1273 *
const unsigned int = 0)
const override
1278 *
virtual void vector_value(
const Point<dim> &p,
1281 *
for (
unsigned int c = 0; c < this->
n_components; ++c)
1288 *
template <
int dim>
1289 *
class TemperatureRightHandSide :
public Function<dim>
1292 * TemperatureRightHandSide()
1297 *
const unsigned int component = 0) const override
1301 *
ExcMessage(
"Invalid operation for a scalar function."));
1305 *
static const Point<dim> source_centers[3] = {
1307 * (dim == 2 ?
Point<dim>(.45, .1) :
Point<dim>(.45, .5, .1)),
1308 * (dim == 2 ?
Point<dim>(.75, .1) :
Point<dim>(.75, .5, .1))};
1309 *
static const double source_radius = (dim == 2 ? 1. / 32 : 1. / 8);
1311 *
return ((source_centers[0].distance(p) < source_radius) ||
1312 * (source_centers[1].
distance(p) < source_radius) ||
1313 * (source_centers[2].distance(p) < source_radius) ?
1321 *
for (
unsigned int c = 0; c < this->
n_components; ++c)
1332 * <a name=
"Linearsolversandpreconditioners"></a>
1333 * <h3>Linear solvers and preconditioners</h3>
1337 * This section introduces some objects that are used
for the solution of
1338 * the linear equations of the Stokes system that we need to solve in each
1339 * time step. Many of the ideas used here are the same as in @ref step_20
"step-20", where
1340 * Schur complement based preconditioners and solvers have been introduced,
1341 * with the actual
interface taken from @ref step_22 "step-22" (in particular the
1342 * discussion in the
"Results" section of @ref step_22
"step-22", in which we introduce
1343 * alternatives to the direct Schur complement approach). Note, however,
1344 * that here we don
't use the Schur complement to solve the Stokes
1345 * equations, though an approximate Schur complement (the mass matrix on the
1346 * pressure space) appears in the preconditioner.
1349 * namespace LinearSolvers
1354 * <a name="ThecodeInverseMatrixcodeclasstemplate"></a>
1355 * <h4>The <code>InverseMatrix</code> class template</h4>
1359 * This class is an interface to calculate the action of an "inverted"
1360 * matrix on a vector (using the <code>vmult</code> operation) in the same
1361 * way as the corresponding class in @ref step_22 "step-22": when the product of an
1362 * object of this class is requested, we solve a linear equation system
1363 * with that matrix using the CG method, accelerated by a preconditioner
1364 * of (templated) class <code>PreconditionerType</code>.
1368 * In a minor deviation from the implementation of the same class in
1369 * @ref step_22 "step-22", we make the <code>vmult</code> function take any
1370 * kind of vector type (it will yield compiler errors, however, if the
1371 * matrix does not allow a matrix-vector product with this kind of
1376 * Secondly, we catch any exceptions that the solver may have thrown. The
1377 * reason is as follows: When debugging a program like this one
1378 * occasionally makes a mistake of passing an indefinite or nonsymmetric
1379 * matrix or preconditioner to the current class. The solver will, in that
1380 * case, not converge and throw a run-time exception. If not caught here
1381 * it will propagate up the call stack and may end up in
1382 * <code>main()</code> where we output an error message that will say that
1383 * the CG solver failed. The question then becomes: Which CG solver? The
1384 * one that inverted the mass matrix? The one that inverted the top left
1385 * block with the Laplace operator? Or a CG solver in one of the several
1386 * other nested places where we use linear solvers in the current code? No
1387 * indication about this is present in a run-time exception because it
1388 * doesn't store the stack of calls through which we got to the place
1389 * where the exception was generated.
1393 * So rather than letting the exception propagate freely up to
1394 * <code>main()</code> we realize that there is little that an outer
1395 *
function can
do if the inner solver fails and rather convert the
1396 *
run-time exception into an assertion that fails and triggers a
call to
1397 * <code>
abort()</code>, allowing us to
trace back in a debugger how we
1398 * got to the current place.
1401 *
template <
class MatrixType,
class PreconditionerType>
1405 * InverseMatrix(
const MatrixType & m,
1406 *
const PreconditionerType &preconditioner);
1409 *
template <
typename VectorType>
1414 *
const PreconditionerType & preconditioner;
1418 *
template <
class MatrixType,
class PreconditionerType>
1419 * InverseMatrix<MatrixType, PreconditionerType>::InverseMatrix(
1420 *
const MatrixType & m,
1421 *
const PreconditionerType &preconditioner)
1423 * , preconditioner(preconditioner)
1428 *
template <
class MatrixType,
class PreconditionerType>
1429 *
template <
typename VectorType>
1430 *
void InverseMatrix<MatrixType, PreconditionerType>::vmult(
1434 *
SolverControl solver_control(src.size(), 1
e-7 * src.l2_norm());
1441 * cg.solve(*
matrix, dst, src, preconditioner);
1443 *
catch (std::exception &
e)
1452 * <a name=
"Schurcomplementpreconditioner"></a>
1453 * <h4>Schur complement preconditioner</h4>
1457 * This is the implementation of the Schur complement preconditioner as
1458 * described in detail in the introduction. As opposed to @ref step_20
"step-20" and
1459 * @ref step_22
"step-22", we solve the block system all-at-once
using GMRES, and use the
1460 * Schur complement of the block structured
matrix to build a good
1461 * preconditioner instead.
1465 * Let
's have a look at the ideal preconditioner matrix
1466 * @f$P=\left(\begin{array}{cc} A & 0 \\ B & -S \end{array}\right)@f$
1467 * described in the introduction. If we apply this matrix in the solution
1468 * of a linear system, convergence of an iterative GMRES solver will be
1469 * governed by the matrix @f{eqnarray*} P^{-1}\left(\begin{array}{cc} A &
1470 * B^T \\ B & 0 \end{array}\right) = \left(\begin{array}{cc} I & A^{-1}
1471 * B^T \\ 0 & I \end{array}\right), @f} which indeed is very simple. A
1472 * GMRES solver based on exact matrices would converge in one iteration,
1473 * since all eigenvalues are equal (any Krylov method takes at most as
1474 * many iterations as there are distinct eigenvalues). Such a
1475 * preconditioner for the blocked Stokes system has been proposed by
1476 * Silvester and Wathen ("Fast iterative solution of stabilised Stokes
1477 * systems part II. Using general block preconditioners", SIAM
1478 * J. Numer. Anal., 31 (1994), pp. 1352-1367).
1482 * Replacing @f$P@f$ by @f$\tilde{P}@f$ keeps that spirit alive: the product
1483 * @f$P^{-1} A@f$ will still be close to a matrix with eigenvalues 1 with a
1484 * distribution that does not depend on the problem size. This lets us
1485 * hope to be able to get a number of GMRES iterations that is
1486 * problem-size independent.
1490 * The deal.II users who have already gone through the @ref step_20 "step-20" and @ref step_22 "step-22"
1491 * tutorials can certainly imagine how we're going to implement this. We
1492 * replace the exact inverse matrices in @f$P^{-1}@f$ by some
approximate
1493 * inverses built from the InverseMatrix
class, and the inverse Schur
1494 * complement will be approximated by the pressure mass
matrix @f$M_p@f$
1495 * (weighted by @f$\eta^{-1}@f$ as mentioned in the introduction). As pointed
1496 * out in the results section of @ref step_22
"step-22", we can replace the exact inverse
1497 * of @f$A@f$ by just the application of a preconditioner, in
this case
1498 * on a vector Laplace
matrix as was explained in the introduction. This
1499 * does increase the number of (outer) GMRES iterations, but is still
1500 * significantly cheaper than an exact inverse, which would require
1501 * between 20 and 35 CG iterations for <em>each</em> outer solver step
1502 * (
using the AMG preconditioner).
1506 * Having the above explanations in mind, we define a preconditioner
class
1507 * with a <code>vmult</code> functionality, which is all we need
for the
1508 * interaction with the usual solver
functions further below in the
1513 * First the declarations. These are similar to the definition of the
1514 * Schur complement in @ref step_20
"step-20", with the difference that we need some more
1515 * preconditioners in the constructor and that the matrices we use here
1516 * are built upon Trilinos:
1519 * template <class PreconditionerTypeA, class PreconditionerTypeMp>
1520 *
class BlockSchurPreconditioner :
public Subscriptor
1523 * BlockSchurPreconditioner(
1526 * PreconditionerTypeMp> &Mpinv,
1527 *
const PreconditionerTypeA & Apreconditioner);
1536 * PreconditionerTypeMp>>
1538 *
const PreconditionerTypeA &a_preconditioner;
1554 * program can only be
run sequentially and will throw an exception if used
1558 * template <class PreconditionerTypeA, class PreconditionerTypeMp>
1559 * BlockSchurPreconditioner<PreconditionerTypeA, PreconditionerTypeMp>::
1560 * BlockSchurPreconditioner(
1563 * PreconditionerTypeMp> &Mpinv,
1564 * const PreconditionerTypeA & Apreconditioner)
1565 * : stokes_matrix(&S)
1566 * , m_inverse(&Mpinv)
1567 * , a_preconditioner(Apreconditioner)
1574 * Next is the <code>vmult</code>
function. We implement the action of
1575 * @f$P^{-1}@f$ as described above in three successive steps. In formulas, we
1576 * want to compute @f$Y=P^{-1}X@f$ where @f$X,Y@f$ are both vectors with two block
1581 * The
first step multiplies the velocity part of the vector by a
1582 * preconditioner of the
matrix @f$A@f$, i.e., we compute @f$Y_0={\tilde
1583 *
A}^{-1}X_0@f$. The resulting velocity vector is then multiplied by @f$B@f$
1584 * and subtracted from the pressure, i.e., we want to compute @f$X_1-BY_0@f$.
1585 * This
second step only acts on the pressure vector and is accomplished
1586 * by the residual
function of our
matrix classes, except that the
sign is
1587 * wrong. Consequently, we change the
sign in the temporary pressure
1588 * vector and
finally multiply by the inverse pressure mass
matrix to get
1589 * the
final pressure vector, completing our work on the Stokes
1593 *
template <
class PreconditionerTypeA,
class PreconditionerTypeMp>
1595 * BlockSchurPreconditioner<PreconditionerTypeA, PreconditionerTypeMp>::vmult(
1599 * a_preconditioner.vmult(dst.block(0), src.block(0));
1600 * stokes_matrix->block(1, 0).residual(tmp, dst.block(0), src.block(1));
1602 * m_inverse->vmult(dst.block(1), tmp);
1611 * <a name=
"ThecodeBoussinesqFlowProblemcodeclasstemplate"></a>
1612 * <h3>The <code>BoussinesqFlowProblem</code>
class template</h3>
1616 * The definition of the
class that defines the top-
level logic of solving
1617 * the time-dependent Boussinesq problem is mainly based on the @ref step_22
"step-22"
1618 * tutorial program. The main differences are that now we also have to solve
1619 *
for the temperature equation, which forces us to have a
second DoFHandler
1620 *
object for the temperature variable as well as matrices, right hand
1621 * sides, and solution vectors
for the current and previous time steps. As
1622 * mentioned in the introduction, all linear algebra objects are going to
1623 * use wrappers of the corresponding Trilinos functionality.
1627 * The member
functions of
this class are reminiscent of @ref step_21 "step-21", where we
1628 * also used a staggered scheme that
first solve the flow equations (here
1629 * the Stokes equations, in @ref step_21
"step-21" Darcy flow) and then update the advected
1630 * quantity (here the temperature, there the saturation). The
functions that
1631 * are
new are mainly concerned with determining the time step, as well as
1632 * the proper size of the artificial viscosity stabilization.
1636 * The last three variables indicate whether the various matrices or
1637 * preconditioners need to be rebuilt the next time the corresponding build
1638 *
functions are called. This allows us to move the corresponding
1639 * <code>
if</code> into the respective
function and thereby keeping our main
1640 * <code>
run()</code>
function clean and easy to read.
1643 *
template <
int dim>
1644 *
class BoussinesqFlowProblem
1647 * BoussinesqFlowProblem();
1651 *
void setup_dofs();
1652 *
void assemble_stokes_preconditioner();
1653 *
void build_stokes_preconditioner();
1654 *
void assemble_stokes_system();
1655 *
void assemble_temperature_system(
const double maximal_velocity);
1656 *
void assemble_temperature_matrix();
1657 *
double get_maximal_velocity()
const;
1658 * std::pair<double, double> get_extrapolated_temperature_range()
const;
1660 *
void output_results()
const;
1661 *
void refine_mesh(
const unsigned int max_grid_level);
1663 *
double compute_viscosity(
1664 *
const std::vector<double> & old_temperature,
1665 *
const std::vector<double> & old_old_temperature,
1668 *
const std::vector<double> & old_temperature_laplacians,
1669 *
const std::vector<double> & old_old_temperature_laplacians,
1672 *
const std::vector<double> & gamma_values,
1673 *
const double global_u_infty,
1674 *
const double global_T_variation,
1675 *
const double cell_diameter)
const;
1679 *
double global_Omega_diameter;
1681 *
const unsigned int stokes_degree;
1686 * std::vector<IndexSet> stokes_partitioning;
1695 *
const unsigned int temperature_degree;
1711 *
double old_time_step;
1712 *
unsigned int timestep_number;
1714 * std::shared_ptr<TrilinosWrappers::PreconditionAMG> Amg_preconditioner;
1715 * std::shared_ptr<TrilinosWrappers::PreconditionIC> Mp_preconditioner;
1717 *
bool rebuild_stokes_matrix;
1718 *
bool rebuild_temperature_matrices;
1719 *
bool rebuild_stokes_preconditioner;
1726 * <a name=
"BoussinesqFlowProblemclassimplementation"></a>
1727 * <h3>BoussinesqFlowProblem
class implementation</h3>
1732 * <a name=
"BoussinesqFlowProblemBoussinesqFlowProblem"></a>
1733 * <h4>BoussinesqFlowProblem::BoussinesqFlowProblem</h4>
1737 * The constructor of
this class is an extension of the constructor in
1738 * @ref step_22
"step-22". We need to add the various variables that concern the
1739 * temperature. As discussed in the introduction, we are going to use
1740 * @f$Q_2\times Q_1@f$ (Taylor-Hood) elements again
for the Stokes part, and
1741 * @f$Q_2@f$ elements
for the temperature. However, by
using variables that
1742 * store the polynomial degree of the Stokes and temperature finite
1743 * elements, it is easy to consistently modify the degree of the elements as
1744 * well as all quadrature formulas used on them
downstream. Moreover, we
1745 * initialize the time stepping as well as the options
for matrix assembly
1746 * and preconditioning:
1749 * template <int dim>
1750 * BoussinesqFlowProblem<dim>::BoussinesqFlowProblem()
1752 * , global_Omega_diameter(std::numeric_limits<double>::quiet_NaN())
1753 * , stokes_degree(1)
1758 * temperature_degree(2)
1759 * , temperature_fe(temperature_degree)
1764 * , old_time_step(0)
1765 * , timestep_number(0)
1766 * , rebuild_stokes_matrix(
true)
1767 * , rebuild_temperature_matrices(
true)
1768 * , rebuild_stokes_preconditioner(
true)
1776 * <a name=
"BoussinesqFlowProblemget_maximal_velocity"></a>
1777 * <h4>BoussinesqFlowProblem::get_maximal_velocity</h4>
1781 * Starting the real functionality of
this class is a helper function that
1782 * determines the maximum (@f$L_\infty@f$) velocity in the domain (at the
1783 * quadrature points, in fact). How it works should be relatively obvious to
1784 * all who have gotten to
this point of the tutorial. Note that since we are
1785 * only interested in the velocity, rather than
using
1786 * <code>stokes_fe_values.get_function_values</code> to get the values of
1787 * the entire Stokes solution (velocities and pressures) we use
1788 * <code>stokes_fe_values[velocities].get_function_values</code> to
extract
1789 * only the velocities part. This has the additional benefit that we get it
1791 * allowing us to process it right away
using the <code>
norm()</code>
1792 *
function to get the magnitude of the velocity.
1796 * The only
point worth thinking about a bit is how to choose the quadrature
1797 * points we use here. Since the goal of
this function is to find the
1798 * maximal velocity over a domain by looking at quadrature points on each
1799 * cell. So we should ask how we should best choose these quadrature points
1800 * on each cell. To
this end, recall that
if we had a single @f$Q_1@f$ field
1801 * (rather than the vector-valued field of higher order) then the maximum
1802 * would be attained at a vertex of the mesh. In other words, we should use
1803 * the
QTrapez class that has quadrature points only at the
vertices of
1808 * For higher order shape
functions, the situation is more complicated: the
1809 * maxima and minima may be attained at points between the support points of
1810 * shape
functions (
for the usual @f$Q_p@f$ elements the support points are the
1811 * equidistant Lagrange interpolation points); furthermore, since we are
1812 * looking
for the maximum magnitude of a vector-valued quantity, we can
1813 * even less say with certainty where the
set of potential maximal points
1814 * are. Nevertheless, intuitively
if not provably, the Lagrange
1815 * interpolation points appear to be a better choice than the Gauss points.
1819 * There are now different methods to produce a quadrature formula with
1820 * quadrature points
equal to the interpolation points of the finite
1821 * element. One option would be to use the
1823 * unique
set of points to avoid duplicate function evaluations, and create
1824 * a
Quadrature object using these points. Another option, chosen here, is
1826 * repeats the
QTrapez formula on a number of sub-cells in each coordinate
1827 * direction. To cover all support points, we need to iterate it
1828 * <code>stokes_degree+1</code> times since this is the polynomial degree of
1829 * the Stokes element in use:
1832 * template <
int dim>
1833 *
double BoussinesqFlowProblem<dim>::get_maximal_velocity() const
1836 *
const unsigned int n_q_points = quadrature_formula.size();
1839 * std::vector<Tensor<1, dim>> velocity_values(n_q_points);
1840 *
double max_velocity = 0;
1844 *
for (
const auto &cell : stokes_dof_handler.active_cell_iterators())
1846 * fe_values.reinit(cell);
1847 * fe_values[velocities].get_function_values(stokes_solution,
1850 *
for (
unsigned int q = 0; q < n_q_points; ++q)
1851 * max_velocity =
std::max(max_velocity, velocity_values[q].
norm());
1854 *
return max_velocity;
1862 * <a name=
"BoussinesqFlowProblemget_extrapolated_temperature_range"></a>
1863 * <h4>BoussinesqFlowProblem::get_extrapolated_temperature_range</h4>
1867 * Next a
function that determines the minimum and maximum temperature at
1868 * quadrature points inside @f$\Omega@f$ when extrapolated from the two previous
1869 * time steps to the current
one. We need
this information in the
1870 * computation of the artificial viscosity parameter @f$\nu@f$ as discussed in
1875 * The formula
for the extrapolated temperature is
1876 * @f$\left(1+\frac{k_n}{k_{n-1}} \right)
T^{n-1} + \frac{k_n}{k_{n-1}}
1877 *
T^{n-2}@f$. The way to compute it is to
loop over all quadrature points and
1878 * update the maximum and minimum
value if the current
value is
1879 * bigger/smaller than the previous
one. We initialize the variables that
1880 * store the
max and
min before the
loop over all quadrature points by the
1881 * smallest and the largest number representable as a
double. Then we know
1882 *
for a fact that it is larger/smaller than the minimum/maximum and that
1883 * the
loop over all quadrature points is ultimately going to update the
1888 * The only other complication worth mentioning here is that in the
first
1889 * time step, @f$T^{k-2}@f$ is not yet available of course. In that
case, we can
1890 * only use @f$T^{k-1}@f$ which we have from the
initial temperature. As
1891 * quadrature points, we use the same choice as in the previous
function
1892 * though with the difference that now the number of repetitions is
1893 * determined by the polynomial degree of the temperature field.
1896 *
template <
int dim>
1897 * std::pair<double, double>
1898 * BoussinesqFlowProblem<dim>::get_extrapolated_temperature_range() const
1901 *
const unsigned int n_q_points = quadrature_formula.size();
1904 * std::vector<double> old_temperature_values(n_q_points);
1905 * std::vector<double> old_old_temperature_values(n_q_points);
1907 *
if (timestep_number != 0)
1912 *
for (
const auto &cell : temperature_dof_handler.active_cell_iterators())
1914 * fe_values.reinit(cell);
1915 * fe_values.get_function_values(old_temperature_solution,
1916 * old_temperature_values);
1917 * fe_values.get_function_values(old_old_temperature_solution,
1918 * old_old_temperature_values);
1920 *
for (
unsigned int q = 0; q < n_q_points; ++q)
1922 *
const double temperature =
1923 * (1. + time_step / old_time_step) * old_temperature_values[q] -
1924 * time_step / old_time_step * old_old_temperature_values[q];
1926 * min_temperature =
std::min(min_temperature, temperature);
1927 * max_temperature =
std::max(max_temperature, temperature);
1931 *
return std::make_pair(min_temperature, max_temperature);
1938 *
for (
const auto &cell : temperature_dof_handler.active_cell_iterators())
1940 * fe_values.reinit(cell);
1941 * fe_values.get_function_values(old_temperature_solution,
1942 * old_temperature_values);
1944 *
for (
unsigned int q = 0; q < n_q_points; ++q)
1946 *
const double temperature = old_temperature_values[q];
1948 * min_temperature =
std::min(min_temperature, temperature);
1949 * max_temperature =
std::max(max_temperature, temperature);
1953 *
return std::make_pair(min_temperature, max_temperature);
1962 * <a name=
"BoussinesqFlowProblemcompute_viscosity"></a>
1963 * <h4>BoussinesqFlowProblem::compute_viscosity</h4>
1967 * The last of the tool
functions computes the artificial viscosity
1968 * parameter @f$\nu|_K@f$ on a cell @f$K@f$ as a
function of the extrapolated
1969 * temperature, its gradient and Hessian (
second derivatives), the velocity,
1970 * the right hand side @f$
\gamma@f$ all on the quadrature points of the current
1971 * cell, and various other parameters as described in detail in the
1976 * There are some universal constants worth mentioning here. First, we need
1977 * to fix @f$\beta@f$; we choose @f$\beta=0.017\cdot dim@f$, a choice discussed in
1978 * detail in the results section of
this tutorial program. The
second is the
1979 * exponent @f$\alpha@f$; @f$\alpha=1@f$ appears to work fine
for the current
1980 * program, even though some additional benefit might be expected from
1981 * choosing @f$\alpha = 2@f$. Finally, there is
one thing that requires special
1982 * casing: In the
first time step, the velocity equals
zero, and the formula
1983 *
for @f$\nu|_K@f$ is not defined. In that
case, we
return @f$\nu|_K=5\cdot 10^3
1984 * \cdot h_K@f$, a choice admittedly more motivated by heuristics than
1985 * anything
else (it is in the same order of magnitude, however, as the
1986 *
value returned
for most cells on the
second time step).
1990 * The rest of the
function should be mostly obvious based on the material
1991 * discussed in the introduction:
1994 *
template <
int dim>
1995 *
double BoussinesqFlowProblem<dim>::compute_viscosity(
1996 *
const std::vector<double> & old_temperature,
1997 *
const std::vector<double> & old_old_temperature,
2000 *
const std::vector<double> & old_temperature_laplacians,
2001 *
const std::vector<double> & old_old_temperature_laplacians,
2004 *
const std::vector<double> & gamma_values,
2005 *
const double global_u_infty,
2006 *
const double global_T_variation,
2007 *
const double cell_diameter)
const
2009 * constexpr
double beta = 0.017 * dim;
2010 * constexpr
double alpha = 1.0;
2012 *
if (global_u_infty == 0)
2013 *
return 5
e-3 * cell_diameter;
2015 *
const unsigned int n_q_points = old_temperature.size();
2017 *
double max_residual = 0;
2018 *
double max_velocity = 0;
2020 *
for (
unsigned int q = 0; q < n_q_points; ++q)
2023 * (old_velocity_values[q] + old_old_velocity_values[q]) / 2;
2025 *
const double dT_dt =
2026 * (old_temperature[q] - old_old_temperature[q]) / old_time_step;
2027 *
const double u_grad_T =
2028 * u * (old_temperature_grads[q] + old_old_temperature_grads[q]) / 2;
2030 *
const double kappa_Delta_T =
2031 * EquationData::kappa *
2032 * (old_temperature_laplacians[q] + old_old_temperature_laplacians[q]) /
2035 *
const double residual =
2036 *
std::abs((dT_dt + u_grad_T - kappa_Delta_T - gamma_values[q]) *
2037 * std::pow((old_temperature[q] + old_old_temperature[q]) / 2,
2040 * max_residual =
std::max(residual, max_residual);
2041 * max_velocity =
std::max(std::sqrt(u * u), max_velocity);
2044 *
const double c_R =
std::pow(2., (4. - 2 * alpha) / dim);
2045 *
const double global_scaling = c_R * global_u_infty * global_T_variation *
2046 *
std::pow(global_Omega_diameter, alpha - 2.);
2049 * beta * max_velocity *
2051 * std::pow(cell_diameter, alpha) * max_residual / global_scaling));
2059 * <a name=
"BoussinesqFlowProblemsetup_dofs"></a>
2060 * <h4>BoussinesqFlowProblem::setup_dofs</h4>
2064 * This is the
function that sets up the
DoFHandler objects we have here
2065 * (
one for the Stokes part and
one for the temperature part) as well as
set
2066 * to the right sizes the various objects required
for the linear algebra in
2067 *
this program. Its basic operations are similar to what we
do in @ref step_22
"step-22".
2071 * The body of the
function first enumerates all degrees of freedom
for the
2072 * Stokes and temperature systems. For the Stokes part, degrees of freedom
2073 * are then sorted to ensure that velocities precede pressure DoFs so that
2075 * difference to @ref step_22
"step-22", we
do not perform any additional DoF
2076 * renumbering. In that program, it paid off since our solver was heavily
2077 * dependent on ILU
's, whereas we use AMG here which is not sensitive to the
2078 * DoF numbering. The IC preconditioner for the inversion of the pressure
2079 * mass matrix would of course take advantage of a Cuthill-McKee like
2080 * renumbering, but its costs are low compared to the velocity portion, so
2081 * the additional work does not pay off.
2085 * We then proceed with the generation of the hanging node constraints that
2086 * arise from adaptive grid refinement for both DoFHandler objects. For the
2087 * velocity, we impose no-flux boundary conditions @f$\mathbf{u}\cdot
2088 * \mathbf{n}=0@f$ by adding constraints to the object that already stores the
2089 * hanging node constraints matrix. The second parameter in the function
2090 * describes the first of the velocity components in the total dof vector,
2091 * which is zero here. The variable <code>no_normal_flux_boundaries</code>
2092 * denotes the boundary indicators for which to set the no flux boundary
2093 * conditions; here, this is boundary indicator zero.
2097 * After having done so, we count the number of degrees of freedom in the
2101 * template <int dim>
2102 * void BoussinesqFlowProblem<dim>::setup_dofs()
2104 * std::vector<unsigned int> stokes_sub_blocks(dim + 1, 0);
2105 * stokes_sub_blocks[dim] = 1;
2108 * stokes_dof_handler.distribute_dofs(stokes_fe);
2109 * DoFRenumbering::component_wise(stokes_dof_handler, stokes_sub_blocks);
2111 * stokes_constraints.clear();
2112 * DoFTools::make_hanging_node_constraints(stokes_dof_handler,
2113 * stokes_constraints);
2114 * std::set<types::boundary_id> no_normal_flux_boundaries;
2115 * no_normal_flux_boundaries.insert(0);
2116 * VectorTools::compute_no_normal_flux_constraints(stokes_dof_handler,
2118 * no_normal_flux_boundaries,
2119 * stokes_constraints);
2120 * stokes_constraints.close();
2123 * temperature_dof_handler.distribute_dofs(temperature_fe);
2125 * temperature_constraints.clear();
2126 * DoFTools::make_hanging_node_constraints(temperature_dof_handler,
2127 * temperature_constraints);
2128 * temperature_constraints.close();
2131 * const std::vector<types::global_dof_index> stokes_dofs_per_block =
2132 * DoFTools::count_dofs_per_fe_block(stokes_dof_handler, stokes_sub_blocks);
2134 * const unsigned int n_u = stokes_dofs_per_block[0],
2135 * n_p = stokes_dofs_per_block[1],
2136 * n_T = temperature_dof_handler.n_dofs();
2138 * std::cout << "Number of active cells: " << triangulation.n_active_cells()
2139 * << " (on " << triangulation.n_levels() << " levels)" << std::endl
2140 * << "Number of degrees of freedom: " << n_u + n_p + n_T << " ("
2141 * << n_u << '+
' << n_p << '+
' << n_T << ')
' << std::endl
2146 * The next step is to create the sparsity pattern for the Stokes and
2147 * temperature system matrices as well as the preconditioner matrix from
2148 * which we build the Stokes preconditioner. As in @ref step_22 "step-22", we choose to
2149 * create the pattern by
2150 * using the blocked version of DynamicSparsityPattern.
2154 * So, we first release the memory stored in the matrices, then set up an
2155 * object of type BlockDynamicSparsityPattern consisting of
2156 * @f$2\times 2@f$ blocks (for the Stokes system matrix and preconditioner) or
2157 * DynamicSparsityPattern (for the temperature part). We then
2158 * fill these objects with the nonzero pattern, taking into account that
2159 * for the Stokes system matrix, there are no entries in the
2160 * pressure-pressure block (but all velocity vector components couple with
2161 * each other and with the pressure). Similarly, in the Stokes
2162 * preconditioner matrix, only the diagonal blocks are nonzero, since we
2163 * use the vector Laplacian as discussed in the introduction. This
2164 * operator only couples each vector component of the Laplacian with
2165 * itself, but not with the other vector components. (Application of the
2166 * constraints resulting from the no-flux boundary conditions will couple
2167 * vector components at the boundary again, however.)
2171 * When generating the sparsity pattern, we directly apply the constraints
2172 * from hanging nodes and no-flux boundary conditions. This approach was
2173 * already used in @ref step_27 "step-27", but is different from the one in early
2174 * tutorial programs where we first built the original sparsity pattern
2175 * and only then added the entries resulting from constraints. The reason
2176 * for doing so is that later during assembly we are going to distribute
2177 * the constraints immediately when transferring local to global
2178 * dofs. Consequently, there will be no data written at positions of
2179 * constrained degrees of freedom, so we can let the
2180 * DoFTools::make_sparsity_pattern function omit these entries by setting
2181 * the last Boolean flag to <code>false</code>. Once the sparsity pattern
2182 * is ready, we can use it to initialize the Trilinos matrices. Since the
2183 * Trilinos matrices store the sparsity pattern internally, there is no
2184 * need to keep the sparsity pattern around after the initialization of
2188 * stokes_partitioning.resize(2);
2189 * stokes_partitioning[0] = complete_index_set(n_u);
2190 * stokes_partitioning[1] = complete_index_set(n_p);
2192 * stokes_matrix.clear();
2194 * BlockDynamicSparsityPattern dsp(2, 2);
2196 * dsp.block(0, 0).reinit(n_u, n_u);
2197 * dsp.block(0, 1).reinit(n_u, n_p);
2198 * dsp.block(1, 0).reinit(n_p, n_u);
2199 * dsp.block(1, 1).reinit(n_p, n_p);
2201 * dsp.collect_sizes();
2203 * Table<2, DoFTools::Coupling> coupling(dim + 1, dim + 1);
2205 * for (unsigned int c = 0; c < dim + 1; ++c)
2206 * for (unsigned int d = 0; d < dim + 1; ++d)
2207 * if (!((c == dim) && (d == dim)))
2208 * coupling[c][d] = DoFTools::always;
2210 * coupling[c][d] = DoFTools::none;
2212 * DoFTools::make_sparsity_pattern(
2213 * stokes_dof_handler, coupling, dsp, stokes_constraints, false);
2215 * stokes_matrix.reinit(dsp);
2219 * Amg_preconditioner.reset();
2220 * Mp_preconditioner.reset();
2221 * stokes_preconditioner_matrix.clear();
2223 * BlockDynamicSparsityPattern dsp(2, 2);
2225 * dsp.block(0, 0).reinit(n_u, n_u);
2226 * dsp.block(0, 1).reinit(n_u, n_p);
2227 * dsp.block(1, 0).reinit(n_p, n_u);
2228 * dsp.block(1, 1).reinit(n_p, n_p);
2230 * dsp.collect_sizes();
2232 * Table<2, DoFTools::Coupling> coupling(dim + 1, dim + 1);
2233 * for (unsigned int c = 0; c < dim + 1; ++c)
2234 * for (unsigned int d = 0; d < dim + 1; ++d)
2236 * coupling[c][d] = DoFTools::always;
2238 * coupling[c][d] = DoFTools::none;
2240 * DoFTools::make_sparsity_pattern(
2241 * stokes_dof_handler, coupling, dsp, stokes_constraints, false);
2243 * stokes_preconditioner_matrix.reinit(dsp);
2248 * The creation of the temperature matrix (or, rather, matrices, since we
2249 * provide a temperature mass matrix and a temperature stiffness matrix,
2250 * that will be added together for time discretization) follows the
2251 * generation of the Stokes matrix – except that it is much easier
2252 * here since we do not need to take care of any blocks or coupling
2253 * between components. Note how we initialize the three temperature
2254 * matrices: We only use the sparsity pattern for reinitialization of the
2255 * first matrix, whereas we use the previously generated matrix for the
2256 * two remaining reinits. The reason for doing so is that reinitialization
2257 * from an already generated matrix allows Trilinos to reuse the sparsity
2258 * pattern instead of generating a new one for each copy. This saves both
2259 * some time and memory.
2263 * temperature_mass_matrix.clear();
2264 * temperature_stiffness_matrix.clear();
2265 * temperature_matrix.clear();
2267 * DynamicSparsityPattern dsp(n_T, n_T);
2268 * DoFTools::make_sparsity_pattern(temperature_dof_handler,
2270 * temperature_constraints,
2273 * temperature_matrix.reinit(dsp);
2274 * temperature_mass_matrix.reinit(temperature_matrix);
2275 * temperature_stiffness_matrix.reinit(temperature_matrix);
2280 * Lastly, we set the vectors for the Stokes solutions @f$\mathbf u^{n-1}@f$
2281 * and @f$\mathbf u^{n-2}@f$, as well as for the temperatures @f$T^{n}@f$,
2282 * @f$T^{n-1}@f$ and @f$T^{n-2}@f$ (required for time stepping) and all the system
2283 * right hand sides to their correct sizes and block structure:
2286 * IndexSet temperature_partitioning = complete_index_set(n_T);
2287 * stokes_solution.reinit(stokes_partitioning, MPI_COMM_WORLD);
2288 * old_stokes_solution.reinit(stokes_partitioning, MPI_COMM_WORLD);
2289 * stokes_rhs.reinit(stokes_partitioning, MPI_COMM_WORLD);
2291 * temperature_solution.reinit(temperature_partitioning, MPI_COMM_WORLD);
2292 * old_temperature_solution.reinit(temperature_partitioning, MPI_COMM_WORLD);
2293 * old_old_temperature_solution.reinit(temperature_partitioning,
2296 * temperature_rhs.reinit(temperature_partitioning, MPI_COMM_WORLD);
2304 * <a name="BoussinesqFlowProblemassemble_stokes_preconditioner"></a>
2305 * <h4>BoussinesqFlowProblem::assemble_stokes_preconditioner</h4>
2309 * This function assembles the matrix we use for preconditioning the Stokes
2310 * system. What we need are a vector Laplace matrix on the velocity
2311 * components and a mass matrix weighted by @f$\eta^{-1}@f$ on the pressure
2312 * component. We start by generating a quadrature object of appropriate
2313 * order, the FEValues object that can give values and gradients at the
2314 * quadrature points (together with quadrature weights). Next we create data
2315 * structures for the cell matrix and the relation between local and global
2316 * DoFs. The vectors <code>grad_phi_u</code> and <code>phi_p</code> are
2317 * going to hold the values of the basis functions in order to faster build
2318 * up the local matrices, as was already done in @ref step_22 "step-22". Before we start
2319 * the loop over all active cells, we have to specify which components are
2320 * pressure and which are velocity.
2323 * template <int dim>
2324 * void BoussinesqFlowProblem<dim>::assemble_stokes_preconditioner()
2326 * stokes_preconditioner_matrix = 0;
2328 * const QGauss<dim> quadrature_formula(stokes_degree + 2);
2329 * FEValues<dim> stokes_fe_values(stokes_fe,
2330 * quadrature_formula,
2331 * update_JxW_values | update_values |
2332 * update_gradients);
2334 * const unsigned int dofs_per_cell = stokes_fe.dofs_per_cell;
2335 * const unsigned int n_q_points = quadrature_formula.size();
2337 * FullMatrix<double> local_matrix(dofs_per_cell, dofs_per_cell);
2338 * std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
2340 * std::vector<Tensor<2, dim>> grad_phi_u(dofs_per_cell);
2341 * std::vector<double> phi_p(dofs_per_cell);
2343 * const FEValuesExtractors::Vector velocities(0);
2344 * const FEValuesExtractors::Scalar pressure(dim);
2346 * for (const auto &cell : stokes_dof_handler.active_cell_iterators())
2348 * stokes_fe_values.reinit(cell);
2353 * The creation of the local matrix is rather simple. There are only a
2354 * Laplace term (on the velocity) and a mass matrix weighted by
2355 * @f$\eta^{-1}@f$ to be generated, so the creation of the local matrix is
2356 * done in two lines. Once the local matrix is ready (loop over rows
2357 * and columns in the local matrix on each quadrature point), we get
2358 * the local DoF indices and write the local information into the
2359 * global matrix. We do this as in @ref step_27 "step-27", i.e., we directly apply the
2360 * constraints from hanging nodes locally. By doing so, we don't have
2361 * to
do that afterwards, and we don
't also write into entries of the
2362 * matrix that will actually be set to zero again later when
2363 * eliminating constraints.
2366 * for (unsigned int q = 0; q < n_q_points; ++q)
2368 * for (unsigned int k = 0; k < dofs_per_cell; ++k)
2370 * grad_phi_u[k] = stokes_fe_values[velocities].gradient(k, q);
2371 * phi_p[k] = stokes_fe_values[pressure].value(k, q);
2374 * for (unsigned int i = 0; i < dofs_per_cell; ++i)
2375 * for (unsigned int j = 0; j < dofs_per_cell; ++j)
2376 * local_matrix(i, j) +=
2377 * (EquationData::eta *
2378 * scalar_product(grad_phi_u[i], grad_phi_u[j]) +
2379 * (1. / EquationData::eta) * phi_p[i] * phi_p[j]) *
2380 * stokes_fe_values.JxW(q);
2383 * cell->get_dof_indices(local_dof_indices);
2384 * stokes_constraints.distribute_local_to_global(
2385 * local_matrix, local_dof_indices, stokes_preconditioner_matrix);
2394 * <a name="BoussinesqFlowProblembuild_stokes_preconditioner"></a>
2395 * <h4>BoussinesqFlowProblem::build_stokes_preconditioner</h4>
2399 * This function generates the inner preconditioners that are going to be
2400 * used for the Schur complement block preconditioner. Since the
2401 * preconditioners need only to be regenerated when the matrices change,
2402 * this function does not have to do anything in case the matrices have not
2403 * changed (i.e., the flag <code>rebuild_stokes_preconditioner</code> has
2404 * the value <code>false</code>). Otherwise its first task is to call
2405 * <code>assemble_stokes_preconditioner</code> to generate the
2406 * preconditioner matrices.
2410 * Next, we set up the preconditioner for the velocity-velocity matrix
2411 * @f$A@f$. As explained in the introduction, we are going to use an AMG
2412 * preconditioner based on a vector Laplace matrix @f$\hat{A}@f$ (which is
2413 * spectrally close to the Stokes matrix @f$A@f$). Usually, the
2414 * TrilinosWrappers::PreconditionAMG class can be seen as a good black-box
2415 * preconditioner which does not need any special knowledge. In this case,
2416 * however, we have to be careful: since we build an AMG for a vector
2417 * problem, we have to tell the preconditioner setup which dofs belong to
2418 * which vector component. We do this using the function
2419 * DoFTools::extract_constant_modes, a function that generates a set of
2420 * <code>dim</code> vectors, where each one has ones in the respective
2421 * component of the vector problem and zeros elsewhere. Hence, these are the
2422 * constant modes on each component, which explains the name of the
2426 * template <int dim>
2427 * void BoussinesqFlowProblem<dim>::build_stokes_preconditioner()
2429 * if (rebuild_stokes_preconditioner == false)
2432 * std::cout << " Rebuilding Stokes preconditioner..." << std::flush;
2434 * assemble_stokes_preconditioner();
2436 * Amg_preconditioner = std::make_shared<TrilinosWrappers::PreconditionAMG>();
2438 * std::vector<std::vector<bool>> constant_modes;
2439 * FEValuesExtractors::Vector velocity_components(0);
2440 * DoFTools::extract_constant_modes(stokes_dof_handler,
2441 * stokes_fe.component_mask(
2442 * velocity_components),
2444 * TrilinosWrappers::PreconditionAMG::AdditionalData amg_data;
2445 * amg_data.constant_modes = constant_modes;
2449 * Next, we set some more options of the AMG preconditioner. In
2450 * particular, we need to tell the AMG setup that we use quadratic basis
2451 * functions for the velocity matrix (this implies more nonzero elements
2452 * in the matrix, so that a more robust algorithm needs to be chosen
2453 * internally). Moreover, we want to be able to control how the coarsening
2454 * structure is build up. The way the Trilinos smoothed aggregation AMG
2455 * does this is to look which matrix entries are of similar size as the
2456 * diagonal entry in order to algebraically build a coarse-grid
2457 * structure. By setting the parameter <code>aggregation_threshold</code>
2458 * to 0.02, we specify that all entries that are more than two percent of
2459 * size of some diagonal pivots in that row should form one coarse grid
2460 * point. This parameter is rather ad hoc, and some fine-tuning of it can
2461 * influence the performance of the preconditioner. As a rule of thumb,
2462 * larger values of <code>aggregation_threshold</code> will decrease the
2463 * number of iterations, but increase the costs per iteration. A look at
2464 * the Trilinos documentation will provide more information on these
2465 * parameters. With this data set, we then initialize the preconditioner
2466 * with the matrix we want it to apply to.
2470 * Finally, we also initialize the preconditioner for the inversion of the
2471 * pressure mass matrix. This matrix is symmetric and well-behaved, so we
2472 * can chose a simple preconditioner. We stick with an incomplete Cholesky
2473 * (IC) factorization preconditioner, which is designed for symmetric
2474 * matrices. We could have also chosen an SSOR preconditioner with
2475 * relaxation factor around 1.2, but IC is cheaper for our example. We
2476 * wrap the preconditioners into a <code>std::shared_ptr</code>
2477 * pointer, which makes it easier to recreate the preconditioner next time
2478 * around since we do not have to care about destroying the previously
2482 * amg_data.elliptic = true;
2483 * amg_data.higher_order_elements = true;
2484 * amg_data.smoother_sweeps = 2;
2485 * amg_data.aggregation_threshold = 0.02;
2486 * Amg_preconditioner->initialize(stokes_preconditioner_matrix.block(0, 0),
2489 * Mp_preconditioner = std::make_shared<TrilinosWrappers::PreconditionIC>();
2490 * Mp_preconditioner->initialize(stokes_preconditioner_matrix.block(1, 1));
2492 * std::cout << std::endl;
2494 * rebuild_stokes_preconditioner = false;
2502 * <a name="BoussinesqFlowProblemassemble_stokes_system"></a>
2503 * <h4>BoussinesqFlowProblem::assemble_stokes_system</h4>
2507 * The time lag scheme we use for advancing the coupled Stokes-temperature
2508 * system forces us to split up the assembly (and the solution of linear
2509 * systems) into two step. The first one is to create the Stokes system
2510 * matrix and right hand side, and the second is to create matrix and right
2511 * hand sides for the temperature dofs, which depends on the result of the
2512 * linear system for the velocity.
2516 * This function is called at the beginning of each time step. In the first
2517 * time step or if the mesh has changed, indicated by the
2518 * <code>rebuild_stokes_matrix</code>, we need to assemble the Stokes
2519 * matrix; on the other hand, if the mesh hasn't changed and the
matrix is
2520 * already available,
this is not necessary and all we need to
do is
2521 *
assemble the right hand side vector which changes in each time step.
2525 * Regarding the technical details of implementation, not much has changed
2526 * from @ref step_22
"step-22". We reset
matrix and vector, create a quadrature formula on
2527 * the cells, and then create the respective
FEValues object. For the update
2528 * flags, we require basis
function derivatives only in
case of a full
2529 * assembly, since they are not needed
for the right hand side; as
always,
2530 * choosing the minimal
set of flags depending on what is currently needed
2536 * There is
one thing that needs to be commented – since we have a
2537 * separate finite element and
DoFHandler for the temperature, we need to
2538 * generate a
second FEValues object for the proper evaluation of the
2539 * temperature solution. This isn
't too complicated to realize here: just
2540 * use the temperature structures and set an update flag for the basis
2541 * function values which we need for evaluation of the temperature
2542 * solution. The only important part to remember here is that the same
2543 * quadrature formula is used for both FEValues objects to ensure that we
2544 * get matching information when we loop over the quadrature points of the
2549 * The declarations proceed with some shortcuts for array sizes, the
2550 * creation of the local matrix and right hand side as well as the vector
2551 * for the indices of the local dofs compared to the global system.
2554 * template <int dim>
2555 * void BoussinesqFlowProblem<dim>::assemble_stokes_system()
2557 * std::cout << " Assembling..." << std::flush;
2559 * if (rebuild_stokes_matrix == true)
2560 * stokes_matrix = 0;
2564 * const QGauss<dim> quadrature_formula(stokes_degree + 2);
2565 * FEValues<dim> stokes_fe_values(
2567 * quadrature_formula,
2568 * update_values | update_quadrature_points | update_JxW_values |
2569 * (rebuild_stokes_matrix == true ? update_gradients : UpdateFlags(0)));
2571 * FEValues<dim> temperature_fe_values(temperature_fe,
2572 * quadrature_formula,
2575 * const unsigned int dofs_per_cell = stokes_fe.dofs_per_cell;
2576 * const unsigned int n_q_points = quadrature_formula.size();
2578 * FullMatrix<double> local_matrix(dofs_per_cell, dofs_per_cell);
2579 * Vector<double> local_rhs(dofs_per_cell);
2581 * std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
2585 * Next we need a vector that will contain the values of the temperature
2586 * solution at the previous time level at the quadrature points to
2587 * assemble the source term in the right hand side of the momentum
2588 * equation. Let's
call this vector <code>old_solution_values</code>.
2592 * The
set of vectors we create next hold the evaluations of the basis
2593 *
functions as well as their gradients and symmetrized gradients that
2594 * will be used
for creating the matrices. Putting these into their own
2595 * arrays rather than asking the
FEValues object for this information each
2596 * time it is needed is an optimization to accelerate the assembly
2597 * process, see @ref step_22
"step-22" for details.
2601 * The last two declarations are used to
extract the individual blocks
2602 * (velocity, pressure, temperature) from the total FE system.
2605 * std::vector<double> old_temperature_values(n_q_points);
2607 * std::vector<Tensor<1, dim>> phi_u(dofs_per_cell);
2608 * std::vector<SymmetricTensor<2, dim>> grads_phi_u(dofs_per_cell);
2609 * std::vector<double> div_phi_u(dofs_per_cell);
2610 * std::vector<double> phi_p(dofs_per_cell);
2617 * Now start the
loop over all cells in the problem. We are working on two
2618 * different DoFHandlers
for this assembly routine, so we must have two
2619 * different cell iterators
for the two objects in use. This might seem a
2620 * bit peculiar, since both the Stokes system and the temperature system
2621 * use the same grid, but that
's the only way to keep degrees of freedom
2622 * in sync. The first statements within the loop are again all very
2623 * familiar, doing the update of the finite element data as specified by
2624 * the update flags, zeroing out the local arrays and getting the values
2625 * of the old solution at the quadrature points. Then we are ready to loop
2626 * over the quadrature points on the cell.
2629 * auto cell = stokes_dof_handler.begin_active();
2630 * const auto endc = stokes_dof_handler.end();
2631 * auto temperature_cell = temperature_dof_handler.begin_active();
2633 * for (; cell != endc; ++cell, ++temperature_cell)
2635 * stokes_fe_values.reinit(cell);
2636 * temperature_fe_values.reinit(temperature_cell);
2641 * temperature_fe_values.get_function_values(old_temperature_solution,
2642 * old_temperature_values);
2644 * for (unsigned int q = 0; q < n_q_points; ++q)
2646 * const double old_temperature = old_temperature_values[q];
2650 * Next we extract the values and gradients of basis functions
2651 * relevant to the terms in the inner products. As shown in
2652 * @ref step_22 "step-22" this helps accelerate assembly.
2656 * Once this is done, we start the loop over the rows and columns
2657 * of the local matrix and feed the matrix with the relevant
2658 * products. The right hand side is filled with the forcing term
2659 * driven by temperature in direction of gravity (which is
2660 * vertical in our example). Note that the right hand side term
2661 * is always generated, whereas the matrix contributions are only
2662 * updated when it is requested by the
2663 * <code>rebuild_matrices</code> flag.
2666 * for (unsigned int k = 0; k < dofs_per_cell; ++k)
2668 * phi_u[k] = stokes_fe_values[velocities].value(k, q);
2669 * if (rebuild_stokes_matrix)
2672 * stokes_fe_values[velocities].symmetric_gradient(k, q);
2674 * stokes_fe_values[velocities].divergence(k, q);
2675 * phi_p[k] = stokes_fe_values[pressure].value(k, q);
2679 * if (rebuild_stokes_matrix)
2680 * for (unsigned int i = 0; i < dofs_per_cell; ++i)
2681 * for (unsigned int j = 0; j < dofs_per_cell; ++j)
2682 * local_matrix(i, j) +=
2683 * (EquationData::eta * 2 * (grads_phi_u[i] * grads_phi_u[j]) -
2684 * div_phi_u[i] * phi_p[j] - phi_p[i] * div_phi_u[j]) *
2685 * stokes_fe_values.JxW(q);
2687 * const Point<dim> gravity =
2688 * -((dim == 2) ? (Point<dim>(0, 1)) : (Point<dim>(0, 0, 1)));
2689 * for (unsigned int i = 0; i < dofs_per_cell; ++i)
2690 * local_rhs(i) += (-EquationData::density * EquationData::beta *
2691 * gravity * phi_u[i] * old_temperature) *
2692 * stokes_fe_values.JxW(q);
2697 * The last step in the loop over all cells is to enter the local
2698 * contributions into the global matrix and vector structures to the
2699 * positions specified in <code>local_dof_indices</code>. Again, we
2700 * let the AffineConstraints class do the insertion of the cell
2701 * matrix elements to the global matrix, which already condenses the
2702 * hanging node constraints.
2705 * cell->get_dof_indices(local_dof_indices);
2707 * if (rebuild_stokes_matrix == true)
2708 * stokes_constraints.distribute_local_to_global(local_matrix,
2710 * local_dof_indices,
2714 * stokes_constraints.distribute_local_to_global(local_rhs,
2715 * local_dof_indices,
2719 * rebuild_stokes_matrix = false;
2721 * std::cout << std::endl;
2729 * <a name="BoussinesqFlowProblemassemble_temperature_matrix"></a>
2730 * <h4>BoussinesqFlowProblem::assemble_temperature_matrix</h4>
2734 * This function assembles the matrix in the temperature equation. The
2735 * temperature matrix consists of two parts, a mass matrix and the time step
2736 * size times a stiffness matrix given by a Laplace term times the amount of
2737 * diffusion. Since the matrix depends on the time step size (which varies
2738 * from one step to another), the temperature matrix needs to be updated
2739 * every time step. We could simply regenerate the matrices in every time
2740 * step, but this is not really efficient since mass and Laplace matrix do
2741 * only change when we change the mesh. Hence, we do this more efficiently
2742 * by generating two separate matrices in this function, one for the mass
2743 * matrix and one for the stiffness (diffusion) matrix. We will then sum up
2744 * the matrix plus the stiffness matrix times the time step size once we
2745 * know the actual time step.
2749 * So the details for this first step are very simple. In case we need to
2750 * rebuild the matrix (i.e., the mesh has changed), we zero the data
2751 * structures, get a quadrature formula and a FEValues object, and create
2752 * local matrices, local dof indices and evaluation structures for the basis
2756 * template <int dim>
2757 * void BoussinesqFlowProblem<dim>::assemble_temperature_matrix()
2759 * if (rebuild_temperature_matrices == false)
2762 * temperature_mass_matrix = 0;
2763 * temperature_stiffness_matrix = 0;
2765 * QGauss<dim> quadrature_formula(temperature_degree + 2);
2766 * FEValues<dim> temperature_fe_values(temperature_fe,
2767 * quadrature_formula,
2768 * update_values | update_gradients |
2769 * update_JxW_values);
2771 * const unsigned int dofs_per_cell = temperature_fe.dofs_per_cell;
2772 * const unsigned int n_q_points = quadrature_formula.size();
2774 * FullMatrix<double> local_mass_matrix(dofs_per_cell, dofs_per_cell);
2775 * FullMatrix<double> local_stiffness_matrix(dofs_per_cell, dofs_per_cell);
2777 * std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
2779 * std::vector<double> phi_T(dofs_per_cell);
2780 * std::vector<Tensor<1, dim>> grad_phi_T(dofs_per_cell);
2785 * to
zero out the local matrices, update the finite element evaluations,
2786 * and then
loop over the rows and columns of the matrices on each
2787 * quadrature
point, where we then create the mass
matrix and the
2788 * stiffness
matrix (Laplace terms times the diffusion
2789 * <code>EquationData::kappa</code>. Finally, we let the constraints
2790 *
object insert these values into the global
matrix, and directly
2791 * condense the constraints into the
matrix.
2794 * for (
const auto &cell : temperature_dof_handler.active_cell_iterators())
2796 * local_mass_matrix = 0;
2797 * local_stiffness_matrix = 0;
2799 * temperature_fe_values.reinit(cell);
2801 * for (unsigned int q = 0; q < n_q_points; ++q)
2803 * for (unsigned int k = 0; k < dofs_per_cell; ++k)
2805 * grad_phi_T[k] = temperature_fe_values.shape_grad(k, q);
2806 * phi_T[k] = temperature_fe_values.shape_value(k, q);
2809 * for (unsigned int i = 0; i < dofs_per_cell; ++i)
2810 * for (unsigned int j = 0; j < dofs_per_cell; ++j)
2812 * local_mass_matrix(i, j) +=
2813 * (phi_T[i] * phi_T[j] * temperature_fe_values.JxW(q));
2814 * local_stiffness_matrix(i, j) +=
2815 * (EquationData::kappa * grad_phi_T[i] * grad_phi_T[j] *
2816 * temperature_fe_values.JxW(q));
2820 * cell->get_dof_indices(local_dof_indices);
2822 * temperature_constraints.distribute_local_to_global(
2823 * local_mass_matrix, local_dof_indices, temperature_mass_matrix);
2824 * temperature_constraints.distribute_local_to_global(
2825 * local_stiffness_matrix,
2826 * local_dof_indices,
2827 * temperature_stiffness_matrix);
2830 * rebuild_temperature_matrices =
false;
2838 * <a name=
"BoussinesqFlowProblemassemble_temperature_system"></a>
2839 * <h4>BoussinesqFlowProblem::assemble_temperature_system</h4>
2843 * This
function does the
second part of the assembly work on the
2844 * temperature
matrix, the actual addition of pressure mass and stiffness
2845 *
matrix (where the time step size comes into play), as well as the
2846 * creation of the velocity-dependent right hand side. The declarations
for
2847 * the right hand side assembly in
this function are pretty much the same as
2848 * the ones used in the other assembly routines, except that we restrict
2849 * ourselves to vectors
this time. We are going to calculate residuals on
2850 * the temperature system, which means that we have to evaluate
second
2851 * derivatives, specified by the update flag <code>
update_hessians</code>.
2855 * The temperature equation is coupled to the Stokes system by means of the
2856 * fluid velocity. These two parts of the solution are associated with
2857 * different DoFHandlers, so we again need to create a
second FEValues
2858 *
object for the evaluation of the velocity at the quadrature points.
2861 *
template <
int dim>
2862 *
void BoussinesqFlowProblem<dim>::assemble_temperature_system(
2863 *
const double maximal_velocity)
2865 *
const bool use_bdf2_scheme = (timestep_number != 0);
2867 *
if (use_bdf2_scheme ==
true)
2869 * temperature_matrix.copy_from(temperature_mass_matrix);
2870 * temperature_matrix *=
2871 * (2 * time_step + old_time_step) / (time_step + old_time_step);
2872 * temperature_matrix.add(time_step, temperature_stiffness_matrix);
2876 * temperature_matrix.copy_from(temperature_mass_matrix);
2877 * temperature_matrix.add(time_step, temperature_stiffness_matrix);
2880 * temperature_rhs = 0;
2882 *
const QGauss<dim> quadrature_formula(temperature_degree + 2);
2884 * quadrature_formula,
2890 * quadrature_formula,
2893 *
const unsigned int dofs_per_cell = temperature_fe.dofs_per_cell;
2894 *
const unsigned int n_q_points = quadrature_formula.size();
2898 * std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
2902 * Next comes the declaration of vectors to hold the old and older
2903 * solution values (as a notation
for time levels @f$n-1@f$ and
2904 * @f$n-2@f$, respectively) and gradients at quadrature points of the
2905 * current cell. We also declare an
object to hold the temperature right
2906 * hand side values (<code>gamma_values</code>), and we again use
2907 * shortcuts
for the temperature basis
functions. Eventually, we need to
2908 * find the temperature extrema and the
diameter of the computational
2909 * domain which will be used
for the definition of the stabilization
2910 * parameter (we got the maximal velocity as an input to
this function).
2913 * std::vector<Tensor<1, dim>> old_velocity_values(n_q_points);
2914 * std::vector<Tensor<1, dim>> old_old_velocity_values(n_q_points);
2915 * std::vector<double> old_temperature_values(n_q_points);
2916 * std::vector<double> old_old_temperature_values(n_q_points);
2917 * std::vector<Tensor<1, dim>> old_temperature_grads(n_q_points);
2918 * std::vector<Tensor<1, dim>> old_old_temperature_grads(n_q_points);
2919 * std::vector<double> old_temperature_laplacians(n_q_points);
2920 * std::vector<double> old_old_temperature_laplacians(n_q_points);
2922 * EquationData::TemperatureRightHandSide<dim> temperature_right_hand_side;
2923 * std::vector<double> gamma_values(n_q_points);
2925 * std::vector<double> phi_T(dofs_per_cell);
2926 * std::vector<Tensor<1, dim>> grad_phi_T(dofs_per_cell);
2928 *
const std::pair<double, double> global_T_range =
2929 * get_extrapolated_temperature_range();
2935 * Now, let
's start the loop over all cells in the triangulation. Again,
2936 * we need two cell iterators that walk in parallel through the cells of
2937 * the two involved DoFHandler objects for the Stokes and temperature
2938 * part. Within the loop, we first set the local rhs to zero, and then get
2939 * the values and derivatives of the old solution functions at the
2940 * quadrature points, since they are going to be needed for the definition
2941 * of the stabilization parameters and as coefficients in the equation,
2942 * respectively. Note that since the temperature has its own DoFHandler
2943 * and FEValues object we get the entire solution at the quadrature point
2944 * (which is the scalar temperature field only anyway) whereas for the
2945 * Stokes part we restrict ourselves to extracting the velocity part (and
2946 * ignoring the pressure part) by using
2947 * <code>stokes_fe_values[velocities].get_function_values</code>.
2950 * auto cell = temperature_dof_handler.begin_active();
2951 * const auto endc = temperature_dof_handler.end();
2952 * auto stokes_cell = stokes_dof_handler.begin_active();
2954 * for (; cell != endc; ++cell, ++stokes_cell)
2958 * temperature_fe_values.reinit(cell);
2959 * stokes_fe_values.reinit(stokes_cell);
2961 * temperature_fe_values.get_function_values(old_temperature_solution,
2962 * old_temperature_values);
2963 * temperature_fe_values.get_function_values(old_old_temperature_solution,
2964 * old_old_temperature_values);
2966 * temperature_fe_values.get_function_gradients(old_temperature_solution,
2967 * old_temperature_grads);
2968 * temperature_fe_values.get_function_gradients(
2969 * old_old_temperature_solution, old_old_temperature_grads);
2971 * temperature_fe_values.get_function_laplacians(
2972 * old_temperature_solution, old_temperature_laplacians);
2973 * temperature_fe_values.get_function_laplacians(
2974 * old_old_temperature_solution, old_old_temperature_laplacians);
2976 * temperature_right_hand_side.value_list(
2977 * temperature_fe_values.get_quadrature_points(), gamma_values);
2979 * stokes_fe_values[velocities].get_function_values(stokes_solution,
2980 * old_velocity_values);
2981 * stokes_fe_values[velocities].get_function_values(
2982 * old_stokes_solution, old_old_velocity_values);
2986 * Next, we calculate the artificial viscosity for stabilization
2987 * according to the discussion in the introduction using the dedicated
2988 * function. With that at hand, we can get into the loop over
2989 * quadrature points and local rhs vector components. The terms here
2990 * are quite lengthy, but their definition follows the time-discrete
2991 * system developed in the introduction of this program. The BDF-2
2992 * scheme needs one more term from the old time step (and involves
2993 * more complicated factors) than the backward Euler scheme that is
2994 * used for the first time step. When all this is done, we distribute
2995 * the local vector into the global one (including hanging node
3000 * compute_viscosity(old_temperature_values,
3001 * old_old_temperature_values,
3002 * old_temperature_grads,
3003 * old_old_temperature_grads,
3004 * old_temperature_laplacians,
3005 * old_old_temperature_laplacians,
3006 * old_velocity_values,
3007 * old_old_velocity_values,
3010 * global_T_range.second - global_T_range.first,
3011 * cell->diameter());
3013 * for (unsigned int q = 0; q < n_q_points; ++q)
3015 * for (unsigned int k = 0; k < dofs_per_cell; ++k)
3017 * grad_phi_T[k] = temperature_fe_values.shape_grad(k, q);
3018 * phi_T[k] = temperature_fe_values.shape_value(k, q);
3021 * const double T_term_for_rhs =
3022 * (use_bdf2_scheme ?
3023 * (old_temperature_values[q] * (1 + time_step / old_time_step) -
3024 * old_old_temperature_values[q] * (time_step * time_step) /
3025 * (old_time_step * (time_step + old_time_step))) :
3026 * old_temperature_values[q]);
3028 * const Tensor<1, dim> ext_grad_T =
3029 * (use_bdf2_scheme ?
3030 * (old_temperature_grads[q] * (1 + time_step / old_time_step) -
3031 * old_old_temperature_grads[q] * time_step / old_time_step) :
3032 * old_temperature_grads[q]);
3034 * const Tensor<1, dim> extrapolated_u =
3035 * (use_bdf2_scheme ?
3036 * (old_velocity_values[q] * (1 + time_step / old_time_step) -
3037 * old_old_velocity_values[q] * time_step / old_time_step) :
3038 * old_velocity_values[q]);
3040 * for (unsigned int i = 0; i < dofs_per_cell; ++i)
3042 * (T_term_for_rhs * phi_T[i] -
3043 * time_step * extrapolated_u * ext_grad_T * phi_T[i] -
3044 * time_step * nu * ext_grad_T * grad_phi_T[i] +
3045 * time_step * gamma_values[q] * phi_T[i]) *
3046 * temperature_fe_values.JxW(q);
3049 * cell->get_dof_indices(local_dof_indices);
3050 * temperature_constraints.distribute_local_to_global(local_rhs,
3051 * local_dof_indices,
3061 * <a name="BoussinesqFlowProblemsolve"></a>
3062 * <h4>BoussinesqFlowProblem::solve</h4>
3066 * This function solves the linear systems of equations. Following the
3067 * introduction, we start with the Stokes system, where we need to generate
3068 * our block Schur preconditioner. Since all the relevant actions are
3069 * implemented in the class <code>BlockSchurPreconditioner</code>, all we
3070 * have to do is to initialize the class appropriately. What we need to pass
3071 * down is an <code>InverseMatrix</code> object for the pressure mass
3072 * matrix, which we set up using the respective class together with the IC
3073 * preconditioner we already generated, and the AMG preconditioner for the
3074 * velocity-velocity matrix. Note that both <code>Mp_preconditioner</code>
3075 * and <code>Amg_preconditioner</code> are only pointers, so we use
3076 * <code>*</code> to pass down the actual preconditioner objects.
3080 * Once the preconditioner is ready, we create a GMRES solver for the block
3081 * system. Since we are working with Trilinos data structures, we have to
3082 * set the respective template argument in the solver. GMRES needs to
3083 * internally store temporary vectors for each iteration (see the discussion
3084 * in the results section of @ref step_22 "step-22") – the more vectors it can use,
3085 * the better it will generally perform. To keep memory demands in check, we
3086 * set the number of vectors to 100. This means that up to 100 solver
3087 * iterations, every temporary vector can be stored. If the solver needs to
3088 * iterate more often to get the specified tolerance, it will work on a
3089 * reduced set of vectors by restarting at every 100 iterations.
3093 * With this all set up, we solve the system and distribute the constraints
3094 * in the Stokes system, i.e., hanging nodes and no-flux boundary condition,
3095 * in order to have the appropriate solution values even at constrained
3096 * dofs. Finally, we write the number of iterations to the screen.
3099 * template <int dim>
3100 * void BoussinesqFlowProblem<dim>::solve()
3102 * std::cout << " Solving..." << std::endl;
3105 * const LinearSolvers::InverseMatrix<TrilinosWrappers::SparseMatrix,
3106 * TrilinosWrappers::PreconditionIC>
3107 * mp_inverse(stokes_preconditioner_matrix.block(1, 1),
3108 * *Mp_preconditioner);
3110 * const LinearSolvers::BlockSchurPreconditioner<
3111 * TrilinosWrappers::PreconditionAMG,
3112 * TrilinosWrappers::PreconditionIC>
3113 * preconditioner(stokes_matrix, mp_inverse, *Amg_preconditioner);
3115 * SolverControl solver_control(stokes_matrix.m(),
3116 * 1e-6 * stokes_rhs.l2_norm());
3118 * SolverGMRES<TrilinosWrappers::MPI::BlockVector> gmres(
3120 * SolverGMRES<TrilinosWrappers::MPI::BlockVector>::AdditionalData(100));
3122 * for (unsigned int i = 0; i < stokes_solution.size(); ++i)
3123 * if (stokes_constraints.is_constrained(i))
3124 * stokes_solution(i) = 0;
3126 * gmres.solve(stokes_matrix, stokes_solution, stokes_rhs, preconditioner);
3128 * stokes_constraints.distribute(stokes_solution);
3130 * std::cout << " " << solver_control.last_step()
3131 * << " GMRES iterations for Stokes subsystem." << std::endl;
3136 * Once we know the Stokes solution, we can determine the new time step
3137 * from the maximal velocity. We have to do this to satisfy the CFL
3138 * condition since convection terms are treated explicitly in the
3139 * temperature equation, as discussed in the introduction. The exact form
3140 * of the formula used here for the time step is discussed in the results
3141 * section of this program.
3145 * There is a snatch here. The formula contains a division by the maximum
3146 * value of the velocity. However, at the start of the computation, we
3147 * have a constant temperature field (we start with a constant
3148 * temperature, and it will be nonconstant only after the first time step
3149 * during which the source acts). Constant temperature means that no
3150 * buoyancy acts, and so the velocity is zero. Dividing by it will not
3151 * likely lead to anything good.
3155 * To avoid the resulting infinite time step, we ask whether the maximal
3156 * velocity is very small (in particular smaller than the values we
3157 * encounter during any of the following time steps) and if so rather than
3158 * dividing by zero we just divide by a small value, resulting in a large
3159 * but finite time step.
3162 * old_time_step = time_step;
3163 * const double maximal_velocity = get_maximal_velocity();
3165 * if (maximal_velocity >= 0.01)
3166 * time_step = 1. / (1.7 * dim * std::sqrt(1. * dim)) / temperature_degree *
3167 * GridTools::minimal_cell_diameter(triangulation) /
3170 * time_step = 1. / (1.7 * dim * std::sqrt(1. * dim)) / temperature_degree *
3171 * GridTools::minimal_cell_diameter(triangulation) / .01;
3174 * << "Time step: " << time_step << std::endl;
3176 * temperature_solution = old_temperature_solution;
3180 * Next we set up the temperature system and the right hand side using the
3181 * function <code>assemble_temperature_system()</code>. Knowing the
3182 * matrix and right hand side of the temperature equation, we set up a
3183 * preconditioner and a solver. The temperature matrix is a mass matrix
3184 * (with eigenvalues around one) plus a Laplace matrix (with eigenvalues
3185 * between zero and @f$ch^{-2}@f$) times a small number proportional to the
3186 * time step @f$k_n@f$. Hence, the resulting symmetric and positive definite
3187 * matrix has eigenvalues in the range @f$[1,1+k_nh^{-2}]@f$ (up to
3188 * constants). This matrix is only moderately ill conditioned even for
3189 * small mesh sizes and we get a reasonably good preconditioner by simple
3190 * means, for example with an incomplete Cholesky decomposition
3191 * preconditioner (IC) as we also use for preconditioning the pressure
3192 * mass matrix solver. As a solver, we choose the conjugate gradient
3193 * method CG. As before, we tell the solver to use Trilinos vectors via
3194 * the template argument <code>TrilinosWrappers::MPI::Vector</code>.
3195 * Finally, we solve, distribute the hanging node constraints and write out
3196 * the number of iterations.
3199 * assemble_temperature_system(maximal_velocity);
3201 * SolverControl solver_control(temperature_matrix.m(),
3202 * 1e-8 * temperature_rhs.l2_norm());
3203 * SolverCG<TrilinosWrappers::MPI::Vector> cg(solver_control);
3205 * TrilinosWrappers::PreconditionIC preconditioner;
3206 * preconditioner.initialize(temperature_matrix);
3208 * cg.solve(temperature_matrix,
3209 * temperature_solution,
3213 * temperature_constraints.distribute(temperature_solution);
3215 * std::cout << " " << solver_control.last_step()
3216 * << " CG iterations for temperature." << std::endl;
3220 * At the end of this function, we step through the vector and read out
3221 * the maximum and minimum temperature value, which we also want to
3222 * output. This will come in handy when determining the correct constant
3223 * in the choice of time step as discuss in the results section of this
3227 * double min_temperature = temperature_solution(0),
3228 * max_temperature = temperature_solution(0);
3229 * for (unsigned int i = 0; i < temperature_solution.size(); ++i)
3232 * std::min<double>(min_temperature, temperature_solution(i));
3234 * std::max<double>(max_temperature, temperature_solution(i));
3237 * std::cout << " Temperature range: " << min_temperature << ' '
3238 * << max_temperature << std::endl;
3247 * <a name="BoussinesqFlowProblemoutput_results"></a>
3248 * <h4>BoussinesqFlowProblem::output_results</h4>
3252 * This function writes the solution to a VTK output file for visualization,
3253 * which is done every tenth time step. This is usually quite a simple task,
3254 * since the deal.II library provides functions that do almost all the job
3255 * for us. There is one new function compared to previous examples: We want
3256 * to visualize both the Stokes solution and the temperature as one data
3257 * set, but we have done all the calculations based on two different
3258 * DoFHandler objects. Luckily, the DataOut class is prepared to deal with
3259 * it. All we have to do is to not attach one single DoFHandler at the
3260 * beginning and then use that for all added vector, but specify the
3261 * DoFHandler to each vector separately. The rest is done as in @ref step_22 "step-22". We
3262 * create solution names (that are going to appear in the visualization
3263 * program for the individual components). The first <code>dim</code>
3264 * components are the vector velocity, and then we have pressure for the
3265 * Stokes part, whereas temperature is scalar. This information is read out
3266 * using the DataComponentInterpretation helper class. Next, we actually
3267 * attach the data vectors with their DoFHandler objects, build patches
3268 * according to the degree of freedom, which are (sub-) elements that
3269 * describe the data for visualization programs. Finally, we open a file
3270 * (that includes the time step number) and write the vtk data into it.
3273 * template <int dim>
3274 * void BoussinesqFlowProblem<dim>::output_results() const
3276 * if (timestep_number % 10 != 0)
3279 * std::vector<std::string> stokes_names(dim, "velocity");
3280 * stokes_names.emplace_back("p");
3281 * std::vector<DataComponentInterpretation::DataComponentInterpretation>
3282 * stokes_component_interpretation(
3283 * dim + 1, DataComponentInterpretation::component_is_scalar);
3284 * for (unsigned int i = 0; i < dim; ++i)
3285 * stokes_component_interpretation[i] =
3286 * DataComponentInterpretation::component_is_part_of_vector;
3288 * DataOut<dim> data_out;
3289 * data_out.add_data_vector(stokes_dof_handler,
3292 * stokes_component_interpretation);
3293 * data_out.add_data_vector(temperature_dof_handler,
3294 * temperature_solution,
3296 * data_out.build_patches(std::min(stokes_degree, temperature_degree));
3298 * std::ofstream output("solution-" +
3299 * Utilities::int_to_string(timestep_number, 4) + ".vtk");
3300 * data_out.write_vtk(output);
3308 * <a name="BoussinesqFlowProblemrefine_mesh"></a>
3309 * <h4>BoussinesqFlowProblem::refine_mesh</h4>
3313 * This function takes care of the adaptive mesh refinement. The three tasks
3314 * this function performs is to first find out which cells to
3315 * refine/coarsen, then to actually do the refinement and eventually
3316 * transfer the solution vectors between the two different grids. The first
3317 * task is simply achieved by using the well-established Kelly error
3318 * estimator on the temperature (it is the temperature we're mainly
3319 * interested in
for this program, and we need to be accurate in regions of
3320 * high temperature gradients, also to not have too much numerical
3321 * diffusion). The
second task is to actually
do the remeshing. That
3322 * involves only basic
functions as well, such as the
3323 * <code>refine_and_coarsen_fixed_fraction</code> that refines those cells
3324 * with the largest estimated error that together make up 80 per cent of the
3325 * error, and coarsens those cells with the smallest error that make up
for
3326 * a combined 10 per cent of the error.
3330 * If implemented like
this, we would get a program that will not make much
3331 * progress: Remember that we expect temperature fields that are nearly
3332 * discontinuous (the diffusivity @f$\kappa@f$ is very small after all) and
3333 * consequently we can expect that a freely adapted mesh will
refine further
3334 * and further into the areas of large gradients. This decrease in mesh size
3335 * will then be accompanied by a decrease in time step, requiring an
3336 * exceedingly large number of time steps to solve to a given
final time. It
3337 * will also lead to meshes that are much better at resolving
3338 * discontinuities after several mesh refinement cycles than in the
3343 * In particular to prevent the decrease in time step size and the
3344 * correspondingly large number of time steps, we limit the maximal
3345 * refinement
depth of the mesh. To
this end, after the refinement indicator
3346 * has been applied to the cells, we simply
loop over all cells on the
3347 * finest
level and unselect them from refinement
if they would result in
3348 * too high a mesh
level.
3351 *
template <
int dim>
3353 * BoussinesqFlowProblem<dim>::refine_mesh(
const unsigned int max_grid_level)
3360 * temperature_solution,
3361 * estimated_error_per_cell);
3364 * estimated_error_per_cell,
3369 *
triangulation.active_cell_iterators_on_level(max_grid_level))
3370 * cell->clear_refine_flag();
3374 * As part of mesh refinement we need to transfer the solution vectors
3375 * from the old mesh to the
new one. To
this end we use the
3377 * should be transferred to the
new grid (we will lose the old grid once
3378 * we have done the refinement so the transfer has to happen concurrently
3379 * with refinement). What we definitely need are the current and the old
3380 * temperature (BDF-2 time stepping requires two old solutions). Since the
3382 * handler, we need to collect the two temperature solutions in
one data
3383 * structure. Moreover, we choose to transfer the Stokes solution, too,
3384 * since we need the velocity at two previous time steps, of which only
3385 *
one is calculated on the fly.
3390 * and temperature
DoFHandler objects, by attaching them to the old dof
3391 * handlers. With
this at place, we can prepare the
triangulation and the
3392 * data vectors
for refinement (in
this order).
3395 * std::vector<TrilinosWrappers::MPI::Vector> x_temperature(2);
3396 * x_temperature[0] = temperature_solution;
3397 * x_temperature[1] = old_temperature_solution;
3401 * temperature_dof_handler);
3403 * stokes_dof_handler);
3406 * temperature_trans.prepare_for_coarsening_and_refinement(x_temperature);
3407 * stokes_trans.prepare_for_coarsening_and_refinement(x_stokes);
3411 * Now everything is ready, so
do the refinement and recreate the dof
3412 * structure on the
new grid, and initialize the
matrix structures and the
3413 *
new vectors in the <code>setup_dofs</code>
function. Next, we actually
3414 * perform the interpolation of the solutions between the grids. We create
3415 * another
copy of temporary vectors
for temperature (now corresponding to
3416 * the
new grid), and let the
interpolate function do the job. Then, the
3417 * resulting array of vectors is written into the respective vector member
3422 * Remember that the
set of constraints will be updated
for the
new
3430 * tmp[0].
reinit(temperature_solution);
3431 * tmp[1].
reinit(temperature_solution);
3432 * temperature_trans.
interpolate(x_temperature, tmp);
3434 * temperature_solution = tmp[0];
3435 * old_temperature_solution = tmp[1];
3439 * After the solution has been transferred we then enforce the constraints
3440 * on the transferred solution.
3443 * temperature_constraints.distribute(temperature_solution);
3444 * temperature_constraints.distribute(old_temperature_solution);
3448 * For the Stokes vector, everything is just the same – except that
3449 * we do not need another temporary vector since we just
interpolate a
3450 * single vector. In the
end, we have to tell the program that the matrices
3451 * and preconditioners need to be regenerated, since the mesh has changed.
3454 * stokes_trans.
interpolate(x_stokes, stokes_solution);
3456 * stokes_constraints.distribute(stokes_solution);
3458 * rebuild_stokes_matrix = true;
3459 * rebuild_temperature_matrices = true;
3460 * rebuild_stokes_preconditioner = true;
3468 * <a name="BoussinesqFlowProblemrun"></a>
3469 * <h4>BoussinesqFlowProblem::
run</h4>
3473 * This function performs all the essential steps in the Boussinesq
3474 * program. It starts by setting up a grid (depending on the spatial
3475 * dimension, we choose some different
level of
initial refinement and
3476 * additional adaptive refinement steps, and then create a cube in
3477 * <code>dim</code> dimensions and
set up the dofs for the
first time. Since
3478 * we want to start the time stepping already with an adaptively refined
3479 * grid, we perform some pre-refinement steps, consisting of all assembly,
3480 * solution and refinement, but without actually advancing in time. Rather,
3481 * we use the vilified <code>goto</code> statement to jump out of the time
3482 *
loop right after mesh refinement to start all over again on the new mesh
3483 * beginning at the <code>start_time_iteration</code> label. (The use of the
3484 * <code>goto</code> is discussed in @ref step_26 "step-26".)
3488 * Before we start, we
project the
initial values to the grid and obtain the
3489 *
first data for the <code>old_temperature_solution</code> vector. Then, we
3490 * initialize time step number and time step and start the time
loop.
3493 * template <
int dim>
3494 *
void BoussinesqFlowProblem<dim>::
run()
3496 *
const unsigned int initial_refinement = (dim == 2 ? 4 : 2);
3497 *
const unsigned int n_pre_refinement_steps = (dim == 2 ? 4 : 3);
3507 *
unsigned int pre_refinement_step = 0;
3509 * start_time_iteration:
3512 * temperature_constraints,
3514 * EquationData::TemperatureInitialValues<dim>(),
3515 * old_temperature_solution);
3517 * timestep_number = 0;
3518 * time_step = old_time_step = 0;
3524 * std::cout <<
"Timestep " << timestep_number <<
": t=" << time
3529 * The
first steps in the time
loop are all obvious – we
3530 *
assemble the Stokes system, the preconditioner, the temperature
3531 *
matrix (matrices and preconditioner
do actually only change in
case
3532 * we
've remeshed before), and then do the solve. Before going on with
3533 * the next time step, we have to check whether we should first finish
3534 * the pre-refinement steps or if we should remesh (every fifth time
3535 * step), refining up to a level that is consistent with initial
3536 * refinement and pre-refinement steps. Last in the loop is to advance
3537 * the solutions, i.e., to copy the solutions to the next "older" time
3541 * assemble_stokes_system();
3542 * build_stokes_preconditioner();
3543 * assemble_temperature_matrix();
3549 * std::cout << std::endl;
3551 * if ((timestep_number == 0) &&
3552 * (pre_refinement_step < n_pre_refinement_steps))
3554 * refine_mesh(initial_refinement + n_pre_refinement_steps);
3555 * ++pre_refinement_step;
3556 * goto start_time_iteration;
3558 * else if ((timestep_number > 0) && (timestep_number % 5 == 0))
3559 * refine_mesh(initial_refinement + n_pre_refinement_steps);
3561 * time += time_step;
3562 * ++timestep_number;
3564 * old_stokes_solution = stokes_solution;
3565 * old_old_temperature_solution = old_temperature_solution;
3566 * old_temperature_solution = temperature_solution;
3570 * Do all the above until we arrive at time 100.
3573 * while (time <= 100);
3575 * } // namespace Step31
3582 * <a name="Thecodemaincodefunction"></a>
3583 * <h3>The <code>main</code> function</h3>
3587 * The main function looks almost the same as in all other programs.
3591 * There is one difference we have to be careful about. This program uses
3592 * Trilinos and, typically, Trilinos is configured so that it can run in
3593 * %parallel using MPI. This doesn't
mean that it <i>has</i> to
run in
3594 * %
parallel, and in fact
this program (unlike @ref step_32
"step-32") makes no attempt at
3595 * all to
do anything in %
parallel using MPI. Nevertheless, Trilinos wants the
3596 * MPI system to be initialized. We
do that be creating an
object of type
3598 * the arguments given to main() (i.e., <code>argc</code> and
3599 * <code>argv</code>) and de-initializes it again when the
object goes out of
3603 *
int main(
int argc,
char *argv[])
3607 *
using namespace dealii;
3608 *
using namespace Step31;
3615 * This program can only be
run in serial. Otherwise,
throw an exception.
3620 *
"This program can only be run in serial, use ./step-31"));
3622 * BoussinesqFlowProblem<2> flow_problem;
3623 * flow_problem.run();
3625 *
catch (std::exception &exc)
3627 * std::cerr << std::endl
3629 * <<
"----------------------------------------------------"
3631 * std::cerr <<
"Exception on processing: " << std::endl
3632 * << exc.what() << std::endl
3633 * <<
"Aborting!" << std::endl
3634 * <<
"----------------------------------------------------"
3641 * std::cerr << std::endl
3643 * <<
"----------------------------------------------------"
3645 * std::cerr <<
"Unknown exception!" << std::endl
3646 * <<
"Aborting!" << std::endl
3647 * <<
"----------------------------------------------------"
3655 <a name=
"Results"></a><h1>Results</h1>
3658 <a name=
"Resultsin2d"></a><h3> Results in 2
d </h3>
3661 When you
run the program in 2
d, the output will look something like
3665 Number of active cells: 256 (on 5 levels)
3666 Number of degrees of freedom: 3556 (2178+289+1089)
3670 Rebuilding Stokes preconditioner...
3672 0 GMRES iterations
for Stokes subsystem.
3674 9 CG iterations
for temperature.
3675 Temperature range: -0.16687 1.30011
3677 Number of active cells: 280 (on 6 levels)
3678 Number of degrees of freedom: 4062 (2490+327+1245)
3682 Rebuilding Stokes preconditioner...
3684 0 GMRES iterations
for Stokes subsystem.
3686 9 CG iterations
for temperature.
3687 Temperature range: -0.0982971 0.598503
3689 Number of active cells: 520 (on 7 levels)
3690 Number of degrees of freedom: 7432 (4562+589+2281)
3694 Rebuilding Stokes preconditioner...
3696 0 GMRES iterations
for Stokes subsystem.
3698 9 CG iterations
for temperature.
3699 Temperature range: -0.0551098 0.294493
3701 Number of active cells: 1072 (on 8 levels)
3702 Number of degrees of freedom: 15294 (9398+1197+4699)
3706 Rebuilding Stokes preconditioner...
3708 0 GMRES iterations
for Stokes subsystem.
3710 9 CG iterations
for temperature.
3711 Temperature range: -0.0273524 0.156861
3713 Number of active cells: 2116 (on 9 levels)
3714 Number of degrees of freedom: 30114 (18518+2337+9259)
3718 Rebuilding Stokes preconditioner...
3720 0 GMRES iterations
for Stokes subsystem.
3721 Time step: 0.0574449
3722 9 CG iterations
for temperature.
3723 Temperature range: -0.014993 0.0738328
3725 Timestep 1: t=0.0574449
3728 56 GMRES iterations
for Stokes subsystem.
3729 Time step: 0.0574449
3730 9 CG iterations
for temperature.
3731 Temperature range: -0.0273934 0.14488
3737 In the beginning we
refine the mesh several times adaptively and
3738 always return to time step
zero to restart on the newly refined
3739 mesh. Only then
do we start the actual time iteration.
3741 The program runs
for a
while. The temperature field
for time steps 0,
3742 500, 1000, 1500, 2000, 3000, 4000, and 5000 looks like
this (note that
3743 the color
scale used
for the temperature is not
always the same):
3745 <table align=
"center" class=
"doxtable">
3748 <img src=
"https://www.dealii.org/images/steps/developer/step-31.2d.solution.00.png" alt=
"">
3751 <img src=
"https://www.dealii.org/images/steps/developer/step-31.2d.solution.01.png" alt=
"">
3754 <img src=
"https://www.dealii.org/images/steps/developer/step-31.2d.solution.02.png" alt=
"">
3757 <img src=
"https://www.dealii.org/images/steps/developer/step-31.2d.solution.03.png" alt=
"">
3762 <img src=
"https://www.dealii.org/images/steps/developer/step-31.2d.solution.04.png" alt=
"">
3765 <img src=
"https://www.dealii.org/images/steps/developer/step-31.2d.solution.05.png" alt=
"">
3768 <img src=
"https://www.dealii.org/images/steps/developer/step-31.2d.solution.06.png" alt=
"">
3771 <img src=
"https://www.dealii.org/images/steps/developer/step-31.2d.solution.07.png" alt=
"">
3776 The visualizations shown here were generated
using a version of the example
3777 which did not enforce the constraints after transferring the mesh.
3779 As can be seen, we have three heat sources that heat fluid and
3780 therefore produce a buoyancy effect that lets hots pockets of fluid
3781 rise up and swirl around. By a chimney effect, the three streams are
3782 pressed together by fluid that comes from the outside and wants to
3783 join the updraft party. Note that because the fluid is initially at
3784 rest, those parts of the fluid that were initially over the sources
3785 receive a longer heating time than that fluid that is later dragged
3786 over the source by the fully developed flow field. It is therefore
3787 hotter, a fact that can be seen in the red tips of the three
3788 plumes. Note also the relatively fine features of the flow field, a
3789 result of the sophisticated transport stabilization of the temperature
3790 equation we have chosen.
3792 In addition to the pictures above, the following ones show the
3793 adaptive mesh and the flow field at the same time steps:
3795 <table align=
"center" class=
"doxtable">
3798 <img src=
"https://www.dealii.org/images/steps/developer/step-31.2d.grid.00.png" alt=
"">
3801 <img src=
"https://www.dealii.org/images/steps/developer/step-31.2d.grid.01.png" alt=
"">
3804 <img src=
"https://www.dealii.org/images/steps/developer/step-31.2d.grid.02.png" alt=
"">
3807 <img src=
"https://www.dealii.org/images/steps/developer/step-31.2d.grid.03.png" alt=
"">
3812 <img src=
"https://www.dealii.org/images/steps/developer/step-31.2d.grid.04.png" alt=
"">
3815 <img src=
"https://www.dealii.org/images/steps/developer/step-31.2d.grid.05.png" alt=
"">
3818 <img src=
"https://www.dealii.org/images/steps/developer/step-31.2d.grid.06.png" alt=
"">
3821 <img src=
"https://www.dealii.org/images/steps/developer/step-31.2d.grid.07.png" alt=
"">
3827 <a name=
"Resultsin3d"></a><h3> Results in 3d </h3>
3830 The same thing can of course be done in 3
d by changing the
template
3831 parameter to the BoussinesqFlowProblem
object in <code>main()</code>
3832 from 2 to 3, so that the output now looks like follows:
3836 Number of active cells: 64 (on 3 levels)
3837 Number of degrees of freedom: 3041 (2187+125+729)
3841 Rebuilding Stokes preconditioner...
3843 0 GMRES iterations
for Stokes subsystem.
3845 9 CG iterations
for temperature.
3846 Temperature range: -0.675683 4.94725
3848 Number of active cells: 288 (on 4 levels)
3849 Number of degrees of freedom: 12379 (8943+455+2981)
3853 Rebuilding Stokes preconditioner...
3855 0 GMRES iterations
for Stokes subsystem.
3857 9 CG iterations
for temperature.
3858 Temperature range: -0.527701 2.25764
3860 Number of active cells: 1296 (on 5 levels)
3861 Number of degrees of freedom: 51497 (37305+1757+12435)
3865 Rebuilding Stokes preconditioner...
3867 0 GMRES iterations
for Stokes subsystem.
3869 10 CG iterations
for temperature.
3870 Temperature range: -0.496942 0.847395
3872 Number of active cells: 5048 (on 6 levels)
3873 Number of degrees of freedom: 192425 (139569+6333+46523)
3877 Rebuilding Stokes preconditioner...
3879 0 GMRES iterations
for Stokes subsystem.
3881 10 CG iterations
for temperature.
3882 Temperature range: -0.267683 0.497739
3884 Timestep 1: t=0.306373
3887 27 GMRES iterations
for Stokes subsystem.
3889 10 CG iterations
for temperature.
3890 Temperature range: -0.461787 0.958679
3896 Visualizing the temperature isocontours at time steps 0,
3897 50, 100, 150, 200, 300, 400, 500, 600, 700, and 800 yields the
3900 <table align=
"center" class=
"doxtable">
3903 <img src=
"https://www.dealii.org/images/steps/developer/step-31.3d.solution.00.png" alt=
"">
3906 <img src=
"https://www.dealii.org/images/steps/developer/step-31.3d.solution.01.png" alt=
"">
3909 <img src=
"https://www.dealii.org/images/steps/developer/step-31.3d.solution.02.png" alt=
"">
3912 <img src=
"https://www.dealii.org/images/steps/developer/step-31.3d.solution.03.png" alt=
"">
3917 <img src=
"https://www.dealii.org/images/steps/developer/step-31.3d.solution.04.png" alt=
"">
3920 <img src=
"https://www.dealii.org/images/steps/developer/step-31.3d.solution.05.png" alt=
"">
3923 <img src=
"https://www.dealii.org/images/steps/developer/step-31.3d.solution.06.png" alt=
"">
3926 <img src=
"https://www.dealii.org/images/steps/developer/step-31.3d.solution.07.png" alt=
"">
3931 <img src=
"https://www.dealii.org/images/steps/developer/step-31.3d.solution.08.png" alt=
"">
3934 <img src=
"https://www.dealii.org/images/steps/developer/step-31.3d.solution.09.png" alt=
"">
3937 <img src=
"https://www.dealii.org/images/steps/developer/step-31.3d.solution.10.png" alt=
"">
3944 That the
first picture looks like three hedgehogs stems from the fact that our
3945 scheme essentially projects the source times the
first time step size onto the
3946 mesh to obtain the temperature field in the
first time step. Since the source
3947 function is discontinuous, we need to expect over- and undershoots from
this
3948 project. This is in fact what happens (it
's easier to check this in 2d) and
3949 leads to the crumpled appearance of the isosurfaces. The visualizations shown
3950 here were generated using a version of the example which did not enforce the
3951 constraints after transferring the mesh.
3955 <a name="Numericalexperimentstodetermineoptimalparameters"></a><h3> Numerical experiments to determine optimal parameters </h3>
3958 The program as is has three parameters that we don't have much of a
3959 theoretical handle on how to choose in an optimal way. These are:
3961 <li>The time step must satisfy a CFL condition
3962 @f$k\le \min_K \frac{c_kh_K}{\|\mathbf{u}\|_{L^\infty(K)}}@f$. Here, @f$c_k@f$ is
3963 dimensionless, but what is the right
value?
3964 <li>In the computation of the artificial viscosity,
3969 \|\mathbf{u}\|_{
L^\infty(K)}
3973 \frac{\|R_\alpha(
T)\|_{
L^\infty(K)}}{c(\mathbf{u},
T)}
3976 with @f$c(\mathbf{u},
T) =
3977 c_R\ \|\mathbf{u}\|_{
L^\infty(\Omega)} \ \mathrm{var}(
T)
3978 \ |\mathrm{diam}(\Omega)|^{\alpha-2}@f$.
3979 Here, the choice of the dimensionless %
numbers @f$\beta,c_R@f$ is of
3982 In all of these cases, we will have to expect that the correct choice of each
3983 value depends on that of the others, and most likely also on the space
3984 dimension and polynomial degree of the finite element used
for the
3985 temperature. Below we
'll discuss a few numerical experiments to choose
3986 constants @f$c_k@f$ and @f$\beta@f$.
3988 Below, we will not discuss the choice of @f$c_R@f$. In the program, we set
3989 it to @f$c_R=2^{\frac{4-2\alpha}{d}}@f$. The reason for this value is a
3990 bit complicated and has more to do with the history of the program
3991 than reasoning: while the correct formula for the global scaling
3992 parameter @f$c(\mathbf{u},T)@f$ is shown above, the program (including the
3993 version shipped with deal.II 6.2) initially had a bug in that we
3995 @f$c(\mathbf{u},T) =
3996 \|\mathbf{u}\|_{L^\infty(\Omega)} \ \mathrm{var}(T)
3997 \ \frac{1}{|\mathrm{diam}(\Omega)|^{\alpha-2}}@f$ instead, where
3998 we had set the scaling parameter to one. Since we only computed on the
3999 unit square/cube where @f$\mathrm{diam}(\Omega)=2^{1/d}@f$, this was
4000 entirely equivalent to using the correct formula with
4001 @f$c_R=\left(2^{1/d}\right)^{4-2\alpha}=2^{\frac{4-2\alpha}{d}}@f$. Since
4002 this value for @f$c_R@f$ appears to work just fine for the current
4003 program, we corrected the formula in the program and set @f$c_R@f$ to a
4004 value that reproduces exactly the results we had before. We will,
4005 however, revisit this issue again in @ref step_32 "step-32".
4007 Now, however, back to the discussion of what values of @f$c_k@f$ and
4008 @f$\beta@f$ to choose:
4011 <a name="Choosingicsubksubiandbeta"></a><h4> Choosing <i>c<sub>k</sub></i> and beta </h4>
4014 These two constants are definitely linked in some way. The reason is easy to
4015 see: In the case of a pure advection problem,
4016 @f$\frac{\partial T}{\partial t} + \mathbf{u}\cdot\nabla T = \gamma@f$, any
4017 explicit scheme has to satisfy a CFL condition of the form
4018 @f$k\le \min_K \frac{c_k^a h_K}{\|\mathbf{u}\|_{L^\infty(K)}}@f$. On the other hand,
4019 for a pure diffusion problem,
4020 @f$\frac{\partial T}{\partial t} + \nu \Delta T = \gamma@f$,
4021 explicit schemes need to satisfy a condition
4022 @f$k\le \min_K \frac{c_k^d h_K^2}{\nu}@f$. So given the form of @f$\nu@f$ above, an
4023 advection diffusion problem like the one we have to solve here will result in
4024 a condition of the form
4026 k\le \min_K \min \left\{
4027 \frac{c_k^a h_K}{\|\mathbf{u}\|_{L^\infty(K)}},
4028 \frac{c_k^d h_K^2}{\beta \|\mathbf{u}\|_{L^\infty(K)} h_K}\right\}
4030 \min_K \left( \min \left\{
4032 \frac{c_k^d}{\beta}\right\}
4033 \frac{h_K}{\|\mathbf{u}\|_{L^\infty(K)}} \right)
4035 It follows that we have to face the fact that we might want to choose @f$\beta@f$
4036 larger to improve the stability of the numerical scheme (by increasing the
4037 amount of artificial diffusion), but we have to pay a price in the form of
4038 smaller, and consequently more time steps. In practice, one would therefore
4039 like to choose @f$\beta@f$ as small as possible to keep the transport problem
4040 sufficiently stabilized while at the same time trying to choose the time step
4041 as large as possible to reduce the overall amount of work.
4043 The find the right balance, the only way is to do a few computational
4044 experiments. Here's what we did: We modified the program slightly to allow
4045 less mesh refinement (so we don
't always have to wait that long) and to choose
4050 \|\mathbf{u}\|_{L^\infty(K)} h_K
4051 @f$ to eliminate the effect of the constant @f$c_R@f$ (we know that
4052 solutions are stable by using this version of @f$\nu(T)@f$ as an artificial
4053 viscosity, but that we can improve things -- i.e. make the solution
4054 sharper -- by using the more complicated formula for this artificial
4055 viscosity). We then run the program
4056 for different values @f$c_k,\beta@f$ and observe maximal and minimal temperatures
4057 in the domain. What we expect to see is this: If we choose the time step too
4058 big (i.e. choose a @f$c_k@f$ bigger than theoretically allowed) then we will get
4059 exponential growth of the temperature. If we choose @f$\beta@f$ too small, then
4060 the transport stabilization becomes insufficient and the solution will show
4061 significant oscillations but not exponential growth.
4064 <a name="ResultsforQsub1subelements"></a><h5>Results for Q<sub>1</sub> elements</h5>
4067 Here is what we get for
4068 @f$\beta=0.01, \beta=0.1@f$, and @f$\beta=0.5@f$, different choices of @f$c_k@f$, and
4069 bilinear elements (<code>temperature_degree=1</code>) in 2d:
4071 <table align="center" class="doxtable">
4074 <img src="https://www.dealii.org/images/steps/developer/step-31.timestep.q1.beta=0.01.png" alt="">
4077 <img src="https://www.dealii.org/images/steps/developer/step-31.timestep.q1.beta=0.03.png" alt="">
4082 <img src="https://www.dealii.org/images/steps/developer/step-31.timestep.q1.beta=0.1.png" alt="">
4085 <img src="https://www.dealii.org/images/steps/developer/step-31.timestep.q1.beta=0.5.png" alt="">
4090 The way to interpret these graphs goes like this: for @f$\beta=0.01@f$ and
4091 @f$c_k=\frac 12,\frac 14@f$, we see exponential growth or at least large
4092 variations, but if we choose
4093 @f$k=\frac 18\frac{h_K}{\|\mathbf{u}\|_{L^\infty(K)}}@f$
4094 or smaller, then the scheme is
4095 stable though a bit wobbly. For more artificial diffusion, we can choose
4096 @f$k=\frac 14\frac{h_K}{\|\mathbf{u}\|_{L^\infty(K)}}@f$
4097 or smaller for @f$\beta=0.03@f$,
4098 @f$k=\frac 13\frac{h_K}{\|\mathbf{u}\|_{L^\infty(K)}}@f$
4099 or smaller for @f$\beta=0.1@f$, and again need
4100 @f$k=\frac 1{15}\frac{h_K}{\|\mathbf{u}\|_{L^\infty(K)}}@f$
4101 for @f$\beta=0.5@f$ (this time because much diffusion requires a small time
4104 So how to choose? If we were simply interested in a large time step, then we
4105 would go with @f$\beta=0.1@f$ and
4106 @f$k=\frac 13\frac{h_K}{\|\mathbf{u}\|_{L^\infty(K)}}@f$.
4107 On the other hand, we're also interested in accuracy and here it may be of
4108 interest to actually investigate what these curves show. To
this end note that
4109 we start with a
zero temperature and that our sources are positive — so
4110 we would intuitively expect that the temperature can never drop below
4111 zero. But it does, a consequence of Gibb
's phenomenon when using continuous
4112 elements to approximate a discontinuous solution. We can therefore see that
4113 choosing @f$\beta@f$ too small is bad: too little artificial diffusion leads to
4114 over- and undershoots that aren't diffused away. On the other hand,
for large
4115 @f$\beta@f$, the minimum temperature drops below
zero at the beginning but then
4116 quickly diffuses back to
zero.
4118 On the other hand, let
's also look at the maximum temperature. Watching the
4119 movie of the solution, we see that initially the fluid is at rest. The source
4120 keeps heating the same volume of fluid whose temperature increases linearly at
4121 the beginning until its buoyancy is able to move it upwards. The hottest part
4122 of the fluid is therefore transported away from the solution and fluid taking
4123 its place is heated for only a short time before being moved out of the source
4124 region, therefore remaining cooler than the initial bubble. If @f$\kappa=0@f$
4125 (in the program it is nonzero but very small) then the hottest part of the
4126 fluid should be advected along with the flow with its temperature
4127 constant. That's what we can see in the graphs with the smallest @f$\beta@f$: Once
4128 the maximum temperature is reached, it hardly changes any more. On the other
4129 hand, the larger the artificial diffusion, the more the hot spot is
4130 diffused. Note that
for this criterion, the time step size does not play a
4133 So to
sum up, likely the best choice would appear to be @f$\beta=0.03@f$
4134 and @f$k=\frac 14\frac{h_K}{\|\mathbf{u}\|_{L^\infty(K)}}@f$. The curve is
4135 a bit wobbly, but overall pictures looks pretty reasonable with the
4136 exception of some over and undershoots close to the start time due to
4140 <a name="ResultsforQsub2subelements"></a><h5>Results for Q<sub>2</sub> elements</h5>
4143 One can repeat the same sequence of experiments for higher order
4144 elements as well. Here are the graphs for bi-quadratic shape functions
4145 (<code>temperature_degree=2</code>) for the temperature, while we
4146 retain the @f$Q_2/Q_1@f$ stable Taylor-Hood element for the Stokes system:
4148 <table align="center" class="doxtable">
4151 <img src="https://www.dealii.org/images/steps/developer/step-31.timestep.q2.beta=0.01.png" alt="">
4154 <img src="https://www.dealii.org/images/steps/developer/step-31.timestep.q2.beta=0.03.png" alt="">
4159 <img src="https://www.dealii.org/images/steps/developer/step-31.timestep.q2.beta=0.1.png" alt="">
4164 Again, small values of @f$\beta@f$ lead to less diffusion but we have to
4165 choose the time step very small to keep things under control. Too
4166 large values of @f$\beta@f$ make for more diffusion, but again require
4167 small time steps. The best value would appear to be @f$\beta=0.03@f$, as
4168 for the @f$Q_1@f$ element, and then we have to choose
4169 @f$k=\frac 18\frac{h_K}{\|\mathbf{u}\|_{L^\infty(K)}}@f$ — exactly
4170 half the size for the @f$Q_1@f$ element, a fact that may not be surprising
4171 if we state the CFL condition as the requirement that the time step be
4172 small enough so that the distance transport advects in each time step
4173 is no longer than one <i>grid point</i> away (which for @f$Q_1@f$ elements
4174 is @f$h_K@f$, but for @f$Q_2@f$ elements is @f$h_K/2@f$). It turns out that @f$\beta@f$
4175 needs to be slightly larger for obtaining stable results also late in
4176 the simulation at times larger than 60, so we actually choose it as
4177 @f$\beta = 0.034@f$ in the code.
4180 <a name="Resultsfor3d"></a><h5>Results for 3d</h5>
4183 One can repeat these experiments in 3d and find the optimal time step
4184 for each value of @f$\beta@f$ and find the best value of @f$\beta@f$. What one
4185 finds is that for the same @f$\beta@f$ already used in 2d, the time steps
4186 needs to be a bit smaller, by around a factor of 1.2 or so. This is
4187 easily explained: the time step restriction is
4188 @f$k=\min_K \frac{ch_K}{\|\mathbf{u}\|_{L^\infty(K)}}@f$ where @f$h_K@f$ is
4189 the <i>diameter</i> of the cell. However, what is really needed is the
4190 distance between mesh points, which is @f$\frac{h_K}{\sqrt{d}}@f$. So a
4191 more appropriate form would be
4192 @f$k=\min_K \frac{ch_K}{\|\mathbf{u}\|_{L^\infty(K)}\sqrt{d}}@f$.
4194 The second find is that one needs to choose @f$\beta@f$ slightly bigger
4195 (about @f$\beta=0.05@f$ or so). This then again reduces the time step we
4201 <a name="Conclusions"></a><h5>Conclusions</h5>
4204 Concluding, from the simple computations above, @f$\beta=0.034@f$ appears to be a
4205 good choice for the stabilization parameter in 2d, and @f$\beta=0.05@f$ in 3d. In
4206 a dimension independent way, we can model this as @f$\beta=0.017d@f$. If one does
4207 longer computations (several thousand time steps) on finer meshes, one
4208 realizes that the time step size is not quite small enough and that for
4209 stability one will have to reduce the above values a bit more (by about a
4210 factor of @f$\frac 78@f$).
4212 As a consequence, a formula that reconciles 2d, 3d, and variable polynomial
4213 degree and takes all factors in account reads as follows:
4216 \frac 1{2 \cdot 1.7} \frac 1{\sqrt{d}}
4219 \frac{h_K}{\|\mathbf{u}\|_{L^\infty(K)}}
4221 \frac 1{1.7 d\sqrt{d}}
4223 \frac{h_K}{\|\mathbf{u}\|_{L^\infty(K)}}.
4225 In the first form (in the center of the equation), @f$\frac
4226 1{2 \cdot 1.7}@f$ is a universal constant, @f$\frac 1{\sqrt{d}}@f$
4227 is the factor that accounts for the difference between cell diameter
4228 and grid point separation,
4229 @f$\frac 2d@f$ accounts for the increase in @f$\beta@f$ with space dimension,
4230 @f$\frac 1{q_T}@f$ accounts for the distance between grid points for
4231 higher order elements, and @f$\frac{h_K}{\|\mathbf{u}\|_{L^\infty(K)}}@f$
4232 for the local speed of transport relative to the cell size. This is
4233 the formula that we use in the program.
4235 As for the question of whether to use @f$Q_1@f$ or @f$Q_2@f$ elements for the
4236 temperature, the following considerations may be useful: First,
4237 solving the temperature equation is hardly a factor in the overall
4238 scheme since almost the entire compute time goes into solving the
4239 Stokes system in each time step. Higher order elements for the
4240 temperature equation are therefore not a significant drawback. On the
4241 other hand, if one compares the size of the over- and undershoots the
4242 solution produces due to the discontinuous source description, one
4243 notices that for the choice of @f$\beta@f$ and @f$k@f$ as above, the @f$Q_1@f$
4244 solution dips down to around @f$-0.47@f$, whereas the @f$Q_2@f$ solution only
4245 goes to @f$-0.13@f$ (remember that the exact solution should never become
4246 negative at all. This means that the @f$Q_2@f$ solution is significantly
4247 more accurate; the program therefore uses these higher order elements,
4248 despite the penalty we pay in terms of smaller time steps.
4251 <a name="Possibleextensions"></a><h3> Possible extensions </h3>
4254 There are various ways to extend the current program. Of particular interest
4255 is, of course, to make it faster and/or increase the resolution of the
4256 program, in particular in 3d. This is the topic of the @ref step_32 "step-32"
4257 tutorial program which will implement strategies to solve this problem in
4258 %parallel on a cluster. It is also the basis of the much larger open
4259 source code ASPECT (see https://aspect.geodynamics.org/ ) that can solve realistic
4260 problems and that constitutes the further development of @ref step_32 "step-32".
4262 Another direction would be to make the fluid flow more realistic. The program
4263 was initially written to simulate various cases simulating the convection of
4264 material in the earth's mantle, i.e. the zone between the outer earth core and
4265 the solid earth crust: there, material is heated from below and cooled from
4266 above, leading to thermal convection. The physics of
this fluid are much more
4267 complicated than shown in
this program, however: The viscosity of mantle
4268 material is strongly dependent on the temperature, i.e. @f$\eta=\eta(
T)@f$, with
4269 the dependency frequently modeled as a viscosity that is reduced exponentially
4270 with rising temperature. Secondly, much of the dynamics of the mantle is
4271 determined by chemical reactions, primarily phase changes of the various
4272 crystals that make up the mantle; the buoyancy term on the right hand side of
4273 the Stokes equations then depends not only on the temperature, but also on the
4274 chemical composition at a given location which is advected by the flow field
4275 but also changes as a
function of pressure and temperature. We will
4276 investigate some of these effects in later tutorial programs as well.
4279 <a name=
"PlainProg"></a>
4280 <h1> The plain program</h1>
4281 @include
"step-31.cc"