Generalized de-homogenization via sawtooth-function-based mapping and its demonstration on data-driven frequency response optimization

De-homogenization is becoming an effective method to significantly expedite the design of high-resolution multiscale structures, but existing methods have thus far been confined to simple static compliance minimization problems. There are two critical issues in accommodating general design cases: enabling the design of unit-cell orientation and using free-form microstructures. In this paper, we propose a generalized de-homogenization method to address these two issues, significantly increasing its applicability in various multiscale design cases. Instead of using conventional square cells with rectangular holes, we devise a parameterized microstructure composed of bars in different directions to provide more diversity in stiffness while retaining geometrical simplicity. The microstructural geometry-property relationship is then surrogated by a multi-layer neural network to avoid costly homogenization analysis during optimization. A Cartesian representation of the rotation angle is incorporated into homogenization-based optimization to design the unit-cell orientation. Corresponding high-resolution multiscale structures are obtained from the homogenization-based designs through a conformal mapping constructed with sawtooth function fields. This allows us to morph complex microstructures into an oriented and compatible tiling pattern, while preserving the local homogenized properties. To demonstrate our method with a specific application, we optimize the frequency response of structures under harmonic excitations within a given frequency range. It is the first time that the de-homogenization framework, enhanced by the sawtooth function, is applied for complex design scenarios beyond static compliance minimization. The examples illustrate that high-resolution multiscale structures can be generated with high efficiency and much better dynamic performance compared with the macroscale-only optimization. Beyond frequency response design, our proposed framework can be applied to general static and dynamic problems.