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.

This work introduces a new method to deal with design dependent pressure loads in Topology Optimisation (TO) using the SIMP material model. A pronounced focus is on optimising pressure actuated compliant mechanisms. The difficulty herein is the interpretation of the pressure boundary in a TO design. In TO the boundaries are blurry, because of the filtering of the design variables, which is necessary to prevent checkerboarding. Another reason why the boundary is poorly defined is that the optimisation starts from an equally distributed grey, where black and white respectively are solid and void, so the boundary is either the domain boundary or not defined in early iterations of the optimisation.
The methods proposed in literature often try to find the void-solid interface exposed to the pressure source to apply the loading from a pressure line directly. The method proposed in this work, appropriately called the Darcy method, first calculates the pressure field by using Darcy's law governing the flow through porous media and the associated pressure drop. A flow coefficient is introduced that decreases if the virtual element density increases. This results in a design dependent pressure field that can be solved using the Finite Element Method (FEM), which can then be translated to consistent nodal forces that are applied to the TO problem. A drainage coefficient has also been introduced to make sure that the pressure is drained entirely to the environment pressure over the first encountered void-solid interface exposed to the pressure source. The Darcy method has proven to function well in several test cases. The method has been thoroughly tested using several parameter sweeps on a clamping problem objective. The parameters whose influence is examined are: the initial condition, the density threshold value, flow coefficient gradient at the threshold, output spring stiffness and the volume fraction.
Subsequently, some alternative TO problems are solved showing the diversity of the method. In perspective of future research, the Darcy method can function as a great tool to research load sensitivities by tuning the pressure field control parameters. The extension of the Darcy method to 3D or to several load cases comes naturally but has not been tested in this work, this is also recommended for future research.