1071 * sparsity_pattern.copy_from(dsp);
1073 * system_matrix.reinit(sparsity_pattern);
1075 * solution.reinit(dof_handler.n_dofs());
1076 * system_rhs.reinit(dof_handler.n_dofs());
1084 * assembled. As stated in the documentation of the main
class above,
it
1085 * does not
do this itself, but rather delegates to the
function following
1086 * next, utilizing the
WorkStream concept discussed in @ref threads .
1090 * If you have looked through the @ref threads module, you will have
1091 * seen that assembling in
parallel does not take an incredible
1092 * amount of extra code as long as you diligently describe what the
1093 * scratch and
copy data objects are, and
if you define suitable
1095 * contributions to global objects. This done, the following will do
1096 * all the heavy lifting to get these operations done on multiple
1097 * threads on as many cores as you have in your system:
1100 * template <
int dim>
1101 * void AdvectionProblem<dim>::assemble_system()
1104 * dof_handler.
end(),
1106 * &AdvectionProblem::local_assemble_system,
1107 * &AdvectionProblem::copy_local_to_global,
1108 * AssemblyScratchData(fe),
1109 * AssemblyCopyData());
1116 * As already mentioned above, we need to have scratch objects
for
1117 * the
parallel computation of local contributions. These objects
1119 * we will need to have constructors and
copy constructors that allow us to
1120 * create them. For the cell terms we need the
values
1122 * order to determine the source density and the advection field at
1123 * a given
point, and the weights of the quadrature points times the
1124 *
determinant of the Jacobian at these points. In contrast,
for the
1125 * boundary integrals, we don't need the
gradients, but rather the
1126 * normal vectors to the cells. This determines which update flags
1127 * we will have to pass to the constructors of the members of the
1131 * template <
int dim>
1132 * AdvectionProblem<dim>::AssemblyScratchData::AssemblyScratchData(
1135 *
QGauss<dim>(fe.degree + 1),
1138 * , fe_face_values(fe,
1139 *
QGauss<dim - 1>(fe.degree + 1),
1142 * , rhs_values(fe_values.get_quadrature().size())
1143 * , advection_directions(fe_values.get_quadrature().size())
1144 * , face_boundary_values(fe_face_values.get_quadrature().size())
1145 * , face_advection_directions(fe_face_values.get_quadrature().size())
1150 * template <
int dim>
1151 * AdvectionProblem<dim>::AssemblyScratchData::AssemblyScratchData(
1152 * const AssemblyScratchData &scratch_data)
1153 * : fe_values(scratch_data.fe_values.get_fe(),
1154 * scratch_data.fe_values.get_quadrature(),
1157 * , fe_face_values(scratch_data.fe_face_values.get_fe(),
1158 * scratch_data.fe_face_values.get_quadrature(),
1161 * , rhs_values(scratch_data.rhs_values.size())
1162 * , advection_directions(scratch_data.advection_directions.size())
1163 * , face_boundary_values(scratch_data.face_boundary_values.size())
1164 * , face_advection_directions(scratch_data.face_advection_directions.size())
1171 * Now, this is the
function that does the actual work. It is not very
1172 * different from the <code>assemble_system</code>
functions of previous
1173 * example programs, so we will again only comment on the differences. The
1174 * mathematical stuff closely follows what we have said in the introduction.
1178 * There are a number of points worth mentioning here, though. The
1180 * objects into the ScratchData object. We have done so because the
1181 * alternative would have been to simply create one every time we
1182 * get into this
function -- i.
e., on every cell. It now turns
out
1183 * that the
FEValues classes were written with the explicit goal of
1184 * moving everything that remains the same from cell to cell into
1185 * the construction of the object, and only do as little work as
1187 * cell. What this means is that
it would be very expensive to
1188 * create a new object of this kind in this
function as we would
1189 * have to do
it for every cell -- exactly the thing we wanted to
1190 * avoid with the
FEValues class. Instead, what we do is create
it
1191 * only once (or a small number of times) in the scratch objects and
1192 * then re-use
it as often as we can.
1196 * This begs the question of whether there are other objects we
1197 * create in this
function whose creation is expensive compared to
1198 * its use. Indeed, at the top of the
function, we declare all sorts
1199 * of objects. The <code>AdvectionField</code>,
1200 * <code>RightHandSide</code> and <code>BoundaryValues</code> do not
1201 * cost much to create, so there is no harm here. However,
1202 * allocating memory in creating the <code>rhs_values</code> and
1203 * similar variables below typically costs a significant amount of
1204 * time, compared to just accessing the (temporary)
values we store
1205 * in them. Consequently, these would be candidates
for moving into
1206 * the <code>AssemblyScratchData</code> class. We will leave this as
1210 * template <
int dim>
1211 * void AdvectionProblem<dim>::local_assemble_system(
1212 * const typename
DoFHandler<dim>::active_cell_iterator &cell,
1213 * AssemblyScratchData &scratch_data,
1214 * AssemblyCopyData ©_data)
1218 * We define some abbreviations to avoid unnecessarily long lines:
1221 * const unsigned
int dofs_per_cell = fe.n_dofs_per_cell();
1222 *
const unsigned int n_q_points =
1223 * scratch_data.fe_values.get_quadrature().size();
1224 *
const unsigned int n_face_q_points =
1225 * scratch_data.fe_face_values.get_quadrature().size();
1229 * We declare cell
matrix and cell right hand side...
1232 * copy_data.cell_matrix.reinit(dofs_per_cell, dofs_per_cell);
1233 * copy_data.cell_rhs.reinit(dofs_per_cell);
1237 * ... an array to hold the global indices of the degrees of freedom of
1238 * the cell on which we are presently working...
1241 * copy_data.local_dof_indices.resize(dofs_per_cell);
1245 * ... then initialize the <code>
FEValues</code>
object...
1248 * scratch_data.fe_values.
reinit(cell);
1252 * ... obtain the
values of right hand side and advection directions
1253 * at the quadrature points...
1256 * scratch_data.advection_field.value_list(
1257 * scratch_data.fe_values.get_quadrature_points(),
1258 * scratch_data.advection_directions);
1259 * scratch_data.right_hand_side.value_list(
1260 * scratch_data.fe_values.get_quadrature_points(), scratch_data.rhs_values);
1264 * ...
set the
value of the streamline diffusion parameter as
1265 * described in the introduction...
1268 *
const double delta = 0.1 * cell->diameter();
1272 * ... and
assemble the local contributions to the system
matrix and
1273 * right hand side as also discussed above:
1276 *
for (
unsigned int q_point = 0; q_point < n_q_points; ++q_point)
1277 *
for (
unsigned int i = 0; i < dofs_per_cell; ++i)
1281 * Alias the AssemblyScratchData
object to keep the lines from
1285 *
const auto &sd = scratch_data;
1286 *
for (
unsigned int j = 0; j < dofs_per_cell; ++j)
1287 * copy_data.cell_matrix(i, j) +=
1288 * ((sd.fe_values.shape_value(i, q_point) +
1289 * delta * (sd.advection_directions[q_point] *
1290 * sd.fe_values.shape_grad(i, q_point))) *
1291 * sd.advection_directions[q_point] *
1292 * sd.fe_values.shape_grad(j, q_point)) *
1293 * sd.fe_values.JxW(q_point);
1295 * copy_data.cell_rhs(i) +=
1296 * (sd.fe_values.shape_value(i, q_point) +
1297 * delta * (sd.advection_directions[q_point] *
1298 * sd.fe_values.shape_grad(i, q_point))) *
1299 * sd.rhs_values[q_point] *
1300 * sd.fe_values.JxW(q_point);
1305 * Besides the cell terms which we have built up now, the bilinear
1306 * form of the present problem also contains terms on the boundary of
1307 * the domain. Therefore, we have to
check whether any of the faces of
1308 *
this cell are on the boundary of the domain, and
if so
assemble the
1309 * contributions of
this face as well. Of course, the bilinear form
1310 * only contains contributions from the <code>inflow</code> part of
1311 * the boundary, but to find
out whether a certain part of a face of
1312 * the present cell is part of the inflow boundary, we have to have
1313 * information on the exact location of the quadrature points and on
1314 * the direction of flow at
this point; we obtain
this information
1316 * whether a quadrature
point is on the inflow boundary.
1319 *
for (
const auto &face : cell->face_iterators())
1320 *
if (face->at_boundary())
1324 * Ok, this face of the present cell is on the boundary of the
1325 * domain. Just as
for the usual
FEValues object which we have
1326 *
used in previous examples and also above, we have to
1330 * scratch_data.fe_face_values.
reinit(cell, face);
1334 * For the quadrature points at hand, we ask
for the
values of
1335 * the inflow
function and
for the direction of flow:
1339 * scratch_data.fe_face_values.get_quadrature_points(),
1340 * scratch_data.face_boundary_values);
1341 * scratch_data.advection_field.value_list(
1342 * scratch_data.fe_face_values.get_quadrature_points(),
1343 * scratch_data.face_advection_directions);
1347 * Now
loop over all quadrature points and see whether
this face is on
1348 * the inflow or outflow part of the boundary. The normal
1349 * vector points
out of the cell: since the face is at
1350 * the boundary, the normal vector points
out of the domain,
1351 * so
if the advection direction points into the domain, its
1352 *
scalar product with the normal vector must be
negative (to see why
1353 *
this is
true, consider the scalar product definition that uses a
1357 *
for (unsigned
int q_point = 0; q_point < n_face_q_points; ++q_point)
1358 *
if (scratch_data.fe_face_values.normal_vector(q_point) *
1359 * scratch_data.face_advection_directions[q_point] <
1363 * If the face is part of the inflow boundary, then compute the
1364 * contributions of
this face to the global matrix and right
1365 * hand side,
using the values obtained from the
1366 *
FEFaceValues object and the formulae discussed in the
1370 *
for (unsigned
int i = 0; i < dofs_per_cell; ++i)
1372 *
for (
unsigned int j = 0; j < dofs_per_cell; ++j)
1373 * copy_data.cell_matrix(i, j) -=
1374 * (scratch_data.face_advection_directions[q_point] *
1375 * scratch_data.fe_face_values.normal_vector(q_point) *
1376 * scratch_data.fe_face_values.shape_value(i, q_point) *
1377 * scratch_data.fe_face_values.shape_value(j, q_point) *
1378 * scratch_data.fe_face_values.JxW(q_point));
1380 * copy_data.cell_rhs(i) -=
1381 * (scratch_data.face_advection_directions[q_point] *
1382 * scratch_data.fe_face_values.normal_vector(q_point) *
1383 * scratch_data.face_boundary_values[q_point] *
1384 * scratch_data.fe_face_values.shape_value(i, q_point) *
1385 * scratch_data.fe_face_values.JxW(q_point));
1391 * The
final piece of information the
copy routine needs is the global
1392 * indices of the degrees of freedom on
this cell, so we
end by writing
1393 * them to the local array:
1396 * cell->get_dof_indices(copy_data.local_dof_indices);
1404 * the local contributions the previous
function computed (and
1405 * put into the AssemblyCopyData
object) into the global
matrix and right
1406 * hand side vector objects. This is essentially what we
always had
1407 * as the last block of code when assembling something on every
1408 * cell. The following should therefore be pretty obvious:
1411 *
template <
int dim>
1413 * AdvectionProblem<dim>::copy_local_to_global(
const AssemblyCopyData ©_data)
1415 * hanging_node_constraints.distribute_local_to_global(
1416 * copy_data.cell_matrix,
1417 * copy_data.cell_rhs,
1418 * copy_data.local_dof_indices,
1425 * Here comes the linear solver routine. As the system is no longer
1427 * use the Conjugate Gradient method anymore. Rather, we use a solver that
1428 * is more
general and does not rely on any special properties of the
1429 *
matrix: the GMRES method. GMRES, like the conjugate
gradient method,
1430 *
requires a decent preconditioner: we use a Jacobi preconditioner here,
1431 * which works well enough
for this problem.
1434 *
template <
int dim>
1435 *
void AdvectionProblem<dim>::solve()
1438 * system_rhs.size() / 10),
1439 * 1e-10 * system_rhs.l2_norm());
1442 * preconditioner.
initialize(system_matrix, 1.0);
1443 * solver.solve(system_matrix, solution, system_rhs, preconditioner);
1447 * system_matrix.vmult(residual, solution);
1448 * residual -= system_rhs;
1449 * std::cout <<
" Iterations required for convergence: "
1450 * << solver_control.last_step() <<
'\n'
1451 * <<
" Max norm of residual: "
1452 * << residual.linfty_norm() <<
'\n';
1454 * hanging_node_constraints.distribute(solution);
1459 * The following
function refines the grid according to the quantity
1460 * described in the introduction. The respective computations are made in
1461 * the class <code>GradientEstimation</code>.
1464 *
template <
int dim>
1465 *
void AdvectionProblem<dim>::refine_grid()
1469 * GradientEstimation::estimate(dof_handler,
1471 * estimated_error_per_cell);
1474 * estimated_error_per_cell,
1483 * This
function is similar to the one in step 6, but since we use a higher
1484 * degree finite element we save the solution in a different
1485 * way. Visualization programs like VisIt and Paraview typically only
1486 * understand data that is associated with nodes: they cannot plot
1487 * fifth-degree basis
functions, which results in a very inaccurate picture
1488 * of the solution we computed. To get around
this we save multiple
1489 * <em>patches</em> per cell: in 2
d we save 64 bilinear `cells
' to the VTU
1490 * file for each cell, and in 3d we save 512. The end result is that the
1491 * visualization program will use a piecewise linear interpolation of the
1492 * cubic basis functions: this captures the solution detail and, with most
1493 * screen resolutions, looks smooth. We save the grid in a separate step
1494 * with no extra patches so that we have a visual representation of the cell
1499 * Version 9.1 of deal.II gained the ability to write higher degree
1500 * polynomials (i.e., write piecewise bicubic visualization data for our
1501 * piecewise bicubic solution) VTK and VTU output: however, not all recent
1502 * versions of ParaView and VisIt (as of 2018) can read this format, so we
1503 * use the older, more general (but less efficient) approach here.
1506 * template <int dim>
1507 * void AdvectionProblem<dim>::output_results(const unsigned int cycle) const
1511 * const std::string filename = "grid-" + std::to_string(cycle) + ".vtu";
1512 * std::ofstream output(filename);
1513 * grid_out.write_vtu(triangulation, output);
1514 * std::cout << "Grid written to " << filename << std::endl;
1518 * DataOut<dim> data_out;
1519 * data_out.attach_dof_handler(dof_handler);
1520 * data_out.add_data_vector(solution, "solution");
1521 * data_out.build_patches(8);
1525 * VTU output can be expensive, both to compute and to write to
1526 * disk. Here we ask ZLib, a compression library, to compress the data
1527 * in a way that maximizes throughput.
1530 * DataOutBase::VtkFlags vtk_flags;
1531 * vtk_flags.compression_level = DataOutBase::CompressionLevel::best_speed;
1532 * data_out.set_flags(vtk_flags);
1534 * const std::string filename = "solution-" + std::to_string(cycle) + ".vtu";
1535 * std::ofstream output(filename);
1536 * data_out.write_vtu(output);
1537 * std::cout << "Solution written to " << filename << std::endl;
1544 * ... as is the main loop (setup -- solve -- refine), aside from the number
1545 * of cycles and the initial grid:
1548 * template <int dim>
1549 * void AdvectionProblem<dim>::run()
1551 * for (unsigned int cycle = 0; cycle < 10; ++cycle)
1553 * std::cout << "Cycle " << cycle << ':
' << std::endl;
1557 * GridGenerator::hyper_cube(triangulation, -1, 1);
1558 * triangulation.refine_global(3);
1566 * std::cout << " Number of active cells: "
1567 * << triangulation.n_active_cells() << std::endl;
1571 * std::cout << " Number of degrees of freedom: "
1572 * << dof_handler.n_dofs() << std::endl;
1574 * assemble_system();
1576 * output_results(cycle);
1585 * <a name="step_9-GradientEstimationclassimplementation"></a>
1586 * <h3>GradientEstimation class implementation</h3>
1590 * Now for the implementation of the <code>GradientEstimation</code> class.
1591 * Let us start by defining constructors for the
1592 * <code>EstimateScratchData</code> class used by the
1593 * <code>estimate_cell()</code> function:
1596 * template <int dim>
1597 * GradientEstimation::EstimateScratchData<dim>::EstimateScratchData(
1598 * const FiniteElement<dim> &fe,
1599 * const Vector<double> &solution,
1600 * Vector<float> &error_per_cell)
1601 * : fe_midpoint_value(fe,
1603 * update_values | update_quadrature_points)
1604 * , solution(solution)
1605 * , error_per_cell(error_per_cell)
1606 * , cell_midpoint_value(1)
1607 * , neighbor_midpoint_value(1)
1611 * We allocate a vector to hold iterators to all active neighbors of
1612 * a cell. We reserve the maximal number of active neighbors in order to
1613 * avoid later reallocations. Note how this maximal number of active
1614 * neighbors is computed here.
1617 * active_neighbors.reserve(GeometryInfo<dim>::faces_per_cell *
1618 * GeometryInfo<dim>::max_children_per_face);
1622 * template <int dim>
1623 * GradientEstimation::EstimateScratchData<dim>::EstimateScratchData(
1624 * const EstimateScratchData &scratch_data)
1625 * : fe_midpoint_value(scratch_data.fe_midpoint_value.get_fe(),
1626 * scratch_data.fe_midpoint_value.get_quadrature(),
1627 * update_values | update_quadrature_points)
1628 * , solution(scratch_data.solution)
1629 * , error_per_cell(scratch_data.error_per_cell)
1630 * , cell_midpoint_value(1)
1631 * , neighbor_midpoint_value(1)
1637 * Next comes the implementation of the <code>GradientEstimation</code>
1638 * class. The first function does not much except for delegating work to the
1639 * other function, but there is a bit of setup at the top.
1643 * Before starting with the work, we check that the vector into
1644 * which the results are written has the right size, using the `Assert`
1645 * macro and the exception class we declared above. Programming
1646 * mistakes in which one forgets to size arguments correctly at the
1647 * calling site are quite common. Because the resulting damage from
1648 * not catching such errors is often subtle (e.g., corruption of
1649 * data somewhere in memory, or non-reproducible results), it is
1650 * well worth the effort to check for such things.
1653 * template <int dim>
1654 * void GradientEstimation::estimate(const DoFHandler<dim> &dof_handler,
1655 * const Vector<double> &solution,
1656 * Vector<float> &error_per_cell)
1659 * error_per_cell.size() == dof_handler.get_triangulation().n_active_cells(),
1660 * ExcInvalidVectorLength(error_per_cell.size(),
1661 * dof_handler.get_triangulation().n_active_cells()));
1663 * WorkStream::run(dof_handler.begin_active(),
1664 * dof_handler.end(),
1665 * &GradientEstimation::template estimate_cell<dim>,
1666 * std::function<void(const EstimateCopyData &)>(),
1667 * EstimateScratchData<dim>(dof_handler.get_fe(),
1670 * EstimateCopyData());
1676 * Here comes the function that estimates the local error by computing the
1677 * finite difference approximation of the gradient. The function first
1678 * computes the list of active neighbors of the present cell and then
1679 * computes the quantities described in the introduction for each of
1680 * the neighbors. The reason for this order is that it is not a one-liner
1681 * to find a given neighbor with locally refined meshes. In principle, an
1682 * optimized implementation would find neighbors and the quantities
1683 * depending on them in one step, rather than first building a list of
1684 * neighbors and in a second step their contributions but we will gladly
1685 * leave this as an exercise. As discussed before, the worker function
1686 * passed to WorkStream::run works on "scratch" objects that keep all
1687 * temporary objects. This way, we do not need to create and initialize
1688 * objects that are expensive to initialize within the function that does
1689 * the work every time it is called for a given cell. Such an argument is
1690 * passed as the second argument. The third argument would be a "copy-data"
1691 * object (see @ref threads for more information) but we do not actually use
1692 * any of these here. Since WorkStream::run() insists on passing three
1693 * arguments, we declare this function with three arguments, but simply
1694 * ignore the last one.
1698 * (This is unsatisfactory from an aesthetic perspective. It can be avoided
1699 * by using an anonymous (lambda) function. If you allow, let us here show
1700 * how. First, assume that we had declared this function to only take two
1701 * arguments by omitting the unused last one. Now, WorkStream::run still
1702 * wants to call this function with three arguments, so we need to find a
1703 * way to "forget" the third argument in the call. Simply passing
1704 * WorkStream::run the pointer to the function as we do above will not do
1705 * this -- the compiler will complain that a function declared to have two
1706 * arguments is called with three arguments. However, we can do this by
1707 * passing the following as the third argument to WorkStream::run():
1708 * <div class=CodeFragmentInTutorialComment>
1710 * [](const typename DoFHandler<dim>::active_cell_iterator &cell,
1711 * EstimateScratchData<dim> & scratch_data,
1712 * EstimateCopyData &)
1714 * GradientEstimation::estimate_cell<dim>(cell, scratch_data);
1718 * This is not much better than the solution implemented below: either the
1719 * routine itself must take three arguments or it must be wrapped by
1720 * something that takes three arguments. We don't use
this since adding the
1721 * unused argument at the beginning is simpler.
1725 * Now
for the details:
1728 *
template <
int dim>
1729 *
void GradientEstimation::estimate_cell(
1731 * EstimateScratchData<dim> &scratch_data,
1732 *
const EstimateCopyData &)
1736 * We need space
for the tensor <code>Y</code>, which is the
sum of
1737 * outer products of the y-vectors.
1744 * First initialize the <code>
FEValues</code> object, as well as the
1745 * <code>Y</code> tensor:
1748 * scratch_data.fe_midpoint_value.
reinit(cell);
1752 * Now, before we go on, we
first compute a list of all active neighbors
1753 * of the present cell. We
do so by
first looping over all faces and see
1754 * whether the neighbor there is active, which would be the
case if it
1755 * is on the same
level as the present cell or one
level coarser (note
1756 * that a neighbor can only be once coarser than the present cell, as
1757 * we only allow a maximal difference of one refinement over a face in
1758 * deal.II). Alternatively, the neighbor could be on the same
level
1759 * and be further refined; then we have to find which of its children
1760 * are next to the present cell and select these (note that
if a child
1761 * of a neighbor of an active cell that is next to
this active cell,
1762 * needs necessarily be active itself, due to the one-refinement rule
1767 * Things are slightly different in one space dimension, as there the
1768 * one-refinement rule does not exist: neighboring active cells may
1769 * differ in as many refinement levels as they like. In
this case, the
1770 * computation becomes a little more difficult, but we will explain
1775 * Before starting the
loop over all neighbors of the present cell, we
1776 * have to clear the array storing the iterators to the active
1777 * neighbors, of course.
1780 * scratch_data.active_neighbors.clear();
1781 *
for (
const auto face_n : cell->face_indices())
1782 *
if (!cell->at_boundary(face_n))
1786 * First define an abbreviation
for the iterator to the face and
1790 * const auto face = cell->face(face_n);
1791 *
const auto neighbor = cell->neighbor(face_n);
1795 * Then
check whether the neighbor is active. If
it is, then
it
1796 * is on the same
level or one
level coarser (
if we are not in
1797 * 1d), and we are interested in
it in any
case.
1800 *
if (neighbor->is_active())
1801 * scratch_data.active_neighbors.push_back(neighbor);
1806 * If the neighbor is not active, then
check its children.
1813 * To find the child of the neighbor which bounds to the
1814 * present cell, successively go to its right child
if
1815 * we are left of the present cell (n==0), or go to the
1816 * left child
if we are on the right (n==1), until we
1817 * find an active cell.
1820 *
auto neighbor_child = neighbor;
1821 *
while (neighbor_child->has_children())
1822 * neighbor_child = neighbor_child->child(face_n == 0 ? 1 : 0);
1826 * As
this used some non-trivial geometrical intuition,
1827 * we might want to
check whether we did
it right,
1828 * i.e.,
check whether the neighbor of the cell we found
1829 * is indeed the cell we are presently working
1830 * on. Checks like
this are often useful and have
1831 * frequently uncovered errors both in algorithms like
1832 * the line above (where it is simple to involuntarily
1833 * exchange <code>n==1</code> for <code>n==0</code> or
1834 * the like) and in the library (the assumptions
1835 * underlying the algorithm above could either be wrong,
1836 * wrongly documented, or are violated due to an error
1837 * in the library). One could in principle remove such
1838 * checks after the program works
for some time, but
it
1839 * might be a good things to leave
it in anyway to
check
1840 *
for changes in the library or in the algorithm above.
1844 * Note that
if this check fails, then
this is certainly
1845 * an error that is irrecoverable and probably qualifies
1846 * as an
internal error. We therefore use a predefined
1847 * exception
class to throw here.
1850 *
Assert(neighbor_child->neighbor(face_n == 0 ? 1 : 0) == cell,
1851 * ExcInternalError());
1855 * If the
check succeeded, we push the active neighbor
1856 * we just found to the stack we keep:
1859 * scratch_data.active_neighbors.push_back(neighbor_child);
1864 * If we are not in 1
d, we collect all neighbor children
1865 * `behind
' the subfaces of the current face and move on:
1868 * for (unsigned int subface_n = 0; subface_n < face->n_children();
1870 * scratch_data.active_neighbors.push_back(
1871 * cell->neighbor_child_on_subface(face_n, subface_n));
1877 * OK, now that we have all the neighbors, lets start the computation
1878 * on each of them. First we do some preliminaries: find out about the
1879 * center of the present cell and the solution at this point. The
1880 * latter is obtained as a vector of function values at the quadrature
1881 * points, of which there are only one, of course. Likewise, the
1882 * position of the center is the position of the first (and only)
1883 * quadrature point in real space.
1886 * const Point<dim> this_center =
1887 * scratch_data.fe_midpoint_value.quadrature_point(0);
1889 * scratch_data.fe_midpoint_value.get_function_values(
1890 * scratch_data.solution, scratch_data.cell_midpoint_value);
1894 * Now loop over all active neighbors and collect the data we
1898 * Tensor<1, dim> projected_gradient;
1899 * for (const auto &neighbor : scratch_data.active_neighbors)
1903 * Then get the center of the neighbor cell and the value of the
1904 * finite element function at that point. Note that for this
1905 * information we have to reinitialize the <code>FEValues</code>
1906 * object for the neighbor cell.
1909 * scratch_data.fe_midpoint_value.reinit(neighbor);
1910 * const Point<dim> neighbor_center =
1911 * scratch_data.fe_midpoint_value.quadrature_point(0);
1913 * scratch_data.fe_midpoint_value.get_function_values(
1914 * scratch_data.solution, scratch_data.neighbor_midpoint_value);
1918 * Compute the vector <code>y</code> connecting the centers of the
1919 * two cells. Note that as opposed to the introduction, we denote
1920 * by <code>y</code> the normalized difference vector, as this is
1921 * the quantity used everywhere in the computations.
1924 * Tensor<1, dim> y = neighbor_center - this_center;
1925 * const double distance = y.norm();
1930 * Then add up the contribution of this cell to the Y matrix...
1933 * for (unsigned int i = 0; i < dim; ++i)
1934 * for (unsigned int j = 0; j < dim; ++j)
1935 * Y[i][j] += y[i] * y[j];
1939 * ... and update the sum of difference quotients:
1942 * projected_gradient += (scratch_data.neighbor_midpoint_value[0] -
1943 * scratch_data.cell_midpoint_value[0]) /
1949 * If now, after collecting all the information from the neighbors, we
1950 * can determine an approximation of the gradient for the present
1951 * cell, then we need to have passed over vectors <code>y</code> which
1952 * span the whole space, otherwise we would not have all components of
1953 * the gradient. This is indicated by the invertibility of the matrix.
1957 * If the matrix is not invertible, then the present
1958 * cell had an insufficient number of active neighbors. In contrast to
1959 * all previous cases (where we raised exceptions) this is, however,
1960 * not a programming error: it is a runtime error that can happen in
1961 * optimized mode even if it ran well in debug mode, so it is
1962 * reasonable to try to catch this error also in optimized mode. For
1963 * this case, there is the <code>AssertThrow</code> macro: it checks
1964 * the condition like the <code>Assert</code> macro, but not only in
1965 * debug mode; it then outputs an error message, but instead of
1966 * aborting the program as in the case of the <code>Assert</code>
1967 * macro, the exception is thrown using the <code>throw</code> command
1968 * of C++. This way, one has the possibility to catch this error and
1969 * take reasonable counter actions. One such measure would be to
1970 * refine the grid globally, as the case of insufficient directions
1971 * can not occur if every cell of the initial grid has been refined at
1975 * AssertThrow(determinant(Y) != 0, ExcInsufficientDirections());
1979 * If, on the other hand, the matrix is invertible, then invert it,
1980 * multiply the other quantity with it, and compute the estimated error
1981 * using this quantity and the correct powers of the mesh width:
1984 * const Tensor<2, dim> Y_inverse = invert(Y);
1986 * const Tensor<1, dim> gradient = Y_inverse * projected_gradient;
1990 * The last part of this function is the one where we write into
1991 * the element of the output vector what we have just
1992 * computed. The address of this vector has been stored in the
1993 * scratch data object, and all we have to do is know how to get
1994 * at the correct element inside this vector -- but we can ask the
1995 * cell we're on the how-manyth active cell
it is
for this:
1998 * scratch_data.error_per_cell(cell->active_cell_index()) =
2007 * <a name=
"step_9-Mainfunction"></a>
2012 * The <code>main</code>
function is similar to the previous examples. The
2014 * number of threads (see the documentation module @ref threads
2015 *
"Parallel computing with multiple processors accessing shared memory"
2016 *
for more information). The number of threads
used is the minimum of the
2017 * environment variable DEAL_II_NUM_THREADS and the parameter of
2018 * <code>set_thread_limit</code>. If no
value is given to
2019 * <code>set_thread_limit</code>, the
default value from the Intel Threading
2020 * Building Blocks (TBB) library is
used. If the call to
2021 * <code>set_thread_limit</code> is omitted, the number of threads will be
2022 * chosen by TBB independently of DEAL_II_NUM_THREADS.
2027 *
using namespace dealii;
2032 * Step9::AdvectionProblem<2> advection_problem_2d;
2033 * advection_problem_2d.run();
2035 *
catch (std::exception &exc)
2037 * std::cerr << std::endl
2039 * <<
"----------------------------------------------------"
2041 * std::cerr <<
"Exception on processing: " << std::endl
2042 * << exc.what() << std::endl
2043 * <<
"Aborting!" << std::endl
2044 * <<
"----------------------------------------------------"
2050 * std::cerr << std::endl
2052 * <<
"----------------------------------------------------"
2054 * std::cerr <<
"Unknown exception!" << std::endl
2055 * <<
"Aborting!" << std::endl
2056 * <<
"----------------------------------------------------"
2064<a name=
"step_9-Results"></a><h1>Results</h1>
2068The results of
this program are not particularly spectacular. They
2069consist of the console output, some grid files, and the solution on
2070each of these grids. First
for the console output:
2073 Number of active cells: 64
2074 Number of degrees of freedom: 1681
2075 Iterations required
for convergence: 298
2076 Max
norm of residual: 3.60316e-12
2078 Number of active cells: 124
2079 Number of degrees of freedom: 3537
2080 Iterations required
for convergence: 415
2081 Max
norm of residual: 3.70682e-12
2083 Number of active cells: 247
2084 Number of degrees of freedom: 6734
2085 Iterations required
for convergence: 543
2086 Max
norm of residual: 7.19716e-13
2088 Number of active cells: 502
2089 Number of degrees of freedom: 14105
2090 Iterations required
for convergence: 666
2091 Max
norm of residual: 3.45628e-13
2093 Number of active cells: 1003
2094 Number of degrees of freedom: 27462
2095 Iterations required
for convergence: 1064
2096 Max
norm of residual: 1.86495e-13
2098 Number of active cells: 1993
2099 Number of degrees of freedom: 55044
2100 Iterations required
for convergence: 1251
2101 Max
norm of residual: 1.28765e-13
2103 Number of active cells: 3985
2104 Number of degrees of freedom: 108492
2105 Iterations required
for convergence: 2035
2106 Max
norm of residual: 6.78085e-14
2108 Number of active cells: 7747
2109 Number of degrees of freedom: 210612
2110 Iterations required
for convergence: 2187
2111 Max
norm of residual: 2.61457e-14
2113 Number of active cells: 15067
2114 Number of degrees of freedom: 406907
2115 Iterations required
for convergence: 3079
2116 Max
norm of residual: 2.9932e-14
2118 Number of active cells: 29341
2119 Number of degrees of freedom: 780591
2120 Iterations required
for convergence: 3913
2121 Max
norm of residual: 8.15689e-15
2124Quite a number of cells are
used on the finest
level to resolve the features of
2125the solution. Here are the fourth and tenth grids:
2126<div
class=
"twocolumn" style=
"width: 80%">
2128 <img src=
"https://www.dealii.org/images/steps/developer/step-9-grid-3.png"
2129 alt=
"Fourth grid in the refinement cycle, showing some adaptivity to features."
2130 width=
"400" height=
"400">
2133 <img src=
"https://www.dealii.org/images/steps/developer/step-9-grid-9.png"
2134 alt=
"Tenth grid in the refinement cycle, showing that the waves are fully captured."
2135 width=
"400" height=
"400">
2138and the fourth and tenth solutions:
2139<div
class=
"twocolumn" style=
"width: 80%">
2141 <img src=
"https://www.dealii.org/images/steps/developer/step-9-solution-3.png"
2142 alt=
"Fourth solution, showing that we resolve most features but some
2143 are sill unresolved and appear blury."
2144 width=
"400" height=
"400">
2147 <img src=
"https://www.dealii.org/images/steps/developer/step-9-solution-9.png"
2148 alt=
"Tenth solution, showing a fully resolved flow."
2149 width=
"400" height=
"400">
2152and both the grid and solution zoomed in:
2153<div
class=
"twocolumn" style=
"width: 80%">
2155 <img src=
"https://www.dealii.org/images/steps/developer/step-9-solution-3-zoom.png"
2156 alt=
"Detail of the fourth solution, showing that we resolve most
2157 features but some are sill unresolved and appear blury. In particular,
2158 the larger cells need to be refined."
2159 width=
"400" height=
"400">
2162 <img src=
"https://www.dealii.org/images/steps/developer/step-9-solution-9-zoom.png"
2163 alt=
"Detail of the tenth solution, showing that we needed a lot more
2164 cells than were present in the fourth solution."
2165 width=
"400" height=
"400">
2169The solution is created by that part that is transported along the wiggly
2170advection field from the left and lower boundaries to the top right, and the
2171part that is created by the source in the lower left corner, and the results of
2172which are also transported along. The grid shown above is well-adapted to
2173resolve these features. The comparison between plots shows that, even though we
2174are
using a high-order approximation, we still need adaptive
mesh refinement to
2175fully resolve the wiggles.
2178<a name=
"step_9-PlainProg"></a>
2179<h1> The plain program</h1>
void reinit(const TriaIterator< DoFCellAccessor< dim, spacedim, level_dof_access > > &cell)
virtual void value_list(const std::vector< Point< dim > > &points, std::vector< RangeNumberType > &values, const unsigned int component=0) const
static void set_thread_limit(const unsigned int max_threads=numbers::invalid_unsigned_int)
void initialize(const MatrixType &A, const AdditionalData ¶meters=AdditionalData())
__global__ void set(Number *val, const Number s, const size_type N)
#define Assert(cond, exc)
typename ActiveSelector::active_cell_iterator active_cell_iterator
void loop(IteratorType begin, std_cxx20::type_identity_t< IteratorType > end, DOFINFO &dinfo, INFOBOX &info, const std::function< void(DOFINFO &, typename INFOBOX::CellInfo &)> &cell_worker, const std::function< void(DOFINFO &, typename INFOBOX::CellInfo &)> &boundary_worker, const std::function< void(DOFINFO &, DOFINFO &, typename INFOBOX::CellInfo &, typename INFOBOX::CellInfo &)> &face_worker, AssemblerType &assembler, const LoopControl &lctrl=LoopControl())
@ update_values
Shape function values.
@ update_normal_vectors
Normal vectors.
@ update_JxW_values
Transformed quadrature weights.
@ update_gradients
Shape function gradients.
@ update_quadrature_points
Transformed quadrature points.
void refine_and_coarsen_fixed_number(Triangulation< dim, spacedim > &triangulation, const Vector< Number > &criteria, const double top_fraction_of_cells, const double bottom_fraction_of_cells, const unsigned int max_n_cells=std::numeric_limits< unsigned int >::max())
@ matrix
Contents is actually a matrix.
@ symmetric
Matrix is symmetric.
@ general
No special properties.
double norm(const FEValuesBase< dim > &fe, const ArrayView< const std::vector< Tensor< 1, dim > > > &Du)
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
SymmetricTensor< 2, dim, Number > e(const Tensor< 2, dim, Number > &F)
SymmetricTensor< 2, dim, Number > d(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
T sum(const T &t, const MPI_Comm mpi_communicator)
void run(const std::vector< std::vector< Iterator > > &colored_iterators, Worker worker, Copier copier, const ScratchData &sample_scratch_data, const CopyData &sample_copy_data, const unsigned int queue_length=2 *MultithreadInfo::n_threads(), const unsigned int chunk_size=8)
bool check(const ConstraintKinds kind_in, const unsigned int dim)
void copy(const T *begin, const T *end, U *dest)
int(&) functions(const void *v1, const void *v2)
void assemble(const MeshWorker::DoFInfoBox< dim, DOFINFO > &dinfo, A *assembler)
void reinit(MatrixBlock< MatrixType > &v, const BlockSparsityPattern &p)
const InputIterator OutputIterator out
const Iterator const std_cxx20::type_identity_t< Iterator > & end
const InputIterator OutputIterator const Function & function
::VectorizedArray< Number, width > pow(const ::VectorizedArray< Number, width > &, const Number p)
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
DEAL_II_HOST constexpr Number determinant(const SymmetricTensor< 2, dim, Number > &)