  We present a new framework for the simultaneous optimiziation of both the
topology as well as the relative density grading of cellular structures and
materials, also known as lattices. Due to manufacturing constraints, the
optimization problem falls into the class of NP-complete mixed-integer
nonlinear programming problems. To tackle this difficulty, we obtain a relaxed
problem from a multiplicative split of the relative density and a penalization
approach. The sensitivities of the objective function are derived such that any
gradient-based solver might be applied for the iterative update of the design
variables. In a next step, we introduce a material model that is parametric in
the design variables of interest and suitable to describe the isotropic
deformation behavior of quasi-stochastic lattices. For that, we derive and
implement further physical constraints and enhance a physics-augmented neural
network from the literature that was formulated initially for rhombic
materials. Finally, to illustrate the applicability of the method, we
incorporate the material model into our computational framework and exemplary
optimize two-and three-dimensional benchmark structures as well as a complex
aircraft component.
