Creating a proper quad-mesh in GMSH for an "I"-shaped geometry

Question

I'm trying to make a 2-D quad-only mesh of an "I"-shaped cross-section, with elements of size 5. So far I've managed to make it quad only, but some elements are very distorted when they shouldn't be. Also, the total number of elements is incorrect. Can you please help me out?

Here's my .geo code:

Point(1) = {0, 0, 0, 1.0};
Point(2) = {45, 0, 0, 1.0};
Point(3) = {55, 0, 0, 1.0};
Point(4) = {100, 0, 0, 1.0};
Point(5) = {100, 10, 0, 1.0};
Point(6) = {55, 10, 0, 1.0};
Point(7) = {45, 10, 0, 1.0};
Point(8) = {0, 10, 0, 1.0};
Point(9) = {0, 90, 0, 1.0};
Point(10) = {45, 90, 0, 1.0};
Point(11) = {55, 90, 0, 1.0};
Point(12) = {100, 90, 0, 1.0};
Point(13) = {100, 100, 0, 1.0};
Point(14) = {55, 100, 0, 1.0};
Point(15) = {45, 100, 0, 1.0};
Point(16) = {0, 100, 0, 1.0};

Line(1) = {1, 2};
Line(2) = {2, 3};
Line(3) = {3, 4};
Line(4) = {4, 5};
Line(5) = {5, 6};
Line(6) = {6, 11};
Line(7) = {11, 12};
Line(8) = {12, 13};
Line(9) = {13, 14};
Line(10) = {14, 15};
Line(11) = {15, 16};
Line(12) = {16, 9};
Line(13) = {9, 10};
Line(14) = {10, 7};
Line(15) = {7, 8};
Line(16) = {8, 1};

Line Loop(1) = {14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13};

Plane Surface(1) = {1};

Recombine Surface {1};

Transfinite Line {2, 4, 16, 12, 10, 8} = 3 Using Progression 1;
Transfinite Line {14, 6} = 17 Using Progression 1;
Transfinite Line {1, 15, 3, 5, 7, 9, 13, 11} = 10 Using Progression 1;

Answer 1

So, by default and without recombination to quads (comment out Recombine Surface {1}), GMSH creates a mesh like the one in the left part of the picture. And by recombining surfaces into quad-elements you will obtain the mesh in the right part of the picture. Naturally, if your recombination comes from the original triangular mesh, you would expect the "distorted" quad elements you mentioned, as well as, "incorrect" number of elements in general.

By default, GMSH uses Blossom recombination algorithm that will be unable to produce your desired mesh from a "messy" mesh like on the left. Notice, that the original mesh on the left satisfies all your requirements in terms of triangles: it actually places the required number of elements over each side (specified by your Transfinite Line definitions); however, it is free to do whatever it wants inside your I-shaped geometry - and it does.

One of your options, is to change the meshing algorithm for the original triangular mesh. Now, GMSH supports (experimentally) Delaunay for quads 2-D meshing algorithm that should produce the mesh you desire (if I undesrstood your intentions correctly).

You can select this algorithm either in the GUI: Tools->Options->Mesh->General->2-D Meshing Algorithm or by adding the following line to the beginning of your GEO-file:

Mesh.Algorithm = 8; // Delaunay for quads

To have the correct output mesh, you should also add the physical surface in the end of your file:

Physical Surface(1) = {1};