Non-singular theories aim at regularizing the unrealistic stress singularity around a dislocation line predicted by elasticity by spreading the Burgers vector around the core. The use of these approaches in discrete dislocation models or field dislocation mechanics requires very fine discretization around the dislocation lines. FFT solvers commonly used in these problems rely on a regular grid which enforces the use of very fine discretizations or very small domains. In this work a methodology is proposed to solve the boundary value problems associated with dislocation modeling using FFT with an adaptive grid which is refined around dislocation cores. The framework introduces a regular domain to compute Fourier derivatives mapped to a physical domain with more points concentrated in the areas of interest. The linear differential operators involved are obtained by the product of the Fourier derivatives in the regular space and the Jacobian of the map defining the transformation between computational and physical domains. The periodic boundary value problem involved is transformed into a linear system that can be efficiently solved using Krylov solvers. The method for adaptive FFT grids is fully general and can be applied to any other micromechanical problem. It is demonstrated that the method allows to use several grid points within the core region of 2D and 3D dislocations even using coarse discretizations, allowing to resolve the non-singular stress fields within this region and also strongly reducing the numerical noise. Moreover, the method allows to preserve the accuracy of the results in fine meshes by reducing the grid spacing up to four times.

中文翻译：

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基于 FFT 的自适应网格框架来表示非奇异位错

非奇异理论旨在通过围绕核心传播伯格斯矢量来规范由弹性预测的位错线周围不切实际的应力奇异性。在离散位错模型或场位错力学中使用这些方法需要位错线周围非常精细的离散化。这些问题中常用的 FFT 求解器依赖于规则网格，该网格强制使用非常精细的离散化或非常小的域。在这项工作中，提出了一种方法来解决与位错建模相关的边值问题，该方法使用 FFT 和围绕位错核心细化的自适应网格。该框架引入了一个常规域来计算映射到物理域的傅里叶导数，其中更多的点集中在感兴趣的区域。所涉及的线性微分算子是通过正则空间中的傅里叶导数与定义计算域和物理域之间的变换的映射的雅可比行列式的乘积获得的。所涉及的周期性边值问题被转换为可以使用 Krylov 求解器有效求解的线性系统。自适应 FFT 网格的方法是完全通用的，可以应用于任何其他微机械问题。结果表明，即使使用粗离散化，该方法也允许在 2D 和 3D 位错的核心区域内使用多个网格点，从而可以解析该区域内的非奇异应力场，并大大降低数值噪声。此外，该方法可以通过将网格间距减少四倍来保持精细网格结果的准确性。