10

You'll get everyone to give different answers to this question, and maybe that's alright. Here are some of my favorite ones: Taylor's theorem that a function (of sufficient smoothness) equals its Taylor expansion plus a remainder term. One can consider the Bramble-Hilbert lemma as a variation of Taylor's theorem, but it has different applications and is ...


5

The particular set of constraints you have chosen does not prevent a rigid body rotation about node 1. Thus the stiffness matrix is singular, as you have noted. One way to prevent this rigid body rotation is to set the y-displacement at node 2 to zero. You could also constrain the x-displacement at either node 3 or node 4 to prevent the rotation. One way ...


2

Since this is a question with pretty subjective answers, I'll add a couple to Prof. Bangerth's very good list. the theorem of adjoint/dual operators and spaces is pretty crucial to Computational Science. We know that dual-consistent discretizations of the PDEs can obtain superconvergence which is a nice property. But I think the more commonly used outcomes ...


1

To my knowledge these are the same things. However, this type of thing is common. For example, the proper orthogonal decomposition also has field-specific names. Others call it principal component analysis, the Karhunen--Loeve expansion, or empirical orthogonal functions. It is also no different than an autoencoder with linear activation function. I'm sure ...


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