From: Guido Kanschat Date: Wed, 23 Mar 2011 21:41:54 +0000 (+0000) Subject: add auxiliary functions X-Git-Tag: v8.0.0~4209 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=7519800ec72ff3aef96044e6559cfb93427f284d;p=dealii.git add auxiliary functions git-svn-id: https://svn.dealii.org/trunk@23513 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/include/deal.II/integrators/maxwell.h b/deal.II/include/deal.II/integrators/maxwell.h index 714a9d89fd..1b5136fd8b 100644 --- a/deal.II/include/deal.II/integrators/maxwell.h +++ b/deal.II/include/deal.II/integrators/maxwell.h @@ -1,7 +1,7 @@ //--------------------------------------------------------------------------- // $Id$ // -// Copyright (C) 2010 by the deal.II authors +// Copyright (C) 2010, 2011 by the deal.II authors // // This file is subject to QPL and may not be distributed // without copyright and license information. Please refer @@ -26,7 +26,32 @@ DEAL_II_NAMESPACE_OPEN namespace LocalIntegrators { /** - * @brief Local integrators related to curl operators and their traces. + * @brief Local integrators related to curl operators and their + * traces. + * + * We use the following conventions for curl + * operators. First, in three space dimensions + * + * @f[ + * \nabla\times \mathbf u = \begin{pmatrix} + * \partial_3 u_2 - \partial 2 u_3 \\ + * \partial_1 u_3 - \partial 3 u_1 \\ + * \partial_2 u_1 - \partial 1 u_2 + * \end{pmatrix} + * @f] + * + * In two space dimensions, the curl is obtained by extending a vector + * u to $(u_1, u_2, 0)^T$ and a scalar p to $(0,0,p)^T$. + * Computing the nonzero components, we obtain the scalar + * curl of a vector function and the vector curl of a scalar + * function. The current implementation exchanges the sign and we have: + * + * @f[ + * \nabla \times \mathbf u = \partial_1 u_2 - \partial 2 u_1 + * \nabla \times p = \begin{pmatrix} + * \partial_2 p \\ -\partial_1 p + * \end{pmatrix} + * @f] * * @ingroup Integrators * @author Guido Kanschat @@ -34,13 +59,88 @@ namespace LocalIntegrators */ namespace Maxwell { - /** - * The curl-curl operator - * @f[ - * \int_Z \nabla\!\times\! u \cdot - * \nabla\!\times\! v \,dx - * @f] - */ +/** + * Auxiliary function. Given the tensors of dim second derivatives, + * compute the curl of the curl of a vector function. The result in + * two dimensions is: + * @f[ + * \nabla\times\nabla\times \mathbf u = \brgin{pmatrix} + * \partial_1\partial_2 u_2 - \partial_2^2 u_1 \\ + * \partial_1\partial_2 u_1 - \partial_1^2 u_2 + * \end{pmatrix} + * @f] + * + * @note The third tensor argument is not used in two dimensions and + * can for instance duplicate one of the previous. + * + * @author Guido Kanschat + * @date 2011 + */ + template + Tensor<1,dim> + curl_curl ( + const Tensor<2,dim>& h0, + const Tensor<2,dim>& h1, + const Tensor<2,dim>& h2) + { + Tensor<1,dim> result; + switch (dim) + { + case 2: + result[0] = h1[0][1]-h0[1][1]; + result[1] = h0[0][1]-h1[0][0]; + break; + default: + Assert(false, ExcNotImplemented()); + } + return result; + } + +/** + * Auxiliary function. Given dim tensors of first + * derivatives and a normal vector, compute the tangential curl + * @f[ + * \mathbf n \times \nabla \times u. + * @f] + * + * @note The third tensor argument is not used in two dimensions and + * can for instance duplicate one of the previous. + * + * @author Guido Kanschat + * @date 2011 + */ + template + Tensor<1,dim> + tangential_curl ( + const Tensor<1,dim>& g0, + const Tensor<1,dim>& g1, + const Tensor<1,dim>& g2, + const Tensor<1,dim>& normal) + { + Tensor<1,dim> result; + + switch (dim) + { + case 2: + result[0] = normal[1] * (g1[0]-g0[1]); + result[1] =-normal[0] * (g1[0]-g0[1]); + break; + default: + Assert(false, ExcNotImplemented()); + } + return result; + } + +/** + * The curl-curl operator + * @f[ + * \int_Z \nabla\!\times\! u \cdot + * \nabla\!\times\! v \,dx + * @f] + * + * @author Guido Kanschat + * @date 2011 + */ template void curl_curl_matrix ( FullMatrix& M, @@ -82,16 +182,17 @@ namespace LocalIntegrators } } - /** - * The curl operator - * @f[ - * \int_Z \nabla\!\times\! u \cdot v \,dx. - * @f] - * - * This is the standard curl - * operator in 3D and the scalar - * curl in 2D. - */ +/** + * The curl operator + * @f[ + * \int_Z \nabla\!\times\! u \cdot v \,dx. + * @f] + * + * This is the standard curl operator in 3D and the scalar curl in 2D. + * + * @author Guido Kanschat + * @date 2011 +*/ template void curl_matrix ( FullMatrix& M,