From 826d206ad4a5f13a15144dd6707ec0d750c7a41e Mon Sep 17 00:00:00 2001
From: Zhuoran Wang <zhrwang@rams.colostate.edu>
Date: Sat, 21 Mar 2020 11:15:05 -0600
Subject: [PATCH] Add more details to simply supported plates case.

---
 examples/step-47/doc/results.dox | 22 +++++++++++++---------
 1 file changed, 13 insertions(+), 9 deletions(-)

diff --git a/examples/step-47/doc/results.dox b/examples/step-47/doc/results.dox
index 7883333d77..31af98ee1c 100644
--- a/examples/step-47/doc/results.dox
+++ b/examples/step-47/doc/results.dox
@@ -213,10 +213,10 @@ make sense:
   same spirit as we used for the assembly of the linear system.
 
 
-  <h3> Derivation for the simply supported plates </h3>
+  <h4> Derivation for the simply supported plates </h3>
 
-  Similar to the “clamped” boundary condition addressed in the implementation,
-  we will derive the $C^0$ IP finite element scheme for the simply supported plates:
+  Similar to the "clamped" boundary condition addressed in the implementation,
+  we will derive the $C^0$ IP finite element scheme for simply supported plates:
   @f{align*}{
     \Delta^2 u(\mathbf x) &= f(\mathbf x)
     \qquad \qquad &&\forall \mathbf x \in \Omega,
@@ -225,7 +225,7 @@ make sense:
     \Delta u(\mathbf x) &= h(\mathbf x) \qquad \qquad
     &&\forall \mathbf x \in \partial\Omega.
   @f}
-  We multiply the biharmonic equation by the test function $v_h$ and integrate over $\Omega$ and get:
+  We multiply the biharmonic equation by the test function $v_h$ and integrate over $ K $ and get:
   @f{align*}{
     \int_K v_h (\Delta^2 u_h)
      &= \int_K (D^2 v_h) : (D^2 u_h)
@@ -233,11 +233,12 @@ make sense:
        -\int_{\partial K} (\nabla v_h) \cdot (\frac{\partial \nabla u_h}{\partial \mathbf{n}}).
   @f}
 
-  Summing up over all cells $K \in  \mathbb{T}$,
+  Summing up over all cells $K \in  \mathbb{T}$,since normal directions of $\Delta u_h$ are pointing at
+  opposite directions on each interior edge shared by two cells and $v_h = 0$ on $\partial \Omega$,
   @f{align*}{
   \sum_{K \in \mathbb{T}} \int_{\partial K} v_h \frac{\partial (\Delta u_h)}{\partial \mathbf{n}} = 0,
   @f}
-  and by the definition of jump over cells,
+  and by the definition of jump over cell interfaces,
   @f{align*}{
   -\sum_{K \in \mathbb{T}} \int_{\partial K} (\nabla v_h) \cdot (\frac{\partial \nabla u_h}{\partial \mathbf{n}}) = -\sum_{e \in \mathbb{F}} \int_{e} \jump{\frac{\partial v_h}{\partial \mathbf{n}} (\frac{\partial^2 u_h}{\partial \mathbf{n^2}}).
   @f}
@@ -246,7 +247,7 @@ make sense:
   -\sum_{K \in \mathbb{T}} \int_{\partial K} (\nabla v_h) \cdot (\frac{\partial \nabla u_h}{\partial \mathbf{n}}) = -\sum_{e \in \mathbb{F}^i} \int_{e} \jump{\frac{\partial v_h}{\partial \mathbf{n}} (\frac{\partial^2 u_h}{\partial \mathbf{n^2}})
   - \sum_{e \in \partial \Omega} \int_{e} \jump{\frac{\partial v_h}{\partial \mathbf{n}} h,
   @f}
-  Where $\mathbb{F}^i$ is the set of interior faces.
+  where $\mathbb{F}^i$ is the set of interior faces.
   This leads us to
   @f{align*}{
   \sum_{K \in \mathbb{T}} \int_K (D^2 v_h) : (D^2 u_h) \ dx - \sum_{e \in \mathbb{F}^i} \int_{e} \jump{\frac{\partial v_h}{\partial \mathbf{n}} (\frac{\partial^2 u_h}{\partial \mathbf{n^2}}) \ ds
@@ -285,7 +286,10 @@ make sense:
   and
   @f{align*}{
   \mathcal{F}(v_h)&:=\sum_{K \in \mathbb{T}}\int_{K} v_h f \ dx
-  -
+  +
   \sum_{e\subset\partial\Omega}
-  \int_e \jump{\frac{\partial v_h}{\partial \mathbf n^2}} h \ ds.
+  \int_e \jump{\frac{\partial v_h}{\partial \mathbf n}} h \ ds.
   @f}
+  The implementation of this boundary case is similar to "clamped" version
+  except for `boundary_worker` is no longer needed for system assembling
+  and the right hand side is changed according to the formulation.
-- 
2.39.5