178 karma · joined August 7, 2022
Don't love that I can't read sentences like this without wondering if an LLM was involved.
This describes how I write a new chunk of research code, often. I'll type along until I get to something like "oh, I'll need to calculate the foo of the widget here," and I'll just put a non-existent function call calculateFoo(widget) there until later, when I'll come back and fill it in. I feel like it keeps it manageable; I'm choosing the level of abstraction that I'm drafting code at, and I come in and fill in the details later. I hadn't connected this idea to the journal articles that I am working on; I typically feel somewhat guilty when I add a FIXME in my LaTex document, but with this framing I see now that that is probably the better way to do it than aiming for a finished paragraph from the get-go. The square brackets and placeholders also seem much nicer that the FIXME I was using. Glad to have seen this at a timely moment for me!
If you’re discussing large groups of people, you have to somehow compress the data. On the other hand, yeah, you probably shouldn’t prefer things like this to explain your neighbor/friend/in-laws over personal interactions with them.
That's how it started, yes. The splines used to specify the geometry are trimmed surfaces, and IGA has expanded from there to the use of splines generally as the shape functions, as well as trimming of volumes, etc. This use of smooth splines as shape functions improves the accuracy per degree of freedom.
> If I recall correctly convergence rates are exactly the same
Okay, looks like I remembered wrong here. What we do definitely see is that in IGA you get the convergence rates of higher degrees without drastically increasing your degree of freedom, meaning that there is better accuracy per degree of freedom for any degree above 1. See for example Figures 16 and 18 in this paper: https://www.researchgate.net/profile/Laurens-Coox/publicatio...
> geometry and the fields of quantities of interest do not have the same spatial distributions.
Using the same shape functions doesn't automatically mean that they will have the same spatial distributions. In fact, with hierarchical refinement in splines you can refine the geometry and any single field of interest separately.
> What is left in terms of potential?
The biggest potential other than higher accuracy per degree of freedom is perhaps trimming. In FEM, trimming your shape functions makes the solution unusable. In IGA, you can immerse your model in a "brick" of smooth spline shape functions, trim off the region outside, and run the simulation while still getting optimal convergence properties. This effectively means little to no meshing required. For a company that is readying this for use in industry, take a look at https://coreform.com/ (disclosure, I used to be a software developer there).
Edit: This one also looks good: https://math.uchicago.edu/~may/REU2018/REUPapers/Bixler.pdf
I do kind of wish that the last note corresponded to a game over, though, and I wonder if a smaller screen or faster ball would widen the playing field a little. Maybe I'll fork the code and try some of those out myself.
Of course, I could still benefit from practicing explaining the intuition...
Am I the only one who finds the phrase "manually photoshopped" in an article about the late 1800s to early 1900s amusingly anochronistic? How about "manually doctored" or even "altered"?
Funny thing is, writing it down helps it stick in my brain, so I need the write up less than I would if I didn't write it. That's got to be some kind of contrapositive of Murphy's law or something.