In this contribution, we review our recent numerical and analytical results on the phase structure of the matrix version of the Grosse-Wulkenhaar model and its renormalizability. The main numerical result is that the curvature of the model shifts the phase transition lines and eliminates the stripe phase associated with UV/IR mixing. This is confirmed analytically in the strong interaction regime up to 4th order in the curvature coupling. Here we further calculate the 6th order effective action and also provide explicit expressions for some useful integrals over the unitary group with integrands containing up to 12 unitary matrices.