press SHIFT+D to duplicate those selected faces then enter/left click to validate
press P -> Selection to make those duplicated faces a different object
in Object Mode
select the original object
press M then select one of the layer in the popup to move the original object to another layer, so only the duplicate faces object is on the main layer.
create another object, that will be your basic half sphere at some distance
Make sure your half sphere object origin point is at its bottom :
(to move the origin point, select only the half bottom of the sphere in edit mode, press SHIFT+S -> Cursor to Selected, then in Object Mode click on the “Origin” button of the toolshelf and select "Origin to 3D Cursor)
Make sure the origin of the group of duplicate faces object is at the exact center of the structure
(in edit mode select all, press SHIFT+S -> Cursor to Selected, then in Object Mode click on the “Origin” button of the toolshelf and select "Origin to 3D Cursor)
Leave the 3D Cursor there, at the center of the duplicate faces object, don’t move it anymore
once done, in Object Mode select your half sphere , hold SHIFT then select your duplicated faces object
Press CTRL+P and at the popup select “Object” to parent the half sphere to the other object.
Select the duplicate faces object now and in the Object Data panel, at the “Duplication” tab enable “Face” (dupliface function)
It will do :
dont worry
In the middle of the mess to the left , select the original half sphere (it’s the only thing you are able to select in the messy structure)
Press SHIFT+S and select “Selection to Cursor” , it will move the half sphere object (and the mess too) to the 3D cursor location, that is at the exact center of the duplicated faces object (and why i mentionned to not move the 3D cursor anymore)
In the mess, this time select the duplicate faces object.
At the object Data panel, in the Duplication tab where you already enabled “Faces” , enable “Scale” (you could have done it earlier to help make the messy thing smaller and so select more easily what was in the middle of it)
and change the “Inherit Scale” value until it looks like you want :
-Now in the bottom header, click on Object -> Apply -> Make Duplicate Real (or press CTRL+A -> Make Duplicate Real)
This will make all the small sphere a real object.
You can now select and delete the original big half sphere, and delete the duplicate faces object too.
Then move back the original model on the same layer as the spheres :