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meshing techniques

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If those are 3 noded triangles DON’T use them. Awful.

Any competent preprocessor can auto mesh that area. You may need to specify 5he number of nodes on each edge.
 
I admit I have a complex geometry, but the software is not so smart as I supposed. Any mesh I tried leaded me to a non-convergence result, except for the structured triangular one. However I don't really like it very much cause there are some distortions. The mesh I uploaded according to me is the best one cause is the most regular or, either, a square one with triangular elements on le long edge
 
Triangles are ok if you use midnodes. Without midnodes you can overestimate stiffness.
What is preprocessor you use?
%D0%A1%D0%BD%D0%B8%D0%BC%D0%BE%D0%BA_acdeij.jpg

Or try to make quada at center.
Figure-1_tkikqp.png
 
mesh the area with triangles as the default element.

use the mid-side node option (as others have suggested, 3 noded trias are not very good ... but try this for yourself to see)

as others have suggested, using quads is typically "better".

you also don't "Have" to use a uniform mesh, depending on the loading, etc.

btw, "half square" made me think of rectangles. "triangle" or more precisely "right triangles" would be a better term (IMHO).
 
Depending on your pre-processor you should be able to get the result you want. This is from Femap with mapped meshing and tri-elements. Some tweaking should fix it if needed.

Good Luck

Thomas

Mesh_pkzujc.png
 
You should also be able to partition the face before you mesh it. You can just draw the vertical and horizontal lines.
 
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