Negative linear compressibility (NLC) describes the relative increase or decrease in a material's linear dimension when subjected to an increase or decrease in external pressure, or a decrease or increase in internal pressure, respectively. This is a rare material property found in only a few naturally existing materials. These materials are not only limited by their availability but also by the range, strength, and stability of NLC behavior. This limitation can be addressed through the design of NLC metamaterials, which are engineered materials composed of repeating architectures or material layouts on the microscopic scale, known as base or unit cells. These cells define the macroscopic properties of the material. In this case, negative linear compressibility.

While there are different methods for designing NLC metamaterials, none of them involve the use of topology optimization (TO), which serves as a powerful tool for designing optimized metamaterial structures. Materials can be designed for different parameters such as base material and pressure applied, while also considering different constraints that may be application-specific.

This study aims to create isotropic NLC metamaterials by designing NLC metamaterial unit cells using a systematic design methodology. We achieve this goal using a density-based TO approach, incorporating different constraints and selecting appropriate parameters to obtain different metamaterial unit cells that exhibit NLC behavior in both two and three dimensions.

The resulting 2D designs exhibited an NLC value of -2.370 %/bar and -3.367 %/bar. While the 3D designs exhibited an NLC value of -2.212 %/bar and -3.534 %/bar. The obtained NLC value and the efficacy of the design method are validated through numerical analysis and experimental testing, with the experimental design showing a maximum deviation of 26.099% from the NLC value obtained through TO. Finally, comparative and parameter studies helped better elucidate the advantages, limitations, and areas for improvement of the methodology used to obtain these designs.