>Basically every advanced/"next generation" thing the article touts has fallen flat on its face when applied to real problems

Even Arnold's work? FEEC seemed quite promising last time I was reading about it, but never seemed to get much traction in the wider FEM world.

I kind of thought Neural Operators were slotting into the some problem domains where FEM is used (based on recent work in weather modelling, cloth modelling, etc) and thought there was some sort of FEM -> NO lineage. Did I completely misunderstand that whole thing?

Those are definitely up next in the flashy-new-thing pipeline and I'm not that up to speed on them yet.

Another group within my company is evaluating them right now and the early results seems to be "not very accurate, but directionally correct and very fast" so there may be some value in non-FEM experts using them to quickly tell if A or B is a better design; but will still need a more proper analysis in more accurate tools.

It's still early though and we're just starting to see the first non-research solvers hitting the market.

Very curious, we are getting good results with PiNN and operators, what's your domain?

I was under the impression that the linear systems that come out of FEM methods are in some cases being solved by neural networks (or partially, e.g. as a preconditioner in an iterative scheme), but I don't know the details.

Could you write a blogpost-style article on how to model the shallow water wave equation on a sphere? The article would start with the simplest possible method, something that could be implemented in short C program, and would continue with a progressively more accurate and complex methods.

If you are interested in this, I'd recommend following an openfoam tutorial, c++ though.

You could do SWE with finite elements, but generally finite volumes would be your choice to handle any potential discontinuities and is more stable and accurate for practical problems.

Here is a tutorial. https://www.tfd.chalmers.se/~hani/kurser/OS_CFD_2010/johanPi...

I'm looking for something like this, but more advanced. The common problem with such tutorials is that they stop with the simplest geometry (square) and the simplest finite difference method.

What's unclear to me is how do I model the spherical geometry without exploding the complexity of the solution. I know that a fully custom mesh with a pile of formulas for something like beltrami-laplace operator would work, but I want something more elegant than this. For a example, can I use the Fibbonacci spiral to generate a uniform spherical mesh, and then somehow compute gradients and the laplacian?

I suspect that the stability of FE or FV methods is rooted in the fact that the FE functions slightly overlap, so computing the next step is a lot like using an implicit FD scheme, or better, a variation of the compact FD scheme. However I'm interested in how an adept in the field would solve this problem in practice. Again, I'm aware that there are methods of solving such systems (Jacobi, etc.), but those make the solution 10x more complex, buggier and slower.

Interesting that this reads almost like an chatgpt prompt.

Lazy people have been lazy forever. I stumbled across an example of this the other day from the 1990s, I think, and was shocked how much the student emails sounded like LLM prompts: https://www.chiark.greenend.org.uk/~martinh/poems/questions....

It reminds me of the old joke that half of the students are below average…

Expect in Lake Woebegone, all of the children are above average

But that's not true, unless by "average" you mean the median.

Normally, it's all the same.

Only if the distribution has zero skewness.

Unless "normally" you mean the normal distribution, which indeed has zero skewness.

Yes, it was a admittedly bad pun.

At least those had some basic politeness. So often I'm blown away not only how people blithely write "I NEED HELP, GIMME XYZ NOW NERDS" but especially how everyone is just falling over themselves to actually help! WTF?

Basic politeness is absolutely dead, nobody has any concept of acknowledging they are asking for a favour; we just blast Instagram/TikTok reels at top volume and smoke next to children and elderly in packed public spaces etc. I'm 100% sure it's not rose-tinted memories of the 90s making me think, it wasn't always like this...

"As an AI language model, I am happy to comply with your request ( https://chatgpt.com/share/6727b644-b2e0-800b-b613-322072d9d3... ), but good luck finding a data set to verify it, LOL."

> Could you write a blogpost-style article on how to model the shallow water wave equation on a sphere?

Typically, Finite Volume Method is used for fluid flow problems. It is possible to use Finite Element Methods, but it is rare.

I'm a mechanical engineer, and I've been wanting to better understand the computational side of the tools I use every day. Do you have any recommendations for learning resources if one wanted to "relearn" FEA from a computer science perspective?

Start with FDM. Solve Bernoulli deflection of a beam

Have a look at FEniCs to start with.

I learned it for the first time from this[0] course; part of the course covers deal.ii[1] where you program the stuff you're learning in C++.

[0]: https://open.umich.edu/find/open-educational-resources/engin...

[1]: https://www.dealii.org/

Have you heard of physics informed neural nets?

It seems like a hot candidate to potentially yield better results in the future

stagnate last 15 years??? Contact elements, bolt preload, modeling individual composite fibers, delamination progressive ply failure, modeling layers of material to a few thousandths of an inch. Design optimization. ANSYS Workbench = FEA For Dummies. The list goes on.