Quasi-smooth manifold element-based parameterized level set method for microstructure design with energy homogenization

This paper presents a unified and efficient framework for microstructure topology optimization by combining the quasi-smooth manifold element (QSME), the parameterized level set method (PLSM) and the energy-based homogenization method (EBHM). The QSME provides a high-accuracy and high-order continuity displacement functions with physically meaningful degrees of freedom (DOFs), which significantly enhances structural analysis accuracy on low-resolution triangular and tetrahedral meshes. By combining QSME with PLSM, the proposed method produces smooth and well-defined structure boundaries, and it can efficiently capture complex topological evolution through hole generation, merging, and elimination. Furthermore, a QSME-based EBHM is developed to evaluate the effective material properties of microstructures. In this method, the physically meaningful DOFs of QSME allows the test strain fields to be directly applied to the element nodes. Based on the unified framework, a series of novel 2D and 3D microstructures are designed for maximum bulk modulus, maximum shear modulus, and negative Poisson’s ratio. Numerical examples show that the proposed method can generate smooth microstructures with desirable effective material properties and demonstrate strong robustness with respect to initial structure, volume fraction, and mesh resolution. Comparative studies confirm that QSME provides much higher computational accuracy than FEM in the mechanical analysis of microstructures. The results verify that the proposed method is a reliable and efficient tool for designing high-performance microstructures with clear geometric features and excellent mechanical behavior.