From: Zhuoran Wang <zhrwang@rams.colostate.edu>
Date: Sun, 23 Feb 2020 22:01:12 +0000 (-0700)
Subject: Add derivation for the simply supported plates to result section.
X-Git-Tag: v9.2.0-rc1~35^2~3
X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=583277fbb618a82b8b1973b1ae86985579bfdbe5;p=dealii.git

Add derivation for the simply supported plates to result section.
---

diff --git a/examples/step-47/doc/results.dox b/examples/step-47/doc/results.dox
index d646e17318..7883333d77 100644
--- a/examples/step-47/doc/results.dox
+++ b/examples/step-47/doc/results.dox
@@ -211,3 +211,81 @@ make sense:
   addition should not be overly difficult using, for example, the
   FEInterfaceValues class combined with MeshWorker::mesh_loop() in the
   same spirit as we used for the assembly of the linear system.
+
+
+  <h3> 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:
+  @f{align*}{
+    \Delta^2 u(\mathbf x) &= f(\mathbf x)
+    \qquad \qquad &&\forall \mathbf x \in \Omega,
+    u(\mathbf x) &= g(\mathbf x) \qquad \qquad
+    &&\forall \mathbf x \in \partial\Omega, \\
+    \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:
+  @f{align*}{
+    \int_K v_h (\Delta^2 u_h)
+     &= \int_K (D^2 v_h) : (D^2 u_h)
+       + \int_{\partial K} v_h \frac{\partial (\Delta u_h)}{\partial \mathbf{n}}
+       -\int_{\partial K} (\nabla v_h) \cdot (\frac{\partial \nabla u_h}{\partial \mathbf{n}}).
+  @f}
+
+  Summing up over all cells $K \in  \mathbb{T}$,
+  @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,
+  @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}
+  We separate interior faces and boundary faces of the domain,
+  @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}^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.
+  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
+  = \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}} h \ ds.
+  @f}
+
+  In order to symmetrize and stabilize the discrete problem,
+  we add symmetrization and stabilization term.
+  We finally get the $C^0$ IP finite element scheme for the biharmonic equation:
+  find $u_h$ such that $u_h =g$ on $\partial \Omega$ and
+  @f{align*}{
+  \mathcal{A}(v_h,u_h)&=\mathcal{F}(v_h) \quad \text{holds for all test functions } v_h,
+  @f}
+  where
+  @f{align*}{
+  \mathcal{A}(v_h,u_h)&=\mathcal{F}(v_h) \quad \text{holds for all test functions } v_h,
+  @f}
+  where
+  @f{align*}{
+  \mathcal{A}(v_h,u_h):=&\sum_{K \in \mathbb{T}}\int_K D^2v_h:D^2u_h \ dx
+  \\
+  &
+   -\sum_{e \in \mathbb{F}^i} \int_{e}
+    \jump{\frac{\partial v_h}{\partial \mathbf n}}
+    \average{\frac{\partial^2 u_h}{\partial \mathbf n^2}} \ ds
+   -\sum_{e \in \mathbb{F}^i} \int_{e}
+   \average{\frac{\partial^2 v_h}{\partial \mathbf n^2}}
+   \jump{\frac{\partial u_h}{\partial \mathbf n}} \ ds
+  \\
+  &+ \sum_{e \in \mathbb{F}^i}
+   \frac{\gamma}{h_e}
+   \int_e
+   \jump{\frac{\partial v_h}{\partial \mathbf n}}
+   \jump{\frac{\partial u_h}{\partial \mathbf n}} \ ds,
+  @f}
+  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.
+  @f}