How to interpolate within defined boundaries of a deformed mesh using R?

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My goal is to create an interpolation map of some output model results that are mapped on a deformed mesh (from a numerical simulation). The model results are in an x-y plane (representing coordinates) with a variable-z to interpolate, however the coordinates are irregularly spaced and have an upper boundary that should be respected.

I am able to plot exactly what I was using a program called Tecplot, however I can only do one image at a time. Using Tecplot is extremely tedious and thus I would like to automate this in R. Here is what I want:



I am currently using this code below in R to do the same thing however my results are less than ideal and I would like to improve my R code.

Here is the link to some example data:link.

Here is my code I am currently using and the non-ideal output of that code:

library(akima)library(ggplot2)library(reshape2)file
 
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