This study introduces a novel methodology, leveraging numerical homogenization and topology optimization, for the systematic design of graded index (GRIN) lenses. These lenses play a crucial role in concentrating wave signals in diverse fields such as electromagnetism and acoustics. The focus here is on the creation of a 2D Luneburg lens tailored for underwater acoustic applications. Typically, the realization of such lenses involves discretizing their volume into a finite lattice. Each cell's refractive index must adhere precisely to a law that varies with the lens's radial coordinate for optimal functionality. Maintaining a constant impedance across the lens to prevent internal reflections is crucial, necessitating stringent conditions on the density and bulk modulus of each cell. Furthermore, achieving a fluid-like behavior in the lens to minimize shear wave propagation requires materials with a shear modulus as close to zero as possible, referred to as pentamodes (3D) or bimodes (2D). Traditional design processes struggle to meet these intricate requirements, especially when dealing with numerous cells. To address this challenge, we propose a fully automated approach leveraging topology optimization and numerical homogenization. This method shapes each cell to meet design specifications, including density and bulk modulus, and minimizes the equivalent shear modulus to approximate bimodal behavior. The lens's geometry is implemented in COMSOL, and its acoustic performance is rigorously evaluated against a finite element simulation incorporating full structural-acoustic coupling. This comprehensive analysis provides insights into how well the lens meets design specifications and highlights its frequency bandwidth. In conclusion, this automated approach not only streamlines the GRIN lens design process but also has the potential to yield devices with enhanced performance.