From e30fe8a8c78f02e6ab8ce84bd3dbf07e1c92263e Mon Sep 17 00:00:00 2001 From: bangerth Date: Thu, 24 Jun 2010 03:02:55 +0000 Subject: [PATCH] Finish comments. Adjust right hand side and solution picture. git-svn-id: https://svn.dealii.org/trunk@21309 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-45/doc/builds-on | 2 +- deal.II/examples/step-45/doc/intro.dox | 4 +- .../examples/step-45/doc/step-45.solution.png | Bin 36656 -> 85370 bytes deal.II/examples/step-45/step-45.cc | 221 ++++++++++++------ 4 files changed, 148 insertions(+), 79 deletions(-) diff --git a/deal.II/examples/step-45/doc/builds-on b/deal.II/examples/step-45/doc/builds-on index beec4c1e1e..5965545848 100644 --- a/deal.II/examples/step-45/doc/builds-on +++ b/deal.II/examples/step-45/doc/builds-on @@ -1 +1 @@ -step-3 step-6 +step-4 step-6 diff --git a/deal.II/examples/step-45/doc/intro.dox b/deal.II/examples/step-45/doc/intro.dox index 06415e6554..9c2eb80b37 100644 --- a/deal.II/examples/step-45/doc/intro.dox +++ b/deal.II/examples/step-45/doc/intro.dox @@ -30,7 +30,7 @@ left and right parts of the boundary are identified. Let $\Omega=(0,1)^2$ and consider the problem @f{align*} -\Delta u &= - \pi^2\sin(\pi x)\sin(\pi y) \qquad &\text{in }\Omega + \cos(2\pi x)e^{-2x}\cos(2\pi y)e^{-2y} \qquad &\text{in }\Omega \\ u(x,0) &= 0 \qquad &\text{for }x\in(0,1)\qquad &&\text{(bottom boundary)} \\ @@ -39,6 +39,8 @@ consider the problem u(0,y) &= u(1,y) \qquad &\text{for }y\in(0,1) \qquad && \text{(left and right boundaries)} @f} +Note that the source term is not symmetric and so the solution would not be +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 diff --git a/deal.II/examples/step-45/doc/step-45.solution.png b/deal.II/examples/step-45/doc/step-45.solution.png index 20dae47f303fa73db0a399249f6f3924747d0f63..d66d77af68e140a46491f419a13803ecffb6eded 100644 GIT binary patch literal 85370 zcmV*DKy1H>P)V>IGcGYOVJ&KPZDMamX?QC|ZfGEH zW*{;!FgGwYGB6MAGNb?iAOJ~3K~#90?7h*JBrD4Wh|&Ge%g*~>&DZH%WG98}4Wbbl zRn@0hvGM?d8VYW;GW)duw>SBrOR0Z$2Y%>>e&~n3HQfwLzyAv`cr!A!?}LTkrJq0a z-RZxu&ioLiZ!z`1<(qVLoBD|t+{TGJvG7yC><2;rv$U5@pAsV9|fY7?#*6L?n9^M9` z?Hm>^(WgM*{f8pC>~DcFUpy~9WeiU?y5#=%Z-n;qt?hjZZQqXWC6bNl zZ*eZX`3(BSbN+{>v>nwq?|?t$HfV$X;U(13ZTF6=i1xd|iEyvUJ=%!TRhMq^;ba~b zeeHnUa4zhrxKYEZ%l&t`Kji4nm9(8W-d(kxZqQc-n@>3|uhM57ulH^oWf9K!iEydy z7Vr^Wa$ushQHr`$Y1i#2i^GC1c;+>i95;3eSG9Gr=^y$Qt&D zYGA5-Gp*Od-* zopQU*RkODa>9*Gy#~+Gld*ghjxuu&Dx~KeE?w^FW&0jDn-Z6J?J#NVgqD@YO^(K*@ zdfjzLqSUqI>c6Vd#Ouu2?hKTk 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ad7ccae831..fb7574972d 100644 --- a/deal.II/examples/step-45/step-45.cc +++ b/deal.II/examples/step-45/step-45.cc @@ -116,9 +116,10 @@ double RightHandSide::value (const Point<2>&p, const unsigned int) const { - return (numbers::PI * numbers::PI * - std::sin (numbers::PI * p (0)) * - std::sin (numbers::PI * p (1))); + return (std::cos (2 * numbers::PI * p(0)) * + std::exp (- 2 * p(0)) * + std::cos (2 * numbers::PI * p(1)) * + std::exp (- 2 * p(1))); } // @sect3{Implementation of the LaplaceProblem class} @@ -213,85 +214,151 @@ void LaplaceProblem::make_grid_and_dofs () // @sect4{LaplaceProblem::make_periodicity_constraints} + // This is the function that provides the new + // material of this tutorial program. The + // general outline of the algorithm is as + // follows: we first loop over all the + // degrees of freedom on the right boundary + // and record their $y$-locations in a map + // together with their global indices. Then + // we go along the left boundary, find + // matching $y$-locations for each degree of + // freedom, and then add constraints that + // identify these matched degrees of freedom. + // + // In this function, we make use of the fact + // that we have a scalar element (i.e. the + // only valid vector component that can be + // passed to DoFAccessor::vertex_dof_index is + // zero) and that we have a $Q_1$ element for + // which all degrees of freedom live in the + // vertices of the cell. Furthermore, we have + // assumed that we are in 2d and that meshes + // were not refined adaptively — the + // latter assumption would imply that there + // may be vertices that aren't matched + // one-to-one and for which we won't be able + // to compute constraints this easily. We + // will discuss in the "outlook" part of the + // results section below other strategies to + // write the current function that can work + // in cases like this as well. void LaplaceProblem::make_periodicity_constraints () { - // matrix. The first constraints we put in - // are the periodic boundary - // conditions. For this let us consider the - // constraints we have to take care of in - // more detail first: We want to identify - // all degrees of freedom located on the - // right part of the boundary with the ones - // located on the left part. Thus, first we - // select a degree of freedom on the right - // part of the boundary. Then we look for - // the corresponding one the left-hand side - // and identify these two. Since we are - // using finite elements of order 1 here, - // finding the corresponding degree of - // freedom on the other side is quite easy: - // All degrees of freedom are located on - // vertices and thus we simply can take the - // degree of freedom on the other side, - // which is located on the vertex with the - // same y-component. Here starts the - // implementation: First we declare a - // vector, which stores the global index of - // the boundary degree of freedom together - // with the y-component of the vertex on - // which it is located. - - std::vector > dof_locations; - - dof_locations.reserve (dof_handler.n_boundary_dofs ()); - - // Then we loop over all active cells and - // check, whether the cell is located at - // the boundary. If this is the case, we - // check, if it is located on the right - // part of the boundary, hence, if face 1 - // is at the boundary. - for (DoFHandler<2>::active_cell_iterator cell = dof_handler.begin_active (); cell != dof_handler.end (); ++cell) - if (cell->at_boundary () && cell->face (1)->at_boundary ()) { - // Since each degree of freedom of the - // face is located on one vertex, we - // can access them directly and store - // the global index of the degree of - // freedom togehter with the - // y-component of the vertex on which - // it is located. - dof_locations.push_back (std::pair (cell->vertex_dof_index (1, 0), cell->vertex (1) (1))); - dof_locations.push_back (std::pair (cell->vertex_dof_index (3, 0), cell->vertex (3) (1))); - } + // To start with the actual implementation, + // we loop over all active cells and check + // whether the cell is located at the right + // boundary (i.e. face 1 — the one at + // the right end of the cell — is at + // the boundary). If that is so, then we + // use that for the currently used finite + // element, each degree of freedom of the + // face is located on one vertex, and store + // their $y$-coordinate along with the + // global number of this degree of freedom + // in the following map: + std::map dof_locations; + + for (DoFHandler<2>::active_cell_iterator cell = dof_handler.begin_active (); + cell != dof_handler.end (); ++cell) + if (cell->at_boundary () + && + cell->face(1)->at_boundary ()) + { + dof_locations[cell->face(1)->vertex_dof_index(0, 0)] + = cell->face(1)->vertex(0)[1]; + dof_locations[cell->face(1)->vertex_dof_index(1, 0)] + = cell->face(1)->vertex(1)[1]; + } + // Note that in the above block, we add + // vertices zero and one of the affected + // face to the map. This means that we will + // add each vertex twice, once from each of + // the two adjacent cells (unless the + // vertex is a corner of the domain). Since + // the coordinates of the vertex are the + // same both times of course, there is no + // harm: we replace one value in the map + // with itself the second time we visit an + // entry. + // + // The same will be true below where we add + // the same constraint twice to the + // ConstraintMatrix — again, we will + // overwrite the constraint with itself, + // and no harm is done. + // Now we have to find the corresponding // degrees of freedom on the left part of // the boundary. Therefore we loop over all - // cells again and choose the ones, where - // face 0 is at the boundary. - for (DoFHandler<2>::active_cell_iterator cell = dof_handler.begin_active (); cell != dof_handler.end (); ++cell) - if (cell->at_boundary () && cell->face (0)->at_boundary ()) { - // For every degree of freedom on this - // face we add a new line to the - // constraint matrix. Then we identify - // it with the corresponding degree of - // freedom on the right part of the - // boundary. - constraints.add_line (cell->vertex_dof_index (0, 0)); - - for (unsigned int i = 0; i < dof_locations.size (); ++i) - if (dof_locations[i].second == cell->vertex (0) (1)) { - constraints.add_entry (cell->vertex_dof_index (0, 0), dof_locations[i].first, 1.0); - break; - } - - constraints.add_line (cell->vertex_dof_index (2, 0)); - - for (unsigned int i = 0; i < dof_locations.size (); ++i) - if (dof_locations[i].second == cell->vertex (2) (1)) { - constraints.add_entry (cell->vertex_dof_index (2, 0), dof_locations[i].first, 1.0); - break; - } - } + // cells again and choose the ones where + // face 0 is at the boundary: + for (DoFHandler<2>::active_cell_iterator cell = dof_handler.begin_active (); + cell != dof_handler.end (); ++cell) + if (cell->at_boundary () + && + cell->face (0)->at_boundary ()) + { + // Every degree of freedom on this + // face needs to have a corresponding + // one on the right side of the face, + // and our goal is to add a + // constraint for the one on the left + // in terms of the one on the + // right. To this end we first add a + // new line to the constraint matrix + // for this one degree of + // freedom. Then we identify it with + // the corresponding degree of + // freedom on the right part of the + // boundary by constraining the + // degree of freedom on the left with + // the one on the right times a + // weight of 1.0. + // + // Consequently, we loop over the two + // vertices of each face we find and + // then loop over all the + // $y$-locations we've previously + // recorded to find which degree of + // freedom on the right boundary + // corresponds to the one we + // currently look at. Note that we + // have entered these into a map, and + // when looping over the iterators + // p of this map, + // p-@>first corresponds + // to the "key" of an entry (the + // global number of the degree of + // freedom), whereas + // p-@>second is the + // "value" (the $y$-location we have + // entered above). + // + // We are quite sure here that we + // should be finding such a + // corresponding degree of + // freedom. However, sometimes stuff + // happens and so the bottom of the + // block contains an assertion that + // our assumption was indeed correct + // and that a vertex was found. + for (unsigned int face_vertex = 0; face_vertex<2; ++face_vertex) + { + constraints.add_line (cell->face(0)->vertex_dof_index (face_vertex, 0)); + + std::map::const_iterator p = dof_locations.begin(); + for (; p != dof_locations.end(); ++p) + if (std::fabs(p->second - cell->face(0)->vertex(face_vertex)[1]) < 1e-8) + { + constraints.add_entry (cell->face(0)->vertex_dof_index (face_vertex, 0), + p->first, 1.0); + break; + } + Assert (p != dof_locations.end(), + ExcMessage ("No corresponding degree of freedom was found!")); + } + } } -- 2.39.5