From 3cade86c6d53e95367988a3bf0bd1d728b498c7e Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Wed, 26 Aug 2009 16:13:09 +0000 Subject: [PATCH] Use better markup for comp(i) and base(i). git-svn-id: https://svn.dealii.org/trunk@19344 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-8/doc/intro.dox | 60 +++++++++++++-------------- 1 file changed, 30 insertions(+), 30 deletions(-) diff --git a/deal.II/examples/step-8/doc/intro.dox b/deal.II/examples/step-8/doc/intro.dox index 264381f3a1..fa25893ab8 100644 --- a/deal.II/examples/step-8/doc/intro.dox +++ b/deal.II/examples/step-8/doc/intro.dox @@ -106,15 +106,15 @@ number of shape functions for the vector element; in two space dimensions, we need $n$ shape functions for each component of the vector, so $N=2n$. Then, the $i$th shape function of the vector element has the form @f[ - \Phi_i({\mathbf x}) = \varphi_{base(i)}({\mathbf x})\ {\mathbf e}_{comp(i)}, + \Phi_i({\mathbf x}) = \varphi_{\text{base}(i)}({\mathbf x})\ {\mathbf e}_{\text{comp}(i)}, @f] -where $e_l$ is the $l$th unit vector, $comp(i)$ is the function that tells +where $e_l$ is the $l$th unit vector, $\text{comp}(i)$ is the function that tells us which component of $\Phi_i$ is the one that is nonzero (for each vector shape function, only one component is nonzero, and all others are -zero). $\varphi_{base(i)}(x)$ describes the space dependence of the shape -function, which is taken to be the $base(i)$-th shape function of the scalar +zero). $\varphi_{\text{base}(i)}(x)$ describes the space dependence of the shape +function, which is taken to be the $\text{base}(i)$-th shape function of the scalar element. Of course, while $i$ is in the range $0,\ldots,N-1$, the functions -$comp(i)$ and $base(i)$ have the ranges $0,1$ (in 2D) and $0,\ldots,n-1$, +$\text{comp}(i)$ and $\text{base}(i)$ have the ranges $0,1$ (in 2D) and $0,\ldots,n-1$, respectively. For example (though this sequence of shape functions is not @@ -144,23 +144,23 @@ the following layout could be used by the library: @f} where here @f[ - comp(0)=0, \quad comp(1)=1, \quad comp(2)=0, \quad comp(3)=1, \quad \ldots + \text{comp}(0)=0, \quad \text{comp}(1)=1, \quad \text{comp}(2)=0, \quad \text{comp}(3)=1, \quad \ldots @f] @f[ - base(0)=0, \quad base(1)=0, \quad base(2)=1, \quad base(3)=1, \quad \ldots + \text{base}(0)=0, \quad \text{base}(1)=0, \quad \text{base}(2)=1, \quad \text{base}(3)=1, \quad \ldots @f] In all but very rare cases, you will not need to know which shape function -$\varphi_{base(i)}$ of the scalar element belongs to a shape function $\Phi_i$ +$\varphi_{\text{base}(i)}$ of the scalar element belongs to a shape function $\Phi_i$ of the vector element. Let us therefore define @f[ - \phi_i = \varphi_{base(i)} + \phi_i = \varphi_{\text{base}(i)} @f] by which we can write the vector shape function as @f[ - \Phi_i({\mathbf x}) = \phi_{i}({\mathbf x})\ {\mathbf e}_{comp(i)}. + \Phi_i({\mathbf x}) = \phi_{i}({\mathbf x})\ {\mathbf e}_{\text{comp}(i)}. @f] -You can now safely forget about the function $base(i)$, at least for the rest +You can now safely forget about the function $\text{base}(i)$, at least for the rest of this example program. Now using this vector shape functions, we can write the discrete finite @@ -233,7 +233,7 @@ where $i,j$ now are local degrees of freedom and therefore $0\le i,j < N$. In these formulas, we always take some component of the vector shape functions $\Phi_i$, which are of course given as follows (see their definition): @f[ - (\Phi_i)_l = \phi_i \delta_{l,comp(i)}, + (\Phi_i)_l = \phi_i \delta_{l,\text{comp}(i)}, @f] with the Kronecker symbol $\delta_{nm}$. Due to this, we can delete some of the sums over $k$ and $l$: @@ -243,26 +243,26 @@ the sums over $k$ and $l$: \sum_{k,l} \Bigl\{ \left( - \lambda \partial_l \phi_i\ \delta_{l,comp(i)}, - \partial_k \phi_j\ \delta_{k,comp(j)} + \lambda \partial_l \phi_i\ \delta_{l,\text{comp}(i)}, + \partial_k \phi_j\ \delta_{k,\text{comp}(j)} \right)_K \\ &\qquad\qquad& + \left( - \mu \partial_l \phi_i\ \delta_{k,comp(i)}, - \partial_l \phi_j\ \delta_{k,comp(j)} + \mu \partial_l \phi_i\ \delta_{k,\text{comp}(i)}, + \partial_l \phi_j\ \delta_{k,\text{comp}(j)} \right)_K + \left( - \mu \partial_l \phi_i\ \delta_{k,comp(i)}, - \partial_k \phi_j\ \delta_{l,comp(j)} + \mu \partial_l \phi_i\ \delta_{k,\text{comp}(i)}, + \partial_k \phi_j\ \delta_{l,\text{comp}(j)} \right)_K \Bigr\} \\ &=& \left( - \lambda \partial_{comp(i)} \phi_i, - \partial_{comp(j)} \phi_j + \lambda \partial_{\text{comp}(i)} \phi_i, + \partial_{\text{comp}(j)} \phi_j \right)_K + \sum_l @@ -270,28 +270,28 @@ the sums over $k$ and $l$: \mu \partial_l \phi_i, \partial_l \phi_j \right)_K - \ \delta_{comp(i),comp(j)} + \ \delta_{\text{comp}(i),\text{comp}(j)} + \left( - \mu \partial_{comp(j)} \phi_i, - \partial_{comp(i)} \phi_j + \mu \partial_{\text{comp}(j)} \phi_i, + \partial_{\text{comp}(i)} \phi_j \right)_K \\ &=& \left( - \lambda \partial_{comp(i)} \phi_i, - \partial_{comp(j)} \phi_j + \lambda \partial_{\text{comp}(i)} \phi_i, + \partial_{\text{comp}(j)} \phi_j \right)_K + \left( \mu \nabla \phi_i, \nabla \phi_j \right)_K - \ \delta_{comp(i),comp(j)} + \ \delta_{\text{comp}(i),\text{comp}(j)} + \left( - \mu \partial_{comp(j)} \phi_i, - \partial_{comp(i)} \phi_j + \mu \partial_{\text{comp}(j)} \phi_i, + \partial_{\text{comp}(i)} \phi_j \right)_K. @f} @@ -309,12 +309,12 @@ Likewise, the contribution of cell $K$ to the right hand side vector is \sum_l \left( f_l, - \phi_j \delta_{l,comp(j)} + \phi_j \delta_{l,\text{comp}(j)} \right)_K \\ &=& \left( - f_{comp(j)}, + f_{\text{comp}(j)}, \phi_j \right)_K. @f} -- 2.39.5