Thermoelectric (TE) effects as a coupling between heat and charge transfer can be described on a classical level in the framework of the Onsager theory. Under isotropic and steady state conditions the conservation equations can be combined to obtain a thermal energy balance containing the temperature distribution as target function. Besides the temperature the balance equation contains material properties represented by the Seebeck coefficient S, the electrical and thermal conductivities σ and κ, respectively. For the sake of simplicity, a 1D scheme has been chosen for the analytical and numerical treatment. Performance investigations are often done within the framework of the Constant Properties Model (CPM) or based on temperature dependent material properties. In the 1D steady state, there is an alternative approach available based on spatial material profiles. Following the approach by Müller and co-workers, the temperature profile T(x) is calculated numerically within a model-free setup directly from the 1D thermal energy balance, e.g., based on continuous monotonous gradient functions for all material profiles, and independent and free variability of the material parameters S(x), σ(x), and κ(x) is assumed initially. Doing so, the optimum electrical current density can be determined from the maximum of the global performance parameter (power output P or efficiency η). We present analytical results for the performance optimization calculating P and η with linear material profiles for S(x), a constant electrical and thermal conductivity, fixed TE element length L and fixed boundary temperatures.