Optimizing membrane–substrate buckling to control surface deformation pattern

Buckling of membrane–substrate structures can result in complex deformation patterns on their top surfaces. Traditionally, control over these patterns has relied on altering the material or thickness ratios of the system, which imposes considerable constraints on design flexibility. Inspired by studies on Winkler foundation optimization, this work presents a novel framework for tailoring membrane–substrate buckling modes via topology optimization. In this approach, the material distribution within the substrate is optimized to tune the mechanical response, while filtering and projection techniques are incorporated to enhance manufacturability. Post-processing of the optimized layout yields a physically realizable structure that preserves the desired mechanical behavior. A two-dimensional case study demonstrates that the optimized design successfully generates the prescribed deformation pattern without modifying material properties or thickness ratios, thus enabling precise control over buckling-induced surface morphologies. Experimental validation using fabricated prototypes subjected to compression further confirms that the observed buckling modes closely match the targeted patterns, underscoring the practical effectiveness of the proposed method.