From: bangerth <bangerth@0785d39b-7218-0410-832d-ea1e28bc413d> Date: Wed, 7 Sep 2011 19:45:49 +0000 (+0000) Subject: Use a way to reference labels that will not lead to them being escaped. X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=e1ed53f0dc2432a2abdd65d5638f0d3b9fce4494;p=dealii-svn.git Use a way to reference labels that will not lead to them being escaped. git-svn-id: https://svn.dealii.org/trunk@24277 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-18/doc/intro.dox b/deal.II/examples/step-18/doc/intro.dox index d912f3b072..55b7f40f23 100644 --- a/deal.II/examples/step-18/doc/intro.dox +++ b/deal.II/examples/step-18/doc/intro.dox @@ -122,7 +122,7 @@ changes with time, with the boundary moving according to the displacements $\mathbf{u}(\mathbf{x},t)$ of the points on the boundary. To complete this system, we have to specify the incremental relationship between the stress and the strain, as follows: -<a name="step-18.stress-strain"></a> +<a name="step_18.stress-strain"></a> @f[ \dot\sigma = C \varepsilon (\dot{\mathbf{u}}), \qquad @@ -166,7 +166,7 @@ The weak form of this set of equations, which as usual is the basis for the finite element formulation, reads as follows: find $\Delta \mathbf{u}^n \in \{v\in H^1(\Omega(t_{n-1}))^d: v|_{\Gamma_D}=\mathbf{d}(\cdot,t_n) - \mathbf{d}(\cdot,t_{n-1})\}$ such that -<a name="step-18.linear-system"></a> +<a name="step_18.linear-system"></a> @f{eqnarray*} (C \varepsilon(\Delta\mathbf{u}^n), \varepsilon(\varphi) )_{\Omega(t_{n-1})} = @@ -219,7 +219,7 @@ complicated and will be discussed in the next section. As indicated above, we need to have the stress variable $\sigma^n$ available when computing time step $n+1$, and we can compute it using -<a name="step-18.stress-update"></a> +<a name="step_18.stress-update"></a> @f[ \sigma^n = \sigma^{n-1} + C \varepsilon (\Delta \mathbf{u}^n). \qquad @@ -290,7 +290,7 @@ the increment describe translations, its divergence the dilational modes, and the curl the rotational modes). Since the exact form of $R$ is cumbersome, we only state it in the program code, and note that the correct updating formula for the stress variable is then -<a name="step-18.stress-update+rot"></a> +<a name="step_18.stress-update+rot"></a> @f[ \sigma^n = @@ -394,7 +394,7 @@ sequence of operations on the present mesh: <li> <code>assemble_system ()</code> [via <code>solve_timestep ()</code>]: This first function is also the most interesting one. It assembles the linear system corresponding to the discretized version of equation - <a href="#step-18.linear-system">[linear-system]</a>. This leads to a system matrix $A_{ij} = \sum_K + <a href="#step_18.linear-system">[linear-system]</a>. This leads to a system matrix $A_{ij} = \sum_K A^K_{ij}$ built up of local contributions on each cell $K$ with entries @f[ A^K_{ij} = (C \varepsilon(\varphi_j), \varepsilon(\varphi_i))_K; @@ -477,7 +477,7 @@ for (unsigned int i=0; i<dofs_per_cell; ++i) \sigma^{n-1}_q : \varepsilon(\varphi_i(\mathbf{x}_q)) \right\} @f} - to the right hand side of <a href="#step-18.linear-system">[linear-system]</a> is equally + to the right hand side of <a href="#step_18.linear-system">[linear-system]</a> is equally straightforward (note that we do not consider any boundary tractions $\mathbf{b}$ here). Remember that we only had to store the old stress in the quadrature points of cells. In the program, we will provide a variable <code>local_quadrature_points_data</code> that allows to access the stress @@ -524,7 +524,7 @@ for (unsigned int i=0; i<dofs_per_cell; ++i) <li> <code>update_quadrature_point_history ()</code> [via <code>solve_timestep ()</code>]: Based on the displacement field $\Delta \mathbf{u}^n$ computed before, we update the stress values in all quadrature points - according to <a href="#step-18.stress-update">[stress-update]</a> and <a href="#step-18.stress-update+rot">[stress-update+rot]</a>, + according to <a href="#step_18.stress-update">[stress-update]</a> and <a href="#step_18.stress-update+rot">[stress-update+rot]</a>, including the rotation of the coordinate system. <li> <code>move_mesh ()</code>: Given the solution computed before, in this