How can we establish mathematical relationship between S11 and arbitrary geometry?
It's complex- you either 1)measure (ie S-parameters) it-in which case you know the result as accurately as you measure it, or 2) set up a finite element model and use a structure simulator (eg HFSS). There are only simple analytic forms for the simplest of antennas. Good luck. Neil
If you don't want to measure the result which is the best solution then there are many commercialy available modelling packages using FDTD, FE, TLM and boundary element algorithms. All will give you answers. Beware, these are sophisticated tools and you really have to know what you are doing when using them. In particular, the way in which you describe the antenna feedpoint is critical for the S11. Never accept a modelled answer at face value. You must always test your model by varying its parameters and looking for convergence. Also, remember that a model is a model and not reality.
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