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Most literature studies dealing with design optimization for RF applications focused to a large extend on size and shape optimization. So far, material and topology optimization has not been pursued primarily due to the challenges associated with the fabrication of inhomogeneous materials and the limited access to analysis tools. In this paper, we focus on optimum topology/material design of dielectric substrates for bandwidth enhancement of a simple patch antenna. First, the possibility of designing arbitrary dielectric constant materials using off-the-shelf dielectrics is presented as is necessary for the practical fabrication of inhomogeneous substrates. Then, a formal design optimization procedure is conducted using the solid isotropic material with penalization (SIMP) method by relying on a fast full wave finite element-boundary integral (FE-BI) simulator. The SIMP method is a mathematically well-posed topology optimization algorithm because a continuous density function is used to relate the cell variable to the actual material properties. This also allows for a formulation in a versatile optimization framework. Sequential linear programming (SLP) is used to solve the nonlinear optimization procedure with the sensitivity analysis based on the adjoint variable method. An important advantage of the proposed design optimization approach is its generality to handle multiple objectives and multidisciplinary problems. Using the proposed automated design procedure, inhomogeneous substrates are designed which allow for 250% bandwidth enhancement of the square patch antenna. Typically, only a few iterations are needed to reach convergence. Finally, the designed substrate is post-processed with image processing and fabricated using thermoplastic green machining.