Gyroids belong to the class of Triply Periodic Minimal Surfaces (TPMS) and are characterized by continuous and smooth curvature, which naturally reduces stress concentrations. Owing to these features, they are attracting growing interest in several engineering fields, including high-performance structural components, additive manufacturing, and biomechanical applications. A significant interest lies in the analysis and design of parameterized TPMS geometries, which often need to be evaluated within automated optimization loops or parametric studies. In such contexts, traditional body-fitted meshing can become a bottleneck due to the difficulties associated with robust and automated re-discretization of evolving surfaces. To address this issue, we employ the Finite Cell Method (FCM) [2], which bypasses the need for body-conforming meshes by embedding the parameterized geometry into a fixed, non-conforming Cartesian grid. The primary advantage of this approach lies in the ability of the FCM to directly process geometries defined implicitly through trigonometric equations, ensuring that the smooth mathematical boundary is preserved throughout the simulation without intermediate discretization steps or geometric approximations. Within this framework, we test and validate hyperelastic constitutive models [3, 4] to capture the nonlinear response and large-deformation behavior of TPMS-based structures. By leveraging the ex act implicit description, the Finite Cell Method provides a robust and mesh-independent framework for the analysis of TPMS geometries, laying the foundation for their automated design and optimization in future engineering applications.