Phase-field based shape optimization of uni- and multiaxially loaded nature-inspired porous structures while maintaining characteristic properties

Triply periodic minimal surfaces (TPMS) are highly versatile porous formations that can be defined by formulas. Computationally based, load-specific shape optimization enables tailoring these structures for their respective application areas and thereby enhance their potential. In this investigation, individual sheet-based gyroid structures with varying porosities are specifically optimized with respect to their stiffness. A modified phase-field method is employed to establish a simulation framework for the shape optimization process. Despite constant volume and the preservation of the periodicity of the unit cells, volume redistribution occurs through displacement of the interfaces. The phase-field-based optimization process is detailed using unidirectional loading on three gyroidal unit cells with porosities of 75 %, 80 %, and 85 %. Subsequently, the gyroidal unit cell with a porosity of 85 % is shape-optimized under multidirectional loading. A subsequent experimental validation of the unidirectionally loaded cells confirms that the shape-optimized structures exhibit, on average, higher stiffness than the non-optimized structures. The highest increase of 40 % in effective modulus is achieved with the gyroid structure having a porosity of 75 %, while maintaining minimal alteration to the surface-to-volume ratio and preserving periodicity. Additionally, the experimental data show that the optimization process resulted in a shift in the linear elasticity and plasticity range. In summary, the phase-field method proves to be a valid optimization technique for complex porous structures, allowing the preservation of characteristic properties.