My PhD work is on modeling a 3D printing process using a commercial FEM software. But I am not confident with basics of finite element theory, non linear Finite element analysis to be accurate, which I learnt by self study. My work was mostly application of established models, so I didn’t have to do any numerical work in my research. It's my final year (starting to write my thesis) and I am not happy with my level of expertise. I am freaking out. What do I do? Sad part is I am in 6th year of PhD.
Nobody knows everything. In many fields it is impossible to know everything about that field. In mathematics it was possible for an individual to know everything up through the end of the 19th century, and a few people did. Henri Poincaré was a likely candidate. But after that it soon became impossible as there was just too much. Computer Science has probably passed out of the era where it was possible to know it all, but it was (possibly) somewhere around the end of the 20th century.
There is a psychological effect called Imposter Syndrome in which you don't believe that you are as good as others know you to be. It affects many people, including many academics.
Learning is a life-long activity. I suggest that in the short term you just get on with it and write the thesis, working as you normally do, which I'm sure includes study. In the longer term you can, as you would likely do anyway, work to fill in those bits that you think you are missing.
But don't delay your work now until you think you are ripe. That point may never come, even though you are perfectly capable at each point at which it matters.
The feeling may never go away. But if you know it for what it is, it needn't inhibit you or become debilitating.
It's not optimal but quite natural that during your PhD you always push forward and only learn what you really need at that moment. I assume you are going to write an introduction and a theory chapter. Use this chance to catch up with what you feel you missed. Do some exercises, work on proofs of important theorems in your field. I guess few weeks of concentrated labour will make you feel much more confident regarding the basics. It will pay off when it's time to defend your thesis!
Just write up the details of the applied work you did. Sure, you see a gap in your fundamental understanding, but so be it.
There are analytical chemists who do good applied work but don't understand all the hairy physics math behind their instruments also. Sure, ideally they would. And to some extent they should. But that's just not the case.
Don't make excuses. Write up what you did. Sit down and type. ;-)
Some elements you do need for your thesis are:
- To be able to identify that you have done something novel
- Why it is novel, i.e. understand and articulate what has been done before
- Ideally some notion as to why it works.
- Where there is room for improvement
So, I suggest you focus on the main story you have to tell and make sure you can defend the arguments presented. That does not require an in-depth understanding of all the foregoing theory, but might well require articulation of the core principles.
If you have a chapter of background material, clearly you need to understand it. You do not need to understand all the related theory work. You might well now see that there are various areas to further study later if you stay in academia.
You say you're about to write your dissertation: do you really know how LaTeX works? Or the TeX that underlies it? Or the OS of your computer under that? Or do you just know how to use them? Same applies to your FEM program, or to any tool in any field: you don't need to know how the tool works, internally.or how to build one. You just need to know how to use it to get results.
A little humility at the start of what may be your first real research project may not be a bad thing. Crossing the bridge from hard problems in textbooks to a research problem can be a difficult journey. With textbook problems, it is at least reasonable to assume that the question makes sense and has an answer. With a research problem it may take a few months before you make sense of the problem, and you don't know whether there is an answer. "If you know what you're doing, how long it will take, or how much it will cost, then you're not doing research."
But you should try to get over feeling "unworthy" of a Ph.D. Presumably, some group of people feels you have the potential to write a decent thesis, or you wouldn't be a position to try. It would be a fine thing if you win a Nobel Prize for your work, but the objective for now should be less ambitious. You need to find a worthwhile unsolved problem, get to understand what is known about the general area that's relevant, and start figuring out how you can know what's true and what's not. The amazing thing is that it's not usually possible to know how important your discoveries may turn out to be.
As long as you show you have a general understanding of Finite Element Method and build a story around your application of it, you should be in a strong position to defend your work.
To be more specific - you already mentioned that your work is application of established models. This is absolutely fine if you provide appropriate references. Also, as others have mentioned, you are not expected to know everything so don't be so hard on yourself.
Your adviser's job is to tell you if you are not ready to do this. That is the adviser's job -- we're not even talking about a "psychological support" role but a simple professional judgement about whether you are capable.
If your adviser cannot even do that, find to another adviser. A good adviser is fact one thing that you cannot do without.
Freaking out might be a healthy feeling at this stage. I am not sure if that helps, but I was in your position once and I got my PhD after three years. I remember telling a friend the same thing as you just said (except the technical details were different), and he told me that everything would be alright because you're feeling that way. I am sure the things you just mentioned are not simple concepts and reaching the level to make novel contributions is not an easy road, but you'll get there eventually.