Using surface asperities for efficient random particle overlap detection in the generation of randomly oriented and located particle arrangements
Graphical abstract
Introduction
The precursor to many numerical simulations is a virtual recreation or development of a statistically representative surrogate of the material or structure to be studied. Particles and particulate composite materials present challenges when depicted virtually due to the complexity of accurately representing the morphology of the particulate components. Particle shape, size gradation, and spatial distribution have been shown to influence rheology [1], [2], [3], [4], [5], [6], fracture behavior [7], [8], diffusivity [9], [10], [11], rate of reactions [12], [13], [14], and mechanical properties of composites [7], [15], [16], [17]. Thus an accurate representation of shape and distribution is of great interest.
Particle packing and spatial arrangement play important roles in many fields of research and industrial applications. Codes generating virtual representations of particles and composite materials have been increasingly used in simulating mechanical behavior [12], [13], [15], fluid dynamics [5], [18], and thermal conductivity [12], [14], [19], [20], [21]. The random packing and placement of regularly shaped particles such as spheres [20], [21], [22], [23], [24], [25], cubes [26], cylinders [5], [27], [28], and ellipsoids [29], [30], [31], [32] have been widely studied. For most practical virtual representations, portions of two particles cannot exist in the same space at the same time, so prevention of virtual particle-particle overlap is an important aspect of these codes. Particle-particle contact and collision has been extensively studied for the application of discrete element method (DEM) analysis, as DEM simulates the effects of contact-driven interparticle forces and external forces on the motion of collections of particles [33]. DEM random-shape particle representations typically utilize sphere clusters [34], [35] or superquadric functions to describe regular shapes such as ellipsoids, cubes, and cylinders [36], [37]. Detection of overlap between spheres is the most straightforward, warranting only a comparison of the distance between the centers to the sum of the radii. Furthermore, overlap between cubes can be determined by evaluating if any of the vertexes or line segments lie within another cube. In the case of cylinder-cylinder overlap detection, a variety of approaches are used - including projection on a line, separation axis tests, and three-dimensional parallel testing [38], [39], [40]. Contact between two ellipses can be determined by solving the quartic equation derived from combining their two equations. Minimization of the geometric potential of the two ellipses can be used to locate intersection points, as described in [41]. Similar contact and overlap detection algorithms for ellipsoids can be found in the literature [31], [42], [43], [44], [45].
In contrast to the assemblage of particles with simple shapes, when placing or packing irregularly shaped particles, detecting overlap becomes more challenging. Some applications combine regular shapes to form more complex particle morphologies - such as tablets [46], [47], clusters of spheres [34], [35], [36], [48], and sphere-cylinder combinations [38], [39], [46], [47]. Additionally, superquadratic equations can be used to represent more complex regular shapes, such as hyperboloids, that can be deformed by changing exponents in the equation of the surface [49]. Other methods of irregular particle shape representation include X-Ray Computed Tomography (XCT) scans [15], laser scans [50], pseudo potentials [51], random Voronoi polygons [52], [53], [54], and spherical harmonics [55], [56], [57]. A review of published literature has shown the main approach to overlap detection of particles is calculating the minimum distance between the surfaces of two particles [56], [58]. The minimum distance approach provides accurate detection of overlap, but is computationally expensive. For random particle shapes represented as a cluster of component spheres of differing sizes, each sphere composing the particle in question is checked for overlap with the spheres composing the nearby particles. The accuracy of this method is dependent on the conformity of the sphere cluster to the original particle shape [59], [60]. Another method is to represent each particle's surface as a collection of planes, where the space enclosed by the intersecting planes represents the particle. The overlap test then checks if all the points on other particles lie on opposing sides of each of these planes from the particle in question [61]. Voxel based approaches treat particles as 3D images, composed of cubic voxels. Overlap detection is fast, however there is a tradeoff between the resolution of the particle's surface characteristics and reducing the number of voxels used [62]. A mathematically driven approach can be used to determine exactly where the two particle surfaces intersect, but for star-shaped particles is computationally expensive [63], [64], [65]. In this paper, the focus is on determining if two particles overlap, not precisely where they overlap.
The objective of this research is to develop an efficient, computationally inexpensive method of determining if two virtual, random-shape particles, each represented by a series of spherical harmonics functions, overlap. A novel asperity-based overlap detection algorithm is presented in this paper. This approach improves upon existing particle overlap detection algorithms during random particle placement by identifying, a priori, the asperities on the surface of a particle, which allows for more rapid overlap detection while preserving complex particle surface geometries. The method presented in this paper can be applied to any particle system where local maxima can be identified on the particles. A few similar methods exists in the literature. The most similar method utilizes a ''contact candidate list' [66], the primary difference being that the candidate list consists of points identified during each particle-particle contact assessment, while the method presented in this paper uses an optimized database of asperities that are identified only once and remain constant. Another approach is the Orientation Discretization Database Solution (ODDS), where the a set of possible orientations between two particles and the associated minimum distance between the two centers to avoid contact between the two particles is calculated [67]. This method has been demonstrated to have high efficiency and effectiveness for packings of regular shaped particles, however building the database for a large set of highly irregular, complex particle shapes would be very computationally intensive, thus not well suited for this application.
Section snippets
Particle generation
Extensive work characterizing the shape of particles has been performed using X-ray Computed Tomography (XCT) [55]. Spherical harmonics is a mathematical technique that may be used to analyze and represent the shape of star-shaped particles derived from XCT data. A star-shaped particle is defined to be a shape where there exists at least one interior point (e.g. usually the center of mass), such that any point on the surface of the particle can be connected to this special point with a line
Asperity detection
The first and second spatial derivatives of the analytical function describing the surface, given in Eq. (2), are used to identify the local maxima in the particle's radii on the particle's surface, called asperities. These maxima can be detected by first determining the locations where the first derivatives with respect to φ and θ are equal to zero. For brevity, the parameter fnm,is used to represent the factorials common to all the derivative equations [55]. The first
Particle placement
To generate a virtual 3D microstructure, the particles are randomly placed into a cubic domain with an edge length at least 2.5 times the largest particle size, where particle size is determined by the width of the particle. The length (L), width (W), and thickness (T) of the particles are measured using ASTM D4791 [70]. Length is defined as the longest axis that is fully contained in the aggregate. The width is defined as the longest axis that is both perpendicular to the length axis and fully
Overlap detection
Detection of overlap between two virtual particles in this work involves a three-step approach. The overlap check function is divided into three filters. The first, coarse filter compares the distance between the centroid of the particle in question and the centroid of a previously placed particle to the sum of the two particles’ maximum radii. Next, the asperities on the surface of the two particles are assessed, which is the new methodology that comprises the novelty of this paper, followed
Particle shape analysis
Four particle size distributions were studied to determine the impact of the asperity check on computation time in a variety of packing scenarios. The particles are divided into four size groups based on their width. These gradations are seen in Fig. 8. Gradations A and B contain greater proportions of larger particles, whereas gradations C and D contain a finer mix of particles. Gradation curve C is based on the Fuller curve, adapted to consider only particles larger than 4.75 mm width [73].
To
Results & discussion
All microstructure generation was carried out using Matlab R2018a on the Texas A&M University High Performance Computing Center's supercomputer Ada, a hybrid cluster from IBM with Intel Ivy Bridge processors and a Mellanox FDR-10 Infiniband interconnect. The plotting code was carried out serially, placing one particle at a time, on a single node with 2500 MB of allotted memory. For each gradation, three microstructures were generated at three particle volume fractions: 20 %, 30 %, and 40 %.
Summary
As demonstrated, the asperity check substantially reduces the runtime to generate a microstructure of realistic volume fractions, averaging a runtime 10x faster than the control code for a 40% particle volume fraction. The particles used in this study were reconstructions of gravel, however this approach is useful for any type of particle described by any analytical function with a real second derivative or with otherwise identifiable surface asperities. The code used in this work places
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This research was supported by a Nuclear Energy University Partnership (NEUP) grant 16-10457 through US DOE, Office of Nuclear Energy, Light Water Reactor Sustainability (LWRS) Program. Additional support came from the U.S. Army Research and Development Center (ERDC) grant W912HZ19C0042. Portions of this research were conducted with high performance research computing resources provided by Texas A&M University (https://hprc.tamu.edu). Additionally this research was supported in part by the
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2023, Transportation GeotechnicsCitation Excerpt :These algorithms provide convenient and effective approaches to the contact detection of irregular particles in the application of DEM. The geometric shapes of irregular particles were represented by various approaches, such as the cubic function extended from the elliptic equation for egg-shaped particles [40], the Fourier series-based method for convex particles [41] and star-shaped concave particles [42], the inverse discrete Fourier transform for realistic stone particles with random shapes [43], the arc-based creation for arbitrary-shaped particles [44] and the spherical-harmonic-transform for irregular-shaped particles [45–48]. From the literatures mentioned above, three contact detection algorithms for irregular particles can be summarized roughly.
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