From: Wolfgang Bangerth Date: Tue, 17 Aug 2010 19:43:07 +0000 (+0000) Subject: Remove trailing whitespace. X-Git-Tag: v8.0.0~5715 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=6af8f6805224384397e96f83a1f6d341336c6229;p=dealii.git Remove trailing whitespace. git-svn-id: https://svn.dealii.org/trunk@21672 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-22/doc/intro.dox b/deal.II/examples/step-22/doc/intro.dox index 12c0d69072..93ff3a273d 100644 --- a/deal.II/examples/step-22/doc/intro.dox +++ b/deal.II/examples/step-22/doc/intro.dox @@ -1,15 +1,15 @@
This program was contributed by Martin Kronbichler and Wolfgang -Bangerth. +Bangerth.
This material is based upon work partly supported by the National -Science Foundation under Award No. EAR-0426271 and The California Institute of +Science Foundation under Award No. EAR-0426271 and The California Institute of Technology. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the National Science Foundation or of The California Institute of Technology. -
+ @@ -162,7 +162,7 @@ possibilities for imposing boundary conditions: @f} In other words, as usual, strongly imposed boundary values do not appear in the weak form. - + It is noteworthy that if we impose Dirichlet boundary values on the entire boundary, then the pressure is only determined up to a constant. An algorithmic realization of that would use similar tools as have been seen in @@ -180,15 +180,15 @@ possibilities for imposing boundary conditions: \sum_{i,j=1}^d -(n_i v_j,\varepsilon(\textbf{u})_{ij})_{\Gamma_N} + - \sum_{i=1}^d - (n_i v_i, p)_{\Gamma_N} + \sum_{i=1}^d + (n_i v_i, p)_{\Gamma_N} \\ &=& \sum_{i,j=1}^d -(n_i v_j,\varepsilon(\textbf{u})_{ij})_{\Gamma_N} + - \sum_{i,j=1}^d - (n_i v_j, p \delta_{ij})_{\Gamma_N} + \sum_{i,j=1}^d + (n_i v_j, p \delta_{ij})_{\Gamma_N} \\ &=& \sum_{i,j=1}^d @@ -206,7 +206,7 @@ possibilities for imposing boundary conditions: prescribe values for the total stress: @f{eqnarray*} \textbf{n}\cdot [p \textbf{1} - \varepsilon(\textbf{u})] - = + = \textbf g_N \qquad\qquad \textrm{on}\ \Gamma_N. @f} If the boundary is subdivided into Dirichlet and Neumann parts @@ -221,13 +221,13 @@ possibilities for imposing boundary conditions: - (\textbf{v}, \textbf g_N)_{\Gamma_N}. @f} - - + +
  • Robin-type boundary conditions: Robin boundary conditions are a mixture of - Dirichlet and Neumann boundary conditions. They would read + Dirichlet and Neumann boundary conditions. They would read @f{eqnarray*} \textbf{n}\cdot [p \textbf{1} - \varepsilon(\textbf{u})] - = + = \textbf S \textbf u \qquad\qquad \textrm{on}\ \Gamma_R, @f} with a rank-2 tensor (matrix) $\textbf S$. The associated weak form is @@ -249,14 +249,14 @@ possibilities for imposing boundary conditions: boundary, i.e. the tangential component $\textbf u_{\textbf t}=(\textbf 1-\textbf n\otimes\textbf n)\textbf u$ be zero, thereby constraining dim-1 components of the velocity. The remaining component can - be constrained by requiring that the normal component of the normal + be constrained by requiring that the normal component of the normal stress be zero, yielding the following set of boundary conditions: @f{eqnarray*} \textbf u_{\textbf t} &=& 0, \\ \textbf n \cdot \left(\textbf{n}\cdot [p \textbf{1} - \varepsilon(\textbf{u})] \right) - &=& + &=& 0. @f} @@ -269,15 +269,15 @@ possibilities for imposing boundary conditions: friction is exerted by the boundary on the fluid (e.g. at the interface between earth mantle and earth core where two fluids meet that are stratified by different densities but that both have small enough - viscosities to not introduce much tangential stress on each other). - In formulas, this means that + viscosities to not introduce much tangential stress on each other). + In formulas, this means that @f{eqnarray*} \textbf{n}\cdot\textbf u &=& 0, \\ (\textbf 1-\textbf n\otimes\textbf n) \left(\textbf{n}\cdot [p \textbf{1} - \varepsilon(\textbf{u})] \right) - &=& + &=& 0, @f} the first condition (which needs to be imposed strongly) fixing a single @@ -305,7 +305,7 @@ g_D\}, p\in Q=L^2(\Omega)$ so that - (\textbf{v}, \textbf g_N)_{\Gamma_N} @f} -for all test functions +for all test functions $\textbf v\in \textbf V_0 = \{\varphi \in H^1(\Omega)^d: \varphi_{\Gamma_D}=0\},q\in Q$. @@ -323,7 +323,7 @@ For the Stokes equations, there are a number of possible choices to ensure that the finite element spaces are compatible with the LBB condition. A simple and accurate choice that we will use here is $\textbf u_h\in Q_{p+1}^d, p_h\in Q_p$, i.e. use elements one order higher for the velocities than for the -pressures. +pressures. This then leads to the following discrete problem: find $\textbf u_h,p_h$ so that @@ -362,7 +362,7 @@ linear system for the nodal values of the velocity and pressure fields: @f} Like in step-20 and step-21, we will solve this system of equations by forming the Schur complement, i.e. we will first find -the solution $P$ of +the solution $P$ of @f{eqnarray*} BA^{-1}B^T P &=& BA^{-1} F - G, \\ @f} @@ -446,7 +446,7 @@ convenience (because it reflects the structure of the FESystem finite element and associated DoFHandler object) but later only operate on the $(0,0),(0,1)$, and $(1,0)$ blocks of this matrix. In other words, our algorithm so far entirely ignores the $(1,1)$ (pressure-pressure) -block as it is empty anyway. +block as it is empty anyway. Now, as mentioned, we need a pressure mass matrix to precondition the Schur complement and that conveniently the pressure-pressure block of @@ -537,13 +537,13 @@ template struct InnerPreconditioner; template <> -struct InnerPreconditioner<2> +struct InnerPreconditioner<2> { typedef SparseDirectUMFPACK type; }; template <> -struct InnerPreconditioner<3> +struct InnerPreconditioner<3> { typedef SparseILU type; }; @@ -619,7 +619,7 @@ in the boundary values into the ConstraintMatrix object by using the call 1, BoundaryValues(), constraints); -@endcode +@endcode very similar to how we were making the list of boundary nodes before (note that we set Dirichlet conditions only on boundaries with @@ -669,17 +669,17 @@ set them to zero. The fraction of elements that are present in the sparsity pattern, but do not really contain any information, can be up to one fourth of the total number of elements in the matrix for the 3D application considered in this tutorial program. Remember that matrix-vector products or -preconditioners operate on all the elements of a sparse matrix (even those +preconditioners operate on all the elements of a sparse matrix (even those that are zero), which is an inefficiency we will avoid here. An advantage of directly resolving constrained degrees of freedom is that we can avoid having most of the entries that are going to be zero in our sparse -matrix — we do not need constrained entries during matrix construction -(as opposed to the traditional algorithms, which first fill the matrix, and -only resolve constraints afterwards). This will save both memory and time -when forming matrix-vector products. The way we are going to do that is to -pass the information about constraints to the function that generates the -sparsity pattern, and then set a false argument specifying that we +matrix — we do not need constrained entries during matrix construction +(as opposed to the traditional algorithms, which first fill the matrix, and +only resolve constraints afterwards). This will save both memory and time +when forming matrix-vector products. The way we are going to do that is to +pass the information about constraints to the function that generates the +sparsity pattern, and then set a false argument specifying that we do not intend to use constrained entries: @code DoFTools::make_sparsity_pattern (dof_handler, sparsity_pattern,