Daniel N. Wilke

Development of a convex polyhedral discrete element simulation framework for NVIDIA Kepler based GPUs

Abstract Understanding the dynamical behavior of Granular Media (GM) is extremely important to many industrial processes. Thus simulating the dynamics of GM is critical in the design and optimization of such processes. However, the dynamics of GM is complex in nature and cannot be described by a closed form solution for more than a few particles. A popular and successful approach in simulating the underlying dynamics of GM is by using the Discrete Element Method (DEM). Computational viable simulations are typically restricted to a few particles with realistic complex interactions or a larger number of particles with simplified interactions. This paper introduces a novel DEM based particle simulation code (BLAZE-DEM) that is capable of simulating millions of particles on a desktop computer utilizing a NVIDIA Kepler Graphical Processor Unit (GPU) via the CUDA programming model. The GPU framework of BLAZE-DEM is limited to applications that require large numbers of particles with simplified interactions such as hopper flow which exhibits task level parallelism that can be exploited on the GPU. BLAZE-DEM also performs real-time visualization with interactive capabilities. In this paper we discuss our GPU framework and validate our code by comparison between experimental and numerical hopper flow.

Collision detection of convex polyhedra on the NVIDIA GPU architecture for the discrete element method

Convex polyhedra represent granular media well. This geometric representation may be critical in obtaining realistic simulations of many industrial processes using the discrete element method (DEM). However detecting collisions between the polyhedra and surfaces that make up the environment and the polyhedra themselves is computationally expensive. This paper demonstrates the significant computational benefits that the graphical processor unit (GPU) offers DEM. As we show, this requires careful consideration due to the architectural differences between CPU and GPU platforms. This paper describes the DEM algorithms and heuristics that are optimized for the parallel NVIDIA Kepler GPU architecture in detail. This includes a GPU optimized collision detection algorithm for convex polyhedra based on the separating plane (SP) method. In addition, we present heuristics optimized for the parallel NVIDIA Kepler GPU architecture. Our algorithms have minimalistic memory requirements, which enables us to store data in the limited but high bandwidth constant memory on the GPU. We systematically verify the DEM implementation, where after we demonstrate the computational scaling on two large-scale simulations. We are able achieve a new performance level in DEM by simulating 34 million polyhedra on a single NVIDIA K6000 GPU. We show that by using the GPU with algorithms tailored for the architecture, large scale industrial simulations are possible on a single graphics card.

A study of shape non-uniformity and poly-dispersity in hopper discharge of spherical and polyhedral particle systems using the Blaze-DEM GPU code

The importance of shape non-uniformity and the polydispersed nature of granular media in industrial hopper discharge applications has been well established experimentally. Although numerous hopper discharge simulations have been conducted over the last thirty years, the investigations into the non-uniformity of particle shape and the polydisperse nature of particle systems remains limited. These studies are usually limited to a single hopper configuration, while the number of polyhedral particles considered are usually limited to a maximum of 5000 particles. In this study we consider the polydispersed particle systems for hoppers at various angles, particle systems with non-uniform shape for hoppers at various angles and polyhedral particle systems up to 1 million particles. This is made possible by extensively utilizing the graphical processing unit (GPU) computing platform via the BlazeDEM3D-GPU code. We first perform an experimental validation of the code for mono-sized spherical and convex polyhedral shaped particles for lab-scale hoppers at three half angles using 3D printed polylactic acid material (PLA) particles. We found good agreement between the experimental, Meyrs and Sellers empirical relation and simulation discrete element method (DEM) discharge rates. We then simulate three larger square hoppers with varying half-angles to study the effect polydispersity and non-uniformity of particle shape have on the mass discharge rate. Again, good agreement between the DEM simulated mono-dispersed spherical particle systems and the Meyrs and Sellers empirical relation is obtained to verify the simulations. Finally we simulate an industrial sized silo for which we compare mono-dispersed spheres against mono-dispersed polyhedra using over a million polyhedral shaped particles. Finally, we briefly comment on the effect that the polydisperse nature of particle systems has on the loading of the supporting structure.

On rotationally invariant continuous-parameter genetic algorithms

We show that the standard CPGA is rotationally variant.We then construct a rotationally invariant CPGA.We ensure diversity using a modified mutation scheme.We also ensure diversity by adding a self-scaling random vector. We examine the rotational (in)variance of the continuous-parameter genetic algorithm (CPGA). We show that a standard CPGA, using blend crossover and standard mutation, is rotationally variant.To construct a rotationally invariant CPGA it is possible to modify the crossover operation to be rotationally invariant. This however results in a loss of diversity. Hence we introduce diversity in two ways: firstly using a modified mutation scheme, and secondly by adding a self-scaling random vector with a standard normal distribution, sampled uniformly from the surface of a n-dimensional unit sphere to the offspring vector. This formulation is strictly invariant, albeit in a stochastic sense only.We compare the three formulations in terms of numerical efficiency for a modest set of test problems; the intention not being the contribution of yet another competitive and/or superior CPGA variant, but rather to present formulations that are both diverse and invariant, in the hope that this will stimulate additional future contributions, since rotational invariance in general is a desirable, salient feature for an optimization algorithm.

Computing with Non-convex Polyhedra on the GPU

We recently introduced Blaze-DEMGPU, a GPU based computing framework for convex polyhedral shaped particles (Govender et al. Appl. Math. Comp. 267, 810–829, 2015). The computing framework was validated against numerous industrial applications that include particulate discharge and estimating power draw for a ball mill in comminution applications. In this study we extend the computing capabilities of the convex polyhedral Blaze-DEMGPU computing platform to include non-convex polyhedral particles. We follow a similar philosophy to the well known clumping, clustering or fusing of spheres (Chong et al. Gran. Mat. 17, 377–387, 2015), but instead we fuse convex polyhedral particles. This allows for fused or super polyhedral particles that constitute effective physical properties for the fused particle e.g. the inertia tensor. The major benefit of fused polyhedral particles as opposed to clustered spherical particles is that the number of particles required to fuse fairly complex particle shapes is small. In addition, numerous decompositions exist to exactly decompose a non-convex particle in a number of convex particles. The main complexity of non-convex polyhedral particles is to resolve contact effectively and efficiently on the GPU. In this paper we outline our approach.

SAFE-3D analysis of a piezoelectric transducer to excite guided waves in a rail web

Our existing Ultrasonic Broken Rail Detection system detects complete breaks and primarily uses a propagating mode with energy concentrated in the head of the rail. Previous experimental studies have demonstrated that a mode with energy concentrated in the head of the rail, is capable of detecting weld reflections at long distances. Exploiting a mode with energy concentrated in the web of the rail would allow us to effectively detect defects in the web of the rail and could also help to distinguish between reflections from welds and cracks. In this paper, we will demonstrate the analysis of a piezoelectric transducer attached to the rail web. The forced response at different frequencies is computed by the Semi-Analytical Finite Element (SAFE) method and compared to a full three-dimensional finite element method using ABAQUS. The SAFE method only requires the rail track cross-section to be meshed using two-dimensional elements. The ABAQUS model in turn requires a full three-dimensional discretisation of th...

Numerical Strategies to Reduce the Effect of Ill-Conditioned Correlation Matrices and Underflow Errors in Kriging

Kriging is used extensively as a metamodel in multidisciplinary design optimization. The correlation matrix used in Kriging metamodeling frequently becomes ill-conditioned. Therefore different numerical methods used to solve the Kriging equations affect the search for the optimum Kriging parameters and the ability of the Kriging surface to accurately interpolate known data points. We illustrate this by firstly computing the inverse of the correlation matrix in the Kriging equations, and secondly by solving the systems of equations using decomposition and back substitution, thereby avoiding the inversion of the correlation matrix. Our results clearly show that by decomposing and back substituting, the interpolation accuracy is maintained for significantly higher condition numbers. We then show that computing the natural logarithm of the determinant using additive calculations as opposed to multiplicative calculations significantly reduces numerical underflow errors encountered when searching for the optimum Kriging parameters. Although the effect of decomposition and back substitution are known, and the underflow difficulties when computing the natural logarithm of the determinant of the correlation matrix has been mentioned in passing in Kriging literature, this work clearly quantifies and reinforces these methods, hopefully for the benefit of researchers entering the field.

Steel Alloy Hot Roll Simulations and Through-Thickness Variation Using Dislocation Density-Based Modeling

Different roll pass reduction schedules have different effects on the through-thickness properties of hot-rolled metal slabs. In order to assess or improve a reduction schedule using the finite element method, a material model is required that captures the relevant deformation mechanisms and physics. The model should also report relevant field quantities to assess variations in material state through the thickness of a simulated rolled metal slab. In this paper, a dislocation density-based material model with recrystallization is presented and calibrated on the material response of a high-strength low-alloy steel. The model has the ability to replicate and predict material response to a fair degree thanks to the physically motivated mechanisms it is built on. An example study is also presented to illustrate the possible effect different reduction schedules could have on the through-thickness material state and the ability to assess these effects based on finite element simulations.

An AMG strategy for efficient solution of free-surface flows

Purpose – An area of great interest in current computational fluid dynamics research is that of free-surface modelling (FSM). Semi-implicit pressure-based FSM flow solvers typically involve the solution of a pressure correction equation. The latter being computationally intensive, the purpose of this paper is to involve the implementation and enhancement of an algebraic multigrid (AMG) method for its solution. Design/methodology/approach – All AMG components were implemented via object-oriented C++ in a manner which ensures linear computational scalability and matrix-free storage. The developed technology was evaluated in two- and three-dimensions via application to a dam-break test case. Findings – AMG performance was assessed via comparison of CPU cost to that of several other competitive sparse solvers. The standard AMG implementation proved inferior to other methods in three-dimensions, while the developed Freeze version achieved significant speed-ups and proved to be superior throughout. Originality/...

GPU DEM Simulations and Experimental Studies of Ball Milling Process for Various Particle Shapes

This work describes a comparative study on the milling process modelled by Discrete Element Method and lab-scale experiments. In particular, analogical complex granular media with spherical and polyhedral shaped particles have been used to support the development of Blaze-DEM GPU, which is graphical processor unit (GPU) based computing framework for convex polyhedral particle shapes. DEM simulations and experiments were performed for several filling rate, rotational speed and shaped particles. The experimental and DEM surface profiles are in good agreement that include avalanching, rolling, cascading and cataracting, The various flow regimes have been validated. The GPU allows for the computational efficient simulation of large numbers of particles or more complex polyhedral shaped particles. We consider the dodecahedron in this study. Therefore, GPU based DEM simulation conducted appropriately allows for large-scale industrial investigations to be conducted.