Finite element method

We feed that into a ANSYS basic plane to get the surface mesh.

And then this one, the service mess is fed into the solver, the PiMapDL solver here, which is an FEM solver.

It takes some additional inputs here, and it spits out the compliance and we can eventually calculate the analytical adjoint, and we want to optimize with regard to the design parameters.

But we see a problem here that the analytical adjoint is differentiable with respect to the density, but we want to optimize with respect to the design parameters, and they are lost because answer space is not differentiable.

Well, welcome to the Deep Dive.

Today we are looking at Thomas J.R. Hughes's legendary 1987 textbook, The Finite Element Method, Linear Static and Dynamic Finite Element Analysis.

A

total classic.

Totally.

So our mission today is to translate that dense matrix math into actual physical intuition.

We're breaking down the underlying logic of the finite element method, or FEM.

Yeah, and the whole premise of FEM is taking this massive, impossibly complex reality and chopping it up into tiny, solvable elements.

So the secret to modern engineering is breaking the impossible down into the manageable.

Mathematically, we do that using a process called the finite element method or FEM.

And that is exactly the mission of our deep dive today.

We are exploring the foundational textbook, The Finite Element Method, using MATLAB by Young W. Kwan and Haichung Bang.

It's such a great text.

It really is.

It is incredibly satisfying when those concepts finally click into place.

It really is.

They needed a mesh that could mold to reality rather than forcing reality to mold to a mesh.

And that desperate search is what gave birth to what we now know as the finite element method.

FEM.

FEM.

Did the invention of digital computers basically rescue FEM from being just a really cool but totally unusable idea locked in a mathematician's notebook?

Without a doubt.

The finite element method and the digital computer are entirely symbiotic.

Digital computers were the oxygen that FEM needed to breathe.

And today, for our deep dive, we are finding out why.

We're digging into the core concepts of the book Steel Structures: Design Using FEM by Rolf Kindemann and Matthias Krauss.

It is a fantastic resource.

It really is.

It's just a waste of computational power.

Pretty much.

But the text does mention that 2D area elements, like flat plates, they do have a place in FEM software.

So if we don't use them for the whole building, when do we actually use them?

There are three popular methods in electrodynamics if you wanna simulate things full wave.

It's finite difference, time domain, method of moments, and finite element method.

Um, so this is just one of the three.

Time, finite difference time domain is literally...

Oh, okay.

Method of moments is a very complex thing where I'm not an expert at it and it's pretty hard to explain.

Finite element method is, it's, in this case it's, uh, it's like the frequency domain version of the finite difference time domain.

What that means is you say, "What's the solution at this frequency?" And then you do it at a different frequency, and then at a different, and you pick 10-

And in those days, uh, you know, we were spending hundreds of thousands of dollars a month with, uh, cloud providers like Amazon for their servers that perform, uh, that, that were able to do high-performance computing, lots of RAM, lots of CPU.

Because we were modeling very complex acoustic and electromagnetic phenomena using these mathematical methods called, you know, FEM and FDTD.

Today, y- we can use, uh, uh, these PINN networks that combine physics laws with deep learning and solve these equations exponentially faster.

Mm-hmm, mm-hmm.