From: Wolfgang Bangerth Date: Wed, 26 Aug 2009 16:07:11 +0000 (+0000) Subject: Use a different symbol (U_i) for the expansion coefficients, to X-Git-Tag: v8.0.0~7227 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=a7b92a9410576832cccc20d3e4e42c7e8b9afcd0;p=dealii.git Use a different symbol (U_i) for the expansion coefficients, to differentiate it from the components u_i of the vector-valued solution u. git-svn-id: https://svn.dealii.org/trunk@19341 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-8/doc/intro.dox b/deal.II/examples/step-8/doc/intro.dox index 7822a7bcc5..56f2b8f553 100644 --- a/deal.II/examples/step-8/doc/intro.dox +++ b/deal.II/examples/step-8/doc/intro.dox @@ -8,7 +8,7 @@ vector-valued. The deal.II library supports such problems (see the extensive documentation in the @ref vector_valued module), and we will show that that is mostly rather simple. The only more complicated problems are in assembling matrix and right hand side, but these are easily -understood as well. +understood as well. In the example, we will want to solve the elastic equations. They are an extension to Laplace's equation with a vector-valued solution that @@ -30,9 +30,9 @@ many cases, one knows that the material under consideration is isotropic, in which case by introduction of the two coefficients $\lambda$ and $\mu$ the coefficient tensor reduces to @f[ - c_{ijkl} + c_{ijkl} = - \lambda \delta_{ij} \delta_{kl} + + \lambda \delta_{ij} \delta_{kl} + \mu (\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk}). @f] @@ -115,28 +115,28 @@ 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 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 $0,\ldots,n-1$, -respectively. +respectively. For example (though this sequence of shape functions is not guaranteed, and you should not rely on it), the following layout could be used by the library: @f{eqnarray*} - \Phi_0({\mathbf x}) &=& + \Phi_0({\mathbf x}) &=& \left(\begin{array}{c} \varphi_0({\mathbf x}) \\ 0 \end{array}\right), \\ - \Phi_1({\mathbf x}) &=& + \Phi_1({\mathbf x}) &=& \left(\begin{array}{c} 0 \\ \varphi_0({\mathbf x}) \end{array}\right), \\ - \Phi_2({\mathbf x}) &=& + \Phi_2({\mathbf x}) &=& \left(\begin{array}{c} \varphi_1({\mathbf x}) \\ 0 \end{array}\right), \\ - \Phi_3({\mathbf x}) &=& + \Phi_3({\mathbf x}) &=& \left(\begin{array}{c} 0 \\ \varphi_1({\mathbf x}) \end{array}\right), @@ -166,12 +166,12 @@ of this example program. Now using this vector shape functions, we can write the discrete finite element solution as @f[ - {\mathbf u}_h({\mathbf x}) = - \sum_i \Phi_i({\mathbf x})\ u_i + {\mathbf u}_h({\mathbf x}) = + \sum_i \Phi_i({\mathbf x})\ U_i @f] -with scalar coefficients $u_i$. If we define an analog function ${\mathbf v}_h$ as +with scalar coefficients $U_i$. If we define an analog function ${\mathbf v}_h$ as test function, we can write the discrete problem as follows: Find coefficients -$u_i$ such that +$U_i$ such that @f[ a({\mathbf u}_h, {\mathbf v}_h) = ({\mathbf f}, {\mathbf v}_h) \qquad @@ -182,7 +182,7 @@ If we insert the definition of the bilinear form and the representation of ${\mathbf u}_h$ and ${\mathbf v}_h$ into this formula: @f{eqnarray*} \sum_{i,j} - u_i v_j + U_i V_j \sum_{k,l} \left\{ \left( @@ -191,15 +191,15 @@ ${\mathbf u}_h$ and ${\mathbf v}_h$ into this formula: + \left( \mu \partial_l (\Phi_i)_k, \partial_l (\Phi_j)_k - \right)_\Omega + \right)_\Omega + \left( \mu \partial_l (\Phi_i)_k, \partial_k (\Phi_j)_l - \right)_\Omega + \right)_\Omega \right\} \\ = - \sum_j v_j + \sum_j V_j \sum_l \left( f_l, @@ -229,7 +229,7 @@ The local stiffness matrix on cell $K$ therefore has the following entries: \right)_K \right\}, @f] -where $i,j$ now are local degrees of freedom and therefore $0\le i,j < N$. +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[ @@ -243,7 +243,7 @@ the sums over $k$ and $l$: \sum_{k,l} \Bigl\{ \left( - \lambda \partial_l \phi_i\ \delta_{l,comp(i)}, + \lambda \partial_l \phi_i\ \delta_{l,comp(i)}, \partial_k \phi_j\ \delta_{k,comp(j)} \right)_K \\