If you want to calculate your own illumination you need to make a custom shader and implement the light() method.
The most simple illumination is n•l (read N dot L), which simply means that you can calculate basic illumination by using the result of the dot product of the fragment's normal and the light's direction.
Look up "ray tracing" to find out more. Scratchapixel is an excellent site. I would check out Introduction to Shading.
However...
I don't think that's what you really want. You obviously want illumination but not the harsh shadows, so disabling illumination and recreating it from scratch is almost certainly not what you're after.
Go back to either standard or toon shading, but disable shadows on your mesh as I suggested in response to your first post about this.
And I will repeat my advice from the previous thread about this issue:
Spend some time to study actual real life photographic lighting.
You will continue to struggle until you have a basic grasp on this subject.
Those links again are photographic lighting and 3 point lighting. Youtube is also full of information on the subject. Just search for "3 point lighting" or "portrait lighting".
And if you want a book I suggest [digital] Lighting & Rendering by Jeremy Birn. It's an older book, but still very relevant.
