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

We introduce here adaptive refinement algorithms for the non-local method peridynamics, which was proposed in (J. Mech. Phys. Solids 2000; 48:175–209) as a reformulation of classical elasticity for discontinuities and long-range forces. We use scaling of the micromodulus and horizon and discuss the particular features of adaptivity in peridynamics for which multiscale modeling and grid refinement are closely connected. We discuss three types of numerical convergence for peridynamics and obtain uniform convergence to the classical solutions of static and dynamic elasticity problems in 1D in the limit of the horizon going to zero. Continuous micromoduli lead to optimal rates of convergence independent of the grid used, while discontinuous micromoduli produce optimal rates of convergence only for uniform grids. Examples for static and dynamic elasticity problems in 1D are shown. The relative error for the static and dynamic solutions obtained using adaptive refinement are significantly lower than those obtained using uniform refinement, for the same number of nodes. Copyright © 2008 John Wiley & Sons, Ltd.

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Convergence, adaptive refinement, and scaling in 1D peridynamics

The paper presents an overview of peridynamics, a continuum theory that employs a nonlocal model of force interaction Specifically, the stress/strain relationship of classical elasticity is replaced by an integral operator that sums internal forces separated by a finite distance This integral operator is not a function of the deformation gradient, allowing for a more general notion of deformation than in classical elasticity that is well aligned with the kinematic assumptions of molecular dynamics Peridynamics effectiveness has been demonstrated in several applications, including fracture and failure of composites, nanofiber networks, and polycrystal fracture These suggest that peridynamics is a viable multiscale material model for length scales ranging from molecular dynamics to those of classical elasticity

https://iopscience.iop.org/article/10.1088/1742-6596/125/1/012078/pdf

Peridynamics for multiscale materials modeling

A condition for the emergence of a discontinuity in an elastic peridynamic body is pro- posed, resulting in a material stability condition for crack nucleation. The condition is derived by deter- mining whether a small discontinuity in displace- ment, superposed on a possibly large deformation, grows over time. Stability is shown to be determined by the sign of the eigenvalues of a tensor field that dependsonlyonthelinearizedmaterialproperties.This condition for nucleation of a discontinuity in dis- placement can be interpreted in terms of the dynamic stability of plane waves with very short wavelength. A numerical example illustrates that cracks in a peri- dynamic body form spontaneously as the body is loaded.

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Ebrahim Askari

Papers

A meshfree method based on the peridynamic model of solid mechanics

Peridynamic States and Constitutive Modeling

Convergence, adaptive refinement, and scaling in 1D peridynamics

Peridynamics for multiscale materials modeling

Crack nucleation in a peridynamic solid