Effective response of heterogeneous materials using the recursive projection method
Effective response of heterogeneous materials using the recursive projection method
复制标题
使用递归投影方法有效响应异质材料
DOI：
10.1016/j.cma.2020.112946
复制
发表时间：
2020
影响因子：
7.2
通讯作者：
Dayal, Kaushik
中科院分区：
文献类型：
--
作者：
Peng, Xiaoyao;Nepal, Dhriti;Dayal, Kaushik
本文将递归投影法应用于求解周期性非均匀固体的有效力学响应问题。以前的工作应用快速傅里叶变换（FFT）结合各种固定点的方法来解决问题的周期性单位细胞。这些已被证明是非常强大的一系列问题，从基于图像的建模位错塑性。然而，如果弹性性质具有高对比度，例如在空隙的情况下，则固定点迭代可以非常缓慢地收敛，或者根本不收敛。本文探讨了缓慢的原因，或缺乏收敛，在变量的角度来看。特别是，当材料包含具有零刚度或非常小刚度的区域时，缺乏唯一性，并且能量景观具有平坦或浅的方向。因此，在这项工作中，固定点迭代被RPM迭代取代。RPM使用固定点迭代来自适应地识别固定点迭代不稳定的子空间，并且仅在不稳定的子空间上执行牛顿迭代，而固定点迭代在互补的稳定子空间上执行。这种有效的固定点迭代的组合，在可能的情况下，昂贵的，但收敛良好的牛顿迭代，在需要时，导致强大的和有效的收敛的方法。特别是，RPM-FFT收敛良好的参考介质的选择范围很广，而通常的定点迭代通常是敏感的选择。
This paper applies the Recursive Projection Method (RPM) to the problem of finding the effective mechanical response of a periodic heterogeneous solid. Previous works apply the Fast Fourier Transform (FFT) in combination with various fixed-point methods to solve the problem on the periodic unit cell. These have proven extremely powerful in a range of problems ranging from image-based modeling to dislocation plasticity. However, the fixed-point iterations can converge very slowly, or not at all, if the elastic properties have high contrast, such as in the case of voids. The paper examines the reasons for slow, or lack of convergence, in terms of a variational perspective. In particular, when the material contains regions with zero or very small stiffness, there is lack of uniqueness, and the energy landscape has flat or shallow directions. Therefore, in this work, the fixed-point iteration is replaced by the RPM iteration. The RPM uses the fixed-point iteration to adaptively identify the subspace on which fixed-point iterations are unstable, and performs Newton iterations only on the unstable subspace, while fixed-point iterations are performed on the complementary stable subspace. This combination of efficient fixed-point iterations where possible, and expensive but well-convergent Newton iterations where required, is shown to lead to robust and efficient convergence of the method. In particular, RPM-FFT converges well for a wide range of choices of the reference medium, while usual fixed-point iterations are usually sensitive to this choice.