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Eighty Years of the Finite Element Method

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Re: Eighty Years of the Finite Element Method

#31

Post by jeffreyrogers »

badpun wrote: post #19
kqr wrote: post #12

Earlier quoted context omitted.

For physical processes, you can sometimes lean on conservation laws. Financial, no-arbitrage "laws". Sometimes it can help to run two different models, or different numerical methods, and compare their results. More generally, it's very, very hard. I am fairly sure there are multiple published results in computational fluid dynamics that are subtly wrong.

> I am fairly sure there are multiple published results in computational fluid dynamics that are subtly wrong. I wonder what that means for the accuraccy of the climate models...

They are all wrong. Historically they've overpredicted warming, and then the models are updated to fit the (new) data better. They perform worse in areas with less data, like polar regions.

To forestall knee-jerk downvotes: I'm not saying climate change isn't real or that anthropogenic global warming doesn't exist. I'm saying the models are not yet developed enough to predict very accurately. Early heliocentric models made poor predictions too because they assumed circular rather than elliptical orbits, they were still more "right" than geocentric models.

Re: Eighty Years of the Finite Element Method

#32

Post by hydrogen7800 »

pclmulqdq wrote: post #21

My numerical analysis professor once told the class a story about how he used the finite element method to solve stresses on pieces of metal for the soviet space program. Computers were too expensive, so they did it by hand. They cleared a university classroom of all the chairs and desks, and rolled out pieces of paper to cover the floor. The team took their shoes off proceeded from one corner of the room to the othe…

I once read or heard a similar anecdote about Ludwig Prandtl who set up computation method using a room full of people each doing one step of a calculation. I recall it describing a finite element method, but that would predate the origin of FEM in this article by a few decades. Maybe it was just some numerical method.

Re: Eighty Years of the Finite Element Method

#33

Post by brutusborn »

jeffreyrogers wrote: post #31
badpun wrote: post #19

Earlier quoted context omitted.

> I am fairly sure there are multiple published results in computational fluid dynamics that are subtly wrong. I wonder what that means for the accuraccy of the climate models...

They are all wrong. Historically they've overpredicted warming, and then the models are updated to fit the (new) data better. They perform worse in areas with less data, like polar regions. To forestall knee-jerk downvotes: I'm not saying climate change isn't real or that anthropogenic global warming doesn't exist. I'm saying the models are not yet developed enough to predict very accurately. Early heliocentric model…

Can you suggest good places to look for more info on this?

I have tried a few times to find info on the accuracy of these models and couldn't find much. And most models seem to be closed source.

Re: Eighty Years of the Finite Element Method

#34

Post by pclmulqdq »

hydrogen7800 wrote: post #32
pclmulqdq wrote: post #21

My numerical analysis professor once told the class a story about how he used the finite element method to solve stresses on pieces of metal for the soviet space program. Computers were too expensive, so they did it by hand. They cleared a university classroom of all the chairs and desks, and rolled out pieces of paper to cover the floor. The team took their shoes off proceeded from one corner of the room to the othe…

I once read or heard a similar anecdote about Ludwig Prandtl who set up computation method using a room full of people each doing one step of a calculation. I recall it describing a finite element method, but that would predate the origin of FEM in this article by a few decades. Maybe it was just some numerical method.

I assume it would be a finite differences calculation, which is also very easy to parallelize/vectorize. They solve the same class of problems, but calculations with finite differences don't always converge the way that finite elements do.

Re: Eighty Years of the Finite Element Method

#35

Post by leephillips »

jeffreyrogers wrote: post #31
badpun wrote: post #19

Earlier quoted context omitted.

> I am fairly sure there are multiple published results in computational fluid dynamics that are subtly wrong. I wonder what that means for the accuraccy of the climate models...

They are all wrong. Historically they've overpredicted warming, and then the models are updated to fit the (new) data better. They perform worse in areas with less data, like polar regions. To forestall knee-jerk downvotes: I'm not saying climate change isn't real or that anthropogenic global warming doesn't exist. I'm saying the models are not yet developed enough to predict very accurately. Early heliocentric model…

In fact, climate models going back over fifty years have performed quite well; surprisingly so:

https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/201...

https://www.theguardian.com/environment/climate-consensus-97...

Re: Eighty Years of the Finite Element Method

#36

Post by jeffreyrogers »

leephillips wrote: post #35
jeffreyrogers wrote: post #31

Earlier quoted context omitted.

They are all wrong. Historically they've overpredicted warming, and then the models are updated to fit the (new) data better. They perform worse in areas with less data, like polar regions. To forestall knee-jerk downvotes: I'm not saying climate change isn't real or that anthropogenic global warming doesn't exist. I'm saying the models are not yet developed enough to predict very accurately. Early heliocentric model…

In fact, climate models going back over fifty years have performed quite well; surprisingly so: https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/201... https://www.theguardian.com/environment/climate-consensus-97...

It depends on how you define "quite well". They are definitely directionally correct and in the right ballpark. This is a politically sensitive research area so the researchers are incentivized to make strong claims that you wouldn't make about, say, predicting sports outcomes or the stock market.

The first link you posted is kind of punting on the hard part by saying that the reason the models overpredict warming is because CO2 didn't rise as much as they expected so if you put the actual observed CO2 concentration in then the temperature prediction comes out closer to what was observed. But the CO2 concentration is a parameter of the model, so they didn't capture its dynamics properly and then had to retroactively change it to get the observed data.

Again, I want to reiterate that I'm not disputing the process of climate change or saying it's not a problem. I'm saying that modeling it is hard and historically the models have overestimated warming.

Re: Eighty Years of the Finite Element Method

#37

Post by leephillips »

jeffreyrogers wrote: post #36
leephillips wrote: post #35

Earlier quoted context omitted.

In fact, climate models going back over fifty years have performed quite well; surprisingly so: https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/201... https://www.theguardian.com/environment/climate-consensus-97...

It depends on how you define "quite well". They are definitely directionally correct and in the right ballpark. This is a politically sensitive research area so the researchers are incentivized to make strong claims that you wouldn't make about, say, predicting sports outcomes or the stock market. The first link you posted is kind of punting on the hard part by saying that the reason the models overpredict warming is…

That doesn’t seem like a fair criticism. Most climate models are physics models. They can’t know how much CO2 people are going to add to the atmosphere. But if you tell them how we altered the composition of the atmosphere, they predict the temperature change correctly. That’s what I mean by good performance.

Re: Eighty Years of the Finite Element Method

#38

Post by jeffreyrogers »

leephillips wrote: post #37
jeffreyrogers wrote: post #36

Earlier quoted context omitted.

It depends on how you define "quite well". They are definitely directionally correct and in the right ballpark. This is a politically sensitive research area so the researchers are incentivized to make strong claims that you wouldn't make about, say, predicting sports outcomes or the stock market. The first link you posted is kind of punting on the hard part by saying that the reason the models overpredict warming is…

That doesn’t seem like a fair criticism. Most climate models are physics models. They can’t know how much CO2 people are going to add to the atmosphere. But if you tell them how we altered the composition of the atmosphere, they predict the temperature change correctly. That’s what I mean by good performance.

Part of the job of the modeler is to make good decisions about the parameters and their uncertainties. If they are systematically overestimating CO2 and getting high predictions as a result then they aren't modeling properly, unless there are exogenous reasons why CO2 is lower than expected. Which might be possible for all I know (Global financial crisis maybe? China growth lower than expected?) but the dynamics of CO2 are part of the model they are using for prediction so they can't just punt on it by saying they didn't know what CO2 would be. Prediction's hard, especially about the future, as the saying goes.

Re: Eighty Years of the Finite Element Method

#39

Post by perilunar »

hydrogen7800 wrote: post #32
pclmulqdq wrote: post #21

My numerical analysis professor once told the class a story about how he used the finite element method to solve stresses on pieces of metal for the soviet space program. Computers were too expensive, so they did it by hand. They cleared a university classroom of all the chairs and desks, and rolled out pieces of paper to cover the floor. The team took their shoes off proceeded from one corner of the room to the othe…

I once read or heard a similar anecdote about Ludwig Prandtl who set up computation method using a room full of people each doing one step of a calculation. I recall it describing a finite element method, but that would predate the origin of FEM in this article by a few decades. Maybe it was just some numerical method.

Huh? Theres's a long history of doing this — 'computers' were people long before they are machines.

Re: Eighty Years of the Finite Element Method

#40

Post by kqr »

shoo wrote: post #22
mazokum wrote: post #11

Earlier quoted context omitted.

You can check this paper. Is the one my advisor recommended when I started my PhD. Is a FEM implementation in 50 lines of MATLAB. https://www.math.hu-berlin.de/~cc/cc_homepage/download/1999-...

What I found amazing about FEM is not the detail of how to implement it in code, but all the PDE theory & approximation theory -- how you can express the original continuous-domain infinite dimensional problem in a weak form using an infinite dimensional space of test functions, then approximating the weak form of the original problem with a finite dimensional Galerkin approximation, using a finite dimensional space…

Yup. I'm trying to learn computational fluid dynamics for fun, and realised it's basically all numerical methods for PDEs, with some fluid dynamics-specific shortcuts. I know I've previously seen this stuff in relation to pricing derivatives in finance.

It's clear the underlying techniques are very powerful any time you have a thing whose rate of change varies as other things change. Once I understand everything better I will try it on e.g. capacity planning cloud resources and such.

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