This article deals with the mathematical derivation and the validation over benchmark examples of a numerical method for the solution of a finite-strain holonomic (rate-independent) Cosserat plasticity problem for materials, possibly with microstructure. Two improvements are made in contrast to earlier approaches: First, the micro-rotations are parameterized with the help of an Euler–Rodrigues formula related to quaternions. Secondly, as main result, a novel two-pass preconditioning scheme for searching the energy-minimizing solutions based on the limited memory Broyden–Fletcher–Goldstein–Shanno quasi-Newton method is proposed that consists of a predictor step and a corrector-iteration. After outlining the necessary adaptations to the model, numerical simulations compare the performance and efficiency of the new and the old algorithm. The proposed numerical model can be effectively employed for studying the mechanical response of complicated materials featuring large size effects.

中文翻译：

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Cosserat材料有限应变完整塑性求解方法的数学分析

本文涉及求解材料（可能具有微观结构）的有限应变完整（速率无关）Cosserat 塑性问题的数值方法的数学推导和基准示例的验证。与早期方法相比，有两个改进：首先，在与四元数相关的 Euler-Rodrigues 公式的帮助下，微旋转参数化。其次，作为主要结果，提出了一种基于有限记忆 Broyden-Fletcher-Goldstein-Shanno 拟牛顿法搜索能量最小化解的新型两遍预处理方案，该方案由预测器步骤和校正器迭代组成。在概述了对模型的必要调整后，数值模拟比较了新旧算法的性能和效率。