MeshAC: A 3D Mesh Generation and Adaptation Package for Multiscale Coupling Methods

TL;DR

MeshAC presents a 3D mesh generation and adaptation package tailored for atomistic/continuum (a/c) coupling in crystalline solids. It constructs coupled meshes by generating region-specific atomistic and continuum meshes and evolves the a/c interface through adaptive operations guided by a gradient-based a posteriori estimator, aiming for optimal accuracy and efficiency. The approach is demonstrated with BGFC coupling for straight edge dislocations in BCC W and double voids in FCC Cu, showing robust performance and manageable computational costs. The work highlights the package's ability to handle complex defects and its potential applicability to a broad class of multiscale materials problems, with plans for extending estimators and moving/coarse-grained meshes in future work.

Abstract

This paper introduces the MeshAC package, which generates three-dimensional adaptive meshes tailored for the efficient and robust implementation of multiscale coupling methods. While Delaunay triangulation is commonly used for mesh generation across the entire computational domain, generating meshes for multiscale coupling methods is more challenging due to intrinsic discrete structures such as defects, and the need to match these structures to the continuum domain at the interface. The MeshAC package tackles these challenges by generating meshes that align with fine-level discrete structures. It also incorporates localized modification and reconstruction operations specifically designed for interfaces. These enhancements improve both the implementation efficiency and the quality of the coupled mesh. Furthermore, MeshAC introduces a novel adaptive feature that utilizes gradient-based a posteriori error estimation, which automatically adjusts the atomistic region and continuum mesh, ensuring an optimal balance between accuracy and efficiency. This package can be directly applied to the geometry optimization problems of a/c coupling in static mechanics, with potential extensions to many other scenarios. Its capabilities are demonstrated for complex material defects, including straight edge dislocation in BCC W and double voids in FCC Cu. These results suggest that MeshAC can be a valuable tool for researchers and practitioners in computational mechanics.

Figures (20)

• Figure 1: The figure depicts a 2D slice of the mesh $\mathcal{T}_h$, where the atomistic region $\mathcal{T}^{\rm a}$ is shown in red, and the continuum region $\mathcal{T}^\textrm{c}$ is shown in green. White dots in the red region represent the atoms located within $\Omega^{\rm a}$, and mesh nodes in the continuum region $\Omega^{\rm c}$.
• Figure 2: The workflow for the generation of the a/c coupling mesh $\mathcal{T}_h$ (2D slice). Firstly, given the atom positions and the computational domain $\Omega$, the canonical mesh $\mathcal{T}^{\rm a}$ for the atomistic region is generated using the techniques described in Section \ref{['sec:sub:sub:mesh_atom']}. Next, the mesh $\mathcal{T}^{\rm c}$ for the continuum region is constructed using the methodology introduced in Section \ref{['sec:sub:sub:mesh_continum']}. The combination of the meshes for both regions completes the process of establishing the coupled mesh $\mathcal{T}_h$.
• Figure 3: The illustration of Algorithm \ref{['algorithm_delaunay_convex_hull']} (the Delaunay triangulation of a 3-dimensional point set). (a) 3-dimensional point set; (b)The point set (i.e., atom positions) in $\mathbb{R}^{3}$ is sorted using the BRIO technique amenta2003incremental to optimize the spatial search efficiency; (c) An initial tetrahedron element is then generated; (d) iterative application of the Bowyer-Watson algorithm bowyer1981computingwatson1981computing to construct the Delaunay triangulation incrementally; (e) the pre-processed mesh $\mathcal{T}^{\rm a-pre}$ is outputted.
• Figure 4: The comparison between $\mathcal{T}^{\rm a-pre}$ and $\mathcal{T}^{\rm a}$, where the major difference is the interfaces of the atomistic region for edge dislocation (cf. Section \ref{['sec:sub:edge']}), and the elements colored in green is the continuous mesh constructed in Section \ref{['sec:sub:sub:mesh_continum']}.
• Figure 5: The illustration of the continuum mesh generation (i.e., the construction of $\mathcal{T}^{\textrm{c}}$) in a two-dimensional view. (a) Initialize the boundary mesh in continuum region and the boundary mesh in atomistic region; (b) Generate a Delaunay triangulation for all the given nodes; (c) Recover the boundary mesh and extract elements in atomistic region; (d) Refine the elements in continuum region. CDT: (a-c), QMR: (d)

Theorems & Definitions (2)

• Lemma 1.1
• Lemma 1.2