Adaptive Virtual Element Method of Material‐Field Series Expansion Topology Optimization and Its Application in Metamaterials

ABSTRACT This paper proposes a novel adaptive topology optimization framework that integrates the Virtual Element Method (VEM) with the Material‐Field Series Expansion (MFSE). Within the VEM‐MFSE framework, we propose a material gradient‐driven adaptive strategy, in which elements are refined in regions with higher material density gradients as the MFSE sharpening parameter increases. The proposed topology optimization method fully leverages the VEM's advantages of naturally handling arbitrary polygonal elements with hanging nodes. Additionally, MFSE decouples the design variables from the mesh resolution, leading to notable improvements in both the efficiency and convergence of the adaptive optimization. Numerical examples, including the MBB beam benchmark, demonstrate the accuracy and efficiency compared with the Solid Isotropic Material with Penalty (SIMP) method. In addition, an energy‐based homogenization scheme is employed for the topology optimization of periodic metamaterial unit cells, where the analytical sensitivity of the effective stiffness with respect to the MFSE coefficient is derived. The results confirm that the adaptive VEM‐MFSE method achieves accurate, high‐resolution solutions on coarse meshes. For certain problems, the adaptive strategy significantly enhances the convergence behavior of the VEM‐MFSE algorithm. Some relevant code is available at https://www.vemhub.com/code .