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Let us consider that you already know how to obtain the stiffness matrix in the local configuration, $$[K]\{U\} = \{F\}\, .$$ Now, we want to express our systems of equations in a system rotated by an angle $\alpha$. The (generalized) displacement components would be given by \begin{align} &u_1' = u_1 \cos\alpha + v_1 \sin\alpha\, ,\\ &v_1' = -...


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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 ...


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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 ...


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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 ...


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It looks to me that you don't have a well posed problem, so who knows how the fenics is handling that. I know that you can get some odd results for ill-posed poisson problems when the solver doesn't diverge, but instead returns garbage. This seems to be on of those cases. Relatedly you can have certain fixed point iterations that solve linear systems of ...


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