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2024 | Book

Computational Mechanics of Arbitrarily Shaped Granular Materials

Authors: Siqiang Wang, Shunying Ji

Publisher: Springer Nature Singapore

Book Series : Springer Tracts in Mechanical Engineering


About this book

This book focuses on discrete element methods for arbitrarily shaped granular materials, including super-quadric models, spherical harmonic functions and level set methods, and numerical analysis of the flow characteristics of non-spherical granular materials. This book is used as a reference book for scientific researchers engaged in dynamic analysis of granular materials and optimal design of equipment structures in the fields of engineering mechanics, applied physics, mechanical engineering, and chemical engineering, as well as for graduate students or senior undergraduates of related majors in institutions of higher education.

Table of Contents

Chapter 1. Introduction
Granular materials are widely found in nature or industrial production, and are complex systems composed of a large number of discrete solid particles (Kou et al. in Nature 551:360–363, 2017; Clerc et al. in Nat Phys 4:249–254, 2008). Granular materials have the special mechanical properties of solids or fluids, and solid–liquid-like transformation phenomena occur under certain conditions. The energy of granular systems can be dissipated rapidly by the friction and damping between particles, and the contact force and direction can be adjusted by the particle rearrangement. The local load can be extended in space and then form a stable granular system.
Siqiang Wang, Shunying Ji
Chapter 2. Superquadric DEM Model Based on Functional Representations
The discrete element method (DEM), proposed by Cundall and Strack, has been shown to be a practical approach to studying various granular materials (Cundall and Strack in Géotechnique 29:47–65, 1979). In this approach, two-dimensional disks or three-dimensional spheres were initially employed because they enable simple calculation and efficient operation (Zhu et al. in Chem Eng Sci 62:3378–3396, 2007; Zhu et al. in Chem Eng Sci 63:5728–5770, 2008). However, the granular systems commonly encountered in industry or nature comprise non-spherical grains.
Siqiang Wang, Shunying Ji
Chapter 3. Multi-superquadric and Poly-superquadric DEM Models
The superquadric equation is a general method for describing spherical and irregular particles in a mathematical sense, and this equation can be used to construct particle morphologies with different surface sharpness and aspect ratio. However, the particles constructed by the superquadric equation are geometrically symmetric and strictly convex, which is different from the actual particle shape and limits the further engineering application of the superquadric model. In recent years, the non-spherical discrete element method based on the combined particle method has been widely developed and applied. Tetrahedral particles are formed by combining several spheres, and local cluster structures are randomly distributed in granular materials (Zhao et al. in Soft Matter 15:2260–2268, 2019). Considering the geometric symmetry of the ellipsoid element, one-eighth of the eight ellipsoids are spliced together to form a poly-ellipsoid with a smooth surface.
Siqiang Wang, Shunying Ji
Chapter 4. Smoothed Polyhedral DEM Model Based on Minkowski Sum Algorithm
In recent years, an increasing number of numerical studies have focused on the actual morphology of particles (Nie et al. in Int J Solids Struct 202:1–11, 2020), and DEMs have been developed for arbitrarily shaped particles (Jha et al. in J Mech Phys Solids 151, 2021; Gui et al. in Appl Math Model 40:2485–2499, 2016; Kafashan et al. in Granular Matter 21:1–19, 2019). Ellipsoidal and super-ellipsoidal models are mathematical formulations for constructing non-spherical particles (Zhao et al. in Int J Solids Struct 150:268–281, 2018), and ellipsoidal, cylindrical, and cubic particles with various aspect ratios and sharp surfaces are obtained by varying the function parameters (Kildashti et al. in Comput Methods Appl Mech Eng 360, 2020).
Siqiang Wang, Shunying Ji
Chapter 5. Arbitrarily Shaped DEM Model Based on Level Set Method
The level set method, first introduced by Osher and Sethian (J Comput Phys 78:12–49, 1988), is a common and efficient method for calculating the motion of an interface and tracking its evolution (Osher and Sethian in J Comput Phys 78:12–49, 1988; Caselles et al. in Numer Math 66:1–31, 1993; Osher and Fedkiw in J Comput Phys 169:463–502, 2001; Sethian in J Comput Phys 169:503–555, 2001). Because these interfaces easily form sharp corners, internal cracks, and merge together in a robust way (Sukumar et al. in Comput Methods Appl Mech Eng 190:6183–6200, 2001; Wang et al. in J Comput Phys 221:395–421, 2007; Tran et al. in Int J Numer Methods Eng 85:1436–1459, 2011), the level set method has a wide range of applications, including solid modeling, crack characterization, image processing, and segmentation (Hettich and Ramm in Comp Methods Appl Mech Eng 195:4753–4767, 2006; Hettich et al. in Comput Methods Appl Mech Eng 197:414–424, 2008; Legrain et al. in Int J Numer Methods Eng 86:915–934, 2011). For the complex morphology of non-spherical granular materials, XRCT is used to obtain the image data of arbitrarily shaped particles, and the geometrical morphology of particles is mathematically characterized by the level set method according to the gradient of the X-ray attenuation (Vlahini et al. in Granular Matter 16:9–21, 2014; Macedo et al. in Granular Matter 20:73, 2018). A smooth distance function between two particles is established by the level set method, and the translation and rotation of the particles and the next collision between particles are accurately predicted (Stafford and Jackson in J Comput Phys 229:3295–3315, 2010; Vlahinic et al. in Acta Geotech 12:85–95, 2017). Recently, the level set method combined with the discrete element method can be used to reasonably simulate the motion behaviors of non-spherical granular materials (Tahmasebi in Comput Geotech 100:52–61, 2018; Duriez and Bonelli in Comput Geotech 134:104033, 2021; Duriez and Galusinski in Comput Geosci 157:104936, 2021). The shape parameters of non-spherical particles such as aspect ratio, circularity, and main geometric direction are extracted by the level set method and then input into the DEM simulations (Jerves et al. in Acta Geotech 11:493–503, 2016, Granular Matter 19:30, 2017; Harmon et al. in Comput Methods Appl Mech Eng 365:112961, 2020). The combination of level set methods and discrete element methods is used not only to capture mechanical behaviors as macroscopic scales (such as stress–strain and volume-strain results of the triaxial test), but also to reproduce shear bands in a similar way to experiments, as well as local and particle-scale quantities (such as local deviatoric stress and particle rotation) (Kawamoto et al. in J Mech Phys Solids 91:1–13, 2016; Lim et al. in Acta Geotech 11:243–253, 2016; Kawamoto et al. in J Mech Phys Solids 111:375–392, 2018). Meanwhile, the combination of these two methods has the ability to reflect the contact force distribution within granular systems composed of arbitrarily shaped particles and provides quantitative estimates of the evolution of force chains and fabric orientations (Li et al. in Granular Matter 21:43, 2019; Bhattacharya et al. in Acta Geotech 16:113–132, 2021; Harmon et al. in Comput Methods Appl Mech Eng 373:113486, 2021).
Siqiang Wang, Shunying Ji
Chapter 6. High Performance Computation and DEM Software Development
The high-performance parallel computing approaches mainly include the open multi-processing method (OpenMP) (Amritkar et al. in J Comput Phys 256:501–519, 2014), the message passing interface (MPI) (Kačianauskas et al. in Adv Eng Softw 41:52–63, 2010), and the MPI-OpenMP algorithm (Berger et al. in Powder Technol 278:234–247, 2015). The above three methods use the CPU as the computing core, and the cores cooperate with each other to improve the calculation efficiency (Kačeniauskas et al. in Adv Eng Softw 42:237–246, 2011). In addition, the parallel algorithms based on a graphics processing unit (GPU) have been successfully used for large-scale computations in the discrete element method (Zhang et al. in Adv Eng Softw 60–61:70–80, 2013). Compared to a CPU, the number of computing cores in a GPU is significantly increased, and the hardware costs of the computer are reduced, which makes the GPU have more powerful parallel computing capabilities (Xu et al. in Particuology 9:446–450, 2011). The GPU parallel computing method for spheres was applied to simulate large-scale granular systems with a wide range of sizes (He et al. in Powder Technol 333:219–228, 2018). Another GPU parallel algorithm for the multi-sphere model was used to analyze the effects of the process parameters on large-scale tablet coating processes (Boehling et al. in Eur J Pharm Sci off J Eur Feder Pharm Sci 90:14–24, 2016). Moreover, considering the computational requirements of non-spherical DEM simulations, the ellipsoidal GPU parallel approach was developed and used to compare the speedup ratio of a GPU to a CPU (Gan et al. in Powder Technol 301:1172–1182, 2016). The polyhedral GPU parallel method was validated and successfully applied to the hopper discharge (Govender et al. in J Comput Appl Math 270:386–400, 2014, in Appl Math Comput 319:318–336, 2018). The coating process of 20 million biconvex tablets in a rotating drum was simulated by a GPU parallel algorithm (Kureck et al. in Chem Eng Sci 202:462–480, 2019). However, thus far, few efforts have been made to contribute to the detailed description of GPU parallel algorithms for superquadric elements.
Siqiang Wang, Shunying Ji
Chapter 7. DEM Analysis of Flow Characteristics of Non-spherical Particles
The flow characteristics of granular materials have attracted extensive attention, and the research focus has gradually expanded from spherical granular materials to irregular granular materials. Zeng et al. (Powder Technol 313:112–121, 2017) used the combined sphere method to construct rice-shaped particles, and also analyzed the pulsation behavior of granular materials during the flow process through the total contact force and velocity fluctuation. Govender et al. (Appl Math Comput 319:318–336, 2018) compared the flow process of polyhedral and spherical particles in hoppers, and found that the polyhedral shapes significantly affected the flow rate of granular materials. Besides, granular materials exhibit more unique mechanical behaviors under external driving force. Liu and Nagel (Nature 396:21–22, 1998) firstly proposed the phase transition mode of granular materials, which is achieved by applying external stresses and then transforming the granular materials from blockage to flow. Meanwhile, the external driving force changes the structural form of the granular material, leading to more strain intensification and expansion effects. In localized regions, the granular materials exhibit fluid-like properties and incomplete Taylor vortices (Huang et al. in Acta Geotech 10:389–397, 2014). In addition, numerical simulation of the mixing process of granular materials in a rotating drum is another important research to analyze the granular flow, and the mechanistic study of its mixing and segregation characteristics is beneficial for the application of granular materials in industrial production. Particle friction and rotational speed significantly affect the flow characteristics of particles inside the rotating drum, and the dynamic angle of repose of the granular material increased with the increase of friction coefficient and rotational speed (Chou et al. in Adv Powder Technol 27:1912–1921, 2016). The mixing behavior of polyhedral particles inside a two-dimensional rotating drum was simulated by the SIPHPM method, and the results showed that triangular, quadrilateral, and hexagonal particles had higher mixing degrees compared to spherical particles (Gui et al. in Powder Technol 314:140–147, 2017). Superquadric equations were used to construct ellipsoidal particles for investigating the effects of different aspect ratios on the mixing process of non-regular granular materials (You and Zhao in Powder Technol 331:179–191, 2018). The results showed that the main axis direction of particles with high aspect ratios was basically parallel to the flow direction of the particles.
Siqiang Wang, Shunying Ji
Computational Mechanics of Arbitrarily Shaped Granular Materials
Siqiang Wang
Shunying Ji
Copyright Year
Springer Nature Singapore
Electronic ISBN
Print ISBN

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