Auxetic metamaterials have garnered significant attention in advanced industrial manufacturing due to their superior mechanical properties, which are characterized by a counterintuitive Negative Poisson’s Ratio (NPR). However, traditional design approaches often rely on linear elasticity assumptions, which fail to accurately predict structural behavior under substantial strain. To satisfy stringent engineering requirements, it is imperative to integrate geometric nonlinearity into the topology optimization framework. In this work, we propose an isogeometric level set topology optimization method specifically tailored for designing geometrically nonlinear auxetic metamaterials. Firstly, the isogeometric level set method is introduced, which seamlessly integrates Computer-Aided Design (CAD) with Computer-Aided Engineering (CAE) to ensure smooth boundary representation and numerical stability. Secondly, the geometric nonlinearity isogeometric analysis (IGA) formulation is derived. By adopting the Total Lagrangian approach, we accurately model the behavior of structures under the large deformation assumption. Thirdly, the generation theory of auxetic metamaterials based on the stretching-dominated assumption is analyzed, and a robust objective function is formulated to maximize the auxetic effect. Moreover, the corresponding topology optimization model is developed, and a comprehensive sensitivity analysis scheme is utilized to guide the structural evolution. Optimization results demonstrate robust performance under large deformation conditions, while the integration of the level set method with isogeometric analysis yields smooth and well-defined structural boundaries. Comprehensive CAE analyses are performed to evaluate the mechanical responses of the optimized configurations. Finally, the effectiveness of the proposed framework is validated through a combination of high-fidelity numerical simulations and experimental testing of models fabricated via Additive Manufacturing (AM), where the experimental results demonstrate strong consistency with the simulation predictions, confirming its superior performance and practical feasibility in geometric nonlinear design.
Yang et al. (Mon,) studied this question.