Recent content by Vanilla Gorilla

Hi, I am a newcomer to FEA/FEM. I am aware that for any practical purpose, software is used to solve problems. However, before I utilize software, I want to understand the process the computer goes about doing. This document is simply an attempt to summarize the solving procedure in a concise...

That makes more sense, but I still have a feq questions Is ##\phi## representative of the shape functions? How would the final system, shown below, end up looking in matrix form? $$ \begin{split} K_{m,n} &= \int_a^b \phi'_n(x) v'_m(x)\,dx \\ F_m &= \int_a^b f(x)v_m(x)\,dx\end{split}$$ In the...

I believe I understand that, but how do we get the source term ##F##? That is where my confusion lies.

I have been watching Mike Foster's video series of the Finite Element Method for Differential Equations (FEM). In this episode, he solves a DE relating to temperature. As the final step, he gives the following equation: $$[K] [T] = [F]$$ In this equation, I understand that ##[K]## is the...

Alright, thank you for your help!

So basically just use the Hodge formula you gave previously, since we already constructed an orthogonal basis, so we'd just be overcomplicating by putting it into a non-orthogonal basis?

"We compute in terms of tensor index notation with respect to a (not necessarily orthonormal) basis..." Our original formula gave $$\star\alpha =-6 \cdot \frac {\sqrt { | det \left [ g_{ij} \right ] | }} {(7-3)!} \;\epsilon^{123}_{4567} \;dx^{4567}$$ The only difference between our formula and...

https://en.wikipedia.org/wiki/Hodge_star_operator#Computation_in_index_notation (I think this is generalized to non-orthonormal? I'm unsure.) "Since Hodge star takes an orthonormal basis to an orthonormal basis, it is an isometry on the exterior algebra" -Link; I'm not sure if this is...

That makes sense, but I thought the square root of the metric tensor term, ##\sqrt {|g|}## in the original equation, already accounted for the change in basis. Is this incorrect? Also, again, thank you so much for your replies, they are incredibly helpful, and I really appreciate it :)

So we would get a slightly differently defined NEW operator if I understand correctly? Also, I'm not just asking if it's defined on non-orthogonal basis, but if it is, how to calculate it on them

Is there a way to generalize to non-orthogonal basis systems? Sorry, I thought this WAS the way to generalize to non-orthogonal systems, which is why I pursued it specifically, and not other definitions.

OK, so this definition requires both a canonical ordering of basis vectors (An oriented volume) and orthogonal (But not necessarily orthonormal) basis vectors? Is that correct? Also, thank you so much, this is so helpful!

By orthogonal I take it you mean basis vectors are perpendicular? And what do you mean by an orientation or order? Just the way in which we write the basis vectors "canonically", like how we write xyz rather than zyx?

Alright, that makes sense; does this apply even in a non-orthonormal basis?

how do you calculate with that formula? (I just don't understand the 4 indices, I know how to calculate using vanilla product notation.)