This article treats finite element simulations of controlled wrinkle formation experiments of a soft bulk material with a thin, stiff layer on top. The wrinkling process is triggered by a stress mismatch between the bulk material and the thin layer. For the finite element simulations, we model the bulk material using a three-dimensional hyperelastic material and the thin layer with a geometrically nonlinear elastic Cosserat shell. For the finite element simulations, we model the bulk material using a three-dimensional hyperelastic material and the thin layer with a geometrically nonlinear elastic Cosserat shell. We use Lagrange finite elements for the bulk material and geodesic finite elements for the shell. The resulting minimization problem is nonlinear and nonconvex. We prove existence of minimizers in the continuous and the discrete function space. Finally, we solve the resulting nonconvex minimization problem numerically using a Riemannian trust-region algorithm and compare our simulations to real experiments.