From 826d206ad4a5f13a15144dd6707ec0d750c7a41e Mon Sep 17 00:00:00 2001 From: Zhuoran Wang 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. -

Derivation for the simply supported plates

+

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: + 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