论文标题
多尺度多时间有限元法| $ \ text {m}^2 $ -fem：整体结构理论的层次结构网格结构
Multiscale multimesh finite element method | $\text{M}^2$-FEM: Hierarchical mesh-decoupling for integral structural theories
论文作者
论文摘要
这项研究提出了一种广义的多尺度有限元方法（$ \ text {m}^2 $ -FEM），该方法在整体结构理论的数值模拟中解决了一些长期存在的挑战，通常用于模拟多尺度和非局部效应。 The major challenges in the numerical simulation of integral boundary value problems are primarily rooted in the coupling of the spatial discretization of the global (parent) and integral (child) domains which severely restricts the computational efficiency of existing algorithms by imposing an implicit trade-off in the accuracy achieved by the child domain and in the resources dedicated to the simulation of the overall parent domain.这项研究最定义的贡献之一是开发网状偶联技术，该技术生成孤立的网格集，以便可以独立地将父和子女域独立化和近似。 This mesh-decoupling has a multi-fold impact on the simulation of integral theories such that, when compared to existing state-of-the-art techniques, the proposed algorithm achieves simultaneously better numerical accuracy and efficiency (hence allowing a greater flexibility in both mesh size and computational cost trade-off decisions), greater ability to adopt generalized integral kernel functions, and the ability to handle non-regular (non-rectangular)通过非结构化的网格划分。在这项研究中，我们根据Eringen的非局部弹性理论的扩展版本选择基准问题（隐含地是一种多尺度理论），该理论利用了通用的衰减内核和非局部性的非稳定范围的使用。尽管如此，提出的$ \ text {m}^2 $ -FEM算法是非常笼统的，它可以应用于各种整体理论，甚至超出结构弹性。
This study presents a generalized multiscale multimesh finite element method ($\text{M}^2$-FEM) that addresses several long-standing challenges in the numerical simulation of integral structural theories, often used to model multiscale and nonlocal effects. The major challenges in the numerical simulation of integral boundary value problems are primarily rooted in the coupling of the spatial discretization of the global (parent) and integral (child) domains which severely restricts the computational efficiency of existing algorithms by imposing an implicit trade-off in the accuracy achieved by the child domain and in the resources dedicated to the simulation of the overall parent domain. One of the most defining contributions of this study consists in the development of a mesh-decoupling technique that generates isolated sets of meshes such that the parent and child domains can be discretized and approximated independently. This mesh-decoupling has a multi-fold impact on the simulation of integral theories such that, when compared to existing state-of-the-art techniques, the proposed algorithm achieves simultaneously better numerical accuracy and efficiency (hence allowing a greater flexibility in both mesh size and computational cost trade-off decisions), greater ability to adopt generalized integral kernel functions, and the ability to handle non-regular (non-rectangular) domains via unstructured meshing. In this study, we choose a benchmark problem based on an extended version of the Eringen's nonlocal elasticity theory (implicitly, a multiscale theory) that leverages the use of generalized attenuation kernels and non-constant horizons of nonlocality. Nonetheless, the proposed $\text{M}^2$-FEM algorithm is very general and it can be applied to a variety of integral theories, even beyond structural elasticity.
