Select Mesh->Unstructured->Assign size on points. A window appears in which to enter the element size around the points to be chosen. Enter 0.1 and click Assign.
Select the point indicated in the figure. Press <Esc> ⁴ to indicate that the selection of points is finished, and Close the window.
Select Mesh->Generate mesh.
A window opens asking whether the previous mesh should be eliminated. Click Yes.
GiD then asks you to enter the general maximum element size. Leave the default value unaltered and click OK.
Click View mesh in the pop up window to see the result.

Geometry of the model. The point around which the mesh will be concentrated.

The mesh with a concentration of elements around the point.
A concentration of elements appears around the chosen point, given the selected size (0.1) of these elements.

One can control the way the size of the elements changes from a finer to a coarser region:

Go to Utilities and open Preferences window. In the Meshing - General branch there is the option Size transitions. This option defines the transition gradient of element sizes (size gradient), whose values are between 0 and 1. The greater the size gradient is, the sharper change of sizes in space. To test this, enter the value 0.4 and click Apply.
Again, generate the mesh (Mesh->Generate mesh) with the same general mesh size.

Mesh with the elements concentrated around a point, with a size gradient of 0.4.

The size gradient (0.4) results in a higher density around the point (see figure).
Now go back and enter 0.6 in Size transitions. This will result in a mesh more suitable for our objectives. Click Apply.

⁴ Instead of pressing the <Esc> key, the center mouse button or the mouse wheel can also be used.

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