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Plotting the load-displacement curve using Ritz approach for the non linear equations of a plate

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Hello! I am using Matlab and I am trying to obtain the correct shape for the load displacement curve for a square plate in which two near sides are clamped, the other two are simply supported, and loaded along one of the simply supported sides. I am using the non linear Mindlin model for a plate with imperfections, and I used the Ritz approach with legendre polynomials to find the approximate solution with a legendre order of n=1. (it should be similar to a cubic function) but instead at the beginning is vertical, and then suddenly has an upwards concavity, similarly to a perfect plate curve. The equations have been written by computing the gradient of the total potential energy and putting it equal to a zero vector.
I think the problem could be in the Newton method, since I have doubts on the choice of the inital guess, or the definition of the imperfections in the deformation tensor. If you need clarifications on the code, ask me.

Any suggestion is appreciated, for reparing / improving the code.

Thank you in advance.

Here's the code:

Consider Paul's rocket mass heater.
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