Hans-Georg Matuttis

Discrete element simulations of dense packings and heaps made of spherical and non-spherical particles

The discrete element method (DEM) in the simulation of static packings allows one to investigate the behavior of granular materials by modeling the forces on the particle level. No macroscopic parameters like the angle of repose enter the simulation, but they can be extracted as a result of the particle properties like friction, roughness or shape. One of the issues of static packings recently discussed is the stress distribution under granular heaps. This problem is used to highlight the possibilities of modeling at the particle level using DEM. Phenomena like arching or stress-chains are observed even for spherical particles in a regular pile in the absence of friction if the bottom is rough. The situation does not change much if polygonal, frictional particles are used without disturbing the regular piling. For more realistic situations, when the pile is built by pouring grains from above, the packing and the stresses are influenced by the creation history. The more eccentric the polygons are, the more pronounced a dip is observed in the vertical stress under the apex of the sand-pile.

Fractal space, cosmic strings and spontaneous symmetry breaking

Abstract The present paper is conceived within the framework of El Naschies fractal-Cantorian program and proposes to develop a model of the fractal properties of spacetime. We show that, starting from the most fundamental level of elementary particles and rising up to the largest scale structure of the Universe, the fractal signature of spacetime is imprinted onto matter and fields via the common concept for all scales emanating from the physical spacetime vacuum fluctuations. The fractal structure of matter, field and spacetime (i.e. the nature and the Universe) possesses a universal character and can encompass also the well-known geometric structures of spacetime as Riemannian curvature and torsion and includes also, deviations from Newtonian or Einsteinian gravity (e.g. the Rossler conjecture). The leitmotiv of the paper is generated by cosmic strings as a fractal evidence of cosmic structures which are directly related to physical properties of a vacuum state of matter (VSM). We present also some physical aspects of a spontaneous breaking of symmetry and the Higgs mechanism in their relation with cosmic string phenomenology. Superconducting cosmic strings and the presence of cosmic inhomogeneities can induce to cosmic Josephson junctions (weak links) along a cosmic string or in connection with a cosmic string (self) interactions and thus some intermittency routes to a cosmic chaos can be explored. The key aspect of fractals in physics and of fractal geometry is to understand why nature gives rise to fractal structures. Our present answer is: because a fractal structure is a manifestation of the universality of self-organisation processes, as a result of a sequence of spontaneous symmetry breaking (SSB). Our conclusion is that it is very difficult to prescribe a certain type of fractal within an empty spacetime. Possibly, a random fractal (like a Brownian motion) characterises the structure of free space. The presence of matter will decide the concrete form of fractalisation. But, what does it mean the presence of matter? Can there exist a spacetime without matter or matter without spacetime? Possibly not, but consider on the other hand a space far removed from usual matter, or a space containing isolated small particles in which a very low density matter can exist. Very low density matter might be influenced by a fractal structure of space, for example in the sense that it is subject also to fluctuations structured by random fractals. Diffraction and diffusion experiments in an empty space and very low density matter could provide evidence of a fractal structure of space. However, at very high (Planck) densities, and a spacetime in which fluctuations represent also the source of matter and fields (which is very resonable within the context of a quantum gravity), we can assert that Einsteins dream of geometrising physics and El Naschies hope to prove the fractalisation (or Cantorisation) of spacetime are fully realised.

PARTICLE SIMULATION OF COHESIVE GRANULAR MATERIALS

We present two-dimensional molecular dynamics simulations of cohesive regular polygonal particles. The cohesive part of the force-law for the particle–particle interaction is validated by the agreement with existing experimental data. We investigate microscopic parameters, which are not accessible to experiments such as contact length, raggedness of the surface and correlation time. With increasing cohesion, the particles move in clusters for long times.

Simulation of the Dependence of the Bulk-Stress-Strain Relations of Granular Materials on the Particle Shape

We investigate the effect of particle shape and interparticle friction on the stress–strain-relation using the discrete element method (DEM) in two dimensions. Elongated particles show a significantly higher shear strength than non-elongated particles. The relative maximum which is characteristic for experimental stress–strain curves of granular materials is found only for elongated particles with finite Coulomb friction, which indicates that the particle elongation is an important parameter in the statistical physics of granular materials. An earlier simulation result from another group which showed a maximum for non-elongated particles could be identified to be due to the formation of a shear band.

Acoustic of Sound Propagation in Granular Materials in One, Two, and Three Dimensions

We investigate the sound velocity of assemblies of granular particles. Computationally, we investigate regular polygons with various corner numbers in two dimensions with the discrete element method and compare the results for large corner numbers with experiments on soft air-gun beads. The sound velocity for one-dimensional granular chains of spherical particles is about one order of magnitude less than the sound velocity of the bulk material, both in the simulation as well as in the experiment. For twodimensional simulations, the results are comparable to that for one-dimensional chains, but vary with the packing. For the experiments in three dimensions, we find that the sound velocity is two orders of magnitude less than that of the bulk, or one order of magnitude less than that for one- or two-dimensional systems. This result is consistent with the group velocity reported by Liu and Nagel, but well below their reported ‘‘sound velocity’’. The latter was in all likelihood not a linear (amplitude-independent) sound wave but one for which the sound-velocity was already affected by non-linear effects, as we elaborate on experimental, theoretical and computational considerations.

Efficient parallelization of the 2D Swendsen-Wang algorithm

We established a fast Swendsen-Wang algorithm for the two-dimensional Ising model on parallel computers with a high efficiency. On an Intel paragon with 140 processors we reached spin update times of only 14 ns with an efficiency of 89%. This algorithm was used to examine the non-equilibrium relaxation of magnetization and energy in large Ising systems of a size up to 17920 × 17920 spins. Nevertheless we observed still a strong finite-size effect for the magnetization. We assume both magnetization and energy decay to behave like (t + Δ)-λe-bt in an infinitely large system. Thus, for long times magnetization and energy show an exponential, asymtotic time-dependence, implying a critical dynamic exponent z of zero.

3D Ising model with Swendsen-Wang dynamics: a parallel approach

We propose an efficient parallel implementation of the Swendsen-Wang algorithm for a 3D Ising system. A modified relaxation method was used for the parallelization. The simulations were performed on the Intel Paragon. We discuss the implementation in detail.

Parallelization of the 2D Swendsen-Wang Algorithm

We implemented a parallel Swendsen–Wang algorithm for a 2D Ising system without magnetization in a host–node programming model. The simulations were performed on the Intel Hypercube IPSC/860. Our maximum number of updates/s on 32 nodes ist three times as high as in the implementation by Stauffer and Kertesz on the same machine. With 32 processors we reach half the speed of the simulations by Tamayo and Flanigan on 256 nodes of a CM5. We discuss the non–equilibrium relaxation for the energy and the magnetization.

OPTIMIZATION AND OPENMP PARALLELIZATION OF A DISCRETE ELEMENT CODE FOR CONVEX POLYHEDRA ON MULTI-CORE MACHINES

We report our experiences with the optimization and parallelization of a discrete element code for convex polyhedra on multi-core machines and introduce a novel variant of the sort-and-sweep neighborhood algorithm. While in theory the whole code in itself parallelizes ideally, in practice the results on different architectures with different compilers and performance measurement tools depend very much on the particle number and optimization of the code. After difficulties with the interpretation of the data for speedup and efficiency are overcome, respectable parallelization speedups could be obtained.

The Effect of the Shape of Granular Particles on Density

We investigate the packing densities of non-elongated regular polyhedra and spheres via discrete element simulations. It turns out that the limit of many (hundred) corners does not correspond to that of spheres with the same Young’s modulus. However, if the Young’s modulus is simultaneously decreased, the packing density of polyhedra approaches that of (harder) spheres. We conclude that the mechanism for the higher packing density is the higher mobility of particles, in the case of round particles due to rolling. While rolling can occur at vanishing energetic costs for perfect spheres, the mobility of polyhedra in packings is determined by the height differences between corners and faces, as well as due to the possibility that corners of neighbouring polyhedra to move through each other via deformation: reducing material strength therefore improves mobility. Additionally, we present finding about the influence of the walls and about the relation between filling height and surface roughness.