三維負泊松比超材料拓撲優化設計

Topology optimization for 3D metamaterials with negative Poisson's ratio

HU Tiannan1， GUO Honghu2 （1. DepartmentofechanicalEngneeringandience，Kyoto UnversityKyoto6553o，Jan;2.Facultyofied Engineering， Waseda University， Kyoto l69-8555， Japan）

Abstract： Negative Poisson's ratio metamaterials have significant application potential in aerospace， biomedical engineering， and flexible electronics due to their anomalous mechanical behavior of expanding laterally under tension and contracting laterally under compression. However， existing research predominantly focuses on 2D or 3D isotropic designs， which struggle to meet practical engineering demands for direction-dependent material properties. This paper proposed a density-based topology optimization method for designing 3D orthotropic negative Poisson's ratio metamaterials. By constructing a novel multi-objective optimization function combined with homogenization theory， negative Poisson's ratio characteristics in three orthogonal directions for unit cell structures was achieved. First， based on the SIMP （solid isotropic material with penalization） material interpolation model， geometric constraints were introduced to ensure unit cell symmetry while eectively reducing computational scale. Second， a new objective function and optimization model were established through penalty functions. Finally， the equivalent mechanical properties were calculated using finite element homogenization under periodic boundary conditions， with design variables updated through sensitivity analysis. Numerical examples demonstrate that the optimized unit cells exhibit negative Poisson's ratio behavior in all three principal directions. This study provides a theoretical foundation for the controllable fabrication of 3D orthotropic auxetic metamaterials and expands the design and application scope of mechanical metamaterials.

Keywords： topology optimization; negative Poisson's ratio; metamaterial; homogenization method; density method

負泊松比材料作為一種力學超材料，因其在單軸載荷下呈現反常的橫向變形行為（拉伸時橫向膨脹，壓縮時橫向收縮），近年來在航空航天、生物醫學、防護裝備及智能傳感器等領域展現出廣闊的應用前景[1]。與傳統材料相比，負泊松比材料在抗剪切性、抗斷裂性和能量吸收率等方面更有優勢[2]。然而，現有研究多集中于二維結構或基于經驗設計的周期性胞元，三維負泊松比超材料的設計仍面臨幾何自由度受限、力學性能可調性不足及制造成本高昂等挑戰[3]。針對這些問題，提出結合拓撲優化方法與均勻化理論，系統研究三維負泊松比超材料的優化設計方法，旨在突破傳統設計框架，實現高性能、可定制化的負泊松比超材料結構設計。

拓撲優化作為一種基于數學模型的材料分布優化方法，能夠通過目標函數與約束條件的靈活配置，高效探索材料與結構的最優構型[4]。均勻化法通過表征微觀結構與宏觀等效性能的關系，為超材料的多尺度設計提供了理論框架。Sigmund 等[5]系統總結了拓撲優化在多物理場超材料設計中的應用，指出其在實現負泊松比、負熱膨脹等多功能耦合方面的潛力。Anaya-Jaimes等提出了基于雙向漸進結構優化的無約束設計框架，結合鄰近材料相間插值、靈敏度穩定化策略及動態添加比率調控，實現了二維正交各向異性超材料的熱膨脹系數精準定制。Murai等運用拓撲優化技術實現了可捕捉局部共振效應，并精準調控多頻電磁波行為的超材料設計。……