A combined finite‐discrete element method for simulating pharmaceutical powder tableting

The pharmaceutical powder and tableting process is simulated using a combined finite‐discrete element method and contact dynamics for irregular‐shaped particles. The particle‐scale formulation and two‐stage contact detection algorithm which has been developed for the proposed method enhances the overall calculation efficiency for particle interaction characteristics. The irregular particle shapes and random sizes are represented as a pseudo‐particle assembly having a scaled up geometry but based on the variations of real powder particles. Our simulations show that particle size, shapes and material properties have a significant influence on the behaviour of compaction and deformation. Copyright © 2005 John Wiley & Sons, Ltd.     

A rapid path‐length searching procedure for multi‐axial fatigue cycle counting

In this paper, a series of advanced searching algorithms have been examined and implemented for accelerating multi‐axial fatigue cycle counting efforts when dealing with large time histories. In a computerized calculation of the path‐length dependent cycle counting method, most of the central processor unit's (CPU) time is spent on searching for the maximum range or distance in a stress or strain space. A brute‐force search is the simplest to implement, and will always find a solution if it exists. However, its cost, in many practical problems, tends to grow exponentially as the size of the loading spectrum increases with a search time measured in the order of O(n²), where n is the number of spectrum data points. In contrast, a form of Andrew's monotone chain algorithm, as demonstrated in this paper, can remarkably reduce the solution time to the order of O(n log n). The effectiveness of the new path‐length searching procedure is demonstrated by a series of worked examples with a varying degree of non‐proportionality in multi‐axial loading history.     

Parallel load‐balanced simulation for short‐range interaction particle methods with hierarchical particle grouping based on orthogonal recursive bisection

We describe an efficient load‐balancing algorithm for parallel simulations of particle‐based discretization methods such as the discrete element method or smoothed particle hydrodynamics. Our approach is based on an orthogonal recursive bisection of the simulation domain that is the basis for recursive particle grouping and assignment of particle groups to the parallel processors. Particle grouping is carried out based on sampled discrete particle distribution functions. For interaction detection and computation, which is the core part of particle simulations, we employ a hierarchical pruning algorithm for an efficient exclusion of non‐interacting particles via the detection of non‐overlapping bounding boxes. Load balancing is based on a hierarchical PI‐controller approach, where the differences of processor per time step waiting times serve as controller input. Copyright © 2007 John Wiley & Sons, Ltd.     

Polygon‐based contact resolution for superquadrics

The representation of discrete objects in the discrete element modelling is a fundamental issue, which has a direct impact on the efficiency of discrete element implementation and the dynamic behaviour of particulate systems. Disks and spheres are the most commonly used geometric shapes due to their geometric simplicity and computational efficiency, but they are unable to provide resistance to rolling motion. For this reason, some non‐circular/spherical objects, such as polygons/polyhedrons, superquadrics, or the clustering of disks/spheres to form irregular shapes, are introduced. When superquadrics are used as discrete elements, the bottleneck of contact resolution is associated with the searching for intersections of two non‐linear functions, which is a very expensive operation and may sometimes fail in finding the solution. In this work, an efficient and robust algorithm is proposed for contact resolution of 2D superquadrics, in which any superquadric is approximated with a convex polygon through adaptive sampling; then by clipping two polygons, an efficient linear algorithm is performed to search for intersections and overlap area of the polygons; the contact forces and directions are determined by employing a newly established corner/corner contact model. It is important to highlight that the proposed methodology can also be extended to general non‐circular discrete object cases. The performance of the algorithm is demonstrated via numerical examples. Copyright © 2005 John Wiley & Sons, Ltd.     

Analysis of two‐grid methods for reaction‐diffusion equations by expanded mixed finite element methods

We present two efficient methods of two‐grid scheme for the approximation of two‐dimensional semi‐linear reaction‐diffusion equations using an expanded mixed finite element method. To linearize the discretized equations, we use two Newton iterations on the fine grid in our methods. Firstly, we solve an original non‐linear problem on the coarse grid. Then we use twice Newton iterations on the fine grid in our first method, and while in second method we make a correction on the coarse grid between two Newton iterations on the fine grid. These two‐grid ideas are from Xu's work (SIAM J. Sci. Comput. 1994; 15 :231–237; SIAM J. Numer. Anal. 1996; 33 :1759–1777) on standard finite element method. We extend the ideas to the mixed finite element method. Moreover, we obtain the error estimates for two algorithms of two‐grid method. It is showed that coarse space can be extremely coarse and we achieve asymptotically optimal approximation as long as the mesh sizes satisfy H =??(h^¼) in the first algorithm and H =??(h^?) in second algorithm. Copyright © 2006 John Wiley & Sons, Ltd.     

A neural network graph partitioning procedure for grid‐based domain decomposition

This paper describes a neural network graph partitioning algorithm which partitions unstructured finite element/volume meshes as a precursor to a parallel domain decomposition solution method. The algorithm works by first constructing a coarse graph approximation using an automatic graph coarsening method. The coarse graph is partitioned and the results are interpolated onto the original graph to initialize an optimization of the graph partition problem. In practice, a hierarchy of (usually more than two) graphs are used to help obtain the final graph partition. A mean field theorem neural network is used to perform all partition optimization. The partitioning method is applied to graphs derived from unstructured finite element meshes and in this context it can be viewed as a multi‐grid partitioning method. Copyright © 1999 John Wiley & Sons, Ltd.     

An accurate and robust contact detection algorithm for particle‐solid interaction in combined finite‐discrete element analysis

When applying the combined finite‐discrete element method for analysis of dynamic problems, contact is often encountered between the finite elements and discrete elements, and thus an effective contact treatment is essential. In this paper, an accurate and robust contact detection algorithm is proposed to resolve contact problems between spherical particles, which represent rigid discrete elements, and convex quadrilateral mesh facets, which represent finite element boundaries of structural components. Different contact scenarios between particles and mesh facets, or edges, or vertices have been taken into account. For each potential contact pair, the contact search is performed in an hierarchical way starting from mesh facets, possibly going to edges and even further to vertices. The invalid contact pairs can be removed by means of two reasonable priorities defined in terms of geometric primitives and facet identifications. This hierarchical contact searching scheme is effective, and its implementation is straightforward. Numerical examples demonstrated the accuracy and robustness of the proposed algorithm. Copyright © 2015 John Wiley & Sons, Ltd.     

A two‐grid method for expanded mixed finite‐element solution of semilinear reaction–diffusion equations

We present a scheme for solving two‐dimensional semilinear reaction–diffusion equations using an expanded mixed finite element method. To linearize the mixed‐method equations, we use a two‐grid algorithm based on the Newton iteration method. The solution of a non‐linear system on the fine space is reduced to the solution of two small (one linear and one non‐linear) systems on the coarse space and a linear system on the fine space. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h^1/3). As a result, solving such a large class of non‐linear equation will not be much more difficult than solving one single linearized equation. Copyright © 2003 John Wiley & Sons, Ltd.     

Algebraic grid generation on trimmed parametric surface using non‐self‐overlapping planar Coons patch

Using a Coons patch mapping to generate a structured grid in the parametric region of a trimmed surface can avoid the singularity of elliptic PDE methods when only C¹ continuous boundary is given; the error of converting generic parametric C¹ boundary curves into a specified representation form is also avoided. However, overlap may happen on some portions of the algebraically generated grid when a linear or naïve cubic blending function is used in the mapping; this severely limits its usage in most of engineering and scientific applications where a grid system of non‐self‐overlapping is strictly required. To solve the problem, non‐trivial blending functions in a Coons patch mapping should be determined adaptively by the given boundary so that self‐overlapping can be averted. We address the adaptive determination problem by a functional optimization method. The governing equation of the optimization is derived by adding a virtual dimension in the parametric space of the given trimmed surface. Both one‐ and two‐parameter blending functions are studied. To resolve the difficulty of guessing good initial blending functions for the conjugate gradient method used, a progressive optimization algorithm is then proposed which has been shown to be very effective in a variety of practical examples. Also, an extension is added to the objective function to control the element shape. Finally, experiment results are shown to illustrate the usefulness and effectiveness of the presented method. Copyright © 2004 John Wiley & Sons, Ltd.     

Effect of particle size ratio and contribution of particle size fractions on micromechanics of uniaxially compressed binary sphere mixtures

This paper is an extension of the recent work of Wi?cek (Granul Matter 18:42, 2016), wherein geometrical parameters of binary granular mixtures with various particle size ratio and contribution of the particle size fractions were investigated. In this study, a micromechanics of binary mixtures with various ratio of the diameter of small and large spheres and contribution of small particles was analyzed using discrete element simulations of confined uniaxial compression. The study addressed contact normal orientation distributions, global and partial contact force distributions and pressure distribution in packings of frictional spheres. Additionally, the effect of particle size ratio and contribution of particle size fractions on energy dissipation in granular mixtures was investigated. The particle size ratio in binary packings was chosen to prevent small particles from percolating through bedding. The bimodality of mixtures was found to have a strong effect on distribution of contact normal orientation and distribution of normal contact forces in binary mixtures. Stress transfer in binary packing was also determined by both, particle size ratio and volume fraction of small particles. Dissipation of energy was higher in mixtures with higher particle size ratios and decreased with increasing contribution of small spheres in system.     

A contact searching algorithm including bounding volume trees applied to finite sliding mortar formulations

This paper presents a new contact searching algorithm for large deformation mortar-based contact formulations. In this algorithm, a bounding volume hierarchy, defined in the context of a binary tree, is built for each contact surface based on the geometry of the surface. A global contact searching procedure based on these bounding volume trees is first performed to find all candidate contact element pairs, and then a local searching procedure is done to find all the mortar segments having contributions to the mortar integrals that define the contact formulation. The searching algorithm is shown to be very efficient and readily applicable to a variety of large sliding contact problems.     

Development of contact algorithm for three‐dimensional numerical manifold method

This paper customizes a contact detection and enforcing scheme to fit the three‐dimensional (3‐D) numerical manifold method (NMM). A hierarchical contact system is established for efficient contact detection. The mathematical mesh, a unique component in the NMM, is utilized for global searching of possible contact blocks and elements, followed by the local searching to identify primitive hierarchies. All the potential contact pairs are then transformed into one of the two essential entrance modes: point‐to‐plane and crossing‐lines modes, among which real contact pairs are detected through a unified formula. The penalty method is selected to enforce the contact constraints, and a general contact solution procedure in the 3‐D NMM is established. Because of the implicit framework, an open‐close iteration is performed within each time step to determine the correct number of contact pairs among multi‐bodies and to achieve complete convergence of imposed contact force at corresponding position. The proposed contact algorithm extensively utilizes most of the original components of the NMM, namely, the mathematical mesh/cells and the manifold elements, as well as the external components associated with contacts, such as the contact body, the contact facet and the contact vertex. In particular, the utilization of two mutually approaching mathematical cells is efficient in detecting contacting territory, which makes this method particularly effective for both convex and non‐convex bodies. The validity and accuracy of the proposed contact algorithm are verified and demonstrated through three benchmark problems. Copyright © 2013 John Wiley & Sons, Ltd.     

An adaptive multi‐scale approach to the modelling of masonry structures

We present an adaptive multi‐scale approach for predicting the mechanical behaviour of masonry structures modelled as dynamic frictional multi‐body contact problems. In this approach, the iterative splitting of the contact problem into normal contact and frictional contact is combined with a semismooth Newton/primal‐dual active‐set procedure to calculate deformations and openings in the model structures. This algorithm is then coupled with a novel adaptive multi‐scale technique involving a macroscopic scale, which is the size of the masonry structure, and a mesoscopic scale, which is the size of the constituents (bricks, stone‐blocks), to predict appearance of dislocations and stress distribution in large‐scale masonry structures. Comparisons of the numerical results with data from experimental tests and from practical observations illustrate the predictive capability of the multi‐scale algorithm. Copyright © 2008 John Wiley & Sons, Ltd.     

Implementation of the Jäger contact model for discrete element simulations

In three‐dimensional discrete element method (DEM) simulations, the particle motions within a granular assembly can produce bewildering sequences of movements at the contacts between particle pairs. With frictional contacts, the relationship between contact movement and force is non‐linear and path‐dependent, requiring an efficient means of computing the forces and storing their histories. By cleverly applying the principles of Cattaneo, Mindlin, and Deresiewicz, J”urgen Jäger developed an efficient approach for computing the full three‐dimensional force between identical elastic spheres that have undergone difficult movement sequences (J. Jäger, New Solutions in Contact Mechanics. WIT Press: Southampton, U.K.). This paper presents a complete Jäger algorithm that can be incorporated into DEM codes and also describes three special provisions for DEM simulations: (1) a method for handling particle pairs that undergo complex tumbling and twirling motions in three‐dimensions; (2) a compact data structure for storing the loading history of the many contacts in a large assembly; and (3) an approximation of the Jäger algorithm that reduces memory demand. The algorithm addresses contact translations between elastic spheres having identical properties, but it does not resolve the tractions produced by twisting or rolling motions. A performance test demonstrates that the algorithm can be applied in a DEM code with modest increases in computation time but with more substantial increases in required storage. Copyright © 2011 John Wiley & Sons, Ltd.     

Fabrication and Optical Properties of Periodic Ag Nano‐Pore and Nano‐Particle Arrays with Controlled Shape and Size over Macroscopic Length Scales

A facile and economical route to preparation of highly ordered sliver pore or particle arrays with controlled pore‐shape and size extended over cm² areas is described. The substrates are prepared at planar and curved surfaces via sphere‐imprinted polymer (PDMS) templating using polystyrene spheres with diameters of 820, 600, or 430 nm. Nano‐pore arrays are created by sputtering 80 nm of Ag directly onto the templates and nano‐particle arrays are prepared by electrode‐less deposition of Ag from Tollen's reagent. The shape of the nano‐pore or particles in the array conformed to that of the imprint of the sphere on the template. Stretching the flexible template enable creation of cuboid shaped nano‐voids and nano‐particles following Ag deposition. Diffuse reflectance from the spherical Ag nano‐cavity arrays showed absorbance maxima at wavelengths comparable similar to the diameter of the templating sphere, whereas reflectance from the cuboid arrays, showed little correlation with the sphere diameter. The cuboid nano‐particle arrays showed the most intense visible absorption which is red‐shifted compared to the spherical arrays. White light diffraction from the arrays, observed by rotating 1 cm² substrates relative to a fixed light source, reflected exactly the symmetry axes of the periodic nano‐features in the arrays demonstrating the remarkable macroscopic order of the periodic structures. Raman spectra of 1‐benzenethiol adsorbed at the arrays indicated SERS enhancements from the substrates are attributed mainly to surface nano‐roughness with only moderate contributions from the periodically corrugated structures. Despite excitation at the major resonance dip in the reflectance spectrum, a weak, localized rim dipole mode is found to elicit a small increase in the SERS enhancement factor for the 430 nm diameter spherical arrays. FDTD studies of nano‐void arrays provided insights into v arious factors affecting the SERS experiment and confirmed the array's plasmonic spectra are dominated by propagating plasmon modes under microscope excitation/collection angles.     

Vertex‐to‐face contact searching algorithm for three‐dimensional frictionless contact problems

This article presents a new vertex‐to‐face contact searching algorithm for the three‐dimensional (3‐D) discontinuous deformation analysis (DDA). In this algorithm, topology is applied to the contact rule when any two polyhedrons are close to each other. Attempt is made to expand the original contact searching algorithm from two‐dimensional (2‐D) to 3‐D DDA. Examples are provided to demonstrate the new contact rule for vertex‐to‐face contacts between two polyhedrons with planar boundaries. Copyright © 2005 John Wiley & Sons, Ltd.     

A co‐rotational grid‐based model for fabric drapes

Fabric drapes are typical large displacement, large rotation and small strain problems. Compared to conventional geometric non‐linear shell analyses, computational fabric drape analysis is particularly challenging due to the extremely weak bending rigidities of fabrics. Compared to continuum shell finite element methods, grid‐ or particle‐based methods appear to be more successful in high drapeability problems. The latter methods often resort to simple particle mechanics and formulate the elastic energy in terms of the inter‐particle distances and trigonometrical functions of the angles between the straight lines joining adjacent particles. In this paper, the co‐rotational approach and commonly employed assumptions for small strain problems in finite element analysis will be adopted to formulate the elastic energy. It will be seen that the internal force vector and the stiffness matrix are considerably simpler than other grid‐based models, yet the sparsity of the tangential stiffness matrix remains unchanged. A number of examples are considered and the predicted results are promising. Copyright © 2003 John Wiley & Sons, Ltd.     

Scale-up and flow behavior of cohesive granular material in a four-bladed mixer: effect of system and particle size

Flow of cohesive granular materials with different moisture contents was examined in a four-bladed mixer via the discrete element method (DEM). Firstly, the mixer diameter (D) was increased while keeping the particle diameter (d) constant. It was observed that when the mixer diameter to the particle diameter ratio (D/d) was larger than a certain critical size (D/d ≥ 75), granular flow behaviors and mixing kinetics followed simple scaling relations. For D/d ≥ 75, flow patterns and mixing kinetics were found to be independent of system size, and velocities of particles scaled linearly with the tip speed of the impeller blades and particle diffusivities scaled with the tip speed of the blades and mixer diameter. These results suggest that past a certain system size the flow and mixing of cohesive particles in large-scale units can be predicted from smaller systems. Secondly, system size was kept constant and particle diameter was changed and it was observed that by keeping the Bond number constant (by changing the level of cohesion) the flow behavior and mixing patterns did not change, showing that larger particles can be used to simulate flow of smaller cohesive particles in a bladed mixer by matching the Bond numbers.     

Hybrid particle methods in frictionless impact‐contact problems

The dual particle dynamic (DPD) methods which employ two sets of particles have been demonstrated to have better accuracy and stability than the co‐locational particle methods, such as the smooth particle hydrodynamics (SPH). The hybrid particle method (HPM) is an extension of the DPD method. Besides the advantages of the DPD method, the HPM possesses features which better facilitate the simulation of large deformations. This paper presents the continued development of the HPM for the numerical solution of two‐dimensional frictionless contact problems. The interface contact force algorithm which employs a modified kinematic constraints method is used to determine the contact tractions. In this method, both the impenetrability condition and the traction condition are simultaneously enforced. In the original kinematic constraints method, only the former condition is satisfied. A new formulation to find stress derivatives at stress‐free corners by imposing stress‐free boundary conditions is also developed. The results for 1‐D and 2‐D contact problems indicate good accuracy for the contact formulation as well as the corner treatment when compared to analytical solutions and explicit finite element results using the commercial code LS‐DYNA. Copyright © 2004 John Wiley & Sons, Ltd.     

Experimental study on the packing densification of mixtures of spherical and cylindrical particles subjected to 3D vibrations

To identify the dense packing of cylinder–sphere binary mixtures (spheres as filling objects), the densification process of such binary mixtures subjected to three-dimensional (3D) mechanical vibrations was experimentally studied. Various influential factors including vibration parameters (such as vibration time t, vibration amplitude A, frequency ω, vibration acceleration Γ) as well as particle size ratio r (small sphere vs. large cylinder), composition of the binary mixtures X_L (volume fraction of cylinders), and container size D (container diameter) on the packing density ρ were systematically investigated. The results show that the optimal vibration parameters for different binary cylinder–sphere mixtures are different. The smaller the size ratio, the less vibration acceleration is needed to form a stable dense packing. For each binary mixture, high packing density can be obtained when the volume fraction of large cylindrical particles is dominant. Meanwhile, increasing the container size can decrease the container wall effect and get higher packing density. The proposed analytical model has been proved to be valid in predicting the packing densification of current cylinder–sphere binary mixtures.