Shape and thickness optimization of micro-shell structure with triply periodic minimal surfaces embedded microstructure in macrostructure

To meet the demand for light-weight structures, many researchers have investigated TPMS (Triply Periodic Minimal Surface) structures for various micro design applications. This study proposes a multi-scale, non-parametric shape and thickness optimization method for TPMS-based micro-shell structures embedded within a macrostructure. The effective properties of these porous micro-shells are evaluated using the NIAH (New Implementation of Asymptotic Homogenization) method. The distributed thickness and shape of the TPMS micro-shell structures are used as design variables. By minimizing the squared error norm between actual and target displacements at arbitrary macrostructure points, the method controls displacement while satisfying a total volume constraint that includes the micro-shell volume. The optimization problem is formulated as a distributed parameter optimization, with sensitivity functions derived using the numerical derivative and adjoint variable methods. These sensitivities are then integrated into a scalar-type H1 gradient method for thickness optimization and a vector-type H1 gradient method for shape optimization. Numerical examples confirm the method’s effectiveness in determining optimal shape and thickness distributions of TPMS-based micro-shell structures to achieve desired displacement control within a macrostructure.