Chamfer or bevel an edge with a smoothly varying depth, as if whittled or gouged out My goal is to create something like a &quot;lark's tongue chamfer&quot; along part of each of 4 edges of a cuboid. This would involve a bevel that has a variable depth that forms a smooth curve, starting at 0 depth, reaching some peak depth at the middle, and then returning to 0 at the other side. The main thing is that I would like to know different ways to control the depth as a curve.
Here's an example, sort of following a circular curve, but other shapes (&quot;smooth&quot;, &quot;sphere&quot;, &quot;root&quot;, etc) would be desirable:

I've created this mesh, but my procedure was long and convoluted, and I'm sure there's a better way someone can suggest.
Additionally, can someone clarify for me: (a) Is there a name for this sort of bevel, chamfer, or shape in general?; (b) Later I want to round off all of the sharp edges (i.e. with a tiny bevel) and I want to (say) subdivide the surface to smooth it out, and for this I understand it might be important to avoid having the triangles shown in my image above, so what should be done about this?
Anyway, below is what I did, including screenshots of key steps. Apologies if the screenshots are not very clear at their default size, but the image itself is available as a larger version. Note that units are in metres...

Image A: Create a 2-metre plane, and go into Edit mode. Subdivide the face 8 times, creating a sheet as a 9x9 grid
Duplicate and move up by a known fixed amount on Z axis (e.g. 0.2m)
Image B: Repeat, duplicating the sheet copy that was made in step 3
Image C: Select the edge loop of the top-most sheet, then hide all but this selection (SHIFT+H). Go to top view, then select the middle-most edge segment of one side
Image D: Turn on Proportional Editing, set to &quot;Sphere&quot; mode. Move the selected edge inwards (i.e. on the X axis only) by 0.1, setting the proportional size to (say) 0.55
Image E: Repeat with the other 3 sides
Image F: Unhide everything with ALT+H. Select everything in the middle sheet now
Image G: Hide everything else with SHIFT+H. Go to top view, then select and delete all interior edges, leaving just the square outline edges. Select the middle edge of each side of the square
Image H: Go to side view and then move these edges again, this time down by -0.1 on Z axis with proportional size still the same as above
Image I: Unhide everything with ALT+H. Select all edges of the top sheet
Image J: Move the top sheet down to superimpose it on the middle sheet (making sure to turn off proportional editing)
Image K: Select the edge loops of the top sheet, and also the edge loops of the middle sheet
Image L: Click RMB and select &quot;Bridge Edge Loops&quot;
Image M: Select the edge loops of the middle sheet, and also of the bottom sheet, and then Bridge Edge Loops there, too
Image N: Select all, and then Mesh =&gt; Cleanup =&gt; Degenerate dissolve. Fix normals with SHIFT+N. Deselect all, and we're done!
What an amazingly detailed post!
Would something like this work for you?

To save too much adjustment in separate places, starting with 1/8
square tile, open on all the sides you can't see, here...
Assign Edge Bevel weights to the edges shown in shades of blue, all at 1 will do to start with...



Assign 2 Mirror modifiers, one about the Empty rotated  in Z by 45, and another in X,Y,and Z.



Assign a Bevel modifier, by width, limited by Weight, segments about 5, with a profile of about 0.15



Assign a second Bevel modifier, to hold the sharp edges under subdivision,limited by Angle, up around 80 degrees, and a Subdivision Surface modifier.


Tweak the Edge Bevel weights, and other Bevel parameters, to get the scalloped shape you want. If you can't get all the way there, you may have to apply the first Bevel modifier, and tweak that in the geometry, by hand.
Edit: Just noticed your first Bevel should probably have only 1 segment..