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

A meshfree method based on the peridynamic model of solid mechanics

A generalization of the original peridynamic framework for solid mechanics is proposed. This generalization permits the response of a material at a point to depend collectively on the deformation of all bonds connected to the point. This extends the types of material response that can be reproduced by peridynamic theory to include an explicit dependence on such collectively determined quantities as volume change or shear angle. To accomplish this generalization, a mathematical object called a deformation state is defined, a function that maps any bond onto its image under the deformation. A similar object called a force state is defined, which contains the forces within bonds of all lengths and orientation. The relation between the deformation state and force state is the constitutive model for the material. In addition to providing a more general capability for reproducing material response, the new framework provides a means to incorporate a constitutive model from the conventional theory of solid mechanics directly into a peridynamic model. It also allows the condition of plastic incompressibility to be enforced in a peridynamic material model for permanent deformation analogous to conventional plasticity theory.

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Peridynamic States and Constitutive Modeling

Publisher Summary The classical theory of solid mechanics is based on the assumption of a continuous distribution of mass within a body and all internal forces are contact forces that act across zero distance. The mathematical description of a solid that follows from these assumptions relies on PDEs that additionally assume sufficient smoothness of the deformation for the PDEs to make sense in their either strong or weak forms. The classical theory has been demonstrated to provide a good approximation to the response of real materials down to small length scales, particularly in single crystals, provided these assumptions are met. Nevertheless, technology increasingly involves the design and fabrication of devices at smaller and smaller length scales, even interatomic dimensions.

Peridynamic Theory of Solid Mechanics

In this paper we discuss the peridynamic analysis of dynamic crack branching in brittle materials and show results of convergence studies under uniform grid refinement (m-convergence) and under decreasing the peridynamic horizon (δ-convergence) Comparisons with experimentally obtained values are made for the crack-tip propagation speed with three different peridynamic horizons We also analyze the influence of the particular shape of the micro-modulus function and of different materials (Duran 50 glass and soda-lime glass) on the crack propagation behavior We show that the peridynamic solution for this problem captures all the main features, observed experimentally, of dynamic crack propagation and branching, as well as it obtains crack propagation speeds that compare well, qualitatively and quantitatively, with experimental results published in the literature The branching patterns also correlate remarkably well with tests published in the literature that show several branching levels at higher stress levels reached when the initial notch starts propagating We notice the strong influence reflecting stress waves from the boundaries have on the shape and structure of the crack paths in dynamic fracture All these computational solutions are obtained by using the minimum amount of input information: density, elastic stiffness, and constant fracture energy No special criteria for crack propagation, crack curving, or crack branching are used: dynamic crack propagation is obtained here as part of the solution We conclude that peridynamics is a reliable formulation for modeling dynamic crack propagation

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Studies of dynamic crack propagation and crack branching with peridynamics

*† ‡ Nearly all finite element codes and similar methods for the analysis of deformation in structures attempt to solve the partial differential equations of the classical theory of continuum mechanics. Yet these equations, because they require the partial derivatives of displacement to be known throughout the region modeled, are in some ways unsuitable for the modeling of cracks and other discontinuities, in which these derivatives fail to exist. As a means of avoiding this limitation, the peridynamic model of solid mechanics has been developed for applications involving discontinuities. The objective of this method is to treat crack and fracture as just another type of deformation, rather than as a pathology that requires special mathematical treatment. The peridynamic theory is based on integral equations, rather than differential equations, so there is no problem in applying the equations directly on a crack tip or crack surface. In the peridynamic model, displacements and internal forces are permitted to have discontinuities and other singularities. Particles interact with each other directly across finite distances through central forces known as “bonds”. Damage is introduced into the peridynamic model by permitting these bonds to break irreversibly. Breakage occurs when a bond is stretched in tension (or possibly compression) beyond some prescribed critical amount. After a bond breaks, it sustains no force. A distinguishing feature of this approach is its ability to treat the spontaneous formation of cracks together with their mutual interaction and dynamic growth in a consistent framework. A three-dimensional code called EMU implements the peridynamic model on parallel computers. The peridynamic method has been applied successfully to the analysis of material and structural failure in aerospace composites, particularly in graphiteepoxy laminates. For example, the method has been applied to the prediction of failure mode and crack direction in large-notch composite panels under tension loads with different layups and stacking sequences. The results have reproduced the experimentally observed dependence of crack growth direction on the relative percentage of fibers in different directions. The authors also have analyzed the damage occurring in a composite panel due to low velocity impact. The method predicts in detail the delamination and matrix damage process. Although the numerical method in EMU lends itself to parallelization, threedimensional analysis of large problems is computationally intensive. The applications reported here were run on the Columbia supercomputer at NASA Advanced Supercomputing (NAS) division. The Columbia supercomputer is proving to be invaluable in high-resolution modeling of the failure of composite materials.

Peridynamic analysis of damage and failure in composites.

The peridynamic theory is an alternative formulation of continuum mechanics oriented toward modeling discontinuites such as cracks. It differs from the classical theory and most nonlocal theories in that it does not involve spatial derivatives of the displacement field. Instead, it is formulated in terms of integral equations, whose validity is not affected by the presence of discontinuities such as cracks. It may be thought of as a “continuum version of molecular dynamics” in that particles interact directly with each other across a finite distance. This paper outlines the basis of the peridynamic theory and its numerical implementation in a three-dimensional code called EMU. Examples include simulations of a Charpy V-notch test, accumulated damage in concrete due to multiple impacts, and crack fragmentation of a glass plate.Copyright © 2004 by ASME

Peridynamic Modeling of Impact Damage

In this paper, we compare small deformations in an infinite linear elastic body due to the presence of point loads within the classical, local formulation to the corresponding deformations in the peridynamic, non-local formulation. Owing to the linearity of the problem, the response to a point load can be used to obtain the response to general body force loading functions by superposition. Using Laplace and Fourier transforms, we thus obtain an integral representation for the three-dimensional peridynamic solution with the help of Green’s functions. We illustrate this new theoretical result by dynamic and static examples in one and three dimensions. In addition to this main result, we also derive the non-local three-dimensional jump conditions, as well as the weak formulation of peridynamics together with the associated finite element discretization.

Green's functions in non-local three-dimensional linear elasticity

Effect of void content on stiffness and strength of composites by a peridynamic analysis and static indentation test

The effects of nanoparticles on impact stiffness and strength are investigated. Nanoparticles are prepared from nanoclay particles embedded into carbon fiber/epoxy wet lay-up laminates. Specimens are characterized via low-velocity impact tests and following Cscanner damage analyses. The results show that the area of damage is reduced 46% when 5% nanoparticle is used in the composite. Modeling of the nanocomposites is performed by peridynamic simulations and compared with experimental measurements. The mode of damage and the improvement of the impact resistance of nanocomposites are captured successfully via numerical simulations.

Ebrahim Askari

Papers

Peridynamic analysis of damage and failure in composites.

Peridynamic Modeling of Impact Damage

Green's functions in non-local three-dimensional linear elasticity

Effect of void content on stiffness and strength of composites by a peridynamic analysis and static indentation test

Effects of Nanoparticles on Stiffness and Impact Strength of Composites