| The development of new techniques in micro-manufacturing in recent years has enabled the fabrication of material microstructures with essentially arbitrary designs, including those with multiple constituent materials and void space in nearly any geometry. With an essentially open design space, the onus is now on the engineer to design composite materials which are optimal for their purpose. These new materials, called meta-materials or materials with architected microstructures, offer the potential to mix and match properties in a way that exceeds that of traditional composites.;We concentrate on the thermal and elastic properties of isotropic meta-materials, and design microstructures with combinations of Young's modulus, Poisson's ratio, thermal conductivity, thermal expansion, and mass density which are not found among naturally-occurring or traditional composite materials. We also produce designs with thermal expansion far below other materials.;We use homogenization theory to predict the material properties of a bulk meta-material comprised of a periodic lattice of unit cells, then use topology optimization to rearrange two constituent materials and void space within the unit cell in order to extremize an objective function which yields the combinations of properties we seek. This method is quite general and can be extended to consider additional properties of interest. We constrain the design space to satisfy material isotropy directly (2D), or to satisfy cubic symmetry (3D), from which point an isotropy constraint function is easily applied. We develop and use filtering, nonlinear interpolation, and thresholding methods to render the design problem well-posed, and as a result ensure our designs are manufacturable.;We have written two computer implementations of this design methodology. The first is for creating two-dimensional designs, which can run on a serial computer in approximately half an hour. The second is a parallel implementation to allow optimization in three dimensions with a large number of parameters. When running on a high-performance computing cluster, it allows for solutions in a few hours despite the greatly increased computational cost. |
