Higher-order phase-field isogeometric analysis for homogenization of porous media: A multi-patch comparative framework

Computational homogenization of porous media is often hindered by the complexity of mesh generation. While isogeometric analysis (IGA) mitigates geometric errors, multi-patch parameterization for internal voids remains cumbersome. This paper introduces a single-patch, higher-order phase-field isogeometric framework to overcome these constraints for two-dimensional periodic porous materials. An implicit scalar phase-field variable is utilized to capture arbitrary void morphologies without explicit meshing. We develop both second- and fourth-order phasefield formulations, effectively exploiting the high-order continuity of NURBS basis functions. The approach is validated against a conventional multi-patch IGA strategy via Representative Volume Elements (RVEs) under periodic boundary conditions. The results show that the phase-field models achieve high fidelity while significantly reducing pre-processing complexity. The comparison between second- and fourth-order models further clarifies tradeoffs between accuracy, interface smoothness, and computational cost. Ultimately, this unified framework offers a robust and efficient solution for statistical homogenization and the automated design of complex metamaterials.