COMSOL Blog

https://link.springer.com/chapter/10.1007/978-0-387-98098-0_13 states the same, and mentions what I couldn’t find earlier: that they criticized Fourier for his lack of rigour (I remember that as something along the lines of “but if that’s correct, this also is and that can’t be true, can it?”, (with “this” being Gibb’s phenomenon (https://en.wikipedia.org/wiki/Gibbs_phenomenon)), but chances are high that isn’t happen, either.

Edit: https://www.comsol.com/blogs/happy-birthday-joseph-fourier/ has more info. If he did a thesis defense (did he?), I guess there’s a decent chance Laplace and Lagrange (his supervisor, according to https://www.genealogy.math.ndsu.nodak.edu/id.php?id=17981) were there.

Legendre seems doubtful. Weirdly, Legendre doesn’t seem to exist in the Mathematics Genealogy Project.

Edit 2: https://www.gutenberg.org/files/16775/16775-h/16775-h.htm#PART_PLAYED_BY_FOURIER_IN_OUR_REVOLUTION_HIS_ENTRANCE_INTO_THE_CORPS (by https://en.wikipedia.org/wiki/François_Arago, who was only 18 years younger than Fourier) is the best I can find on this. It says:

“The first memoir of Fourier on the theory of heat dates from the year 1807. The Academy, to which it was communicated, being desirous of inducing the author to extend and improve his researches, made the question of the propagation of heat the subject of the great mathematical prize which was to be awarded in the beginning of the year 1812. Fourier did, in effect, compete, and his memoir was crowned. But, alas! as…

> and getting the next better thing...

Implying an improvement, like phased array semiconductor microwaves, or something like that.

Edit: Don't laugh, that actually was 'on the radar' about 10 years ago. IIRC the availability of the parts went away because the producing company has been bought by another, shifted goals, whatnot else.

Anyways, no Sci-Fi:

https://patents.google.com/patent/CN202733981U/en

https://reviewed.usatoday.com/ovens/news/freescale-semiconductor-sage-appliance-concept-could-replace-microwave-ovens

Edit: https://www.microwaves101.com/encyclopedias/solid-state-microave-ovens

https://www.comsol.com/blogs/optimizing-microwave-ovens-with-solid-state-rf-cooking/

https://community.nxp.com/t5/NXP-Designs-Knowledge-Base/Solid-State-Microwave-Oven-using-RF-Technology/ta-p/1122475

https://www.nxp.com/company/blog/how-to-build-a-better-microwave-oven-using-solid-state-rf:BL-HOW-TO-BUILD-BETTER-MICROWAVE-OVEN-RF

https://www.electronicsweekly.com/news/design/the-future-of-microwave-cooking-is-solid-state-2016-01/

What about a hybrid approach, as I think folks do right now more or less -- draw it out by eye, and then validate the design with simulation-software (e.g. https://www.comsol.com/microfluidics-module) and iterate as needed from thereon.

Your inclination toward a computational approach makes me think of what folks in mechanical engineering are up to these days. Creating mechanical designs generatively with topological optimization (https://all3dp.com/2/topology-optimization-simply-explained/). There appear to be clear use-cases and advantages: you reduce the amount of material needed, decide exactly what stress points you want to focus on and work to strengthen those particular parts of the structure in interesting ways, etc. Fluid mechanics is a little more complicated though I think, so you're certainly up for a challenge!

I wish you success with your proposal, and I hope to discover your findings published in a nature paper on the frontpage of HN sometime soon ;).

Here is a simple simulation of a bicycle's stability [1]. The simulation more or less predicts what happens in real life, so we do understand why bicycles are stable.

As you can see from the simulation, this is a multibody dynamics problem. The bicycle and human have lots of moving parts, and as they move there are forces in 3D [2] amongst them. So there can't really be a simplified, short and mostly complete English explanation (like for why planets go around the sun). That doesn't mean we can't predict.

[1] https://www.comsol.com/blogs/simulating-the-motion-of-a-self-stable-bicycle/

[2] i.e. you can't find a 2D plane in which all forces act.

https://www.comsol.com/paper/image/66162/big.png

From the following paper:

"Electromagnetic Simulation of Split-Core Current Transformer for Medium Voltage Applications" by N. Paudel, V. Siddharth, S. Shaw and D. Raschka (2018)

https://www.comsol.com/paper/electromagnetic-simulation-of-split-core-current-transformer-for-medium-voltage-66162

Related:

Learn everything about the Split-Core Current Transformer:

https://innovatorsguru.com/split-core-current-transformer

Video about Current Transformers:

https://www.youtube.com/watch?v=32Vw40nUSwA&t=92s

If you’ve got some money,

https://www.comsol.com/blogs/6-examples-of-simulation-driven-loudspeaker-development

Those examples wouldn’t be differentiable in the sense of automatic differentiation. “Just” parametric CAD + FEM/BEM + optimizer.

Though differentiable FEM has kind of been done for a while under the name “adjoint method” for topology/shape optimization. It’s not super common outside of academia though.

Personally, I’m keeping an eye out for projects like https://github.com/deepmodeling/jax-fem to make some waves in that space.