An improved ordered SIMP approach for multiscale concurrent topology optimization with multiple microstructures

Cellular structures are often used to improve the stiffness, fatigue strength, damage tolerance, and other superior functioning properties of structures in engineering. This paper presents a novel multiscale concurrent topology optimization method to optimize structures which are periodically filled with multiple microstructures and connected by solid interfaces. This method generates better structural performance and does not need to preprocess the initial design domain. It enables to save the computational resources, and improves the freedom of structural optimization design to a certain extent. Solid interface layers are built to connect different microstructure blocks, and hence the full degrees of design freedom of microstructures can be further explored. At the macroscale, a novel piecewise projection and a series of gradient-based filtering operations are proposed to distinguish the microstructure blocks and interface layers, respectively. Then, an improved ordered solid isotropic material with penalization (SIMP) method is proposed to optimize the spatial distribution of different microstructures with an affordable computational cost. At the microscale, the microstructures are generated by the numerical homogenization method. Microstructures with different volume constraints are treated as different materials, and the volume fraction limit value of the microstructure corresponds to the design variable of the improved ordered SIMP method. Finally, the compliance minimization problem under the constraint of material volume fractions is investigated, and sensitivity analysis is derived. Several 2D and 3D numerical examples are provided to demonstrate the effectiveness of the proposed method.