jorgensd / jorgensd/FEniCS-workshop
Small textual corrections in "Coupling PDEs of multiple dimensions"
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Description
While going through the page on *Coupling PDEs of multiple dimensions*, I noticed a few small textual inconsistencies in the mathematical description. The code implementation seems to use the correct forms, but I think these edits could help readability and avoid confusion for readers.
#### 1. Signorini conditions: inequality on the normal stress
The current text writes:
\sigma_n(\mathbf{u}) \mathbf{n} \leq 0 \text{ on } \Gamma
but this is inconsistent with the definition given just above for the contact pressure, so the condition should be:
\sigma_n(\mathbf{u}) \leq 0 \text{ on } \Gamma
#### 2. Functional: wrong variable in the elastic energy term
The current functional is written as:
\frac{1}{2} \int_\Omega (C\epsilon(\mathbf{u}):\epsilon(\mathbf{v}))~\mathrm{d}x
I think the correct expression is:
\frac{1}{2} \int_\Omega (C\epsilon(\mathbf{u}):\epsilon(\mathbf{u}))~\mathrm{d}x
#### 3. Variational formulation: sign in the second equation
The current statement writes:
(\mathbf{u}\cdot \mathbf{n}, w)_\Gamma - (e^{\psi}, w)_\Gamma = (g, w)_\Gamma
but this seems inconsistent with the localized form, and later in the code the corresponding equation is effectively used with a plus sign. I believe the correct version is:
(\mathbf{u}\cdot \mathbf{n}, w)_\Gamma + (e^{\psi}, w)_\Gamma = (g, w)_\Gamma
Also, in the first equation I would use the bold symbol for the displacement test function in
\alpha_k(\mathbf{f}, v)_\Omega
If useful, I can also open a small PR with these text corrections.
Thank you for the very useful material.
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