From 700b6bb93380be28f11194c245770292f7d8c59f Mon Sep 17 00:00:00 2001 From: buerg Date: Thu, 24 Jun 2010 08:23:10 +0000 Subject: [PATCH] Documentation and style changes. git-svn-id: https://svn.dealii.org/trunk@21311 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-45/doc/intro.dox | 16 ++++++++-------- deal.II/examples/step-45/doc/results.dox | 4 ++-- 2 files changed, 10 insertions(+), 10 deletions(-) diff --git a/deal.II/examples/step-45/doc/intro.dox b/deal.II/examples/step-45/doc/intro.dox index 9c2eb80b37..73f5e84f03 100644 --- a/deal.II/examples/step-45/doc/intro.dox +++ b/deal.II/examples/step-45/doc/intro.dox @@ -12,11 +12,11 @@ solve some equation on a representative piece of a larger domain that repeats in one or more direction; an example is the simulation of the electronic structure of photonic crystals, because they have a lattice-like structure and, thus, it often -suffices to do the actual computation on only one cell. To be able to proceed -this way one has to assume that the computation can be periodically extended -to the other cells. This requires the solution to be periodic w.r.t. the -cells. Hence the solution has to obtain the same nodal values on opposite parts of the -boundary. In the figure below we show this +suffices to do the actual computation on only one cell of the lattice. To be +able to proceed this way one has to assume that the computation can be periodically +extended to the other cells. This requires the solution to be periodic with respect +to the cells. Hence the solution has to obtain the same nodal values on opposite +parts of the boundary. In the figure below we show this concept in two space-dimensions. There, all dashed faces with the same color should have the same boundary values: @@ -45,14 +45,14 @@ periodic unless this is explicitly enforced. The way one has to see these periodic boundary conditions $u(x,0) = u(x,1)$ is as follows: Assume for a moment (as we do in this program) that we have a uniformly refined mesh. Then, after discretization there are a number of nodes -(degrees of freedom) with indices $i \in {\cal I}_b$ on the bottom boundary of -the domain, and a second set of nodes at the top boundary $j \in {\cal +(degrees of freedom) with indices $i \in {\cal I}_b$ on the left boundary of +the domain, and a second set of nodes at the right boundary $j \in {\cal I}_t$. Since we have assumed that the mesh is uniformly refined, there is exactly one node $j \in {\cal I}_t$ for each $i \in {\cal I}_b$ so that ${\mathrm x}_j = {\mathrm x}_i + (0,1)^T$, i.e. the two of them match with respect to the periodicity. We will then write that $j=\text{periodic}(i)$ (and, if you want, $i=\text{periodic}(j)$). -If now $U_k, k=0,\ldots,N-1$ are the unknowns of our discretized problem, then +If now $U_k, k=0,\ldots,N-1,$ are the unknowns of our discretized problem, then the periodic boundary condition boils down to the following set of constraints: @f{align*} diff --git a/deal.II/examples/step-45/doc/results.dox b/deal.II/examples/step-45/doc/results.dox index 4fc57f22a3..b4f55e3127 100644 --- a/deal.II/examples/step-45/doc/results.dox +++ b/deal.II/examples/step-45/doc/results.dox @@ -1,8 +1,8 @@

Results

The textual output the program generates is not very surprising. It just -prints out the usual information on degrees of freedom, active cells and -iteration steps of the solver, in much the same way as step-3 did: +prints out the usual information on degrees of freedom and active cells, +in much the same way as step-3 did: @code Number of active cells: 1024 -- 2.39.5