Material design using topology optimization with immersed interface finite element method

Materials can be engineered to exhibit tailored effective properties, including behaviors not commonly found in nature. In this work, we present a computational framework that combines topological shape optimization with the Immersed Interface Finite Element Method (IIFEM) to design materials with prescribed effective mechanical properties. A key advantage of the proposed IIFEM-based approach is that it eliminates the need for remeshing or mesh movement during optimization, significantly reducing computational cost and implementation complexity. The effective properties of the evolving microstructures are computed using numerical homogenization. The material topology within the unit cell is represented using a level set formulation, allowing the internal boundaries to evolve naturally and generate optimal periodic microstructures. The design space is parameterized using radial basis functions, which interpolate the level set function, and a standard gradient-based optimization algorithm is employed to determine the optimal coefficients. This formulation yields geometries with smooth boundaries and well-defined material interfaces. Moreover, the compact support of the level set representation, combined with the IIFEM discretization and the dependence of objective and constraint functions only on interface elements, enables efficient and straightforward gradient calculations. Several numerical examples are presented to illustrate the effectiveness and flexibility of the proposed method.