Length‐Scale Effects in Multiscale Structural Optimization Through a Second‐Order Homogenization Model

ABSTRACT Multiscale structural optimization aims to improve structural performance through the simultaneous design of structural layout and local material properties. Through this expanded design space, the complex interactions between microscale and macroscale physics offer new opportunities to improve structural performance at the cost of modeling and computational complexity. In this work, a second‐order homogenization model is incorporated into multiscale structural optimization to capture length‐scale effects induced by spatially varying microarchitecture. A comprehensive study of the influence of length‐scale effects on structural optimization is performed. The approach uses a mixed finite element scheme to analyze the higher‐order continuum, recovering the second‐order gradients and the higher‐order stresses induced by the second‐order material model. A second‐order homogenization scheme is defined through the method of multiscale virtual power (MMVP). Based on this model, several second‐order microstructural material design models are compared, including a solid isotropic material with penalization (SIMP) model, polynomial interpolation, and a neural network (NN) surrogate model. The NN model, trained via finite element analysis of the second‐order homogenization problem, is applied in several multiscale structural optimization examples using linear elastic materials. The performance of structures optimized for minimum compliance is compared for several first‐order and second‐order material models to inform the selection of second‐order design models and demonstrate the practical effects of length scale in an optimization context.