On finite strain micromorphic elastoplasticity

AbstractIn the micromorphic continuum theory of Eringen, it was proposed that microstructure of materials could be represented in a continuum framework using a micro-deformation tensor governing micro-element deformation, in addition to the deformation gradient governing macro-element deformation. The paper formulates finite strain micromorphic elastoplasticity based on micromorphic continuum mechanics in the sense of Eringen. Multiplicative decomposition into elastic and plastic parts of the deformation gradient and micro-deformation are assumed, and the Clausius–Duhem inequality is formulated in the intermediate configuration B¯ to analyze what stresses, elastic deformation measures, and plastic deformation rates are used/defined in the constitutive equations. The resulting forms of plastic and internal state variable evolution equations can be viewed as phenomenological at their various scales (i.e., micro-continuum and macro-continuum). The phenomenology of inelastic mechanical material response at the various scales can be different, but for demonstration purposes, J2 flow plasticity is assumed for each of three levels of plastic evolution equations identified, with different stress, internal state variables, and material parameters. All evolution equations and a semi-implicit time integration scheme are formulated in the intermediate configuration for future coupled Lagrangian finite element implementation. A simpler two-dimensional model for anti-plane shear kinematics is formulated to demonstrate more clearly how such model equations simplify for future finite element implementation

A Monte Carlo packing algorithm for poly-ellipsoids and its comparison with packing generation using Discrete Element Model

Granular material is showing very often in geotechnical engineering, petroleum engineering, material science and physics. The packings of the granular material play a very important role in their mechanical behaviors, such as stress-strain response, stability, permeability and so on. Although packing is such an important research topic that its generation has been attracted lots of attentions for a long time in theoretical, experimental, and numerical aspects, packing of granular material is still a difficult and active research topic, especially the generation of random packing of non-spherical particles. To this end, we will generate packings of same particles with same shapes, numbers, and same size distribution using geometry method and dynamic method, separately. Specifically, we will extend one of Monte Carlo models for spheres to ellipsoids and poly-ellipsoids

Superlinear Speedup Phenomenon in Parallel 3D Discrete Element Method (DEM) Simulations of Complex-shaped Particles

Strong superlinear speedup has been discovered in large scale simulationsof parallel 3D DEM for complex-shaped particles, which is based on an al-gorithm of spatial domain decomposition, and exhibits the &ldquo;high-CPU-low-memory&rdquo; characteristics. The interpretation of this phenomenon requires acareful examination of the speedup theory and practice in the field of parallelcomputing. The superlinear speedup is investigated from three perspectives:(i) memory footprint per process, (ii) cache miss rates of L1, L2 and L3 levelcaches, and (iii) uniprocessor performance, using a wide range of problemsize (across five orders of magnitude of simulation scale regarding number ofparticles) and number of compute nodes (1 to 2,048 nodes) on DoD super-computers. The Performance-API (PAPI) is employed in the source code tomeasure cache miss rate and FLOPS. The strong scaling measurements showthat cache miss rate is sensitive to the memory consumption shrinkage perprocessor, and the last level cache (LLC) contributes most significantly tothe strong superlinear speedup among all of the three cache levels, and this isalso revealed in the weak scaling measurements. The findings are associatedwith the inherently perfect scalability of 3D DEM: its memory scalabilityfunction is a nonlinearly decreasing function of the number of processors. Inaddition, a constant (non-increasing) uniprocessor FLOPS performance w.r.tproblem size can also contribute to the superlinear speedup. The superlinear speedup is a common phenomenon for large scale 3D DEM simulations of complex-shaped particles, and the larger the scale, thestronger is the superlinear speedup. DEM researchers should take advantageof this effect to speedup their parallel simulations.</p

A Comprehensive Study of MPI Parallelism in Three-dimensional Discrete Element Method (DEM) Simulation of Complex-shaped Granular Particles

A three-dimensional (3D) DEM code for simulating complex-shaped granular particles is parallelized using message-passing interface (MPI). The concepts of link-block, ghost/border layer, and migration layer are put forward for design of the parallel algorithm, and theoretical scalability function of 3-D DEM scalability and memory usage is derived. Many performance-critical implementation details are managed optimally to achieve high performance and scalability, such as: minimizing communication overhead, maintaining dynamic load balance, handling particle migrations across block borders, transmitting C++ dynamic objects of particles between MPI processes efficiently, eliminating redundant contact information between adjacent MPI processes. The code executes on multiple US Department of Defense (DoD) supercomputers and tests up to 2048 compute nodes for simulating 10 million three-axis ellipsoidal particles. Performance analyses of the code including speedup, efficiency, scalability, and granularity across five orders of magnitude of simulation scale (number of particles) are provided, and they demonstrate high speedup and excellent scalability. It is also discovered that communication time is a decreasing function of the number of compute nodes in strong scaling measurements. The code&rsquo;s capability of simulating a large number of complex-shaped particles on modern supercomputers will be of value in both laboratory studies on micromechanical properties of granular materials and many realistic engineering applications involving granular materials.</p

Influence of Particle Shape on Microstructure of Granular Materials under Gravity

The paper presents a study of particle shape effect on microstructure within granular materials in gravitational environments using samples acquired from three-dimensional (3D) discrete element method (DEM) modeling and simulation (M&amp;S) of gravitational pluviations. Three-axis ellipsoidal particles with various aspect ratios are examined for their spatial orientations, and the three-axis probability density functions (PDF) are analyzed, before and after pluviation. A complicated data structure, convex hull, is employed to construct tetrahedra that connect particle centroids, and investigate theinternal topology and fabric of particle assemblies composed of various particle shapes. The Qhull-constructed connected tetrahedra within granular materials may serve as a useful tool to study the geometrical fabric (such as tetrahedron shape, solid angles, void ratio, etc), investigate particle collision requirements in sample preparation, describe strain/deformation/failure mechanisms, and explore stress distribution over granular materials.</p

Comparison between Pure MPI and Hybrid MPI-OpenMP Parallelism for Discrete Element Method (DEM) of Ellipsoidal and Poly-ellipsoidal Particles

Parallel computing of 3D Discrete Element Method (DEM) simulations can be achieved in different modes, and two of them are pure MPI and hybrid MPI-OpenMP. The hybrid MPI-OpenMP mode allows flexibly combined mapping schemes on contemporary multiprocessing supercomputers. This paper profiles computational components and floating-point operation features of complex-shaped 3D DEM, develops a space decomposition-based MPI parallelism and various thread-based OpenMP parallelism, and carries out performance comparison and analysis from intranode to internode scales across four orders of magnitude of problem size (namely, number of particles). The influences of memory/cache hierarchy, processes/threads pinning, variation of hybrid MPI-OpenMP mapping scheme, ellipsoid versus poly-ellipsoid are carefully examined. It is found that OpenMP is able to achieve high efficiency in interparticle contact detection, but the unparallelizable code prevents it from achieving the same high efficiency for overall performance; pure MPI achieves not only lower computational granularity (thus higher spatial locality of particles) but also lower communication granularity (thus faster MPI transmission) than hybrid MPI-OpenMP using the same computational resources; the cache miss rate is sensitive to the memory consumption shrinkage per processor, and the last level cache contributes most significantly to the strong superlinear speedup among all of the three cache levels of modern microprocessors; in hybrid MPI-OpenMPI mode, as the number of MPI processes increases (and the number of threads per MPI processes decreases accordingly), the total execution time decreases, until the maximum performance is obtained at pure MPI mode; the processes/threads pinning on NUMA architectures improves performance significantly when there are multiple threads per process, whereas the improvement becomes less pronounced when the number of threads per process decreases; both the communication time and computation time increase substantially from ellipsoids to poly-ellipsoids. Overall, pure MPI outperforms hybrid MPI-OpenMP in 3D DEM modeling of ellipsoidal and poly-ellipsoidal particles.</p

Three-Dimensional Discrete Element Method Parallel Computation of Cauchy Stress Distribution over Granular Materials

The paper presents Cauchy stress tensor computation over parallel grids of message passing interface (MPI) parallel three-dimensional (3D) discrete element method (DEM) simulations of granular materials, considering spherical and nonspherical particles. The stress tensor computation is studied for quasi-static and dynamic conditions, and its resulting symmetry or asymmetry is discussed within the context of classical continuum mechanics (CCM), granular materials mechanics (GMM), and micropolar continuum mechanics (MCM). The average Cauchy stress tensor computation follows Bagi's and Nicot's formulations and is verified within MPI parallel 3D DEM simulations involving dynamically adaptive compute grids. These grids allow calculation of temporal and spatial distributions of stress across granular materials under static and dynamic conditions. The vertical stress component in gravitationally deposited particle assemblies exhibits nonuniform spatial distributions under static equilibrium, and its zone of maximum value changes during the process of gravitational pluviation and collapse. These phenomena reveal a microstructural effect on stress distribution within granular materials that is attributed to their discrete particulate nature (particle size, shape, gradation, boundary conditions, etc). </div

Comparison Between O(n2) and O(n) Neighbor Search Algorithm and its Influence on Superlinear Speedup in Parallel DEM for Complex-shaped Particles

Purpose This paper aims to present performance comparison between O(n2) and O(n) neighbor search algorithms, studies their effects for different particle shape complexity and computational granularity (CG) and investigates the influence on superlinear speedup of 3D discrete element method (DEM) for complex-shaped particles. In particular, it aims to answer the question: O(n2) or O(n) neighbor search algorithm, which performs better in parallel 3D DEM computational practice? Design/methodology/approach The O(n2) and O(n) neighbor search algorithms are carefully implemented in the code paraEllip3d, which is executed on the Department of Defense supercomputers across five orders of magnitude of simulation scale (2,500; 12,000; 150,000; 1 million and 10 million particles) to evaluate and compare the performance, using both strong and weak scaling measurements. Findings The more complex the particle shapes (from sphere to ellipsoid to poly-ellipsoid), the smaller the neighbor search fraction (NSF); and the lower is the CG, the smaller is the NSF. In both serial and parallel computing of complex-shaped 3D DEM, the O(n2) algorithm is inefficient at coarse CG; however, it executes faster than O(n) algorithm at fine CGs that are mostly used in computational practice to achieve the best performance. This means that O(n2) algorithm outperforms O(n) in parallel 3D DEM generally. Practical implications Taking for granted that O(n) outperforms O(n2) unconditionally, complex-shaped 3D DEM is a misconception commonly encountered in the computational engineering and science literature. Originality/value The paper clarifies that performance of O(n2) and O(n) neighbor search algorithms for complex-shaped 3D DEM is affected by particle shape complexity and CG. In particular, the O(n2) algorithm outperforms the O(n) algorithm in large-scale parallel 3D DEM simulations generally, even though this outperformance is counterintuitive.</p

Definition and Symmetry of Averaged Stress Tensor in Granular Media and its 3D DEM Inspection Under Static and Dynamic Conditions

The paper aims to clarify the stress tensor definition and its symmetry property that applies to granular media, and conducts 3D Discrete Element Method (DEM) inspection of the stress tensor definitions provided in the literature. Various stress tensor formulas under static and dynamic conditions are summarized, compared and numerically inspected through different types of simulation, such as gravitational deposition, isotropic/oedometer compression and high-strain-rate (HSR) oedometer impact. The stress tensor symmetry is particularly discussed from the perspective of applying classical continuum mechanics to granular media. It is proved analytically and numerically that the stress tensor should be calculated by Bagi&rsquo;s formula, not Weber&rsquo;s formula or Drescher&rsquo;s formula, for a particle assembly or representative volume element (RVE) in static equilibrium. We propose to modify the De Saxc&eacute; and Nicot formulas by incorporating the boundary-radius-gap term such that they are consistent with Bagi&rsquo;s formula, which is particularly well-suited for studying granular phenomena that transition between static, quasi-static and dynamic conditions. It is shown from the perspective of stress tensor calculation that the number of particles in the RVE does not need to be large. Symmetry of averaged stress tensor can be accurately satisfied in static equilibrium of the granular DEM RVE, however it cannot in quasi-static or dynamic states due to imbalance of angular momentum of the granular DEM RVE (in comparison to the balance of angular momentum which is always satisfied in classical continuum mechanics). When the stress tensor definition is extended to &ldquo;discontinuous&rdquo; state with regard to discrete granular DEM RVEs, the calculated values need to be treated with caution.</p

Large-scale Dynamic and Static Simulations of Complex-shaped Granular Materials Using Parallel Three-dimensional Discrete Element Method (DEM) on DoD Supercomputers

In this paper we present unprecedented three-dimensional (3D) DEM simu-lations of complex-shaped particles across five orders of magnitude of simu-lation scale (namely, number of particles) using up to 32,768 cores on U.S. Department of Defense (DoD) supercomputers. Firstly, we develop the parallel algorithm based on domain decomposition by following Foster&rsquo;s four-step design methodology and incorporating a number of advances, including concept of link-block (LB), border layer and migration layers, adaptive compute gridding technique, MPI transmission of C++ objects and pointers, etc. Secondly, performance analyses of the code including speedup, efficiency, scalability are provided as benchmarks and guidelines, as well as optimal computational granularity (CG) for each simulation scale. Thirdly, three full-scale simulations of sand pluviation, constrained collapse and particle shape effect are carried out to demonstrate the code capability as well as discover mechanical response of particle assemblies. The parallel code enables us to simulate a wide range of dynamic and static laboratory and field tests in engineering applications that involve alarge number of granular and geotechnical material grains, such as sand pluviation process, buried explosion in various soils, earth penetrator interaction with soil, influence of grain size, shape and gradation on packing density and shear strength, mechanical behavior under different gravity environments such as on the Moon and Mars, etc.</p