From 24bdfe225c36e3fc9a1617d6ce621f27b1d40b09 Mon Sep 17 00:00:00 2001 From: wolf Date: Thu, 11 Sep 2003 19:45:37 +0000 Subject: [PATCH] Fix document. git-svn-id: https://svn.dealii.org/trunk@7975 0785d39b-7218-0410-832d-ea1e28bc413d --- .../step-8.data/intro.tex | 71 ++++++++++++++----- 1 file changed, 53 insertions(+), 18 deletions(-) diff --git a/deal.II/doc/tutorial/chapter-2.step-by-step/step-8.data/intro.tex b/deal.II/doc/tutorial/chapter-2.step-by-step/step-8.data/intro.tex index 3a8fb5f998..3a316e1d9e 100644 --- a/deal.II/doc/tutorial/chapter-2.step-by-step/step-8.data/intro.tex +++ b/deal.II/doc/tutorial/chapter-2.step-by-step/step-8.data/intro.tex @@ -49,9 +49,11 @@ $$ The elastic equations can then be rewritten in much simpler a form: $$ - - \nabla (\lambda+\mu) (\div \vec u) + \nabla \lambda (\div \vec u) - (\nabla \cdot \mu \nabla) \vec u + - + \div \mu (\nabla \vec u)^T = \vec f, $$ @@ -59,25 +61,35 @@ and the respective bilinear form is then $$ a(\vec u, \vec v) = \left( - (\lambda+\mu) \div \vec u, \div \vec v + \lambda \div \vec u, \div \vec v \right)_\Omega + - \sum_k + \sum_{i,j} \left( - \mu \nabla u_k, \nabla v_k + \mu \partial_i u_j, \partial_i v_j + \right)_\Omega, + + + \sum_{i,j} + \left( + \mu \partial_i u_j, \partial_j v_i \right)_\Omega, $$ -or as a sum over components: +or also writing the first term a sum over components: $$ a(\vec u, \vec v) = - \sum_{k,l} + \sum_{i,j} \left( - (\lambda+\mu) \partial_l u_l, \partial_k v_k + \lambda \partial_l u_l, \partial_k v_k \right)_\Omega + \sum_{k,l} \left( - \mu \partial_l u_k, \partial_l v_k + \mu \partial_i u_j, \partial_i v_j + \right)_\Omega, + + + \sum_{i,j} + \left( + \mu \partial_i u_j, \partial_j v_i \right)_\Omega. $$ @@ -176,12 +188,16 @@ $\vec u_h$ and $\vec v_h$ into this formula: \sum_{k,l} \left\{ \left( - (\lambda+\mu) \partial_l (\Phi_i)_l, \partial_k (\Phi_j)_k + \lambda \partial_l (\Phi_i)_l, \partial_k (\Phi_j)_k \right)_\Omega + \left( \mu \partial_l (\Phi_i)_k, \partial_l (\Phi_j)_k \right)_\Omega + + + \left( + \mu \partial_l (\Phi_i)_k, \partial_k (\Phi_j)_l + \right)_\Omega \right\} \\ = @@ -194,7 +210,7 @@ $\vec u_h$ and $\vec v_h$ into this formula: \end{multline*} \end{center} We note that here and in the following, the indices $k,l$ run over spatial -directions, i.e. $0\le k,l \le d-1$, and that indices $i,j$ run over degrees +directions, i.e. $0\le k,l < d$, and that indices $i,j$ run over degrees of freedoms. The local stiffness matrix on cell $K$ therefore has the following entries: @@ -204,15 +220,19 @@ $$ \sum_{k,l} \left\{ \left( - (\lambda+\mu) \partial_l (\Phi_i)_l, \partial_k (\Phi_j)_k + \lambda \partial_l (\Phi_i)_l, \partial_k (\Phi_j)_k \right)_K + \left( \mu \partial_l (\Phi_i)_k, \partial_l (\Phi_j)_k \right)_K + + + \left( + \mu \partial_l (\Phi_i)_k, \partial_k (\Phi_j)_l + \right)_K \right\}, $$ -where $i,j$ now are local degrees of freedom and therefore $0\le i,j \le N-1$. +where $i,j$ now are local degrees of freedom and therefore $0\le i,j < N$. In these formulas, we always take some component of the vector shape functions $\Phi_i$, which are of course given as follows (see their definition): $$ @@ -227,8 +247,8 @@ the sums over $k$ and $l$: \sum_{k,l} \Bigl\{ \left( - (\lambda+\mu) \partial_l \phi_i\ \delta_{l,comp(i)}, - \partial_k \phi_j\ \delta_{k,comp(j)} + \lambda \partial_l \phi_i\ \delta_{l,comp(i)}, + \partial_k \phi_j\ \delta_{k,comp(j)} \right)_K \\ &\qquad\qquad + @@ -236,12 +256,17 @@ the sums over $k$ and $l$: \mu \partial_l \phi_i\ \delta_{k,comp(i)}, \partial_l \phi_j\ \delta_{k,comp(j)} \right)_K + + + \left( + \mu \partial_l \phi_i\ \delta_{k,comp(i)}, + \partial_k \phi_j\ \delta_{l,comp(j)} + \right)_K \Bigr\} \\ &= \left( - (\lambda+\mu) \partial_{comp(i)} \phi_i, - \partial_{comp(j)} \phi_j + \lambda \partial_{comp(i)} \phi_i, + \partial_{comp(j)} \phi_j \right)_K + \sum_l @@ -250,17 +275,27 @@ the sums over $k$ and $l$: \partial_l \phi_j \right)_K \ \delta_{comp(i),comp(j)} + + + \left( + \mu \partial_{comp(j)} \phi_i, + \partial_{comp(i)} \phi_j + \right)_K \\ &= \left( - (\lambda+\mu) \partial_{comp(i)} \phi_i, - \partial_{comp(j)} \phi_j + \lambda \partial_{comp(i)} \phi_i, + \partial_{comp(j)} \phi_j \right)_K + \left( \mu \nabla \phi_i, \nabla \phi_j \right)_K + + + \left( + \mu \partial_{comp(j)} \phi_i, + \partial_{comp(i)} \phi_j + \right)_K \ \delta_{comp(i),comp(j)}. \end{align*} \end{center} -- 2.39.5