Engineered geothermal systems are governed by strongly coupled fluid flow and mechanical deformation processes. Such are characterized by low porosity and permeability as well as complex fractured and faulted structures. A typical practical goal is to enhance the effective permeability of the underlying porous medium through high-pressure injections. This results in stimulation of fractured structure. Fractures show complex behavior, being allowed to be open or closed, and when closed then either sticking or sliding depending on the frictional properties. Consequently, the contact states of fractures strongly impact the fluid flow, which is dominated by fracture flow. To model the coupled flow and deformation in fractured systems, we employ a mixed-dimensional modeling concept. Coupled poromechanics equations govern the multi-physics behavior in the matrix, while the Darcy flow in the fracture takes into account varying apertures as well as a cubic-law aperture-dependent permeability. The interaction between matrix and fracture is separately modeled by flow exchange, as well as frictional contact mechanics. The latter introduces a non-smooth character of the mathematical model in addition to the strong nonlinear coupling through the poromechanical coupling as well as the fracture permeability law. The numerical solution is acknowledged to be challenging, both on nonlinear and linear level. In this talk, we present a solution strategy addressing both components. For a robust nonlinear solution, we employ a semi-smooth Newton method for a novel complementarity formulation of the contact conditions together with a line search strategy adopted to the new formulation. In addition, the block structure arising through both the mixed-dimensional and the multi-physics character of the model is exploited during the solution of the resulting linear systems. We discuss realistic three-dimensional numerical examples with intersecting fractures under stimulation to understand both the strengths as well as limitations of our approach.