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The field of meta-materials engineering has largely expanded mechanical design possibilities over the last two decades; some notable design advances include the systematic engineering of negative Poisson's ratio materials and functionally graded materials, materials designed for optimal electronic and thermo-mechanical performances, and the design of materials under uncertainty. With these innovations, the systematic engineering of materials for design-specific uses is becoming more common in industrial and military uses. The motivation for this body of research is the design of the shear beam for a non-pneumatic wheel. Previously, a design optimization of a nite element model of the non-pneumatic wheel was completed, where a linear elastic material was simulated in the shear beam to reduce hysteretic energy losses. As part of the optimization, a set of optimal orthotropic material properties and other geometric properties were identified for the shear beam. Given that no such natural linear elastic material exists, a meta-material can be engineered that meets these properties using the aforementioned tools. However, manufacturing constraints prevent the use of standard homogenization analysis and optimization tools in the engineering of the shear beam due to limitations in the accuracy of the homogenization process for thin materials. In this research, the more general volume averaging analysis is shown to be an accurate tool for meta-material analysis for engineering thin-layered materials. Given an accurate analysis method, several optimization formulations are proposed, and optimality conditions are derived to determine the most mathematically feasible and numerically reliable formulation for topology optimization of a material design problem using a continuous material interpolation over the design domain. This formulation is implemented to engineer meta-materials for problems using the volume averaging analysis, which includes the use of variable linking and the derivation of rst-order design sensitivities to increase computational efficiency. Inspired by honeycomb materials, a new method of discretizing the material design domain into unit cells with non-simple connectivity is proposed as a way of increasing the solution space of the topology optimization problem. Finally, these methods are used in the meta-material design process to identify several candidate meta-material geometries from a polycarbonate base material for the shear layer of the non-pneumatic wheel; notable geometries include an `x'-like geometry, a bent column-like geometry identified previously as a bristle, and, remarkably, an auxetic honeycomb geometry. This is the first reported result demonstrating the auxetic honeycomb geometry to be a minimum weight structure in shear loading where a general topology optimization method was used.