Help for designing a U shape pach antenna

I am trying to design a inv U shape patch antenna with a modified ground plane of W shape. I have taken a science direct paper as my refrence. I am not able to design the patch. Using the normal design equations for a patch antenna I am not able to arrive at the values. Can anyone please help me with the design equations for a U SHAPED PATCH antenna for a center frequency of 2Ghz and substrate as air.

Is not the first time when I read about U, W, E, M or H shape patch antenna (or whatever letters remain from the alphabet).
My concern is not about these antennas and how they behave, is how somebody can believe that is a formula somewhere dedicated for each shape.

For example from the file below, M-shape patch antenna,

http://ceta.mit.edu/PIERL/pierl03/05.08012801.pdf

my question is: How we know actually this is not a W shape, or E shape? Depends how we hold it in our hand?

Could you please help me abt the references or the design equations for a inverted U shape antenna with W shape ground plane