body and component parts such as printed circuit boards, capacitor and battery forms rectangular or cubic shape. Development of an effective sorting for component parts is challenging and an important technical topic in WEEE recycling. The proposed method can contribute to efficient shape approximation of component parts in DEM simulation.
Moreover, the proposed method can be applied to natural granular materials used in conventional powder processing and mineral separation; the high applicability is an advantage of the proposed method. Moreover, the STL format is a standard input file format to generate the clump particle in the bubble-pack algorithm13) implemented in PFC3D, which is one of the commonly used DEM software. Large STL files require a high computational load in the clump generation. The proposed method can reduce the file size while maintaining the high accuracy of shape approximation, which is another advantage of the proposed method.
The optimum design for structural mechanics and fluid dynamics has been practically used in the industry; however, that for granular dynamics does not. A bottleneck to use of DEM in the optimum design is the long computation time. The proposed method can reduce the computation time while maintaining the high accuracy of DEM simulation, and therefore will be useful method for researchers/practitioners who are involved in computational design using DEM.
This paper is based on results obtained from a project commissioned by the New Energy and Industrial Technology Development Organization (NEDO).
