1499 * cell->get_dof_indices(dof_indices);
1501 * dof_indices.end(),
1502 * dof_indices.begin(),
1504 * return partitioner->global_to_local(index);
1509 *
for (
const auto dof : dof_indices)
1510 * dsp.add_entries(dof, dof_indices.begin(), dof_indices.end());
1513 * sparsity_pattern.copy_from(dsp);
1515 * lumped_mass_matrix.reinit(sparsity_pattern);
1516 * norm_matrix.reinit(sparsity_pattern);
1517 *
for (
auto &
matrix : cij_matrix)
1518 *
matrix.reinit(sparsity_pattern);
1519 *
for (
auto &
matrix : nij_matrix)
1520 *
matrix.reinit(sparsity_pattern);
1527 * Next, we have to
assemble various matrices. We define a number of
1528 * helper
functions and data structures in an anonymous
namespace.
1538 * <code>CopyData</code>
class that will be used to
assemble the
1539 * offline data matrices
using WorkStream. It acts as a container: it
1540 * is just a
struct where
WorkStream stores the local cell
1541 * contributions. Note that it also contains a
class member
1542 * <code>local_boundary_normal_map</code> used to store the local
1543 * contributions required to compute the normals at the boundary.
1549 *
template <
int dim>
1552 *
bool is_artificial;
1553 * std::vector<types::global_dof_index> local_dof_indices;
1554 *
typename OfflineData<dim>::BoundaryNormalMap local_boundary_normal_map;
1556 * std::array<FullMatrix<double>, dim> cell_cij_matrix;
1561 * Next we introduce a number of helper
functions that are all
1562 * concerned about reading and writing
matrix and vector entries. They
1563 * are mainly motivated by providing slightly more efficient code and
1564 * <a href=
"https://en.wikipedia.org/wiki/Syntactic_sugar"> syntactic
1565 * sugar</a>
for otherwise somewhat tedious code.
1569 * The
first function we introduce, <code>get_entry()</code>, will be
1570 * used to read the
value stored at the entry pointed by a
1572 * function works around a small deficiency in the
SparseMatrix
1574 * operations of the sparse
matrix stored in CRS format. As such the
1575 * iterator already knows the global index of the corresponding
matrix
1577 * to the lack of an
interface in the
SparseMatrix for accessing the
1579 * have to create a temporary
SparseMatrix iterator. We simply hide
1580 *
this in the <code>get_entry()</code> function.
1586 *
template <
typename IteratorType>
1591 * &
matrix, it->global_index());
1592 *
return matrix_iterator->value();
1597 * The <code>set_entry()</code> helper is the inverse operation of
1598 * <code>get_value()</code>: Given an iterator and a
value, it sets the
1599 * entry pointed to by the iterator in the
matrix.
1605 *
template <
typename IteratorType>
1608 *
const IteratorType & it,
1612 * it->global_index());
1613 * matrix_iterator->value() =
value;
1618 * <code>gather_get_entry()</code>: we note that @f$\mathbf{c}_{ij} \in
1619 * \mathbb{R}^
d@f$. If @f$d=2@f$ then @f$\mathbf{c}_{ij} =
1620 * [\mathbf{c}_{ij}^1,\mathbf{c}_{ij}^2]^\top@f$. Which basically implies
1621 * that we need
one matrix per space dimension to store the
1622 * @f$\mathbf{c}_{ij}@f$ vectors. Similar observation follows
for the
1623 *
matrix @f$\mathbf{n}_{ij}@f$. The purpose of
1624 * <code>gather_get_entry()</code> is to retrieve those entries and store
1631 *
template <std::
size_t k,
typename IteratorType>
1634 *
const IteratorType it)
1637 *
for (
unsigned int j = 0; j < k; ++j)
1638 * result[j] = get_entry(c_ij[j], it);
1645 * signature, having three input arguments, will be used to retrieve
1646 * the individual components <code>(i,
l)</code> of a
matrix. The
1647 * functionality of <code>gather_get_entry()</code> and
1648 * <code>
gather()</code> is very much the same, but their context is
1649 * different: the function <code>
gather()</code> does not rely on an
1650 * iterator (that actually knows the value pointed to) but rather on the
1651 * indices <code>(i,
l)</code> of the entry in order to retrieve its
1652 * actual
value. We should expect <code>
gather()</code> to be slightly
1653 * more expensive than <code>gather_get_entry()</code>. The use of
1654 * <code>
gather()</code> will be limited to the task of computing the
1655 * algebraic viscosity @f$d_{ij}@f$ in the particular
case that when
1656 * both @f$i@f$ and @f$j@f$ lie at the boundary.
1660 * @note The reader should be aware that accessing an arbitrary
1661 * <code>(i,
l)</code> entry of a
matrix (say
for instance Trilinos or PETSc
1662 * matrices) is in
general unacceptably expensive. Here is where we might
1663 * want to keep an eye on complexity: we want
this operation to have
1664 * constant complexity, which is the
case of the current implementation
1665 *
using deal.II matrices.
1671 *
template <std::
size_t k>
1674 *
const unsigned int i,
1675 *
const unsigned int j)
1678 *
for (
unsigned int l = 0;
l < k; ++
l)
1679 * result[
l] = n_ij[
l](i, j);
1686 * signature having two input arguments will be used to
gather the
1687 * state at a node <code>i</code> and
return it as a
1694 * template <std::size_t k>
1697 *
const unsigned int i)
1700 *
for (
unsigned int j = 0; j < k; ++j)
1701 * result[j] =
U[j].local_element(i);
1707 * <code>scatter()</code>:
this function has three input arguments, the
1708 *
first one is meant to be a
"global object" (say a locally owned or
1709 * locally relevant vector), the
second argument which could be a
1711 * which represents a index of the global
object. This function will be
1712 * primarily used to write the updated nodal
values, stored as
1719 * template <std::size_t k, int k2>
1723 *
const unsigned int i)
1725 *
static_assert(k == k2,
1726 *
"The dimensions of the input arguments must agree");
1727 *
for (
unsigned int j = 0; j < k; ++j)
1728 *
U[j].local_element(i) = tensor[j];
1734 * We are now in a position to
assemble all matrices stored in
1735 * <code>OfflineData</code>: the lumped mass entries @f$m_i@f$, the
1736 * vector-valued matrices @f$\mathbf{c}_{ij}@f$ and @f$\mathbf{n}_{ij} =
1737 * \frac{\mathbf{c}_{ij}}{|\mathbf{c}_{ij}|}@f$, and the boundary normals
1738 * @f$\boldsymbol{\nu}_i@f$.
1742 * In order to exploit thread parallelization we use the
WorkStream approach
1743 * detailed in the @ref threads
"Parallel computing with multiple processors"
1744 * accessing shared memory. As customary
this requires
1746 * - Scratch data (i.e. input info required to carry out computations): in
1747 * this case it is <code>scratch_data</code>.
1748 * - The worker: in our case this is the <code>local_assemble_system()</code>
1750 * actually computes the local (i.
e. current cell) contributions from the
1752 * -
A copy data: a struct that contains all the local assembly
1753 * contributions, in this case <code>CopyData<dim>()</code>.
1754 * -
A copy data routine: in this case it is
1755 * <code>copy_local_to_global()</code> in charge of actually coping these
1756 * local contributions into the global objects (matrices and/or vectors)
1760 * Most of the following lines are spent in the definition of the worker
1761 * <code>local_assemble_system()</code> and the
copy data routine
1762 * <code>copy_local_to_global()</code>. There is not much to say about the
1763 *
WorkStream framework since the vast majority of ideas are reasonably
1764 * well-documented in @ref step_9
"step-9", @ref step_13
"step-13" and @ref step_32
"step-32" among others.
1768 * Finally, assuming that @f$\mathbf{x}_i@f$ is a support
point at the boundary,
1769 * the (nodal) normals are defined as:
1774 * \widehat{\boldsymbol{\nu}}_i \dealcoloneq
1775 * \frac{\int_{\partial\Omega} \phi_i \widehat{\boldsymbol{\nu}} \,
1776 * \, \mathrm{
d}\mathbf{s}}{\big|\int_{\partial\Omega} \phi_i
1777 * \widehat{\boldsymbol{\nu}} \, \mathrm{
d}\mathbf{s}\big|}
1782 * We will compute the numerator of
this expression
first and store it in
1783 * <code>OfflineData<dim>::BoundaryNormalMap</code>. We will normalize these
1784 * vectors in a posterior
loop.
1790 *
template <
int dim>
1793 * lumped_mass_matrix = 0.;
1795 *
for (
auto &
matrix : cij_matrix)
1797 *
for (
auto &
matrix : nij_matrix)
1800 *
unsigned int dofs_per_cell =
1801 * discretization->finite_element.n_dofs_per_cell();
1802 *
unsigned int n_q_points = discretization->quadrature.size();
1806 * What follows is the initialization of the scratch data required by
1814 * discretization->mapping,
1815 * discretization->finite_element,
1816 * discretization->quadrature,
1819 * discretization->face_quadrature,
1825 *
"offline_data - assemble lumped mass matrix, and c_ij");
1827 *
const auto local_assemble_system =
1830 * CopyData<dim> &
copy) {
1831 *
copy.is_artificial = cell->is_artificial();
1832 *
if (
copy.is_artificial)
1835 *
copy.local_boundary_normal_map.clear();
1836 *
copy.cell_lumped_mass_matrix.reinit(dofs_per_cell, dofs_per_cell);
1838 *
matrix.reinit(dofs_per_cell, dofs_per_cell);
1840 *
const auto &fe_values = scratch.
reinit(cell);
1842 *
copy.local_dof_indices.resize(dofs_per_cell);
1843 * cell->get_dof_indices(
copy.local_dof_indices);
1846 *
copy.local_dof_indices.end(),
1847 *
copy.local_dof_indices.begin(),
1849 * return partitioner->global_to_local(index);
1854 * We compute the local contributions
for the lumped mass
matrix
1855 * entries @f$m_i@f$ and and vectors @f$c_{ij}@f$ in the usual fashion:
1858 *
for (
unsigned int q_point = 0; q_point < n_q_points; ++q_point)
1860 *
const auto JxW = fe_values.JxW(q_point);
1862 *
for (
unsigned int j = 0; j < dofs_per_cell; ++j)
1864 *
const auto value_JxW =
1865 * fe_values.shape_value(j, q_point) * JxW;
1866 *
const auto grad_JxW = fe_values.shape_grad(j, q_point) * JxW;
1868 *
copy.cell_lumped_mass_matrix(j, j) += value_JxW;
1870 *
for (
unsigned int i = 0; i < dofs_per_cell; ++i)
1872 *
const auto value = fe_values.shape_value(i, q_point);
1873 *
for (
unsigned int d = 0;
d < dim; ++
d)
1874 *
copy.cell_cij_matrix[
d](i, j) +=
value * grad_JxW[
d];
1882 * Now we have to compute the boundary normals. Note that the
1883 * following
loop does not
do much unless the element has faces on
1884 * the boundary of the domain.
1887 *
for (
const auto f : cell->face_indices())
1889 *
const auto face = cell->face(f);
1890 *
const auto id = face->boundary_id();
1892 *
if (!face->at_boundary())
1895 *
const auto &fe_face_values = scratch.
reinit(cell, f);
1897 *
const unsigned int n_face_q_points =
1898 * fe_face_values.get_quadrature().size();
1900 *
for (
unsigned int j = 0; j < dofs_per_cell; ++j)
1902 *
if (!discretization->finite_element.has_support_on_face(j, f))
1907 * Note that
"normal" will only represent the contributions
1908 * from
one of the faces in the support of the shape
1909 * function phi_j. So we cannot normalize
this local
1910 * contribution right here, we have to take it
"as is",
1911 * store it and pass it to the
copy data routine. The
1912 * proper normalization
requires an additional
loop on
1913 * nodes. This is done in the
copy function below.
1917 *
if (
id == Boundaries::free_slip)
1919 *
for (
unsigned int q = 0; q < n_face_q_points; ++q)
1920 * normal += fe_face_values.normal_vector(q) *
1921 * fe_face_values.shape_value(j, q);
1924 *
const auto index =
copy.local_dof_indices[j];
1927 *
for (
const auto v : cell->vertex_indices())
1928 *
if (cell->vertex_dof_index(v, 0) ==
1929 * partitioner->local_to_global(index))
1931 * position = cell->vertex(v);
1935 *
const auto old_id =
1936 * std::get<1>(
copy.local_boundary_normal_map[index]);
1937 *
copy.local_boundary_normal_map[index] =
1938 * std::make_tuple(normal,
std::max(old_id,
id), position);
1945 * Last, we provide a copy_local_to_global function as required
for
1949 *
const auto copy_local_to_global = [&](
const CopyData<dim> &
copy) {
1950 *
if (
copy.is_artificial)
1953 *
for (
const auto &it :
copy.local_boundary_normal_map)
1955 * std::get<0>(boundary_normal_map[it.first]) +=
1956 * std::get<0>(it.second);
1957 * std::get<1>(boundary_normal_map[it.first]) =
1958 *
std::max(std::get<1>(boundary_normal_map[it.first]),
1959 * std::get<1>(it.second));
1960 * std::get<2>(boundary_normal_map[it.first]) = std::get<2>(it.second);
1963 * lumped_mass_matrix.add(
copy.local_dof_indices,
1964 *
copy.cell_lumped_mass_matrix);
1966 *
for (
int k = 0; k < dim; ++k)
1968 * cij_matrix[k].add(
copy.local_dof_indices,
copy.cell_cij_matrix[k]);
1969 * nij_matrix[k].add(
copy.local_dof_indices,
copy.cell_cij_matrix[k]);
1974 * dof_handler.end(),
1975 * local_assemble_system,
1976 * copy_local_to_global,
1983 * At
this point in time we are done with the computation of @f$m_i@f$ and
1984 * @f$\mathbf{c}_{ij}@f$, but so far the matrix <code>nij_matrix</code>
1985 * contains just a
copy of the matrix <code>cij_matrix</code>.
1986 * That
's not what we really want: we have to normalize its entries. In
1987 * addition, we have not filled the entries of the matrix
1988 * <code>norm_matrix</code> and the vectors stored in the map
1989 * <code>OfflineData<dim>::BoundaryNormalMap</code> are not normalized.
1993 * In principle, this is just offline data, it doesn't make much sense
1994 * to over-optimize their computation, since their cost will get amortized
1995 * over the many time steps that we are going to use. However,
1996 * computing/storing the entries of the
matrix
1997 * <code>norm_matrix</code> and the normalization of <code>nij_matrix</code>
1998 * are perfect to illustrate thread-
parallel node-loops:
1999 * - we want to visit every node @f$i@f$ in the mesh/sparsity graph,
2000 * - and
for every such node we want to visit to every @f$j@f$ such that
2001 * @f$\mathbf{c}_{ij} \not \equiv 0@f$.
2005 * From an algebraic
point of view,
this is equivalent to: visiting
2006 * every row in the
matrix and
for each one of these rows execute a
loop on
2007 * the columns. Node-loops is a core theme of
this tutorial step (see
2008 * the pseudo-code in the introduction) that will repeat over and over
2009 * again. That
's why this is the right time to introduce them.
2013 * We have the thread parallelization capability
2014 * parallel::apply_to_subranges() that is somehow more general than the
2015 * WorkStream framework. In particular, parallel::apply_to_subranges() can
2016 * be used for our node-loops. This functionality requires four input
2017 * arguments which we explain in detail (for the specific case of our
2018 * thread-parallel node loops):
2019 * - The iterator <code>indices.begin()</code> points to a row index.
2020 * - The iterator <code>indices.end()</code> points to a numerically higher
2022 * - The function <code>on_subranges(i1,i2)</code> (where <code>i1</code>
2023 * and <code>i2</code> define a sub-range within the range spanned by
2024 * the end and begin iterators defined in the two previous bullets)
2025 * applies an operation to every iterator in such subrange. We may as
2026 * well call <code>on_subranges</code> the "worker".
2027 * - Grainsize: minimum number of iterators (in this case representing
2028 * rows) processed by each thread. We decided for a minimum of 4096
2033 * A minor caveat here is that the iterators <code>indices.begin()</code>
2034 * and <code>indices.end()</code> supplied to
2035 * parallel::apply_to_subranges() have to be random access iterators:
2036 * internally, parallel::apply_to_subranges() will break the range
2037 * defined by the <code>indices.begin()</code> and
2038 * <code>indices.end()</code> iterators into subranges (we want to be
2039 * able to read any entry in those subranges with constant complexity).
2040 * In order to provide such iterators we resort to
2041 * std_cxx20::ranges::iota_view.
2045 * The bulk of the following piece of code is spent defining
2046 * the "worker" <code>on_subranges</code>: i.e. the operation applied at
2047 * each row of the sub-range. Given a fixed <code>row_index</code>
2048 * we want to visit every column/entry in such row. In order to execute
2049 * such columns-loops we use
2050 * <a href="http://www.cplusplus.com/reference/algorithm/for_each/">
2052 * from the standard library, where:
2053 * - <code>sparsity_pattern.begin(row_index)</code>
2054 * gives us an iterator starting at the first column of the row,
2055 * - <code>sparsity_pattern.end(row_index)</code> is an iterator pointing
2056 * at the last column of the row,
2057 * - the last argument required by `std::for_each` is the operation
2058 * applied at each nonzero entry (a lambda expression in this case)
2063 * We note that, parallel::apply_to_subranges() will operate on
2064 * disjoint sets of rows (the subranges) and our goal is to write into
2065 * these rows. Because of the simple nature of the operations we want
2066 * to carry out (computation and storage of normals, and normalization
2067 * of the @f$\mathbf{c}_{ij}@f$ of entries) threads cannot conflict
2068 * attempting to write the same entry (we do not need a scheduler).
2072 * TimerOutput::Scope scope(computing_timer,
2073 * "offline_data - compute |c_ij|, and n_ij");
2075 * const std_cxx20::ranges::iota_view<unsigned int, unsigned int> indices(
2076 * 0, n_locally_relevant);
2078 * const auto on_subranges =
2079 * [&](const auto i1, const auto i2) {
2080 * for (const auto row_index :
2081 * std_cxx20::ranges::iota_view<unsigned int, unsigned int>(*i1,
2086 * First column-loop: we compute and store the entries of the
2087 * matrix norm_matrix and write normalized entries into the
2088 * matrix nij_matrix:
2092 * sparsity_pattern.begin(row_index),
2093 * sparsity_pattern.end(row_index),
2094 * [&](const ::SparsityPatternIterators::Accessor &jt) {
2095 * const auto c_ij = gather_get_entry(cij_matrix, &jt);
2096 * const double norm = c_ij.norm();
2098 * set_entry(norm_matrix, &jt, norm);
2099 * for (unsigned int j = 0; j < dim; ++j)
2100 * set_entry(nij_matrix[j], &jt, c_ij[j] / norm);
2105 * parallel::apply_to_subranges(indices.begin(),
2112 * Finally, we normalize the vectors stored in
2113 * <code>OfflineData<dim>::BoundaryNormalMap</code>. This operation has
2114 * not been thread parallelized as it would neither illustrate any
2115 * important concept nor lead to any noticeable speed gain.
2118 * for (auto &it : boundary_normal_map)
2120 * auto &normal = std::get<0>(it.second);
2121 * normal /= (normal.norm() + std::numeric_limits<double>::epsilon());
2128 * At this point we are very much done with anything related to offline data.
2133 * <a name="EquationofstateandapproximateRiemannsolver"></a>
2134 * <h4>Equation of state and approximate Riemann solver</h4>
2138 * In this section we describe the implementation of the class members of
2139 * the <code>ProblemDescription</code> class. Most of the code here is
2140 * specific to the compressible Euler's equations with an ideal gas law.
2141 * If we wanted to re-purpose @ref step_69
"step-69" for a different conservation law
2142 * (say
for: instance the shallow water equation) most of the
2143 * implementation of
this class would have to change. But most of the
2144 * other classes (in particular those defining
loop structures) would
2149 * We start by implementing a number of small member
functions for
2150 * computing <code>momentum</code>, <code>internal_energy</code>,
2151 * <code>pressure</code>, <code>speed_of_sound</code>, and the flux
2152 * <code>f</code> of the system. The functionality of each
one of these
2153 *
functions is self-explanatory from their names.
2159 *
template <
int dim>
2161 * ProblemDescription<dim>::momentum(
const state_type &
U)
2164 * std::copy_n(&
U[1], dim, &result[0]);
2168 *
template <
int dim>
2170 * ProblemDescription<dim>::internal_energy(
const state_type &
U)
2172 *
const double &rho =
U[0];
2173 *
const auto m = momentum(
U);
2174 *
const double &
E =
U[dim + 1];
2175 *
return E - 0.5 * m.norm_square() / rho;
2178 *
template <
int dim>
2180 * ProblemDescription<dim>::pressure(
const state_type &
U)
2182 *
return (
gamma - 1.) * internal_energy(
U);
2185 *
template <
int dim>
2187 * ProblemDescription<dim>::speed_of_sound(
const state_type &
U)
2189 *
const double &rho =
U[0];
2190 *
const double p = pressure(
U);
2195 *
template <
int dim>
2197 * ProblemDescription<dim>::flux(
const state_type &
U)
2199 *
const double &rho =
U[0];
2200 *
const auto m = momentum(
U);
2201 *
const auto p = pressure(
U);
2202 *
const double &
E =
U[dim + 1];
2207 *
for (
unsigned int i = 0; i < dim; ++i)
2209 * result[1 + i] = m * m[i] / rho;
2210 * result[1 + i][i] += p;
2212 * result[dim + 1] = m / rho * (
E + p);
2219 * Now we discuss the computation of @f$\lambda_{\text{
max}}
2220 * (\mathbf{
U}_i^{n},\mathbf{
U}_j^{n}, \textbf{n}_{ij})@f$. The analysis
2221 * and derivation of sharp upper-bounds of maximum wavespeeds of Riemann
2222 * problems is a very technical endeavor and we cannot include an
2223 * advanced discussion about it in
this tutorial. In
this portion of the
2224 * documentation we will limit ourselves to sketch the main functionality
2226 * references in order to help the (interested) reader
trace the
2227 * source (and proper mathematical justification) of these ideas.
2231 * In
general, obtaining a sharp guaranteed upper-bound on the maximum
2232 * wavespeed
requires solving a quite expensive scalar nonlinear problem.
2233 * This is typically done with an iterative solver. In order to simplify
2234 * the presentation in
this example step we decided not to include such
2235 * an iterative scheme. Instead, we will just use an
initial guess as a
2236 * guess
for an upper bound on the maximum wavespeed. More precisely,
2237 * equations (2.11) (3.7), (3.8) and (4.3) of @cite GuermondPopov2016b
2238 * are enough to define a guaranteed upper bound on the maximum
2239 * wavespeed. This estimate is returned by a
call to the function
2240 * <code>lambda_max_two_rarefaction()</code>. At its core the
2241 * construction of such an upper bound uses the so-called two-rarefaction
2242 * approximation for the intermediate pressure @f$p^*@f$, see for instance
2243 * Equation (4.46), page 128 in @cite Toro2009.
2247 * The estimate returned by <code>lambda_max_two_rarefaction()</code> is
2248 * guaranteed to be an upper bound, it is in
general quite sharp, and
2249 * overall sufficient for our purposes. However, for some specific
2250 * situations (in particular when
one of states is close to vacuum
2251 * conditions) such an estimate will be overly pessimistic. That's why we
2252 * used a
second estimate to avoid this degeneracy that will be invoked
2253 * by a
call to the function <code>lambda_max_expansion()</code>. The most
2254 * important function here is <code>compute_lambda_max()</code> which
2255 * takes the minimum between the estimates returned by
2256 * <code>lambda_max_two_rarefaction()</code> and
2257 * <code>lambda_max_expansion()</code>.
2261 * We start again by defining a couple of helper
functions:
2265 * The
first function takes a state <code>
U</code> and a unit vector
2266 * <code>n_ij</code> and computes the <i>projected</i> 1D state in
2267 * direction of the unit vector.
2272 *
template <
int dim>
2274 *
const typename ProblemDescription<dim>::state_type
U,
2278 * projected_U[0] =
U[0];
2282 * For
this, we have to change the momentum to @f$\textbf{m}\cdot
2283 * n_{ij}@f$ and have to subtract the kinetic energy of the
2284 * perpendicular part from the total energy:
2287 *
const auto m = ProblemDescription<dim>::momentum(
U);
2288 * projected_U[1] = n_ij * m;
2290 *
const auto perpendicular_m = m - projected_U[1] * n_ij;
2291 * projected_U[2] =
U[1 + dim] - 0.5 * perpendicular_m.norm_square() /
U[0];
2295 * We
return the 1D state in <i>primitive</i> variables instead of
2296 * conserved quantities. The
return array consists of density @f$\rho@f$,
2297 * velocity @f$u@f$, pressure @f$p@f$ and local speed of sound @f$a@f$:
2303 *
return {{projected_U[0],
2304 * projected_U[1] / projected_U[0],
2305 * ProblemDescription<1>::pressure(projected_U),
2306 * ProblemDescription<1>::speed_of_sound(projected_U)}};
2311 * At
this point we also define two small
functions that
return the
2312 * positive and negative part of a
double.
2331 * Next, we need two local wavenumbers that are defined in terms of a
2332 * primitive state @f$[\rho, u, p, a]@f$ and a given pressure @f$p^\ast@f$
2333 * @cite GuermondPopov2016 Eqn. (3.7):
2336 * \left(\frac{p^\ast-p}{p}\right)_+}
2338 * Here, the @f$(\cdot)_{+}@f$ denotes the positive part of the given
2346 * lambda1_minus(
const std::array<double, 4> &riemann_data,
2347 *
const double p_star)
2352 *
const auto u = riemann_data[1];
2353 *
const auto p = riemann_data[2];
2354 *
const auto a = riemann_data[3];
2356 *
const double factor = (
gamma + 1.0) / 2.0 /
gamma;
2357 *
const double tmp = positive_part((p_star - p) / p);
2358 *
return u - a *
std::sqrt(1.0 + factor * tmp);
2363 * Analougously @cite GuermondPopov2016 Eqn. (3.8):
2366 * \left(\frac{p^\ast-p}{p}\right)_+}
2374 * lambda3_plus(
const std::array<double, 4> &riemann_data,
const double p_star)
2379 *
const auto u = riemann_data[1];
2380 *
const auto p = riemann_data[2];
2381 *
const auto a = riemann_data[3];
2383 *
const double factor = (
gamma + 1.0) / 2.0 /
gamma;
2384 *
const double tmp = positive_part((p_star - p) / p);
2385 *
return u + a *
std::sqrt(1.0 + factor * tmp);
2390 * All that is left to
do is to compute the maximum of @f$\lambda^-@f$ and
2391 * @f$\lambda^+@f$ computed from the left and right primitive state
2392 * (@cite GuermondPopov2016 Eqn. (2.11)), where @f$p^\ast@f$ is given by
2393 * @cite GuermondPopov2016 Eqn (4.3):
2400 * lambda_max_two_rarefaction(const
std::array<double, 4> &riemann_data_i,
2401 * const
std::array<double, 4> &riemann_data_j)
2404 *
const auto u_i = riemann_data_i[1];
2405 *
const auto p_i = riemann_data_i[2];
2406 *
const auto a_i = riemann_data_i[3];
2407 *
const auto u_j = riemann_data_j[1];
2408 *
const auto p_j = riemann_data_j[2];
2409 *
const auto a_j = riemann_data_j[3];
2411 *
const double numerator = a_i + a_j - (
gamma - 1.) / 2. * (u_j - u_i);
2413 *
const double denominator =
2418 *
const double p_star =
2421 *
const double lambda1 = lambda1_minus(riemann_data_i, p_star);
2422 *
const double lambda3 = lambda3_plus(riemann_data_j, p_star);
2426 *
return std::max(positive_part(lambda3), negative_part(lambda1));
2431 * We compute the
second upper bound of the maximal wavespeed that is, in
2432 *
general, not as sharp as the two-rarefaction estimate. But it will
2433 * save the day in the context of near vacuum conditions when the
2434 * two-rarefaction approximation might attain extreme
values:
2436 * \lambda_{\text{
exp}} =
\max(u_i,u_j) + 5.
\max(a_i, a_j).
2438 * @note The constant 5.0 multiplying the maximum of the sound speeds
2439 * is <i>neither</i> an ad-hoc constant, <i>nor</i> a tuning parameter.
2440 * It defines an upper bound
for any @f$\gamma \in [0,5/3]@f$. Do not play
2448 * lambda_max_expansion(
const std::array<double, 4> &riemann_data_i,
2449 *
const std::array<double, 4> &riemann_data_j)
2451 *
const auto u_i = riemann_data_i[1];
2452 *
const auto a_i = riemann_data_i[3];
2453 *
const auto u_j = riemann_data_j[1];
2454 *
const auto a_j = riemann_data_j[3];
2462 * The following is the main function that we are going to
call in order to
2463 * compute @f$\lambda_{\text{
max}} (\mathbf{
U}_i^{n},\mathbf{
U}_j^{n},
2464 * \textbf{n}_{ij})@f$. We simply compute both maximal wavespeed estimates
2465 * and
return the minimum.
2471 * template <int dim>
2473 * ProblemDescription<dim>::compute_lambda_max(
const state_type & U_i,
2474 *
const state_type & U_j,
2477 *
const auto riemann_data_i = riemann_data_from_state(U_i, n_ij);
2478 *
const auto riemann_data_j = riemann_data_from_state(U_j, n_ij);
2480 *
const double lambda_1 =
2481 * lambda_max_two_rarefaction(riemann_data_i, riemann_data_j);
2483 *
const double lambda_2 =
2484 * lambda_max_expansion(riemann_data_i, riemann_data_j);
2486 *
return std::min(lambda_1, lambda_2);
2491 * We conclude
this section by defining
static arrays
2492 * <code>component_names</code> that contain strings describing the
2493 * component names of our state vector. We have
template specializations
2494 *
for dimensions
one, two and three, that are used later in
DataOut for
2495 * naming the corresponding components:
2502 *
const std::array<std::string, 3> ProblemDescription<1>::component_names{
2503 * {
"rho",
"m",
"E"}};
2506 *
const std::array<std::string, 4> ProblemDescription<2>::component_names{
2507 * {
"rho",
"m_1",
"m_2",
"E"}};
2510 *
const std::array<std::string, 5> ProblemDescription<3>::component_names{
2511 * {
"rho",
"m_1",
"m_2",
"m_3",
"E"}};
2516 * <a name=
"Initialvalues"></a>
2517 * <h4>Initial
values</h4>
2521 * The last preparatory step, before we discuss the implementation of the
2522 * forward Euler scheme, is to briefly implement the `InitialValues`
class.
2526 * In the constructor we initialize all parameters with
default values,
2528 * <code>parse_parameters_call_back</code> slot to the respective signal.
2532 * The <code>parse_parameters_call_back</code> slot will be invoked from
2534 * that regard, its use is appropriate
for situations where the
2535 * parameters have to be postprocessed (in some sense) or some
2536 * consistency condition between the parameters has to be checked.
2542 *
template <
int dim>
2543 * InitialValues<dim>::InitialValues(
const std::string &subsection)
2549 * std::bind(&InitialValues<dim>::parse_parameters_callback,
this));
2551 * initial_direction[0] = 1.;
2552 * add_parameter(
"initial direction",
2553 * initial_direction,
2554 *
"Initial direction of the uniform flow field");
2557 * initial_1d_state[1] = 3.;
2558 * initial_1d_state[2] = 1.;
2559 * add_parameter(
"initial 1d state",
2561 *
"Initial 1d state (rho, u, p) of the uniform flow field");
2566 * So far the constructor of <code>InitialValues</code> has defined
2567 *
default values for the two
private members
2568 * <code>initial_direction</code> and <code>initial_1d_state</code> and
2569 * added them to the parameter list. But we have not defined an
2570 * implementation of the only
public member that we really care about,
2571 * which is <code>initial_state()</code> (the function that we are going to
2572 *
call to actually evaluate the
initial solution at the mesh nodes). At
2573 * the top of the function, we have to ensure that the provided
initial
2574 * direction is not the
zero vector.
2578 * @note As commented, we could have avoided
using the method
2579 * <code>parse_parameters_call_back </code> and defined a
class member
2580 * <code>setup()</code> in order to define the implementation of
2581 * <code>initial_state()</code>. But
for illustrative purposes we want to
2582 * document a different way here and use the
call back signal from
2589 *
template <
int dim>
2590 *
void InitialValues<dim>::parse_parameters_callback()
2594 *
"Initial shock front direction is set to the zero vector."));
2595 * initial_direction /= initial_direction.norm();
2599 * Next, we implement the <code>initial_state</code> function
object
2600 * with a
lambda function computing a uniform flow field. For
this we
2601 * have to translate a given primitive 1
d state (density @f$\rho@f$,
2602 * velocity @f$u@f$, and pressure @f$p@f$) into a conserved n-dimensional state
2603 * (density @f$\rho@f$, momentum @f$\mathbf{m}@f$, and total energy @f$E@f$).
2609 * initial_state = [
this](
const Point<dim> & ,
double ) {
2610 *
const double rho = initial_1d_state[0];
2611 *
const double u = initial_1d_state[1];
2612 *
const double p = initial_1d_state[2];
2618 *
for (
unsigned int i = 0; i < dim; ++i)
2619 * state[1 + i] = rho * u * initial_direction[i];
2621 * state[dim + 1] = p / (
gamma - 1.) + 0.5 * rho * u * u;
2630 * <a name=
"TheForwardEulerstep"></a>
2631 * <h4>The Forward Euler step</h4>
2635 * The constructor of the <code>%
TimeStepping</code>
class does not contain
2636 * any surprising code:
2642 *
template <
int dim>
2646 *
const OfflineData<dim> & offline_data,
2647 *
const InitialValues<dim> &initial_values,
2648 *
const std::string & subsection )
2650 * , mpi_communicator(mpi_communicator)
2651 * , computing_timer(computing_timer)
2652 * , offline_data(&offline_data)
2653 * , initial_values(&initial_values)
2655 * cfl_update = 0.80;
2656 * add_parameter(
"cfl update",
2658 *
"Relative CFL constant used for update");
2663 * In the
class member <code>prepare()</code> we initialize the temporary
2664 * vector <code>temp</code> and the matrix <code>dij_matrix</code>. The
2665 * vector <code>temp</code> will be used to store the solution update
2666 * temporarily before its contents is swapped with the old vector.
2672 *
template <
int dim>
2673 *
void TimeStepping<dim>::prepare()
2676 *
"time_stepping - prepare scratch space");
2678 *
for (
auto &it : temporary_vector)
2679 * it.reinit(offline_data->partitioner);
2681 * dij_matrix.reinit(offline_data->sparsity_pattern);
2686 * It is now time to implement the forward Euler step. Given a (writable
2687 * reference) to the old state <code>
U</code> at time @f$t@f$ we update the
2688 * state <code>
U</code> in place and
return the chosen time-step size. We
2689 *
first declare a number of read-only references to various different
2690 * variables and data structures. We
do this is mainly to have shorter
2691 * variable names (
e.g., <code>sparsity</code> instead of
2692 * <code>offline_data->sparsity_pattern</code>).
2698 *
template <
int dim>
2699 *
double TimeStepping<dim>::make_one_step(vector_type &
U,
double t)
2701 *
const auto &n_locally_owned = offline_data->n_locally_owned;
2702 *
const auto &n_locally_relevant = offline_data->n_locally_relevant;
2705 * indices_owned(0, n_locally_owned);
2707 * indices_relevant(0, n_locally_relevant);
2709 *
const auto &sparsity = offline_data->sparsity_pattern;
2711 *
const auto &lumped_mass_matrix = offline_data->lumped_mass_matrix;
2712 *
const auto &norm_matrix = offline_data->norm_matrix;
2713 *
const auto &nij_matrix = offline_data->nij_matrix;
2714 *
const auto &cij_matrix = offline_data->cij_matrix;
2716 *
const auto &boundary_normal_map = offline_data->boundary_normal_map;
2720 * <
b>Step 1</
b>: Computing the @f$d_{ij}@f$ graph viscosity
matrix.
2724 * It is important to highlight that the viscosity
matrix has to be
2725 *
symmetric, i.e., @f$d_{ij} = d_{ji}@f$. In
this regard we note here that
2726 * @f$\int_{\Omega} \nabla \phi_j \phi_i \, \mathrm{
d}\mathbf{x}= -
2727 * \int_{\Omega} \nabla \phi_i \phi_j \, \mathrm{
d}\mathbf{x}@f$ (or
2728 * equivalently @f$\mathbf{c}_{ij} = - \mathbf{c}_{ji}@f$) provided either
2729 * @f$\mathbf{x}_i@f$ or @f$\mathbf{x}_j@f$ is a support
point located away
2730 * from the boundary. In
this case we can check that
2731 * @f$\lambda_{\text{
max}} (\mathbf{
U}_i^{n}, \mathbf{
U}_j^{n},
2732 * \textbf{n}_{ij}) = \lambda_{\text{
max}} (\mathbf{
U}_j^{n},
2733 * \mathbf{
U}_i^{n},\textbf{n}_{ji})@f$ by construction, which guarantees
2734 * the property @f$d_{ij} = d_{ji}@f$.
2738 * However,
if both support points @f$\mathbf{x}_i@f$ or @f$\mathbf{x}_j@f$
2739 * happen to lie on the boundary, then, the equalities @f$\mathbf{c}_{ij} =
2740 * - \mathbf{c}_{ji}@f$ and @f$\lambda_{\text{
max}} (\mathbf{
U}_i^{n},
2741 * \mathbf{
U}_j^{n}, \textbf{n}_{ij}) = \lambda_{\text{
max}}
2742 * (\mathbf{
U}_j^{n}, \mathbf{
U}_i^{n}, \textbf{n}_{ji})@f$
do not
2743 * necessarily hold
true. The only mathematically safe solution
for this
2744 * dilemma is to compute both of them @f$d_{ij}@f$ and @f$d_{ji}@f$ and
2749 * Overall, the computation of @f$d_{ij}@f$ is quite expensive. In
2750 * order to save some computing time we exploit the fact that the viscosity
2752 * the upper-triangular entries of @f$d_{ij}@f$ and
copy the
2753 * corresponding entries to the lower-triangular counterpart.
2758 * loops. Pretty much all the ideas for
parallel traversal that we
2759 * introduced when discussing the assembly of the
matrix
2760 * <code>norm_matrix</code> and the normalization of
2761 * <code>nij_matrix</code> above are used here again.
2765 * We define again a
"worker" function <code>on_subranges</code> that
2766 * computes the viscosity @f$d_{ij}@f$ for a subrange [i1, i2) of column
2772 *
"time_stepping - 1 compute d_ij");
2774 *
const auto on_subranges =
2775 * [&](
const auto i1,
const auto i2) {
2776 *
for (
const auto i :
2780 *
const auto U_i =
gather(
U, i);
2784 * For a given column index i we iterate over the columns of the
2785 * sparsity pattern from <code>sparsity.begin(i)</code> to
2786 * <code>sparsity.end(i)</code>:
2789 *
for (
auto jt = sparsity.begin(i); jt != sparsity.end(i); ++jt)
2791 *
const auto j = jt->column();
2795 * We only compute @f$d_{ij}@f$
if @f$j < i@f$ (upper triangular
2796 * entries) and later
copy the
values over to @f$d_{ji}@f$.
2802 *
const auto U_j =
gather(
U, j);
2804 *
const auto n_ij = gather_get_entry(nij_matrix, jt);
2805 *
const double norm = get_entry(norm_matrix, jt);
2807 *
const auto lambda_max =
2808 * ProblemDescription<dim>::compute_lambda_max(U_i, U_j, n_ij);
2810 *
double d =
norm * lambda_max;
2814 * If both support points happen to be at the boundary we
2815 * have to compute @f$d_{ji}@f$ as well and then take
2816 * @
f$\max(d_{ij},d_{ji})@f$. After
this we can
finally set the
2817 * upper triangular and lower triangular entries.
2820 * if (boundary_normal_map.count(i) != 0 &&
2821 * boundary_normal_map.count(j) != 0)
2823 *
const auto n_ji =
gather(nij_matrix, j, i);
2824 *
const auto lambda_max_2 =
2825 * ProblemDescription<dim>::compute_lambda_max(U_j,
2828 *
const double norm_2 = norm_matrix(j, i);
2833 * set_entry(dij_matrix, jt,
d);
2834 * dij_matrix(j, i) =
d;
2840 * indices_relevant.end(),
2847 * <
b>Step 2</
b>: Compute
diagonal entries @f$d_{ii}@f$ and
2848 * @f$\tau_{\text{
max}}@f$.
2853 * <code>dij_matrix</code>. We still have to fill its
diagonal entries
2854 * defined as @f$d_{ii}^n = - \sum_{j \in \mathcal{I}(i)\backslash \{i\}}
2856 * purpose. While computing the @f$d_{ii}@f$s we also determine the
2857 * largest admissible time-step, which is defined as
2859 * \tau_n \dealcoloneq c_{\text{cfl}}\,\min_{i\in\mathcal{
V}}
2860 * \left(\frac{m_i}{-2\,d_{ii}^{n}}\right) \, .
2862 * Note that the operation @f$\min_{i \in \mathcal{
V}}@f$ is intrinsically
2863 * global, it operates on all nodes:
first we have to take the minimum
2864 * over all threads (of a given node) and then we have to take the
2865 * minimum over all MPI processes. In the current implementation:
2866 * - We store <code>tau_max</code> (per node) as
2868 * href=
"http://www.cplusplus.com/reference/atomic/atomic/"><code>std::atomic<double></code></a>.
2869 * The
internal implementation of <code>std::atomic</code> will take
2870 * care of guarding any possible race condition when more than
one
2871 * thread attempts to read and/or write <code>tau_max</code> at the
2873 * - In order to take the minimum over all MPI process we use the utility
2874 * function <code>Utilities::MPI::min</code>.
2880 * std::atomic<double> tau_max{std::numeric_limits<double>::infinity()};
2884 *
"time_stepping - 2 compute d_ii, and tau_max");
2888 * on_subranges() will be executed on every thread individually. The
2889 * variable <code>tau_max_on_subrange</code> is thus stored thread
2896 * const auto on_subranges =
2897 * [&](const auto i1, const auto i2) {
2898 *
double tau_max_on_subrange = std::numeric_limits<double>::infinity();
2900 *
for (
const auto i :
2904 *
double d_sum = 0.;
2906 *
for (
auto jt = sparsity.begin(i); jt != sparsity.end(i); ++jt)
2908 *
const auto j = jt->column();
2913 * d_sum -= get_entry(dij_matrix, jt);
2918 * We store the negative
sum of the d_ij entries at the
2922 * dij_matrix.diag_element(i) = d_sum;
2925 * and compute the maximal local time-step size
2929 *
const double mass = lumped_mass_matrix.diag_element(i);
2930 *
const double tau = cfl_update * mass / (-2. * d_sum);
2931 * tau_max_on_subrange =
std::min(tau_max_on_subrange, tau);
2936 * <code>tau_max_on_subrange</code> contains the largest possible
2937 * time-step size computed
for the (thread local) subrange. At
this
2938 *
point we have to synchronize the
value over all threads. This is
2939 * were we use the <a
2940 * href=
"http://www.cplusplus.com/reference/atomic/atomic/"><code>std::atomic<double></code></a>
2941 * <i>compare exchange</i> update mechanism:
2944 *
double current_tau_max = tau_max.load();
2945 *
while (current_tau_max > tau_max_on_subrange &&
2946 * !tau_max.compare_exchange_weak(current_tau_max,
2947 * tau_max_on_subrange))
2952 * indices_relevant.end(),
2958 * After all threads have finished we can simply synchronize the
2959 *
value over all MPI processes:
2969 * This is a good
point to verify that the computed
2970 * <code>tau_max</code> is indeed a
valid floating
point number.
2974 * !std::isnan(tau_max.load()) && !std::isinf(tau_max.load()) &&
2975 * tau_max.load() > 0.,
2977 *
"I'm sorry, Dave. I'm afraid I can't do that. - We crashed."));
2982 * <
b>Step 3</
b>: Perform update.
2986 * At
this point, we have computed all viscosity coefficients @f$d_{ij}@f$
2987 * and we know the maximal admissible time-step size
2988 * @f$\tau_{\text{
max}}@f$. This means we can now compute the update:
2993 * \mathbf{
U}_i^{n+1} = \mathbf{
U}_i^{n} - \frac{\tau_{\text{
max}} }{m_i}
2994 * \sum_{j \in \mathcal{I}(i)} (\mathbb{f}(\mathbf{
U}_j^{n}) -
2995 * \mathbb{f}(\mathbf{
U}_i^{n})) \cdot \mathbf{c}_{ij} - d_{ij}
2996 * (\mathbf{
U}_j^{n} - \mathbf{
U}_i^{n})
3001 * This update formula is slightly different from what was discussed in
3002 * the introduction (in the pseudo-code). However, it can be shown that
3003 * both equations are algebraically equivalent (they will produce the
3004 * same numerical
values). We favor
this second formula since it has
3005 * natural cancellation properties that might help avoid numerical
3014 *
"time_stepping - 3 perform update");
3016 *
const auto on_subranges =
3017 * [&](
const auto i1,
const auto i2) {
3022 *
const auto U_i =
gather(
U, i);
3024 *
const auto f_i = ProblemDescription<dim>::flux(U_i);
3025 *
const double m_i = lumped_mass_matrix.diag_element(i);
3027 *
auto U_i_new = U_i;
3029 *
for (
auto jt = sparsity.begin(i); jt != sparsity.end(i); ++jt)
3031 *
const auto j = jt->column();
3033 *
const auto U_j =
gather(
U, j);
3034 *
const auto f_j = ProblemDescription<dim>::flux(U_j);
3036 *
const auto c_ij = gather_get_entry(cij_matrix, jt);
3037 *
const auto d_ij = get_entry(dij_matrix, jt);
3039 *
for (
unsigned int k = 0; k < problem_dimension; ++k)
3043 * (-(f_j[k] - f_i[k]) * c_ij + d_ij * (U_j[k] - U_i[k]));
3047 * scatter(temporary_vector, U_i_new, i);
3052 * indices_owned.end(),
3059 * <
b>Step 4</
b>: Fix up boundary states.
3063 * As a last step in the Forward Euler method, we have to fix up all
3064 * boundary states. As discussed in the intro we
3065 * -
advance in time satisfying no boundary condition at all,
3066 * - at the
end of the time step enforce boundary conditions strongly
3067 * in a post-processing step.
3071 * Here, we compute the correction
3073 * \mathbf{m}_i \dealcoloneq \mathbf{m}_i - (\boldsymbol{\nu}_i \cdot
3074 * \mathbf{m}_i) \boldsymbol{\nu}_i,
3076 * which removes the normal component of @f$\mathbf{m}@f$.
3084 *
"time_stepping - 4 fix boundary states");
3086 *
for (
auto it : boundary_normal_map)
3088 *
const auto i = it.first;
3092 * We only iterate over the locally owned subset:
3095 *
if (i >= n_locally_owned)
3098 *
const auto &normal = std::get<0>(it.second);
3099 *
const auto &
id = std::get<1>(it.second);
3100 *
const auto &position = std::get<2>(it.second);
3102 *
auto U_i =
gather(temporary_vector, i);
3106 * On
free slip boundaries we remove the normal component of the
3110 *
if (
id == Boundaries::free_slip)
3112 *
auto m = ProblemDescription<dim>::momentum(U_i);
3113 * m -= (m * normal) * normal;
3114 *
for (
unsigned int k = 0; k < dim; ++k)
3115 * U_i[k + 1] = m[k];
3120 * On Dirichlet boundaries we enforce
initial conditions
3124 *
else if (
id == Boundaries::dirichlet)
3126 * U_i = initial_values->initial_state(position, t + tau_max);
3129 * scatter(temporary_vector, U_i, i);
3135 * <
b>Step 5</
b>: We now update the ghost layer over all MPI ranks,
3136 *
swap the temporary vector with the solution vector <code>
U</code>
3137 * (that will get returned by reference) and
return the chosen
3138 * time-step size @f$\tau_{\text{
max}}@f$:
3144 *
for (
auto &it : temporary_vector)
3145 * it.update_ghost_values();
3147 *
U.swap(temporary_vector);
3155 * <a name=
"Schlierenpostprocessing"></a>
3156 * <h4>Schlieren postprocessing</h4>
3160 * At various intervals we will output the current state <code>
U</code>
3161 * of the solution together with a so-called Schlieren plot.
3162 * The constructor of the <code>SchlierenPostprocessor</code>
class again
3163 * contains no surprises. We simply supply
default values to and
register
3166 * is an ad-hoc positive amplification factor in order to enhance the
3167 * contrast in the visualization. Its actual
value is a matter of
3169 * - schlieren_index: is an integer indicating which component of the
3170 * state @f$[\rho, \mathbf{m},
E]@f$ are we going to use in order to generate
3171 * the visualization.
3177 *
template <
int dim>
3178 * SchlierenPostprocessor<dim>::SchlierenPostprocessor(
3181 *
const OfflineData<dim> &offline_data,
3182 *
const std::string & subsection )
3184 * , mpi_communicator(mpi_communicator)
3185 * , computing_timer(computing_timer)
3186 * , offline_data(&offline_data)
3188 * schlieren_beta = 10.;
3189 * add_parameter(
"schlieren beta",
3191 *
"Beta factor used in Schlieren-type postprocessor");
3193 * schlieren_index = 0;
3194 * add_parameter(
"schlieren index",
3196 *
"Use the corresponding component of the state vector for the "
3197 *
"schlieren plot");
3202 * Again, the <code>prepare()</code> function initializes two temporary
3203 * the vectors (<code>r</code> and <code>schlieren</code>).
3209 *
template <
int dim>
3210 *
void SchlierenPostprocessor<dim>::prepare()
3213 *
"schlieren_postprocessor - prepare scratch space");
3215 * r.reinit(offline_data->n_locally_relevant);
3216 * schlieren.reinit(offline_data->partitioner);
3221 * We now discuss the implementation of the
class member
3222 * <code>SchlierenPostprocessor<dim>::compute_schlieren()</code>, which
3223 * basically takes a component of the state vector <code>
U</code> and
3224 * computes the Schlieren indicator
for such component (the formula of the
3225 * Schlieren indicator can be found just before the declaration of the
class
3226 * <code>SchlierenPostprocessor</code>). We start by noting
3227 * that
this formula
requires the
"nodal gradients" @f$\nabla r_j@f$.
3228 * However, nodal
values of
gradients are not defined
for @f$\mathcal{
C}^0@f$
3231 * simplest technique we can use to recover
gradients at nodes is
3232 * weighted-averaging i.e.
3236 * \f[ \nabla r_j \dealcoloneq \frac{1}{\int_{S_i} \omega_i(\mathbf{x}) \,
3237 * \mathrm{
d}\mathbf{x}}
3238 * \int_{S_i} r_h(\mathbf{x}) \omega_i(\mathbf{x}) \, \mathrm{
d}\mathbf{x}
3239 * \ \ \ \ \ \mathbf{(*)} \f]
3243 * where @f$S_i@f$ is the support of the shape function @f$\phi_i@f$, and
3244 * @f$\omega_i(\mathbf{x})@f$ is the weight. The weight could be any
3245 * positive function such as
3246 * @f$\omega_i(\mathbf{x}) \equiv 1@f$ (that would allow us to recover the usual
3247 * notion of
mean value). But as usual, the goal is to reuse the off-line
3248 * data as much as possible. In
this sense, the most natural
3249 * choice of weight is @f$\omega_i = \phi_i@f$. Inserting
this choice of
3250 * weight and the expansion @f$r_h(\mathbf{x}) = \sum_{j \in \mathcal{
V}}
3251 * r_j \phi_j(\mathbf{x})@f$ into @f$\mathbf{(*)}@f$ we get :
3255 * \f[ \nabla r_j \dealcoloneq \frac{1}{m_i} \sum_{j \in \mathcal{I}(i)} r_j
3256 * \mathbf{c}_{ij} \ \ \ \ \ \mathbf{(**)} \, . \f]
3260 * Using
this last formula we can recover averaged nodal
gradients without
3261 * resorting to any form of quadrature. This idea aligns quite well with
3262 * the whole spirit of edge-based schemes (or algebraic schemes) where
3263 * we want to operate on matrices and vectors as directly as
3264 * it could be possible avoiding by all means assembly of bilinear
3265 * forms, cell-loops, quadrature, or any other
3266 * intermediate construct/operation between the input arguments (the state
3267 * from the previous time-step) and the actual matrices and vectors
3268 * required to compute the update.
3272 * The
second thing to note is that we have to compute global minimum and
3273 * maximum @f$\max_j |\nabla r_j|@f$ and @f$\min_j |\nabla r_j|@f$. Following the
3274 * same ideas used to compute the time step size in the
class member
3275 * <code>%
TimeStepping\<dim>::%step()</code> we define @f$\max_j |\nabla r_j|@f$
3276 * and @f$\min_j |\nabla r_j|@f$ as atomic doubles in order to resolve any
3277 * conflicts between threads. As usual, we use
3280 * among all MPI processes.
3284 * Finally, it is not possible to compute the Schlieren indicator in a single
3285 *
loop over all nodes. The entire operation
requires two loops over nodes:
3289 * - The
first loop computes @f$|\nabla r_i|@f$
for all @f$i \in \mathcal{
V}@f$ in
3290 * the mesh, and the bounds @f$\max_j |\nabla r_j|@f$ and
3291 * @f$\min_j |\nabla r_j|@f$.
3292 * - The
second loop finally computes the Schlieren indicator
using the
3297 * \f[ \text{schlieren}[i] =
e^{\beta \frac{ |\nabla r_i|
3298 * - \min_j |\nabla r_j| }{\max_j |\nabla r_j| - \min_j |\nabla r_j| } }
3303 * This means that we will have to define two workers
3304 * <code>on_subranges</code>
for each one of these stages.
3310 *
template <
int dim>
3311 *
void SchlierenPostprocessor<dim>::compute_schlieren(
const vector_type &
U)
3314 * computing_timer,
"schlieren_postprocessor - compute schlieren plot");
3316 *
const auto &sparsity = offline_data->sparsity_pattern;
3317 *
const auto &lumped_mass_matrix = offline_data->lumped_mass_matrix;
3318 *
const auto &cij_matrix = offline_data->cij_matrix;
3319 *
const auto &boundary_normal_map = offline_data->boundary_normal_map;
3320 *
const auto &n_locally_owned = offline_data->n_locally_owned;
3322 *
const auto indices =
3328 * We define the r_i_max and r_i_min in the current MPI process as
3329 * atomic doubles in order to avoid race conditions between threads:
3332 * std::atomic<double> r_i_max{0.};
3333 * std::atomic<double> r_i_min{std::numeric_limits<double>::infinity()};
3337 * First
loop: compute the averaged
gradient at each node and the
3338 * global maxima and minima of the
gradients.
3342 *
const auto on_subranges =
3343 * [&](
const auto i1,
const auto i2) {
3344 *
double r_i_max_on_subrange = 0.;
3345 *
double r_i_min_on_subrange = std::numeric_limits<double>::infinity();
3353 *
for (
auto jt = sparsity.begin(i); jt != sparsity.end(i); ++jt)
3355 *
const auto j = jt->column();
3360 *
const auto U_js =
U[schlieren_index].local_element(j);
3361 *
const auto c_ij = gather_get_entry(cij_matrix, jt);
3362 * r_i += c_ij * U_js;
3367 * We fix up the
gradient r_i at
free slip boundaries similarly to
3368 * how we fixed up boundary states in the forward Euler step.
3369 * This avoids sharp, artificial
gradients in the Schlieren
3370 * plot at
free slip boundaries and is a purely cosmetic choice.
3376 *
const auto bnm_it = boundary_normal_map.find(i);
3377 *
if (bnm_it != boundary_normal_map.end())
3379 *
const auto &normal = std::get<0>(bnm_it->second);
3380 *
const auto &
id = std::get<1>(bnm_it->second);
3382 *
if (
id == Boundaries::free_slip)
3383 * r_i -= 1. * (r_i * normal) * normal;
3390 * We remind the reader that we are not interested in the nodal
3391 *
gradients per se. We only want their norms in order to
3392 * compute the Schlieren indicator (weighted with the lumped
3393 * mass
matrix @f$m_i@f$):
3396 * const double m_i = lumped_mass_matrix.diag_element(i);
3397 * r[i] = r_i.
norm() / m_i;
3398 * r_i_max_on_subrange =
std::max(r_i_max_on_subrange, r[i]);
3399 * r_i_min_on_subrange =
std::min(r_i_min_on_subrange, r[i]);
3404 * We compare the current_r_i_max and current_r_i_min (in the
3405 * current subrange) with r_i_max and r_i_min (
for the current MPI
3406 * process) and update them
if necessary:
3412 *
double current_r_i_max = r_i_max.load();
3413 *
while (current_r_i_max < r_i_max_on_subrange &&
3414 * !r_i_max.compare_exchange_weak(current_r_i_max,
3415 * r_i_max_on_subrange))
3418 *
double current_r_i_min = r_i_min.load();
3419 *
while (current_r_i_min > r_i_min_on_subrange &&
3420 * !r_i_min.compare_exchange_weak(current_r_i_min,
3421 * r_i_min_on_subrange))
3433 * And synchronize <code>r_i_max</code> and <code>r_i_min</code> over
3434 * all MPI processes.
3445 * Second
loop: we now have the vector <code>r</code> and the scalars
3446 * <code>r_i_max</code> and <code>r_i_min</code> at our disposal. We
3447 * are thus in a position to actually compute the Schlieren indicator.
3454 *
const auto on_subranges =
3455 * [&](
const auto i1,
const auto i2) {
3460 * schlieren.local_element(i) =
3461 * 1. -
std::exp(-schlieren_beta * (r[i] - r_i_min) /
3462 * (r_i_max - r_i_min));
3474 * And
finally, exchange ghost elements.
3477 * schlieren.update_ghost_values();
3483 * <a name=
"Themainloop"></a>
3484 * <h4>The main
loop</h4>
3488 * With all classes implemented it is time to create an instance of
3489 * <code>Discretization<dim></code>, <code>OfflineData<dim></code>,
3490 * <code>InitialValues<dim></code>, <code>%
TimeStepping\<dim></code>, and
3491 * <code>SchlierenPostprocessor<dim></code>, and
run the forward Euler
3496 * In the constructor of <code>MainLoop<dim></code> we now initialize an
3497 * instance of all classes, and declare a number of parameters
3498 * controlling output. Most notable, we declare a
boolean parameter
3499 * <code>resume</code> that will control whether the program attempts to
3500 * restart from an interrupted computation, or not.
3506 *
template <
int dim>
3507 * MainLoop<dim>::MainLoop(
const MPI_Comm mpi_communicator)
3509 * , mpi_communicator(mpi_communicator)
3510 * , computing_timer(mpi_communicator,
3515 * , discretization(mpi_communicator, computing_timer,
"B - Discretization")
3516 * , offline_data(mpi_communicator,
3519 *
"C - OfflineData")
3520 * , initial_values(
"D - InitialValues")
3521 * , time_stepping(mpi_communicator,
3525 *
"E - TimeStepping")
3526 * , schlieren_postprocessor(mpi_communicator,
3529 *
"F - SchlierenPostprocessor")
3531 * base_name =
"test";
3532 * add_parameter(
"basename", base_name,
"Base name for all output files");
3535 * add_parameter(
"final time", t_final,
"Final time");
3537 * output_granularity = 0.02;
3538 * add_parameter(
"output granularity",
3539 * output_granularity,
3540 *
"time interval for output");
3542 * asynchronous_writeback =
true;
3543 * add_parameter(
"asynchronous writeback",
3544 * asynchronous_writeback,
3545 *
"Write out solution in a background thread performing IO");
3548 * add_parameter(
"resume", resume,
"Resume an interrupted computation.");
3553 * We start by implementing a helper function <code>print_head()</code>
3554 * in an anonymous
namespace that is used to output messages in the
3555 * terminal with some nice formatting.
3564 *
const std::string & header,
3565 *
const std::string & secondary =
"")
3567 *
const auto header_size = header.size();
3568 *
const auto padded_header = std::string((34 - header_size) / 2,
' ') +
3570 * std::string((35 - header_size) / 2,
' ');
3572 *
const auto secondary_size = secondary.size();
3573 *
const auto padded_secondary =
3574 * std::string((34 - secondary_size) / 2,
' ') + secondary +
3575 * std::string((35 - secondary_size) / 2,
' ');
3578 * pcout << std::endl;
3579 * pcout <<
" ####################################################" << std::endl;
3580 * pcout <<
" ######### #########" << std::endl;
3581 * pcout <<
" #########" << padded_header <<
"#########" << std::endl;
3582 * pcout <<
" #########" << padded_secondary <<
"#########" << std::endl;
3583 * pcout <<
" ######### #########" << std::endl;
3584 * pcout <<
" ####################################################" << std::endl;
3585 * pcout << std::endl;
3592 * With <code>print_head</code> in place it is now time to implement the
3600 *
template <
int dim>
3605 * We start by reading in parameters and initializing all objects. We
3607 * all parameters from the parameter file (whose name is given as a
3610 * declarations
for all
class instances that are derived from
3611 * ParameterAceptor. The
call to initialize enters the subsection
for
3612 * each each derived
class, and sets all variables that were added
3619 * pcout <<
"Reading parameters and allocating objects... " << std::flush;
3622 * pcout <<
"done" << std::endl;
3627 * scratch space, and initialize the
DataOut<dim> object:
3634 * print_head(pcout,
"create triangulation");
3635 * discretization.setup();
3637 * pcout <<
"Number of active cells: "
3638 * << discretization.triangulation.n_global_active_cells()
3641 * print_head(pcout,
"compute offline data");
3642 * offline_data.setup();
3643 * offline_data.assemble();
3645 * pcout <<
"Number of degrees of freedom: "
3646 * << offline_data.dof_handler.n_dofs() << std::endl;
3648 * print_head(pcout,
"set up time step");
3649 * time_stepping.prepare();
3650 * schlieren_postprocessor.prepare();
3655 * We will store the current time and state in the variable
3656 * <code>t</code> and vector <code>
U</code>:
3663 *
unsigned int output_cycle = 0;
3665 * print_head(pcout,
"interpolate initial values");
3666 * vector_type
U = interpolate_initial_values();
3671 * <a name=
"Resume"></a>
3676 * By
default the boolean <code>resume</code> is
set to
false, i.e. the
3677 * following code snippet is not
run. However,
if
3678 * <code>resume==
true</code> we indicate that we have indeed an
3679 * interrupted computation and the program shall restart by reading in
3680 * an old state consisting of <code>t</code>,
3681 * <code>output_cycle</code>, and <code>
U</code> from a checkpoint
3682 * file. These checkpoint files will be created in the
3683 * <code>output()</code> routine discussed below.
3691 * print_head(pcout,
"restore interrupted computation");
3693 *
const unsigned int i =
3694 * discretization.triangulation.locally_owned_subdomain();
3696 *
const std::string name = base_name +
"-checkpoint-" +
3698 * std::ifstream file(name, std::ios::binary);
3702 * We use a <code>boost::archive</code> to store and read in the
3703 * contents the checkpointed state.
3709 * boost::archive::binary_iarchive ia(file);
3710 * ia >> t >> output_cycle;
3712 *
for (
auto &it1 :
U)
3716 * <code>it1</code> iterates over all components of the state
3717 * vector <code>
U</code>. We read in every entry of the
3718 * component in sequence and update the ghost layer afterwards:
3721 *
for (
auto &it2 : it1)
3723 * it1.update_ghost_values();
3729 * With either the
initial state
set up, or an interrupted state
3730 * restored it is time to enter the main
loop:
3736 * output(
U, base_name, t, output_cycle++);
3738 * print_head(pcout,
"enter main loop");
3740 *
for (
unsigned int cycle = 1; t < t_final; ++cycle)
3744 * We
first print an informative status message
3750 * std::ostringstream head;
3751 * std::ostringstream secondary;
3754 * << std::fixed << std::setprecision(1) << t / t_final * 100
3756 * secondary <<
"at time t = " << std::setprecision(8) << std::fixed << t;
3758 * print_head(pcout, head.str(), secondary.str());
3762 * and then perform a single forward Euler step. Note that the
3763 * state vector <code>
U</code> is updated in place and that
3764 * <code>time_stepping.make_one_step()</code> returns the chosen step
3771 * t += time_stepping.make_one_step(
U, t);
3775 * Post processing, generating output and writing out the current
3776 * state is a CPU and IO intensive task that we cannot afford to
do
3777 * every time step - in particular with
explicit time stepping. We
3778 * thus only schedule output by calling the <code>output()</code>
3779 * function
if we are past a threshold
set by
3780 * <code>output_granularity</code>.
3786 *
if (t > output_cycle * output_granularity)
3788 * output(
U, base_name, t, output_cycle,
true);
3795 * We wait
for any remaining background output thread to finish before
3796 * printing a summary and exiting.
3799 *
if (background_thread_state.valid())
3800 * background_thread_state.wait();
3802 * computing_timer.print_summary();
3803 * pcout << timer_output.str() << std::endl;
3808 * The <code>interpolate_initial_values</code> takes an
initial time
"t"
3809 * as input argument and populates a state vector <code>
U</code> with the
3810 * help of the <code>InitialValues<dim>::initial_state</code>
object.
3816 *
template <
int dim>
3817 *
typename MainLoop<dim>::vector_type
3818 * MainLoop<dim>::interpolate_initial_values(
const double t)
3820 * pcout <<
"MainLoop<dim>::interpolate_initial_values(t = " << t <<
")"
3823 *
"main_loop - setup scratch space");
3827 *
for (
auto &it :
U)
3828 * it.reinit(offline_data.partitioner);
3830 *
constexpr auto problem_dimension =
3831 * ProblemDescription<dim>::problem_dimension;
3835 * The function signature of
3836 * <code>InitialValues<dim>::initial_state</code> is not quite right
3838 * creating a
lambda function that
for a given position <code>x</code>
3839 * returns just the
value of the <code>i</code>th component. This
3840 *
lambda in turn is converted to a ::Function with the help of
3847 *
for (
unsigned int i = 0; i < problem_dimension; ++i)
3851 * return initial_values.initial_state(x, t)[i];
3855 *
for (
auto &it :
U)
3856 * it.update_ghost_values();
3864 * <a name=
"Outputandcheckpointing"></a>
3865 * <h5>Output and checkpointing</h5>
3869 * Writing out the
final vtk files is quite an IO intensive task that can
3870 * stall the main
loop for a
while. In order to avoid
this we use an <a
3871 * href=
"https://en.wikipedia.org/wiki/Asynchronous_I/O">asynchronous
3872 * IO</a> strategy by creating a background thread that will perform IO
3873 *
while the main
loop is allowed to
continue. In order
for this to work
3874 * we have to be mindful of two things:
3875 * - Before running the <code>output_worker</code> thread, we have to create
3876 * a
copy of the state vector <code>
U</code>. We store it in the
3877 * vector <code>output_vector</code>.
3878 * - We have to avoid any MPI communication in the background thread,
3879 * otherwise the program might deadlock. This implies that we have to
3880 *
run the postprocessing outside of the worker thread.
3886 *
template <
int dim>
3887 *
void MainLoop<dim>::output(
const typename MainLoop<dim>::vector_type &
U,
3888 *
const std::string & name,
3890 *
const unsigned int cycle,
3891 *
const bool checkpoint)
3893 * pcout <<
"MainLoop<dim>::output(t = " << t
3894 * <<
", checkpoint = " << checkpoint <<
")" << std::endl;
3898 * If the asynchronous writeback option is
set we launch a background
3899 * thread performing all the slow IO to disc. In that
case we have to
3900 * make sure that the background thread actually finished running. If
3901 * not, we have to wait to
for it to finish. We launch said background
3903 * href=
"https://en.cppreference.com/w/cpp/thread/async"><code>std::async()</code></a>
3905 * href=
"https://en.cppreference.com/w/cpp/thread/future"><code>std::future</code></a>
3906 *
object. This <code>std::future</code>
object contains the
return
3907 *
value of the function, which is in our
case simply
3908 * <code>
void</code>.
3914 *
if (background_thread_state.valid())
3917 * background_thread_state.wait();
3920 *
constexpr auto problem_dimension =
3921 * ProblemDescription<dim>::problem_dimension;
3925 * At
this point we make a
copy of the state vector,
run the schlieren
3927 * output code is standard: We create a
DataOut instance, attach all
3928 * data vectors we want to output and
call
3929 *
DataOut<dim>::build_patches(). There is
one twist, however. In order
3930 * to perform asynchronous IO on a background thread we create the
3931 *
DataOut<dim>
object as a shared pointer that we pass on to the
3932 * worker thread to ensure that once we exit this function and the
3933 * worker thread finishes the
DataOut<dim>
object gets destroyed again.
3939 * for (
unsigned int i = 0; i < problem_dimension; ++i)
3941 * output_vector[i] =
U[i];
3942 * output_vector[i].update_ghost_values();
3945 * schlieren_postprocessor.compute_schlieren(output_vector);
3947 *
auto data_out = std::make_shared<DataOut<dim>>();
3949 * data_out->attach_dof_handler(offline_data.dof_handler);
3951 *
const auto &component_names = ProblemDescription<dim>::component_names;
3953 *
for (
unsigned int i = 0; i < problem_dimension; ++i)
3954 * data_out->add_data_vector(output_vector[i], component_names[i]);
3956 * data_out->add_data_vector(schlieren_postprocessor.schlieren,
3957 *
"schlieren_plot");
3959 * data_out->build_patches(discretization.mapping,
3960 * discretization.finite_element.degree - 1);
3964 * Next we create a
lambda function
for the background thread. We <a
3965 * href=
"https://en.cppreference.com/w/cpp/language/lambda">capture</a>
3966 * the <code>
this</code> pointer as well as most of the arguments of
3967 * the output function by
value so that we have access to them inside
3971 *
const auto output_worker = [
this, name, t, cycle, checkpoint, data_out]() {
3976 * We checkpoint the current state by doing the precise inverse
3977 * operation to what we discussed
for the <a href=
"Resume">resume
3984 *
const unsigned int i =
3985 * discretization.triangulation.locally_owned_subdomain();
3986 * std::string filename =
3989 * std::ofstream file(filename, std::ios::binary | std::ios::trunc);
3991 * boost::archive::binary_oarchive oa(file);
3993 *
for (
const auto &it1 : output_vector)
3994 *
for (
const auto &it2 : it1)
4002 * data_out->set_flags(flags);
4004 * data_out->write_vtu_with_pvtu_record(
4005 *
"", name +
"-solution", cycle, mpi_communicator, 6);
4010 * If the asynchronous writeback option is
set we launch a
new
4011 * background thread with the help of
4013 * href=
"https://en.cppreference.com/w/cpp/thread/async"><code>std::async</code></a>
4014 * function. The function returns a <a
4015 * href=
"https://en.cppreference.com/w/cpp/thread/future"><code>std::future</code></a>
4016 *
object that we can use to query the status of the background thread.
4017 * At
this point we can
return from the <code>output()</code> function
4018 * and resume with the time stepping in the main
loop - the thread will
4019 *
run in the background.
4022 *
if (asynchronous_writeback)
4024 * background_thread_state = std::async(std::launch::async, output_worker);
4036 * And
finally, the main function.
4042 *
int main(
int argc,
char *argv[])
4046 *
constexpr int dim = 2;
4048 *
using namespace dealii;
4049 *
using namespace Step69;
4053 *
MPI_Comm mpi_communicator(MPI_COMM_WORLD);
4054 * MainLoop<dim> main_loop(mpi_communicator);
4058 *
catch (std::exception &exc)
4060 * std::cerr << std::endl
4062 * <<
"----------------------------------------------------"
4064 * std::cerr <<
"Exception on processing: " << std::endl
4065 * << exc.what() << std::endl
4066 * <<
"Aborting!" << std::endl
4067 * <<
"----------------------------------------------------"
4073 * std::cerr << std::endl
4075 * <<
"----------------------------------------------------"
4077 * std::cerr <<
"Unknown exception!" << std::endl
4078 * <<
"Aborting!" << std::endl
4079 * <<
"----------------------------------------------------"
4085<a name=
"Results"></a>
4086<a name=
"Results"></a><h1>Results</h1>
4089Running the program with
default parameters in release mode takes about 1
4090minute on a 4 core machine (with hyperthreading):
4092# mpirun -np 4 ./step-69 | tee output
4093Reading parameters and allocating objects... done
4095 ####################################################
4099 ####################################################
4101Number of active cells: 36864
4103 ####################################################
4105 ######### compute offline data #########
4107 ####################################################
4109Number of degrees of freedom: 37376
4111 ####################################################
4113 #########
set up time step #########
4115 ####################################################
4117 ####################################################
4122 ####################################################
4124TimeLoop<dim>::interpolate_initial_values(t = 0)
4125TimeLoop<dim>::output(t = 0, checkpoint = 0)
4127 ####################################################
4129 ######### enter main
loop #########
4132 ####################################################
4134 ####################################################
4136 ######### Cycle 000001 (0.0%) #########
4137 ######### at time t = 0.00000000 #########
4139 ####################################################
4143 ####################################################
4145 ######### Cycle 007553 (100.0%) #########
4146 ######### at time t = 3.99984036 #########
4148 ####################################################
4150TimeLoop<dim>::output(t = 4.00038, checkpoint = 1)
4152+------------------------------------------------------------------------+------------+------------+
4153| Total CPU time elapsed since start | 357s | |
4155| Section | no. calls | CPU time | % of total |
4156+------------------------------------------------------------+-----------+------------+------------+
4157| discretization - setup | 1 | 0.113s | 0% |
4158| offline_data -
assemble lumped mass
matrix, and c_ij | 1 | 0.167s | 0% |
4159| offline_data - compute |c_ij|, and n_ij | 1 | 0.00255s | 0% |
4160| offline_data - create sparsity pattern and
set up matrices | 1 | 0.0224s | 0% |
4161| offline_data - distribute dofs | 1 | 0.0617s | 0% |
4162| offline_data - fix slip boundary c_ij | 1 | 0.0329s | 0% |
4163| schlieren_postprocessor - compute schlieren plot | 201 | 0.811s | 0.23% |
4164| schlieren_postprocessor - prepare scratch space | 1 | 7.6e-05s | 0% |
4165| time_loop - setup scratch space | 1 | 0.127s | 0% |
4166| time_loop - stalled output | 200 | 0.000685s | 0% |
4167| time_step - 1 compute d_ij | 7553 | 240s | 67% |
4168| time_step - 2 compute d_ii, and tau_max | 7553 | 11.5s | 3.2% |
4169| time_step - 3 perform update | 7553 | 101s | 28% |
4170| time_step - 4 fix boundary states | 7553 | 0.724s | 0.2% |
4171| time_step - prepare scratch space | 1 | 0.00245s | 0% |
4172+------------------------------------------------------------+-----------+------------+------------+
4175One thing that becomes evident is the fact that the program spends two
4176thirds of the execution time computing the graph viscosity d_ij and about a
4177third of the execution time in performing the update, where computing the
4178flux @f$f(
U)@f$ is the expensive operation. The preset default resolution is
4179about 37k gridpoints, which amounts to about 148k spatial degrees of
4180freedom in 2D. An animated schlieren plot of the solution looks as follows:
4182<img src=
"https://www.dealii.org/images/steps/developer/step-69.coarse.gif" alt=
"" height=
"300">
4184It is evident that 37k gridpoints for the
first-order method is nowhere
4185near the resolution needed to resolve any flow features. For comparison,
4186here is a
"reference" computation with a
second-order method and about 9.5M
4187gridpoints (<a href=
"https://github.com/conservation-laws/ryujin">github
4190<img src=
"https://www.dealii.org/images/steps/developer/step-69.2nd-order.t400.jpg" alt=
"" height=
"300">
4192So, we give the
first-order method a
second chance and
run it with about
41932.4M gridpoints on a small compute server:
4196# mpirun -np 16 ./step-69 | tee output
4200 ####################################################
4202 ######### Cycle 070216 (100.0%) #########
4203 ######### at time t = 3.99999231 #########
4205 ####################################################
4207TimeLoop<dim>::output(t = 4.00006, checkpoint = 1)
4211+------------------------------------------------------------------------+------------+------------+
4212| Total wallclock time elapsed since start | 6.75e+03s | |
4214| Section | no. calls | wall time | % of total |
4215+------------------------------------------------------------+-----------+------------+------------+
4216| discretization - setup | 1 | 1.97s | 0% |
4217| offline_data -
assemble lumped mass
matrix, and c_ij | 1 | 1.19s | 0% |
4218| offline_data - compute |c_ij|, and n_ij | 1 | 0.0172s | 0% |
4219| offline_data - create sparsity pattern and
set up matrices | 1 | 0.413s | 0% |
4220| offline_data - distribute dofs | 1 | 1.05s | 0% |
4221| offline_data - fix slip boundary c_ij | 1 | 0.252s | 0% |
4222| schlieren_postprocessor - compute schlieren plot | 201 | 1.82s | 0% |
4223| schlieren_postprocessor - prepare scratch space | 1 | 0.000497s | 0% |
4224| time_loop - setup scratch space | 1 | 1.45s | 0% |
4225| time_loop - stalled output | 200 | 0.00342s | 0% |
4226| time_step - 1 compute d_ij | 70216 | 4.38e+03s | 65% |
4227| time_step - 2 compute d_ii, and tau_max | 70216 | 419s | 6.2% |
4228| time_step - 3 perform update | 70216 | 1.87e+03s | 28% |
4229| time_step - 4 fix boundary states | 70216 | 24s | 0.36% |
4230| time_step - prepare scratch space | 1 | 0.0227s | 0% |
4231+------------------------------------------------------------+-----------+------------+------------+
4234And with the following result:
4236<img src=
"https://www.dealii.org/images/steps/developer/step-69.fine.gif" alt=
"" height=
"300">
4238That
's substantially better, although of course at the price of having run
4239the code for roughly 2 hours on 16 cores.
4243<a name="extensions"></a>
4244<a name="Possibilitiesforextensions"></a><h3>Possibilities for extensions</h3>
4247The program showcased here is really only first-order accurate, as
4248discussed above. The pictures above illustrate how much diffusion that
4249introduces and how far the solution is from one that actually resolves
4250the features we care about.
4252This can be fixed, but it would exceed what a *tutorial* is about.
4253Nevertheless, it is worth showing what one can achieve by adding a
4254second-order scheme. For example, here is a video computed with <a
4255href=https://conservation-laws.43-1.org/>the following research code</a>
4256that shows (with a different color scheme) a 2d simulation that corresponds
4257to the cases shown above:
4261 <iframe width="560" height="315" src="https://www.youtube.com/embed/xIwJZlsXpZ4"
4263 allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture"
4264 allowfullscreen></iframe>
4268This simulation was done with 38 million degrees of freedom
4269(continuous @f$Q_1@f$ finite elements) per component of the solution
4270vector. The exquisite detail of the solution is remarkable for these
4271kinds of simulations, including in the sub-sonic region behind the
4274One can also with relative ease further extend this to the 3d case:
4278 <iframe width="560" height="315" src="https://www.youtube.com/embed/vBCRAF_c8m8"
4280 allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture"
4281 allowfullscreen></iframe>
4285Solving this becomes expensive, however: The simulation was done with
42861,817 million degrees of freedom (continuous @f$Q_1@f$ finite elements)
4287per component (for a total of 9.09 billion spatial degrees of freedom)
4288and ran on 30,720 MPI ranks. The code achieved an average througput of
4289969M grid points per second (0.04M gridpoints per second per CPU). The
4290front and back wall show a "Schlieren plot": the magnitude of the
4291gradient of the density on an exponential scale from white (low) to
4292black (high). All other cutplanes and the surface of the obstacle show
4293the magnitude of the vorticity on a white (low) - yellow (medium) -
4294red (high) scale. (The scales of the individual cutplanes have been
4295adjusted for a nicer visualization.)
4298<a name="PlainProg"></a>
4299<h1> The plain program</h1>
4300@include "step-69.cc"
void swap(BlockIndices &u, BlockIndices &v)
virtual void build_patches(const unsigned int n_subdivisions=0)
void reinit(const Triangulation< dim, spacedim > &tria)
static void initialize(const std::string &filename="", const std::string &output_filename="", const ParameterHandler::OutputStyle output_style_for_output_filename=ParameterHandler::Short, ParameterHandler &prm=ParameterAcceptor::prm, const ParameterHandler::OutputStyle output_style_for_filename=ParameterHandler::DefaultStyle)
boost::signals2::signal< void()> parse_parameters_call_back
void add_parameter(const std::string &entry, ParameterType ¶meter, const std::string &documentation="", ParameterHandler &prm_=prm, const Patterns::PatternBase &pattern= *Patterns::Tools::Convert< ParameterType >::to_pattern())
SymmetricTensor< rank, dim, Number > sum(const SymmetricTensor< rank, dim, Number > &local, const MPI_Comm &mpi_communicator)
constexpr Number trace(const SymmetricTensor< 2, dim, Number > &d)
numbers::NumberTraits< Number >::real_type norm() const
VectorizedArray< Number, width > max(const ::VectorizedArray< Number, width > &x, const ::VectorizedArray< Number, width > &y)
VectorizedArray< Number, width > exp(const ::VectorizedArray< Number, width > &x)
VectorizedArray< Number, width > sqrt(const ::VectorizedArray< Number, width > &x)
#define DEAL_II_ALWAYS_INLINE
@ update_values
Shape function values.
@ update_normal_vectors
Normal vectors.
@ update_JxW_values
Transformed quadrature weights.
@ update_gradients
Shape function gradients.
@ update_quadrature_points
Transformed quadrature points.
__global__ void set(Number *val, const Number s, const size_type N)
#define Assert(cond, exc)
static ::ExceptionBase & ExcInternalError()
static ::ExceptionBase & ExcMessage(std::string arg1)
#define AssertThrow(cond, exc)
typename ActiveSelector::cell_iterator cell_iterator
void loop(ITERATOR begin, typename identity< ITERATOR >::type end, DOFINFO &dinfo, INFOBOX &info, const std::function< void(DOFINFO &, typename INFOBOX::CellInfo &)> &cell_worker, const std::function< void(DOFINFO &, typename INFOBOX::CellInfo &)> &boundary_worker, const std::function< void(DOFINFO &, DOFINFO &, typename INFOBOX::CellInfo &, typename INFOBOX::CellInfo &)> &face_worker, ASSEMBLER &assembler, const LoopControl &lctrl=LoopControl())
IteratorRange< BaseIterator > make_iterator_range(const BaseIterator &begin, const typename identity< BaseIterator >::type &end)
@ valid
Iterator points to a valid object.
static const types::blas_int zero
@ matrix
Contents is actually a matrix.
@ symmetric
Matrix is symmetric.
@ diagonal
Matrix is diagonal.
@ general
No special properties.
static const types::blas_int one
double norm(const FEValuesBase< dim > &fe, const ArrayView< const std::vector< Tensor< 1, dim > > > &Du)
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
SymmetricTensor< 2, dim, Number > C(const Tensor< 2, dim, Number > &F)
SymmetricTensor< 2, dim, Number > e(const Tensor< 2, dim, Number > &F)
Tensor< 2, dim, Number > l(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
SymmetricTensor< 2, dim, Number > b(const Tensor< 2, dim, Number > &F)
SymmetricTensor< 2, dim, Number > d(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
void call(const std::function< RT()> &function, internal::return_value< RT > &ret_val)
VectorType::value_type * end(VectorType &V)
unsigned int this_mpi_process(const MPI_Comm &mpi_communicator)
T min(const T &t, const MPI_Comm &mpi_communicator)
T max(const T &t, const MPI_Comm &mpi_communicator)
std::string int_to_string(const unsigned int value, const unsigned int digits=numbers::invalid_unsigned_int)
void run(const Iterator &begin, const typename identity< Iterator >::type &end, Worker worker, Copier copier, const ScratchData &sample_scratch_data, const CopyData &sample_copy_data, const unsigned int queue_length, const unsigned int chunk_size)
void run(const std::vector< std::vector< Iterator > > &colored_iterators, Worker worker, Copier copier, const ScratchData &sample_scratch_data, const CopyData &sample_copy_data, const unsigned int queue_length=2 *MultithreadInfo::n_threads(), const unsigned int chunk_size=8)
long double gamma(const unsigned int n)
void copy(const T *begin, const T *end, U *dest)
int(&) functions(const void *v1, const void *v2)
void assemble(const MeshWorker::DoFInfoBox< dim, DOFINFO > &dinfo, A *assembler)
static constexpr double E
void apply_to_subranges(const tbb::blocked_range< RangeType > &range, const Function &f)
void apply_to_subranges(const RangeType &begin, const typename identity< RangeType >::type &end, const Function &f, const unsigned int grainsize)
void transform(const InputIterator &begin_in, const InputIterator &end_in, OutputIterator out, const Predicate &predicate, const unsigned int grainsize)
boost::integer_range< IncrementableType > iota_view
::VectorizedArray< Number, width > exp(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > min(const ::VectorizedArray< Number, width > &, const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > max(const ::VectorizedArray< Number, width > &, const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > sqrt(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > pow(const ::VectorizedArray< Number, width > &, const Number p)
::VectorizedArray< Number, width > abs(const ::VectorizedArray< Number, width > &)
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
void advance(std::tuple< I1, I2 > &t, const unsigned int n)
void gather(VectorizedArray< Number, width > &out, const std::array< Number *, width > &ptrs, const unsigned int offset)