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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