Synthesis of auxetic structures using optimization of compliant mechanisms and a micropolar material model

Abstract

Aim of this work is the synthesis of auxetic structures using a topology optimization approach for micropolar (or Cosserat) materials. A distributed compliant mechanism design problem is formulated, adopting a SIMP–like model to approximate the constitutive parameters of 2D micropolar bodies. The robustness of the proposed approach is assessed through numerical examples concerning the optimal design of structures that can expand perpendicularly to an applied tensile stress. The influence of the material characteristic length on the optimal layouts is investigated. Depending on the inherent flexural stiffness of micropolar solids, truss–like solutions typical of Cauchy solids are replaced by curved beam–like material distributions. No homogenization technique is implemented, since the proposed design approach applies to elements made of microstructured material with prescribed properties and not to the material itself

3D auxetic single material periodic structure with ultra-wide tunable bandgap

“…1c ), the ellipsoides which lay on the plane orthogonal to the load move away from each other’s by more than half of the applied displacement, thus showing a negative Poisson’s ratio ( v ≈ − 0.6). It is important to notice that, by proper design of the elastic connection, a Poisson’s ratio near to −1.0 can be reached 42 , 43 , thus resulting in a fully symmetric 3D behaviour. In this work the manufacturability constraints lead to the adopted design.…”

“…In this work, a 3D single-phase PnC structure endowed with ultra-wide complete 3D bandgaps is proposed. The tunability of the first bandgap is obtained by exploiting the negative Poisson’s ratio of its unit cells, whose topology is a mix of oustanding PnC properties 15 and 3D-extension of the results of a proper topology optimization on the auxetic behaviour 42 , 43 . In the first part of the paper, numerical simulations are adopted to prove the bangap tuning as a strict consequence of the expansion in all the orthogonal directions of the auxetic unit cell.…”

Optimal Design of the Band Structure for Beam Lattice Metamaterials

“…For the sake of clarity, it is worth remarking that the present issue differs from other band gap maximization problems, which specifically deal with the topological optimization of phononic materials. Indeed, although pursuing the same objective (the largest gap amplitude), the topological optimization seeks for the optimal distribution of two or more material phases in a sufficiently-fine pixelation of the periodic cell (Cox and Dobson, 2000;Shen et al, 2003;Sigmund and Jensen, 2003;Kaminakis and Stavroulakis, 2012;Bruggi et al, 2017). On the contrary, here, for both the beam lattice material and metamaterial, the topology of the periodic cell is fixed a priori, whereas the parametric optimization is limited to the cellular micro-structural parameters, whose values allow to distinguish among different materials belonging to the same topological class.…”

Topology Optimization of Multiple Materials Compliant Mechanisms Based on Sequence Interpolation Model and Multigrid Method

“…The optimal layout pattern of multi-material compliant mechanism is obtained by using the modified optimality criterion method. Widely studied * 国家自然科学基金(51775308, 51705287, 51475265)、湖北省科技支撑计 [5] 。 现有柔性机构的设计方法大体分为两大类：基 于运动学的方法和拓扑优化方法 [6][7][8] 。基于运动学的 方法主要思想是将所期望的柔性机构的设计等同于 传统的刚性连杆机构运动学设计。这种方法很大程 度上依赖于设计者对最终柔性系统直觉的判断和经 验 [9] 。近年来，拓扑优化技术在理论研究和工程应 用中均取得了长足发展，拓扑优化方法因其在设计 域内材料高效分布、尺度局限性小、设计结果不依 赖于人为经验等方面更具优势，受到众多研究人员 的关注 [10][11][12][13] 。目前，已有多种拓扑优化方法被成功 应用于柔性机构的设计 [14][15][16][17][18] 。多材料柔性机构拓扑 优化是指在限定设计域、给定的约束条件及载荷作 用条件下，寻求各种材料在设计空间内的最优布局 形式，获得最佳传力途径，以实现预定的功能或运 动，并使某种设计目标达到最优。SIGMUND [19] 研 究了几何非线性大变形情况下多材料电热制动器的 拓扑优化设计，在此基础上，SAXENA [20] 基于遗传 算法实现了大变形条件下多材料柔性机构的设计。 YOON [21] 提出了考虑多材料及其独立边界条件下的 几何非线性动态拓扑优化策略。基于水平集的拓扑 优化方法因其能使各种材料边界光滑清晰、方便提 取拓扑构型等独特优势， 受到了许多的关注与研究。 LUO 等 [22] 在参数化水平集框架内研究了电-热耦合 多材料柔性微激励器的拓扑优化问题，实现了多物 理场作用下多材料微机电系统结构的线性和几何非 线性的拓扑形状优化设计。WANG 等 [23] 以力学增益 为优化目标，扩展水平集方法，实现了多材料整体 式柔性机构的设计。同时，多输入多输出的多材料 柔性机构拓扑优化设计是柔性机构的重点研究方向 之一。张宪民等 [24][25] 提出一种并行策略，将多材料 问题离散为单材料子问题，实现了多输入多输出柔 顺机构在热固耦合条件下的多目标拓扑优化。 上述研究基本是将应用于单材料柔性机构的拓 扑优化方法理论直接扩展至多材料柔性机构的设 计，相比于单材料设计，多材料拓扑优化目前仍存 在诸多挑战 [26][27] 。其一是缺少能表征多种材料属性 的插值模型，使设计域内每种材料都能被有效地描 述。应用于单材料的插值模型由于其独特的 0/1 两 端极化特性，每次只能处理两种材料，所以将原多 材料拓扑优化分解成若干子问题进行分层优化，成 倍地增加了设计变量数量，计算开销巨大。目前针 对这方面的问题，有学者开始探索用于多材料拓扑 优化的插值策略。ALONSO 等 [28][29] 提出多材料柔性 机构的序列单元互斥策略，以单元材料增删的不同 标准，实现材料在其预定义序列中重布。YIN 等 [30] 提出独特的峰值函数多材料插值模型，用单一变量 描述多种材料属性。 ZUO 等 [31] 提出序列幂函数插值 方法，实现了质量和材料费用约束下离散变量结构 多材料连续体结构拓扑优化。其二是缺乏高效的数 值方法兼顾多材料柔性机构优化构型清晰度和计算 效率。现有多材料柔性机构的拓扑优化流程中，有 限元分析求解占用了迭代过程中绝大部分的计算和 存储成本，理想的拓扑优化框架是在减少分析和优 化的计算成本条件下得到较高分辨率的优化结果， 而目前采用的传统有限元计算方法，为了得到清晰 的拓扑结构，直接用细密网格将设计域整体离散， 网格数量必然巨大，计算效率低下。多重网格方法 是一种高效的多水平迭代算法，基于 Krylov 子空间 迭代求解器，采用预处理共轭梯度法，利用较少的 网格得到了较高的求解精度，极大地提升了计算效 率。ADMIR 等 [32] 首次将多重网格方法引入拓扑优 化中，旨在解决三维结构拓扑优化等大规模复杂优 化问题。该方法应用于单材料稳健性拓扑优化中， 表现出网格独立收敛性以及良好的并行可扩展性， 实现了柔性机构的拓扑优化设计 [33][34] …”