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I have the following mesh:

mesh

I would like to extrude the face at the center so that a dome forms out of it, such that the dome's boundary would be exactly that of the circle. (Think bubble wrap.)

Is there a quick and easy way to do this?

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Yes, try extruding and scaling it multiple times:

enter image description here

To help make the extrusions more even, you might try using the To Sphere operator (⎇ Alt⇧ ShiftS). You can define where the center of the "sphere" To Sphere uses by using the 3D cursor as the pivot (.).

If you want to be more precise, try adding a sphere and deleting half, then using Bridge edge loops (W, E)

enter image description here

Note that the number of vertices on the perimeter of the sphere should match the number of vertices around the initial circle.

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You can get a dome in the centre without extruding.

  1. Add some extra edge loops CtrlR
  2. Enable proportional editing with falloff set to sphere
  3. Lift centre piece up, adjusting falloff size as needed (with scroll wheel or PageUp/PageDown).

enter image description here

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  • $\begingroup$ That's darn clever. $\endgroup$ – PythonNut Jul 27 '15 at 3:09
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I would not extrude it. Place your cursor in the center of your object. Add a sphere in the middle and do a boolean modifier.

enter image description here

Select your original object and set the boolean modifier to union.

enter image description here

enter image description here

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  • $\begingroup$ The problem with the boolean modifier is that it nukes your topology. $\endgroup$ – wchargin Jan 4 '15 at 16:25
  • $\begingroup$ @WChargin as long as both objects are made of quads there should be no problems topology wise. $\endgroup$ – cegaton Jan 4 '15 at 16:32
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    $\begingroup$ Oh, you're right! I had never noticed that. It looks like it is fine when both objects are locally exclusively quad-based (that is, there can be tris far away without it affecting the modifier), although the modifier may produce output that includes higher-order (but planar) ngons. Good to know. (Not my downvote, by the way.) $\endgroup$ – wchargin Jan 4 '15 at 18:28

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