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Register a new userFast algorithms for evaluating the stress field of dislocation lines in anisotropic elastic media
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In dislocation dynamics (DD) simulations, the most computationally intensive step is the evaluation of the elastic interaction forces among dislocation ensembles. Because the pair-wise interaction between dislocations is long-range, this force calculation step can be significantly accelerated by the fast multipole method (FMM). We implemented and compared four different methods in isotropic and anisotropic elastic media: one based on the Taylor series expansion (Taylor FMM), one based on the spherical harmonics expansion (Spherical FMM), one kernel-independent method based on the Chebyshev interpolation (Chebyshev FMM), and a new kernel-independent method that we call the Lagrange FMM. The Taylor FMM is an existing method, used in ParaDiS, one of the most popular DD simulation softwares. The Spherical FMM employs a more compact multipole representation than the Taylor FMM does and is thus more efficient. However, both the Taylor FMM and the Spherical FMM are difficult to derive in anisotropic elastic media because the interaction force is complex and has no closed analytical formula. The Chebyshev FMM requires only being able to evaluate the interaction between dislocations and thus can be applied easily in anisotropic elastic media. But it has a relatively large memory footprint, which limits its usage. The Lagrange FMM was designed to be a memory-efficient black-box method. Various numerical experiments are presented to demonstrate the convergence and the scalability of the four methods.
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The linked cell list algorithm is an essential part of molecular simulation software, both molecular dynamics and Monte Carlo. Though it scales linearly with the number of particles, there has been a constant interest in increasing its efficiency, because a large part of CPU time is spent to identify the interacting particles. Several recent publications proposed improvements to the algorithm and investigated their efficiency by applying them to particular setups. In this publication we develop a general method to evaluate the efficiency of these algorithms, which is mostly independent of the parameters of the simulation, and test it for a number of linked cell list algorithms. We also propose a combination of linked cell reordering and interaction sorting that shows a good efficiency for a broad range of simulation setups.
In an interesting recent paper [1] (A. Acharya, Stress of a spatially uniform dislocation density field, J. Elasticity 137 (2019), 151--155), Acharya proved that the stress produced by a spatially uniform dislocation density field in a body comprising a nonlinear elastic material may fail to vanish under no loads. The class of counterexamples constructed in [1] is essentially $2$-dimensional: it works with the subgroup $mathcal{O}(2) oplus langle{bf Id}rangle subset mathcal{O}(3)$. The objective of this note is to extend Acharyas result in [1] to $mathcal{O}(3)$, subject to an additional structural assumption and less regularity requirements.
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