From 583277fbb618a82b8b1973b1ae86985579bfdbe5 Mon Sep 17 00:00:00 2001 From: Zhuoran Wang Date: Sun, 23 Feb 2020 15:01:12 -0700 Subject: [PATCH] Add derivation for the simply supported plates to result section. --- examples/step-47/doc/results.dox | 78 ++++++++++++++++++++++++++++++++ 1 file changed, 78 insertions(+) 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. + + +

Derivation 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 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} -- 2.39.5