It is okay to take a shower after wudu, and it doesn’t invalidate it.
The only thing to remember is, when performing wudu, the head and feet that are wiped must be dry prior to wiping, and if they are wet to the extent that if you wipe your wet hand on them, the wetness of the hand will not have any effect, the wudu will be problematic; but if the dampness of the head or feet is so little that when you wipe your wet hand on them, the wetness they have afterwards is attributed to the wetness of the hand only [and not the previous dampness they already had], then it is valid. All of what was said has to do with before one has performed wudu, so if after making wudu, one takes a shower, it will have no effect on the wudu and it will be valid.
 Tawdih al-Masa’el (annotated by Imam Khomeini), vol. 1, pg. 158, under issue 256.