There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Three parallel algorithms for classical molecular dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-atomic forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for molecular dynamics models which can be difficult to parallelize efficiently—those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a standard Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers--the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C9O processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex molecular dynamics simulations are also discussed.

Fast parallel algorithms for short-range molecular dynamics

The American Society for Testing and Materials (ASTM) is an independent organization devoted to the development of standards.

American Society for Testing and Materials

http://mech.spbstu.ru/images/b/bd/Potyondy_Cundall_2004_A_bonded-particle_model_for_rock.pdf

A bonded-particle model for rock

The density-matrix renormalization group (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly become the method of choice for numerical studies of such systems. Its applications to the calculation of static, dynamic, and thermodynamic quantities in these systems are reviewed here. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and nonequilibrium statistical physics, and time-dependent phenomena is also discussed. This review additionally considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by the DMRG.

http://www.physics.udel.edu/~bnikolic/QTTG/NOTES/DMRG/SCHOLLWOCK=RMP_dmrg.pdf

The density-matrix renormalization group

We present a two-dimensional model of heterogeneous cohesive frictional solids where the material structure is idealized by a discrete granular particle assembly. Our discrete model is composed of convex polygons which are linked together by simple beams accounting for cohesive effects. Varying its parameters the model naturally interpolates between a continuous solid state and a discontinuous dry granular state of the material. We performed simulations of the quasi-static uniaxial loading with two different boundary conditions, and shearing of a solid. To provide a geometrical description of the damage state, and of the microscopic fracture events, different characteristic quantities will be introduced and their relevance will be examined. The results obtained from simulations are in satisfactory agreement with experimental observations.

On the application of a discrete model to the fracture process of cohesive granular materials

loss in the flow. Collision of particles also occurs in the solar system in planetary rings. In this case the energy dissipation due to impact damage might also influence the large scale structure formation in the rings @26#. On a larger length scale in the solar system, the so-called collisional evolution of asteroids due to subsequent collisions, and the formation of rubble piles in the asteroid belt, are still challenging problems @6#. Among industrial applications the breakup of agglomerates in chemical processes can be mentioned. Due to experimental difficulties, the computer simulation of microscopic models is an indispensable tool in the study of these impact phenomena @12‐17#. In the present paper we want to elaborate upon the impact fracture and fragmentation of solids at low imparted energy using a two-dimensional dynamical model of breakable granular solids. Simulating collisions of two solid disks, we show that, depending on the imparted energy, the outcome of a collision process can be classified into two states: a damaged state and a fragmented state, with a sharp transition in between. Analyzing the energetics of the impact and the resulting fragment size distributions, we give numerical evidence that the transition point between the damaged and fragmented states behaves as a critical point. The transition proved to be the lower bound for the occurrence of power law size distributions. The possible mechanism of the transition between the two states is discussed. In spite of the specific features of the system studied here, most of our results can be considered generally valid for impact phenomena mentioned above. After giving a short summary of the main ingredients of our model in Sec. II, the numerical results concerning to the energetics of the collision process and to the size distribution of fragments will be presented in Secs. III and IV, respectively. In Sec. V, we discuss the possible mechanism of the transition and some general consequences of our work for other types of fragmentation phenomena.

Transition from damage to fragmentation in collision of solids

A study of fragmentation processes using a discrete element method

Basquin's law of fatigue states that the lifetime of the system has a power-law dependence on the external load amplitude, tf approximately sigma 0- alpha, where the exponent alpha has a strong material dependence. We show that in spite of the broad scatter of the exponent alpha, the fatigue fracture of heterogeneous materials exhibits universal features. We propose a generic scaling form for the macroscopic deformation and show that at the fatigue limit the system undergoes a continuous phase transition. On the microlevel, the fatigue fracture proceeds in bursts characterized by universal power-law distributions. We demonstrate that the system dependent details are contained in Basquin's exponent for time to failure, and once this is taken into account, remaining features of failure are universal.

/pdf/universality-behind-basquin-s-law-of-fatigue-109715sw0p.pdf

Universality behind Basquin's Law of Fatigue.

We study the brittle fragmentation of spheres by using a three-dimensional discrete element model. Large scale computer simulations are performed with a model that consists of agglomerates of many particles, interconnected by beam-truss elements. We focus on the detailed development of the fragmentation process and study several fragmentation mechanisms. The evolution of meridional cracks is studied in detail. These cracks are found to initiate in the inside of the specimen with quasiperiodic angular distribution. The fragments that are formed when these cracks penetrate the specimen surface give a broad peak in the fragment mass distribution for large fragments that can be fitted by a two-parameter Weibull distribution. This mechanism can only be observed in three-dimensional models or experiments. The results prove to be independent of the degree of disorder in the model. Our results significantly improve the understanding of the fragmentation process for impact fracture since besides reproducing the experimental observations of fragment shapes, impact energy dependence, and mass distribution, we also have full access to the failure conditions and evolution.

/pdf/fragmentation-processes-in-impact-of-spheres-50865bqj5y.pdf

Ferenc Kun

Papers

On the application of a discrete model to the fracture process of cohesive granular materials

Transition from damage to fragmentation in collision of solids

A study of fragmentation processes using a discrete element method

Universality behind Basquin's Law of Fatigue.

Fragmentation processes in impact of spheres