Integrated multiscale topology optimization based on the Bi-directional evolutionary method

Traditional multiscale topology optimization methods face a fundamental trade-off: concurrent multiscale topology optimization (CMTO) require predefined volume fractions and limited diversity for representative volume elements (RVEs), hindering optimal multiscale design, while full-scale methods are computationally prohibitive. To overcome these limitations, this paper introduces an integrated multiscale topology optimization (IMTO) method that enhances efficiency without sacrificing performance. The proposed method employs the Direct FE² method to improve the solution efficiency of multiscale structures, thereby avoiding the iterative solution process between macro and micro scales inherent in conventional FE² methods. Additionally, IMTO is implemented based on the bi-directional evolutionary structural optimization (BESO) method, which prevents the emergence of gray elements. The primary contribution of this method lies in its coupling of topologies across different scales by leveraging the intrinsic connections between macroscale and microscale design variables. This not only reduces the number of design variables and simplifies the multiscale optimization workflow but, more importantly, eliminates the need for predefining volume fractions for different RVEs, enabling adaptive material distribution and thereby laying the foundation for high-performance multiscale structural design. A novel dual-scale filtering method is also proposed to eliminate mesh dependency at two scales. Numerical examples demonstrate that the optimized results obtained via IMTO achieve structural stiffness comparable to that of the full-scale approach, while significantly reducing computational time. Furthermore, the numerical examples verify that adjusting the weights in the objective function allows for additional control over the deformation design of multiscale structures.