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Do we need to perform mesh independence study for the stresss analysis of the parts?
This is rarely done for structural analysis and more for CFD it seems to me,you should check with your professor if it is necessary for your analysis.
To find out if the analysis in mesh independent, what we call mesh convergence, you will need to do a study, what can be called a mesh convergence study, here is a good video about such aspects
https://altair-2.wistia.com/medias/rmretoumym