This paper presents a topology optimization framework for designing periodic viscoplastic microstructures under finite deformation. To demonstrate the framework, microstructures with tailored macroscopic mechanical properties, e.g., maximum viscoplastic energy absorption and prescribed zero contraction, are designed. The simulated macroscopic properties are obtained via homogenization wherein the unit cell constitutive model is based on finite strain isotropic hardening viscoplasticity. To solve the coupled equilibrium and constitutive equations, a nested Newton method is used together with an adaptive time-stepping scheme. A well-posed topology optimization problem is formulated by restriction using filtration which is implemented via a periodic version of the Helmholtz partial differential equation filter. The optimization problem is iteratively solved with the method of moving asymptotes, where the path-dependent sensitivities are derived using the adjoint method. The applicability of the framework is demonstrated by optimizing several two-dimensional continuum composites exposed to a wide range of macroscopic strains.

本文提出了一种拓扑优化框架，用于设计有限变形下的周期粘塑性微观结构。为了证明该框架，设计了具有定制的宏观机械性能（例如最大的粘塑性能量吸收和规定的零收缩）的微结构。通过均质化获得模拟的宏观特性，其中，晶胞本构模型基于有限应变各向同性硬化粘塑性。为了求解耦合的平衡方程和本构方程，将嵌套牛顿法与自适应时步法一起使用。通过使用滤波来限制条件，提出了一个恰当的拓扑优化问题，该滤波是通过亥姆霍兹偏微分方程滤波器的周期形式实现的。使用移动渐近线的方法迭代地解决了优化问题，其中使用伴随方法得出与路径相关的灵敏度。该框架的适用性通过优化暴露于各种宏观应变的几种二维连续体复合材料得到了证明。