Dynamic coupling of a finite element solver to large-scale atomistic simulations

Dynamic coupling of a finite element solver to large-scale atomistic simulations Mihkel Veske1, Andreas Kyritsakis1, Kristjan Eimre2, Vahur Zadin2, Alvo Aabloo2 and Flyura Djurabekova1 1Department of Physics and Helsinki Institute of Physics, University of Helsinki, PO Box 43 (Pietari Kalmin katu 2), 00014 Helsinki, Finland 2Intelligent Materials and Systems Lab, Institute of Technology, University of Tartu, Nooruse 1, 50411 Tartu, Estonia E-mail: mihkel.veske@helsinki.fi Abstract We propose a method for efficiently coupling the finite element method with atomistic simulations, while using molecular dynamics or kinetic Monte Carlo techniques. Our method can dynamically build an optimized unstructured mesh that follows the geometry defined by atomistic data. On this mesh, different multiphysics problems can be solved to obtain distributions of physical quantities of interest, which can be fed back to the atomistic system. The simulation flow is optimized to maximize computational efficiency while maintaining good accuracy. This is achieved by providing the modules for a) optimization of the density of the generated mesh according to requirements of a specific geometry and b) efficient extension of the finite element domain without a need to extend the atomistic one. Our method is organized as an open-source C++ code. In the current implementation, an efficient Laplace equation solver for calculating the electric field distribution near a rough atomistic surface demonstrates the capability of the suggested approach. Keywords: multiphysics, multiscale, electric field, Laplace equation, finite element method, atomistic simulation
1. Introduction Achieving atomistic spatial and temporal resolution is still challenging for experimental physics and, in many cases, numerical simulations based on well-motivated physical models are the only tools which can provide interesting insight on the atomic scale. However, due to an unavoidable trade-off between computational efficiency and desired accuracy, often seemingly promising computational models turn out to be impractical.

2 One way to achieve high computational efficiency and numerical accuracy is to combine continuous- space calculations with atomistic simulations like classical molecular dynamics (MD) or kinetic Monte Carlo (KMC). Some such approaches [1]–[4] have shown promising results when simulating the elastoplastic evolution of nanostructures. Others [5]–[9], being especially relevant to the present work, have used such a technique to study the effects of electric field around nanostructured materials. When a strong electric field is applied on the surface of a metal, it induces surface charge and polarization, and under certain circumstances, it triggers field emission (FE) currents with consequent electromigration effects [10]. Thus, the high electric field may significantly affect the evolution of the system and under certain conditions might cause major surface deformations [11]. For that reason, atomistic simulations that take into account the effects of electrostatic field have a wide range of applications in atom probe tomography (APT) [12], nanoelectronics [13] and space technology [14]. Moreover, atomistic modeling is a valuable tool in the investigation of vacuum arcing phenomena (vacuum breakdowns), as the fundamental mechanisms that trigger a breakdown are not entirely clear yet. The breakdown studies are relevant to the development of new-generation linear colliders like CLIC in CERN [15], vacuum interrupters [16], free electron lasers [17] and fusion devices [18]. Simulating electronic processes on material surfaces requires an accurately calculated spatial distribution of the electric field. The common method for calculating the field around any geometry is to build a mesh around the system of interest and solve the Laplace or Poisson equation on it. The solver is usually based on the finite difference method (FDM) [9], [19], finite element method (FEM) [7] or their modifications [6]. Many authors [8], [20] calculate the electric field around nanostructures without building any mesh around it. Although such mesh-free methods might be more flexible and efficient under certain conditions, they are limited in practical applications as they incorporate only the calculation of electric field. The mesh for solving the differential equations can be either static (it does not change during the evolution of the underlying atomistic geometry) or dynamic (the mesh is adjusted with the movement of the atoms). Both can be either structured or unstructured. The main advantage of a structured mesh is its implementation simplicity, while the unstructured one provides higher tolerance to the underlying geometry. Although the generation of an unstructured dynamic mesh requires significant computational effort, it has conside

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