The topology optimization (TO) framework for isotropic and orthotropic multi-material periodic microstructures with auxetic performance is proposed based on element-free Galerkin method (EFGM). Negative Poisson's ratio is chosen as the objective function while satisfying specified volume constraints, and the effective elastic properties of the multi-material microstructures are computed by the energy-based homogenization method under periodic boundary conditions. The alternating active-phase algorithm is integrated into meshless solid isotropic microstructure with penalization (SIMP) scheme, and the optimal topological configuration of the multiphase composite is obtained by solving the subproblem of each two-phase materials' distribution. Several topological optimization examples are presented to evaluate the effects of different material design parameters on the topological configurations, minimum objective functions, and negative Poisson's ratios (NPRs) of microstructures. The results demonstrate that the optimum microstructures with isotropic and orthotropic multiple materials can generate more diversified patterns of re-entrant and chiral topological configurations and wider ranges of NPRs in comparison to those with a single material. The auxetic performance of microstructures will be effectively improved by reasonably adjusting the material design parameters and the obtained NPRs will be easy to get close to or break through -1.

中文翻译：

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基于无单元伽辽金法的各向同性和正交各向异性多元材料拉胀微结构拓扑优化


基于无单元伽辽金法（EFGM），提出了具有拉胀性能的各向同性和正交各向异性多材料周期性微结构的拓扑优化（TO）框架。在满足指定体积约束的情况下，选择负泊松比作为目标函数，采用基于能量的均匀化方法计算周期性边界条件下多材料微结构的有效弹性特性。将交替活性相算法集成到带惩罚的无网格固体各向同性微观结构（SIMP）方案中，通过求解各两相材料分布的子问题获得多相复合材料的最优拓扑构型。提出了几个拓扑优化示例，以评估不同材料设计参数对微观结构的拓扑配置、最小目标函数和负泊松比 (NPR) 的影响。结果表明，与单一材料相比，各向同性和正交各向异性多种材料的最佳微观结构可以产生更加多样化的重入和手性拓扑结构图案以及更广泛的 NPR 范围。通过合理调整材料设计参数，可有效改善微结构的拉胀性能，所得NPR易于接近或突破-1。