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As in the following figure if you combine a cylinder with a cube you will find the meshing is something like as in the left; we would like however to have it as in the right.
(at least in the render window when we use a wireframe material for the above.)

How to make something like that?

note: Face beautifier didn't work.

enter image description here enter image description here

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  • $\begingroup$ There's no straight-enough solution for this. I personally suggest you have the cube triangulated / subdivided BEFORE combining them together. (e.g. combining by Boolean Modifier) $\endgroup$ – Leon Cheung Dec 19 '13 at 5:55
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    $\begingroup$ Note, theres an opensource tool called meshlab which has a lot of advanced operations like this. Its possible these could be integrated into Blender, but you could export models to it. $\endgroup$ – ideasman42 Dec 19 '13 at 18:00
  • $\begingroup$ @ideasman42 Nice point. You just reminded us to use resting meshlab on our system. We will try. $\endgroup$ – Developer Dec 20 '13 at 1:11
  • $\begingroup$ @ideasman42 mehlab does not help as whatever we choose from menus (simplifying, remeshing etc) the highlighted (yellow) lines remain unchanged! If you could we would suggest you to put your idea as another full answer, probably with some snapshots. $\endgroup$ – Developer Dec 20 '13 at 5:09
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While having the magic one button solution is enticing the old manual approach can give more control using basic modelling techniques.

Start by removing the unwanted edges.

enter image description here

See how many vertices you need - you can optionally add corner edges to clarify the targets. In this example I need to add three more vertices to each edge.

enter image description here

Use CtrlR to add more edge loops - use the mouse scroll wheel or + and - on the numpad to change the number of cuts being added.

enter image description here

Select all the vertices in two edge loops to be filled in.

enter image description here

Choose Grid Fill (2.69 or higher) - this is available in File->Faces->Grid Fill or search after pressing space.

enter image description here

For versions before 2.69 you can use Bridge Edge Loops - File->Edges->Bridge Edge Loops or search after pressing space. This is still available in 2.69. You may find this works better in edge select mode without any corner connecting edges being selected, while grid fill works better if you have the corner edges in place.

The grid fill initially uses the order the vertices are selected to fill in the faces, if you initially get a strange fill you may want to press F6 (also available at the bottom of the toolbar T) to adjust the Offset until it fills in nicely.

enter image description here

You can also adjust how the bridge edge loops works

enter image description here

Repeat for each side. This gives you a nice final topology

enter image description here

You can also adjust this along the way by adding lop cuts between the two edges. The bridge edges has the option to do this for you and the grid fill will join these edges when you fill in the last two sides.

If you want the same result on the opposite side, if both are the same, you can select the top vertices and duplicate them, move to the opposite side using snap to vertex to align them and remove doubles to join together.

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    $\begingroup$ Note on these instructions (grid fill), theres an area for a step to remove: Notice there is a white vertex. Before running grid fill -- select a corner vertex last, Then grid fill will know to treat this as a corner vertex and it attempts calculate the best square fill to perform without having to adjust settings. $\endgroup$ – ideasman42 Dec 19 '13 at 17:56
  • $\begingroup$ Accepted, because it is an interesting solution, indeed. More interesting is your effort with passion on finding a working solution which is appreciated. $\endgroup$ – Developer Dec 20 '13 at 1:09
  • $\begingroup$ @ideasman42 - a good tip to know but while I could take out adjusting the gridfill options, if someone just uses box select and gets a bad fill they just think it is useless. I think it helps to show ways to fix odd fills and highlight that you can adjust things not just accept defaults. Remembering or noticing which vertex is active or needs to be active is something that can easily be skipped. $\endgroup$ – sambler Dec 20 '13 at 3:20
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You may try adding a remesh modifier to the object.

In the properties view click the tool icon for the modifiers panel.

Without Remesh

Without Remesh

With Remesh

With Remesh

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  • $\begingroup$ Good tool. However we couldn't apply Remesh modifier to selected faces. It seems it can be applied on a whole object for which the result is very acceptable as in your example. You got our up-vote for this. $\endgroup$ – Developer Dec 21 '13 at 3:58
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What might work is entering sculpt mode, box mask the central cylinder (available in trunk builds) from the top view and then enable dyntopo and stroke on the top surface with a standard draw brush with zero strength.

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