"We conduct a theoretical and numerical study on the second-harmonic (SH) optical response of nanostructured metamaterials. Metamaterials are composed of a periodic array of metallic or dielectric inclusions. With a proper choice of the shape of the inclusions, we may produce a geometrically non-centrosymmetric and efficient SH response. The linear and nonlinear optical responses can be tuned by simple variations in the geometry of the inclusions. A theory for the calculation of the linear and nonlinear optical properties from the geometry of the system is developed and numerically implemented using Haydock’s recursion method."