1462 * cell->get_dof_indices(dof_indices);
1464 * dof_indices.end(),
1465 * dof_indices.begin(),
1467 * return partitioner->global_to_local(index);
1472 *
for (
const auto dof : dof_indices)
1473 * dsp.add_entries(dof, dof_indices.begin(), dof_indices.end());
1476 * sparsity_pattern.copy_from(dsp);
1478 * lumped_mass_matrix.reinit(sparsity_pattern);
1479 * norm_matrix.reinit(sparsity_pattern);
1480 *
for (
auto &
matrix : cij_matrix)
1481 *
matrix.reinit(sparsity_pattern);
1482 *
for (
auto &
matrix : nij_matrix)
1483 *
matrix.reinit(sparsity_pattern);
1490 * Next, we have to
assemble various matrices. We define a number of
1491 * helper
functions and data structures in an anonymous
namespace.
1501 * <code>CopyData</code>
class that will be used to
assemble the
1502 * offline data matrices
using WorkStream. It acts as a container: it
1503 * is just a
struct where
WorkStream stores the local cell
1504 * contributions. Note that it also contains a
class member
1505 * <code>local_boundary_normal_map</code> used to store the local
1506 * contributions required to compute the normals at the boundary.
1512 *
template <
int dim>
1515 *
bool is_artificial;
1516 * std::vector<types::global_dof_index> local_dof_indices;
1517 *
typename OfflineData<dim>::BoundaryNormalMap local_boundary_normal_map;
1519 * std::array<FullMatrix<double>, dim> cell_cij_matrix;
1524 * Next we introduce a number of helper
functions that are all
1525 * concerned about reading and writing
matrix and vector entries. They
1526 * are mainly motivated by providing slightly more efficient code and
1527 * <a href=
"https://en.wikipedia.org/wiki/Syntactic_sugar"> syntactic
1528 * sugar</a>
for otherwise somewhat tedious code.
1532 * The
first function we introduce, <code>get_entry()</code>, will be
1533 * used to read the
value stored at the entry pointed by a
1535 *
function works around a small deficiency in the
SparseMatrix
1537 * operations of the sparse
matrix stored in CRS format. As such the
1538 * iterator already knows the global index of the corresponding
matrix
1540 * to the lack of an
interface in the
SparseMatrix for accessing the
1542 * have to create a temporary
SparseMatrix iterator. We simply hide
1543 *
this in the <code>get_entry()</code>
function.
1549 *
template <
typename IteratorType>
1554 * &
matrix, it->global_index());
1555 *
return matrix_iterator->value();
1560 * The <code>set_entry()</code> helper is the inverse operation of
1561 * <code>get_value()</code>: Given an iterator and a
value, it sets the
1562 * entry pointed to by the iterator in the
matrix.
1568 *
template <
typename IteratorType>
1571 *
const IteratorType & it,
1575 * it->global_index());
1576 * matrix_iterator->value() =
value;
1581 * <code>gather_get_entry()</code>: we note that @f$\mathbf{c}_{ij} \in
1582 * \mathbb{R}^
d@f$. If @f$d=2@f$ then @f$\mathbf{c}_{ij} =
1583 * [\mathbf{c}_{ij}^1,\mathbf{c}_{ij}^2]^\top@f$. Which basically implies
1584 * that we need
one matrix per space dimension to store the
1585 * @f$\mathbf{c}_{ij}@f$ vectors. Similar observation follows
for the
1586 *
matrix @f$\mathbf{n}_{ij}@f$. The purpose of
1587 * <code>gather_get_entry()</code> is to retrieve those entries and store
1594 *
template <std::
size_t k,
typename IteratorType>
1597 *
const IteratorType it)
1600 *
for (
unsigned int j = 0; j < k; ++j)
1601 * result[j] = get_entry(c_ij[j], it);
1608 * signature, having three input arguments, will be used to retrieve
1609 * the individual components <code>(i,
l)</code> of a
matrix. The
1610 * functionality of <code>gather_get_entry()</code> and
1611 * <code>
gather()</code> is very much the same, but their context is
1612 * different: the function <code>
gather()</code> does not rely on an
1613 * iterator (that actually knows the
value pointed to) but rather on the
1614 * indices <code>(i,
l)</code> of the entry in order to retrieve its
1615 * actual
value. We should expect <code>
gather()</code> to be slightly
1616 * more expensive than <code>gather_get_entry()</code>. The use of
1617 * <code>
gather()</code> will be limited to the task of computing the
1618 * algebraic viscosity @f$d_{ij}@f$ in the particular
case that when
1619 * both @f$i@f$ and @f$j@f$ lie at the boundary.
1623 * @note The reader should be aware that accessing an arbitrary
1624 * <code>(i,
l)</code> entry of a
matrix (say
for instance Trilinos or PETSc
1625 * matrices) is in
general unacceptably expensive. Here is where we might
1626 * want to keep an eye on complexity: we want
this operation to have
1627 * constant complexity, which is the
case of the current implementation
1628 *
using deal.II matrices.
1634 *
template <std::
size_t k>
1637 *
const unsigned int i,
1638 *
const unsigned int j)
1641 *
for (
unsigned int l = 0;
l < k; ++
l)
1642 * result[
l] = n_ij[
l](i, j);
1649 * signature having two input arguments will be used to
gather the
1650 * state at a node <code>i</code> and
return it as a
1657 * template <std::size_t k>
1660 *
const unsigned int i)
1663 *
for (
unsigned int j = 0; j < k; ++j)
1664 * result[j] =
U[j].local_element(i);
1670 * <code>scatter()</code>:
this function has three input arguments, the
1671 *
first one is meant to be a
"global object" (say a locally owned or
1672 * locally relevant vector), the
second argument which could be a
1674 * which represents a index of the global
object. This
function will be
1675 * primarily used to write the updated nodal values, stored as
1682 * template <std::size_t k, int k2>
1686 *
const unsigned int i)
1688 * static_assert(k == k2,
1689 *
"The dimensions of the input arguments must agree");
1690 *
for (
unsigned int j = 0; j < k; ++j)
1691 *
U[j].local_element(i) = tensor[j];
1697 * We are now in a position to
assemble all matrices stored in
1698 * <code>OfflineData</code>: the lumped mass entries @f$m_i@f$, the
1699 * vector-valued matrices @f$\mathbf{c}_{ij}@f$ and @f$\mathbf{n}_{ij} =
1700 * \frac{\mathbf{c}_{ij}}{|\mathbf{c}_{ij}|}@f$, and the boundary normals
1701 * @f$\boldsymbol{\nu}_i@f$.
1705 * In order to exploit thread parallelization we use the
WorkStream approach
1706 * detailed in the @ref threads
"Parallel computing with multiple processors"
1707 * accessing shared memory. As customary
this requires
1709 * - Scratch data (i.e. input info required to carry out computations): in
1710 * this case it is <code>scratch_data</code>.
1711 * - The worker: in our case this is the <code>local_assemble_system()</code>
1713 * actually computes the local (i.
e. current cell) contributions from the
1715 * -
A copy data: a struct that contains all the local assembly
1716 * contributions, in this case <code>CopyData<dim>()</code>.
1717 * -
A copy data routine: in this case it is
1718 * <code>copy_local_to_global()</code> in charge of actually coping these
1719 * local contributions into the global objects (matrices and/or vectors)
1723 * Most of the following lines are spent in the definition of the worker
1724 * <code>local_assemble_system()</code> and the
copy data routine
1725 * <code>copy_local_to_global()</code>. There is not much to say about the
1726 *
WorkStream framework since the vast majority of ideas are reasonably
1727 * well-documented in @ref step_9
"step-9", @ref step_13
"step-13" and @ref step_32
"step-32" among others.
1731 * Finally, assuming that @f$\mathbf{x}_i@f$ is a support
point at the boundary,
1732 * the (nodal) normals are defined as:
1737 * \widehat{\boldsymbol{\nu}}_i \dealcoloneq
1738 * \frac{\boldsymbol{\nu}_i}{|\boldsymbol{\nu}_i|} \ \text{ where }
1739 * \boldsymbol{\nu}_i \dealcoloneq \sum_{T \subset \text{supp}(\phi_i)}
1740 * \sum_{F \subset \partial T \cap \partial \Omega}
1741 * \sum_{\mathbf{x}_{q,
F}} \nu(\mathbf{x}_{q,
F})
1742 * \phi_i(\mathbf{x}_{q,
F}).
1747 * Here @f$T@f$ denotes elements,
1748 * @f$\text{supp}(\phi_i)@f$ the support of the shape
function @f$\phi_i@f$,
1749 * @f$F@f$ are faces of the element @f$T@f$, and @f$\mathbf{x}_{q,
F}@f$
1750 * are quadrature points on such face. Note that
this formula
for
1751 * @f$\widehat{\boldsymbol{\nu}}_i@f$ is nothing
else than some form of
1752 * weighted averaging. Other more sophisticated definitions
for @f$\nu_i@f$
1753 * are possible but
none of them have much influence in theory or practice.
1759 *
template <
int dim>
1762 * lumped_mass_matrix = 0.;
1764 *
for (
auto &
matrix : cij_matrix)
1766 *
for (
auto &
matrix : nij_matrix)
1769 *
unsigned int dofs_per_cell = discretization->finite_element.dofs_per_cell;
1770 *
unsigned int n_q_points = discretization->quadrature.size();
1774 * What follows is the initialization of the scratch data required by
1782 * discretization->mapping,
1783 * discretization->finite_element,
1784 * discretization->quadrature,
1787 * discretization->face_quadrature,
1793 *
"offline_data - assemble lumped mass matrix, and c_ij");
1795 *
const auto local_assemble_system =
1798 * CopyData<dim> &
copy) {
1799 *
copy.is_artificial = cell->is_artificial();
1800 *
if (
copy.is_artificial)
1803 *
copy.local_boundary_normal_map.clear();
1804 *
copy.cell_lumped_mass_matrix.reinit(dofs_per_cell, dofs_per_cell);
1806 *
matrix.reinit(dofs_per_cell, dofs_per_cell);
1808 *
const auto &fe_values = scratch.reinit(cell);
1810 *
copy.local_dof_indices.resize(dofs_per_cell);
1811 * cell->get_dof_indices(
copy.local_dof_indices);
1814 *
copy.local_dof_indices.end(),
1815 *
copy.local_dof_indices.begin(),
1817 * return partitioner->global_to_local(index);
1822 * We compute the local contributions
for the lumped mass
matrix
1823 * entries @f$m_i@f$ and and vectors @f$c_{ij}@f$ in the usual fashion:
1826 *
for (
unsigned int q_point = 0; q_point < n_q_points; ++q_point)
1828 *
const auto JxW = fe_values.JxW(q_point);
1830 *
for (
unsigned int j = 0; j < dofs_per_cell; ++j)
1832 *
const auto value_JxW =
1833 * fe_values.shape_value(j, q_point) * JxW;
1834 *
const auto grad_JxW = fe_values.shape_grad(j, q_point) * JxW;
1836 *
copy.cell_lumped_mass_matrix(j, j) += value_JxW;
1838 *
for (
unsigned int i = 0; i < dofs_per_cell; ++i)
1840 *
const auto value = fe_values.shape_value(i, q_point);
1841 *
for (
unsigned int d = 0;
d < dim; ++
d)
1842 *
copy.cell_cij_matrix[
d](i, j) +=
value * grad_JxW[
d];
1850 * Now we have to compute the boundary normals. Note that the
1851 * following
loop does not
do much unless the element has faces on
1852 * the boundary of the domain.
1857 *
const auto face = cell->face(f);
1858 *
const auto id = face->boundary_id();
1860 *
if (!face->at_boundary())
1863 *
const auto &fe_face_values = scratch.reinit(cell, f);
1865 *
const unsigned int n_face_q_points =
1866 * fe_face_values.get_quadrature().size();
1868 *
for (
unsigned int j = 0; j < dofs_per_cell; ++j)
1870 *
if (!discretization->finite_element.has_support_on_face(j, f))
1875 * Note that
"normal" will only represent the contributions
1876 * from
one of the faces in the support of the shape
1877 *
function phi_j. So we cannot normalize
this local
1878 * contribution right here, we have to take it
"as is",
1879 * store it and pass it to the
copy data routine. The
1880 * proper normalization requires an additional
loop on
1881 * nodes. This is done in the
copy function below.
1885 *
if (
id == Boundaries::free_slip)
1887 *
for (
unsigned int q = 0; q < n_face_q_points; ++q)
1888 * normal += fe_face_values.normal_vector(q) *
1889 * fe_face_values.shape_value(j, q);
1892 *
const auto index =
copy.local_dof_indices[j];
1895 *
for (
unsigned int v = 0;
1896 * v < GeometryInfo<dim>::vertices_per_cell;
1898 *
if (cell->vertex_dof_index(v, 0) ==
1899 * partitioner->local_to_global(index))
1901 * position = cell->vertex(v);
1905 *
const auto old_id =
1906 * std::get<1>(
copy.local_boundary_normal_map[index]);
1907 *
copy.local_boundary_normal_map[index] =
1908 * std::make_tuple(normal,
std::max(old_id,
id), position);
1915 * Last, we provide a copy_local_to_global
function as required
for
1919 *
const auto copy_local_to_global = [&](
const CopyData<dim> &
copy) {
1920 *
if (
copy.is_artificial)
1923 *
for (
const auto &it :
copy.local_boundary_normal_map)
1925 * std::get<0>(boundary_normal_map[it.first]) +=
1926 * std::get<0>(it.second);
1927 * std::get<1>(boundary_normal_map[it.first]) =
1928 *
std::max(std::get<1>(boundary_normal_map[it.first]),
1929 * std::get<1>(it.second));
1930 * std::get<2>(boundary_normal_map[it.first]) = std::get<2>(it.second);
1933 * lumped_mass_matrix.add(
copy.local_dof_indices,
1934 *
copy.cell_lumped_mass_matrix);
1936 *
for (
int k = 0; k < dim; ++k)
1938 * cij_matrix[k].add(
copy.local_dof_indices,
copy.cell_cij_matrix[k]);
1939 * nij_matrix[k].add(
copy.local_dof_indices,
copy.cell_cij_matrix[k]);
1944 * dof_handler.end(),
1945 * local_assemble_system,
1946 * copy_local_to_global,
1953 * At
this point in time we are done with the computation of @f$m_i@f$ and
1954 * @f$\mathbf{c}_{ij}@f$, but so far the matrix <code>nij_matrix</code>
1955 * contains just a
copy of the matrix <code>cij_matrix</code>.
1956 * That
's not what we really want: we have to normalize its entries. In
1957 * addition, we have not filled the entries of the matrix
1958 * <code>norm_matrix</code> and the vectors stored in the map
1959 * <code>OfflineData<dim>::BoundaryNormalMap</code> are not normalized.
1963 * In principle, this is just offline data, it doesn't make much sense
1964 * to over-optimize their computation, since their cost will get amortized
1965 * over the many time steps that we are going to use. However,
1966 * computing/storing the entries of the
matrix
1967 * <code>norm_matrix</code> and the normalization of <code>nij_matrix</code>
1968 * are perfect to illustrate thread-
parallel node-loops:
1969 * - we want to visit every node @f$i@f$ in the mesh/sparsity graph,
1970 * - and
for every such node we want to visit to every @f$j@f$ such that
1971 * @f$\mathbf{c}_{ij} \not \equiv 0@f$.
1975 * From an algebraic
point of view,
this is equivalent to: visiting
1976 * every row in the
matrix and
for each one of these rows execute a
loop on
1977 * the columns. Node-loops is a core theme of
this tutorial step (see
1978 * the pseudo-code in the introduction) that will repeat over and over
1979 * again. That
's why this is the right time to introduce them.
1983 * We have the thread parallelization capability
1984 * parallel::apply_to_subranges() that is somehow more general than the
1985 * WorkStream framework. In particular, parallel::apply_to_subranges() can
1986 * be used for our node-loops. This functionality requires four input
1987 * arguments which we explain in detail (for the specific case of our
1988 * thread-parallel node loops):
1989 * - The iterator <code>indices.begin()</code> points to a row index.
1990 * - The iterator <code>indices.end()</code> points to a numerically higher
1992 * - The function <code>on_subranges(i1,i2)</code> (where <code>i1</code>
1993 * and <code>i2</code> define a sub-range within the range spanned by
1994 * the end and begin iterators defined in the two previous bullets)
1995 * applies an operation to every iterator in such subrange. We may as
1996 * well call <code>on_subranges</code> the "worker".
1997 * - Grainsize: minimum number of iterators (in this case representing
1998 * rows) processed by each thread. We decided for a minimum of 4096
2003 * A minor caveat here is that the iterators <code>indices.begin()</code>
2004 * and <code>indices.end()</code> supplied to
2005 * parallel::apply_to_subranges() have to be random access iterators:
2006 * internally, parallel::apply_to_subranges() will break the range
2007 * defined by the <code>indices.begin()</code> and
2008 * <code>indices.end()</code> iterators into subranges (we want to be
2009 * able to read any entry in those subranges with constant complexity).
2010 * In order to provide such iterators we resort to
2011 * std_cxx20::ranges::iota_view.
2015 * The bulk of the following piece of code is spent defining
2016 * the "worker" <code>on_subranges</code>: i.e. the operation applied at
2017 * each row of the sub-range. Given a fixed <code>row_index</code>
2018 * we want to visit every column/entry in such row. In order to execute
2019 * such columns-loops we use
2020 * <a href="http://www.cplusplus.com/reference/algorithm/for_each/">
2022 * from the standard library, where:
2023 * - <code>sparsity_pattern.begin(row_index)</code>
2024 * gives us an iterator starting at the first column of the row,
2025 * - <code>sparsity_pattern.end(row_index)</code> is an iterator pointing
2026 * at the last column of the row,
2027 * - the last argument required by `std::for_each` is the operation
2028 * applied at each nonzero entry (a lambda expression in this case)
2033 * We note that, parallel::apply_to_subranges() will operate on
2034 * disjoint sets of rows (the subranges) and our goal is to write into
2035 * these rows. Because of the simple nature of the operations we want
2036 * to carry out (computation and storage of normals, and normalization
2037 * of the @f$\mathbf{c}_{ij}@f$ of entries) threads cannot conflict
2038 * attempting to write the same entry (we do not need a scheduler).
2042 * TimerOutput::Scope scope(computing_timer,
2043 * "offline_data - compute |c_ij|, and n_ij");
2045 * const std_cxx20::ranges::iota_view<unsigned int, unsigned int> indices(
2046 * 0, n_locally_relevant);
2048 * const auto on_subranges =
2050 * std_cxx20::ranges::iota_view<unsigned int, unsigned int>::iterator i1,
2051 * const std_cxx20::ranges::iota_view<unsigned int,
2052 * unsigned int>::iterator i2) {
2053 * for (const auto row_index :
2054 * std_cxx20::ranges::iota_view<unsigned int, unsigned int>(*i1,
2059 * First column-loop: we compute and store the entries of the
2060 * matrix norm_matrix and write normalized entries into the
2061 * matrix nij_matrix:
2065 * sparsity_pattern.begin(row_index),
2066 * sparsity_pattern.end(row_index),
2067 * [&](const ::SparsityPatternIterators::Accessor &jt) {
2068 * const auto c_ij = gather_get_entry(cij_matrix, &jt);
2069 * const double norm = c_ij.norm();
2071 * set_entry(norm_matrix, &jt, norm);
2072 * for (unsigned int j = 0; j < dim; ++j)
2073 * set_entry(nij_matrix[j], &jt, c_ij[j] / norm);
2078 * parallel::apply_to_subranges(indices.begin(),
2085 * Finally, we normalize the vectors stored in
2086 * <code>OfflineData<dim>::BoundaryNormalMap</code>. This operation has
2087 * not been thread parallelized as it would neither illustrate any
2088 * important concept nor lead to any noticeable speed gain.
2091 * for (auto &it : boundary_normal_map)
2093 * auto &normal = std::get<0>(it.second);
2094 * normal /= (normal.norm() + std::numeric_limits<double>::epsilon());
2100 * As commented in the introduction (see section titled "Stable boundary
2101 * conditions and conservation properties") we use three different types of
2102 * boundary conditions: essential-like boundary conditions (we prescribe a
2103 * state in the left portion of our domain), outflow boundary conditions
2104 * (also called "do-nothing" boundary conditions) in the right portion of
2105 * the domain, and "reflecting" boundary conditions (also called "free
2106 * slip" boundary conditions). With these boundary conditions we should
2107 * not expect any form of conservation to hold.
2111 * However, if we were to use reflecting boundary conditions
2112 * @f$\mathbf{m} \cdot \boldsymbol{\nu}_i =0@f$ on the entirety of the
2113 * boundary we should preserve the density and total (mechanical)
2114 * energy. This requires us to modify the @f$\mathbf{c}_{ij}@f$ vectors at
2115 * the boundary as follows @cite GuermondEtAl2018 :
2120 * \mathbf{c}_{ij} \, +\!\!= \int_{\partial \Omega}
2121 * (\boldsymbol{\nu}_j - \boldsymbol{\nu}(\mathbf{s})) \phi_i \phi_j \,
2122 * \mathrm{d}\mathbf{s} \ \text{whenever} \ \mathbf{x}_i \text{ and }
2123 * \mathbf{x}_j \text{ lie in the boundary.}
2128 * The ideas repeat themselves: we use WorkStream in order to compute
2129 * this correction, most of the following code is about the definition
2130 * of the worker <code>local_assemble_system()</code>.
2134 * TimerOutput::Scope scope(computing_timer,
2135 * "offline_data - fix slip boundary c_ij");
2137 * const auto local_assemble_system =
2138 * [&](const typename DoFHandler<dim>::cell_iterator &cell,
2139 * MeshWorker::ScratchData<dim> & scratch,
2140 * CopyData<dim> & copy) {
2141 * copy.is_artificial = cell->is_artificial();
2143 * if (copy.is_artificial)
2146 * for (auto &matrix : copy.cell_cij_matrix)
2147 * matrix.reinit(dofs_per_cell, dofs_per_cell);
2149 * copy.local_dof_indices.resize(dofs_per_cell);
2150 * cell->get_dof_indices(copy.local_dof_indices);
2151 * std::transform(copy.local_dof_indices.begin(),
2152 * copy.local_dof_indices.end(),
2153 * copy.local_dof_indices.begin(),
2154 * [&](types::global_dof_index index) {
2155 * return partitioner->global_to_local(index);
2158 * for (auto &matrix : copy.cell_cij_matrix)
2161 * for (auto f : GeometryInfo<dim>::face_indices())
2163 * const auto face = cell->face(f);
2164 * const auto id = face->boundary_id();
2166 * if (!face->at_boundary())
2169 * if (id != Boundaries::free_slip)
2172 * const auto &fe_face_values = scratch.reinit(cell, f);
2174 * const unsigned int n_face_q_points =
2175 * fe_face_values.get_quadrature().size();
2177 * for (unsigned int q = 0; q < n_face_q_points; ++q)
2179 * const auto JxW = fe_face_values.JxW(q);
2180 * const auto normal_q = fe_face_values.normal_vector(q);
2182 * for (unsigned int j = 0; j < dofs_per_cell; ++j)
2184 * if (!discretization->finite_element.has_support_on_face(
2188 * const auto &normal_j = std::get<0>(
2189 * boundary_normal_map[copy.local_dof_indices[j]]);
2191 * const auto value_JxW =
2192 * fe_face_values.shape_value(j, q) * JxW;
2194 * for (unsigned int i = 0; i < dofs_per_cell; ++i)
2196 * const auto value = fe_face_values.shape_value(i, q);
2198 * /* This is the correction of the boundary c_ij */
2199 * for (unsigned int d = 0; d < dim; ++d)
2200 * copy.cell_cij_matrix[d](i, j) +=
2201 * (normal_j[d] - normal_q[d]) * (value * value_JxW);
2208 * const auto copy_local_to_global = [&](const CopyData<dim> ©) {
2209 * if (copy.is_artificial)
2212 * for (int k = 0; k < dim; ++k)
2213 * cij_matrix[k].add(copy.local_dof_indices, copy.cell_cij_matrix[k]);
2216 * WorkStream::run(dof_handler.begin_active(),
2217 * dof_handler.end(),
2218 * local_assemble_system,
2219 * copy_local_to_global,
2227 * At this point we are very much done with anything related to offline data.
2232 * <a name="EquationofstateandapproximateRiemannsolver"></a>
2233 * <h4>Equation of state and approximate Riemann solver</h4>
2237 * In this section we describe the implementation of the class members of
2238 * the <code>ProblemDescription</code> class. Most of the code here is
2239 * specific to the compressible Euler's equations with an ideal gas law.
2240 * If we wanted to re-purpose @ref step_69
"step-69" for a different conservation law
2241 * (say
for: instance the shallow water equation) most of the
2242 * implementation of
this class would have to change. But most of the
2243 * other classes (in particular those defining
loop structures) would
2248 * We start by implementing a number of small member
functions for
2249 * computing <code>momentum</code>, <code>internal_energy</code>,
2250 * <code>pressure</code>, <code>speed_of_sound</code>, and the flux
2251 * <code>f</code> of the system. The functionality of each
one of these
2252 *
functions is
self-explanatory from their names.
2258 *
template <
int dim>
2260 * ProblemDescription<dim>::momentum(
const state_type &
U)
2267 *
template <
int dim>
2269 * ProblemDescription<dim>::internal_energy(
const state_type &
U)
2271 *
const double &rho =
U[0];
2272 *
const auto m = momentum(
U);
2273 *
const double &
E =
U[dim + 1];
2274 *
return E - 0.5 * m.norm_square() / rho;
2277 *
template <
int dim>
2279 * ProblemDescription<dim>::pressure(
const state_type &
U)
2281 *
return (
gamma - 1.) * internal_energy(
U);
2284 *
template <
int dim>
2286 * ProblemDescription<dim>::speed_of_sound(
const state_type &
U)
2288 *
const double &rho =
U[0];
2289 *
const double p = pressure(
U);
2294 *
template <
int dim>
2296 * ProblemDescription<dim>::flux(
const state_type &
U)
2298 *
const double &rho =
U[0];
2299 *
const auto m = momentum(
U);
2300 *
const auto p = pressure(
U);
2301 *
const double &
E =
U[dim + 1];
2306 *
for (
unsigned int i = 0; i < dim; ++i)
2308 * result[1 + i] = m * m[i] / rho;
2309 * result[1 + i][i] += p;
2311 * result[dim + 1] = m / rho * (
E + p);
2318 * Now we discuss the computation of @f$\lambda_{\text{
max}}
2319 * (\mathbf{
U}_i^{n},\mathbf{
U}_j^{n}, \textbf{n}_{ij})@f$. The analysis
2320 * and derivation of sharp upper-bounds of maximum wavespeeds of Riemann
2321 * problems is a very technical endeavor and we cannot include an
2322 * advanced discussion about it in
this tutorial. In
this portion of the
2323 * documentation we will limit ourselves to sketch the main functionality
2325 * references in order to help the (interested) reader
trace the
2326 * source (and proper mathematical justification) of these ideas.
2330 * In
general, obtaining a sharp guaranteed upper-bound on the maximum
2331 * wavespeed requires solving a quite expensive scalar nonlinear problem.
2332 * This is typically done with an iterative solver. In order to simplify
2333 * the presentation in
this example step we decided not to include such
2334 * an iterative scheme. Instead, we will just use an
initial guess as a
2335 * guess
for an upper bound on the maximum wavespeed. More precisely,
2336 * equations (2.11) (3.7), (3.8) and (4.3) of @cite GuermondPopov2016b
2337 * are enough to define a guaranteed upper bound on the maximum
2338 * wavespeed. This estimate is returned by a
call to the function
2339 * <code>lambda_max_two_rarefaction()</code>. At its core the
2340 * construction of such an upper bound uses the so-called two-rarefaction
2341 * approximation for the intermediate pressure @f$p^*@f$, see for instance
2342 * Equation (4.46), page 128 in @cite Toro2009.
2346 * The estimate returned by <code>lambda_max_two_rarefaction()</code> is
2347 * guaranteed to be an upper bound, it is in
general quite sharp, and
2348 * overall sufficient for our purposes. However, for some specific
2349 * situations (in particular when
one of states is close to vacuum
2350 * conditions) such an estimate will be overly pessimistic. That's why we
2351 * used a
second estimate to avoid this degeneracy that will be invoked
2352 * by a
call to the function <code>lambda_max_expansion()</code>. The most
2353 * important function here is <code>compute_lambda_max()</code> which
2354 * takes the minimum between the estimates returned by
2355 * <code>lambda_max_two_rarefaction()</code> and
2356 * <code>lambda_max_expansion()</code>.
2360 * We start again by defining a couple of helper
functions:
2364 * The
first function takes a state <code>
U</code> and a unit vector
2365 * <code>n_ij</code> and computes the <i>projected</i> 1D state in
2366 * direction of the unit vector.
2371 *
template <
int dim>
2373 *
const typename ProblemDescription<dim>::state_type
U,
2377 * projected_U[0] =
U[0];
2381 * For
this, we have to change the momentum to @f$\textbf{m}\cdot
2382 * n_{ij}@f$ and have to subtract the kinetic energy of the
2383 * perpendicular part from the total energy:
2386 *
const auto m = ProblemDescription<dim>::momentum(
U);
2387 * projected_U[1] = n_ij * m;
2389 *
const auto perpendicular_m = m - projected_U[1] * n_ij;
2390 * projected_U[2] =
U[1 + dim] - 0.5 * perpendicular_m.norm_square() /
U[0];
2394 * We
return the 1D state in <i>primitive</i> variables instead of
2395 * conserved quantities. The
return array consists of density @f$\rho@f$,
2396 * velocity @f$u@f$, pressure @f$p@f$ and local speed of sound @f$a@f$:
2402 *
return {{projected_U[0],
2403 * projected_U[1] / projected_U[0],
2404 * ProblemDescription<1>::pressure(projected_U),
2405 * ProblemDescription<1>::speed_of_sound(projected_U)}};
2410 * At
this point we also define two small
functions that
return the
2411 * positive and negative part of a
double.
2430 * Next, we need two local wavenumbers that are defined in terms of a
2431 * primitive state @f$[\rho, u, p, a]@f$ and a given pressure @f$p^\ast@f$
2432 * @cite GuermondPopov2016 Eqn. (3.7):
2435 * \left(\frac{p^\ast-p}{p}\right)_+}
2437 * Here, the @f$(\cdot)_{+}@f$ denotes the positive part of the given
2445 * lambda1_minus(
const std::array<double, 4> &riemann_data,
2446 *
const double p_star)
2451 *
const auto u = riemann_data[1];
2452 *
const auto p = riemann_data[2];
2453 *
const auto a = riemann_data[3];
2455 *
const double factor = (
gamma + 1.0) / 2.0 /
gamma;
2456 *
const double tmp = positive_part((p_star - p) / p);
2457 *
return u - a *
std::sqrt(1.0 + factor * tmp);
2462 * Analougously @cite GuermondPopov2016 Eqn. (3.8):
2465 * \left(\frac{p^\ast-p}{p}\right)_+}
2473 * lambda3_plus(
const std::array<double, 4> &riemann_data,
const double p_star)
2478 *
const auto u = riemann_data[1];
2479 *
const auto p = riemann_data[2];
2480 *
const auto a = riemann_data[3];
2482 *
const double factor = (
gamma + 1.0) / 2.0 /
gamma;
2483 *
const double tmp = positive_part((p_star - p) / p);
2484 *
return u + a *
std::sqrt(1.0 + factor * tmp);
2489 * All that is left to
do is to compute the maximum of @f$\lambda^-@f$ and
2490 * @f$\lambda^+@f$ computed from the left and right primitive state
2491 * (@cite GuermondPopov2016 Eqn. (2.11)), where @f$p^\ast@f$ is given by
2492 * @cite GuermondPopov2016 Eqn (4.3):
2499 * lambda_max_two_rarefaction(const std::array<
double, 4> &riemann_data_i,
2500 * const std::array<
double, 4> &riemann_data_j)
2503 *
const auto u_i = riemann_data_i[1];
2504 *
const auto p_i = riemann_data_i[2];
2505 *
const auto a_i = riemann_data_i[3];
2506 *
const auto u_j = riemann_data_j[1];
2507 *
const auto p_j = riemann_data_j[2];
2508 *
const auto a_j = riemann_data_j[3];
2510 *
const double numerator = a_i + a_j - (
gamma - 1.) / 2. * (u_j - u_i);
2512 *
const double denominator =
2517 *
const double p_star =
2520 *
const double lambda1 = lambda1_minus(riemann_data_i, p_star);
2521 *
const double lambda3 = lambda3_plus(riemann_data_j, p_star);
2525 *
return std::max(positive_part(lambda3), negative_part(lambda1));
2530 * We compute the
second upper bound of the maximal wavespeed that is, in
2531 *
general, not as sharp as the two-rarefaction estimate. But it will
2532 * save the day in the context of near vacuum conditions when the
2533 * two-rarefaction approximation might attain extreme values:
2535 * \lambda_{\text{
exp}} =
\max(u_i,u_j) + 5.
\max(a_i, a_j).
2537 * @note The constant 5.0 multiplying the maximum of the sound speeds
2538 * is <i>neither</i> an ad-hoc constant, <i>nor</i> a tuning parameter.
2539 * It defines an upper bound
for any @f$\gamma \in [0,5/3]@f$. Do not play
2547 * lambda_max_expansion(
const std::array<double, 4> &riemann_data_i,
2548 *
const std::array<double, 4> &riemann_data_j)
2550 *
const auto u_i = riemann_data_i[1];
2551 *
const auto a_i = riemann_data_i[3];
2552 *
const auto u_j = riemann_data_j[1];
2553 *
const auto a_j = riemann_data_j[3];
2561 * The following is the main
function that we are going to
call in order to
2562 * compute @f$\lambda_{\text{
max}} (\mathbf{
U}_i^{n},\mathbf{
U}_j^{n},
2563 * \textbf{n}_{ij})@f$. We simply compute both maximal wavespeed estimates
2564 * and
return the minimum.
2570 * template <int dim>
2572 * ProblemDescription<dim>::compute_lambda_max(
const state_type & U_i,
2573 *
const state_type & U_j,
2576 *
const auto riemann_data_i = riemann_data_from_state(U_i, n_ij);
2577 *
const auto riemann_data_j = riemann_data_from_state(U_j, n_ij);
2579 *
const double lambda_1 =
2580 * lambda_max_two_rarefaction(riemann_data_i, riemann_data_j);
2582 *
const double lambda_2 =
2583 * lambda_max_expansion(riemann_data_i, riemann_data_j);
2585 *
return std::min(lambda_1, lambda_2);
2590 * We conclude
this section by defining
static arrays
2591 * <code>component_names</code> that contain strings describing the
2592 * component names of our state vector. We have
template specializations
2593 *
for dimensions
one, two and three, that are used later in
DataOut for
2594 * naming the corresponding components:
2601 *
const std::array<std::string, 3> ProblemDescription<1>::component_names{
2602 * {
"rho",
"m",
"E"}};
2605 *
const std::array<std::string, 4> ProblemDescription<2>::component_names{
2606 * {
"rho",
"m_1",
"m_2",
"E"}};
2609 *
const std::array<std::string, 5> ProblemDescription<3>::component_names{
2610 * {
"rho",
"m_1",
"m_2",
"m_3",
"E"}};
2615 * <a name=
"Initialvalues"></a>
2616 * <h4>Initial values</h4>
2620 * The last preparatory step, before we discuss the implementation of the
2621 * forward Euler scheme, is to briefly implement the `InitialValues`
class.
2625 * In the constructor we initialize all parameters with
default values,
2627 * <code>parse_parameters_call_back</code> slot to the respective signal.
2631 * The <code>parse_parameters_call_back</code> slot will be invoked from
2633 * that regard, its use is appropriate
for situations where the
2634 * parameters have to be postprocessed (in some sense) or some
2635 * consistency condition between the parameters has to be checked.
2641 *
template <
int dim>
2642 * InitialValues<dim>::InitialValues(
const std::string &subsection)
2648 * std::bind(&InitialValues<dim>::parse_parameters_callback,
this));
2650 * initial_direction[0] = 1.;
2651 * add_parameter(
"initial direction",
2652 * initial_direction,
2653 *
"Initial direction of the uniform flow field");
2656 * initial_1d_state[1] = 3.;
2657 * initial_1d_state[2] = 1.;
2658 * add_parameter(
"initial 1d state",
2660 *
"Initial 1d state (rho, u, p) of the uniform flow field");
2665 * So far the constructor of <code>InitialValues</code> has defined
2666 *
default values
for the two
private members
2667 * <code>initial_direction</code> and <code>initial_1d_state</code> and
2668 * added them to the parameter list. But we have not defined an
2669 * implementation of the only
public member that we really care about,
2670 * which is <code>initial_state()</code> (the
function that we are going to
2671 *
call to actually evaluate the
initial solution at the mesh nodes). At
2672 * the top of the
function, we have to ensure that the provided
initial
2673 * direction is not the
zero vector.
2677 * @note As commented, we could have avoided
using the method
2678 * <code>parse_parameters_call_back </code> and defined a
class member
2679 * <code>setup()</code> in order to define the implementation of
2680 * <code>initial_state()</code>. But
for illustrative purposes we want to
2681 * document a different way here and use the
call back signal from
2688 *
template <
int dim>
2689 *
void InitialValues<dim>::parse_parameters_callback()
2693 *
"Initial shock front direction is set to the zero vector."));
2694 * initial_direction /= initial_direction.norm();
2698 * Next, we implement the <code>initial_state</code>
function object
2699 * with a
lambda function computing a uniform flow field. For
this we
2700 * have to translate a given primitive 1
d state (density @f$\rho@f$,
2701 * velocity @f$u@f$, and pressure @f$p@f$) into a conserved n-dimensional state
2702 * (density @f$\rho@f$, momentum @f$\mathbf{m}@f$, and total energy @f$E@f$).
2708 * initial_state = [
this](
const Point<dim> & ,
double ) {
2709 *
const double rho = initial_1d_state[0];
2710 *
const double u = initial_1d_state[1];
2711 *
const double p = initial_1d_state[2];
2717 *
for (
unsigned int i = 0; i < dim; ++i)
2718 * state[1 + i] = rho * u * initial_direction[i];
2720 * state[dim + 1] = p / (
gamma - 1.) + 0.5 * rho * u * u;
2729 * <a name=
"TheForwardEulerstep"></a>
2730 * <h4>The Forward Euler step</h4>
2734 * The constructor of the <code>%
TimeStepping</code>
class does not contain
2735 * any surprising code:
2741 *
template <
int dim>
2745 *
const OfflineData<dim> & offline_data,
2746 *
const InitialValues<dim> &initial_values,
2747 *
const std::string & subsection )
2749 * , mpi_communicator(mpi_communicator)
2750 * , computing_timer(computing_timer)
2751 * , offline_data(&offline_data)
2752 * , initial_values(&initial_values)
2754 * cfl_update = 0.80;
2755 * add_parameter(
"cfl update",
2757 *
"Relative CFL constant used for update");
2762 * In the
class member <code>prepare()</code> we initialize the temporary
2763 * vector <code>temp</code> and the matrix <code>dij_matrix</code>. The
2764 * vector <code>temp</code> will be used to store the solution update
2765 * temporarily before its contents is swapped with the old vector.
2771 *
template <
int dim>
2772 *
void TimeStepping<dim>::prepare()
2775 *
"time_stepping - prepare scratch space");
2777 *
for (
auto &it : temporary_vector)
2778 * it.reinit(offline_data->partitioner);
2780 * dij_matrix.reinit(offline_data->sparsity_pattern);
2785 * It is now time to implement the forward Euler step. Given a (writable
2786 * reference) to the old state <code>
U</code> at time @f$t@f$ we update the
2787 * state <code>
U</code> in place and
return the chosen time-step size. We
2788 *
first declare a number of read-only references to various different
2789 * variables and data structures. We
do this is mainly to have shorter
2790 * variable names (
e.g., <code>sparsity</code> instead of
2791 * <code>offline_data->sparsity_pattern</code>).
2797 *
template <
int dim>
2798 *
double TimeStepping<dim>::make_one_step(vector_type &
U,
double t)
2800 *
const auto &n_locally_owned = offline_data->n_locally_owned;
2801 *
const auto &n_locally_relevant = offline_data->n_locally_relevant;
2804 * indices_owned(0, n_locally_owned);
2806 * indices_relevant(0, n_locally_relevant);
2808 *
const auto &sparsity = offline_data->sparsity_pattern;
2810 *
const auto &lumped_mass_matrix = offline_data->lumped_mass_matrix;
2811 *
const auto &norm_matrix = offline_data->norm_matrix;
2812 *
const auto &nij_matrix = offline_data->nij_matrix;
2813 *
const auto &cij_matrix = offline_data->cij_matrix;
2815 *
const auto &boundary_normal_map = offline_data->boundary_normal_map;
2819 * <
b>Step 1</
b>: Computing the @f$d_{ij}@f$ graph viscosity
matrix.
2823 * It is important to highlight that the viscosity
matrix has to be
2824 *
symmetric, i.e., @f$d_{ij} = d_{ji}@f$. In
this regard we note here that
2825 * @f$\int_{\Omega} \nabla \phi_j \phi_i \, \mathrm{
d}\mathbf{x}= -
2826 * \int_{\Omega} \nabla \phi_i \phi_j \, \mathrm{
d}\mathbf{x}@f$ (or
2827 * equivalently @f$\mathbf{c}_{ij} = - \mathbf{c}_{ji}@f$) provided either
2828 * @f$\mathbf{x}_i@f$ or @f$\mathbf{x}_j@f$ is a support
point located away
2829 * from the boundary. In
this case we can check that
2830 * @f$\lambda_{\text{
max}} (\mathbf{
U}_i^{n}, \mathbf{
U}_j^{n},
2831 * \textbf{n}_{ij}) = \lambda_{\text{
max}} (\mathbf{
U}_j^{n},
2832 * \mathbf{
U}_i^{n},\textbf{n}_{ji})@f$ by construction, which guarantees
2833 * the property @f$d_{ij} = d_{ji}@f$.
2837 * However,
if both support points @f$\mathbf{x}_i@f$ or @f$\mathbf{x}_j@f$
2838 * happen to lie on the boundary, then, the equalities @f$\mathbf{c}_{ij} =
2839 * - \mathbf{c}_{ji}@f$ and @f$\lambda_{\text{
max}} (\mathbf{
U}_i^{n},
2840 * \mathbf{
U}_j^{n}, \textbf{n}_{ij}) = \lambda_{\text{
max}}
2841 * (\mathbf{
U}_j^{n}, \mathbf{
U}_i^{n}, \textbf{n}_{ji})@f$
do not
2842 * necessarily hold
true. The only mathematically safe solution
for this
2843 * dilemma is to compute both of them @f$d_{ij}@f$ and @f$d_{ji}@f$ and
2848 * Overall, the computation of @f$d_{ij}@f$ is quite expensive. In
2849 * order to save some computing time we exploit the fact that the viscosity
2851 * the upper-triangular entries of @f$d_{ij}@f$ and
copy the
2852 * corresponding entries to the lower-triangular counterpart.
2857 * loops. Pretty much all the ideas for
parallel traversal that we
2858 * introduced when discussing the assembly of the
matrix
2859 * <code>norm_matrix</code> and the normalization of
2860 * <code>nij_matrix</code> above are used here again.
2864 * We define again a "worker" function <code>on_subranges</code> that
2865 * computes the viscosity @f$d_{ij}@f$
for a subrange [i1, i2) of column
2871 *
"time_stepping - 1 compute d_ij");
2873 *
const auto on_subranges =
2877 *
unsigned int>::iterator i2) {
2878 *
for (
const auto i :
2882 *
const auto U_i =
gather(
U, i);
2886 * For a given column index i we iterate over the columns of the
2887 * sparsity pattern from <code>sparsity.begin(i)</code> to
2888 * <code>sparsity.end(i)</code>:
2891 *
for (
auto jt = sparsity.begin(i); jt != sparsity.end(i); ++jt)
2893 *
const auto j = jt->column();
2897 * We only compute @f$d_{ij}@f$
if @f$j < i@f$ (upper triangular
2898 * entries) and later
copy the values over to @f$d_{ji}@f$.
2904 *
const auto U_j =
gather(
U, j);
2906 *
const auto n_ij = gather_get_entry(nij_matrix, jt);
2907 *
const double norm = get_entry(norm_matrix, jt);
2909 *
const auto lambda_max =
2910 * ProblemDescription<dim>::compute_lambda_max(U_i, U_j, n_ij);
2912 *
double d =
norm * lambda_max;
2916 * If both support points happen to be at the boundary we
2917 * have to compute @f$d_{ji}@f$ as well and then take
2918 * @f$
\max(d_{ij},d_{ji})@f$. After
this we can
finally set the
2919 * upper triangular and lower triangular entries.
2922 * if (boundary_normal_map.count(i) != 0 &&
2923 * boundary_normal_map.count(j) != 0)
2925 *
const auto n_ji =
gather(nij_matrix, j, i);
2926 *
const auto lambda_max_2 =
2927 * ProblemDescription<dim>::compute_lambda_max(U_j,
2930 *
const double norm_2 = norm_matrix(j, i);
2935 * set_entry(dij_matrix, jt,
d);
2936 * dij_matrix(j, i) =
d;
2942 * indices_relevant.end(),
2949 * <
b>Step 2</
b>: Compute
diagonal entries @f$d_{ii}@f$ and
2950 * @f$\tau_{\text{
max}}@f$.
2955 * <code>dij_matrix</code>. We still have to fill its
diagonal entries
2956 * defined as @f$d_{ii}^n = - \sum_{j \in \mathcal{I}(i)\backslash \{i\}}
2958 * purpose. While computing the @f$d_{ii}@f$s we also determine the
2959 * largest admissible time-step, which is defined as
2961 * \tau_n \dealcoloneq c_{\text{cfl}}\,\min_{i\in\mathcal{
V}}
2962 * \left(\frac{m_i}{-2\,d_{ii}^{n}}\right) \, .
2964 * Note that the operation @f$\min_{i \in \mathcal{
V}}@f$ is intrinsically
2965 * global, it operates on all nodes:
first we have to take the minimum
2966 * over all threads (of a given node) and then we have to take the
2967 * minimum over all MPI processes. In the current implementation:
2968 * - We store <code>tau_max</code> (per node) as
2970 * href=
"http://www.cplusplus.com/reference/atomic/atomic/"><code>std::atomic<double></code></a>.
2971 * The
internal implementation of <code>std::atomic</code> will take
2972 * care of guarding any possible race condition when more than
one
2973 * thread attempts to read and/or write <code>tau_max</code> at the
2975 * - In order to take the minimum over all MPI process we use the utility
2976 * function <code>Utilities::MPI::min</code>.
2982 * std::atomic<double> tau_max{std::numeric_limits<double>::infinity()};
2986 *
"time_stepping - 2 compute d_ii, and tau_max");
2990 * on_subranges() will be executed on every thread individually. The
2991 * variable <code>tau_max_on_subrange</code> is thus stored thread
2998 * const auto on_subranges =
3002 *
unsigned int>::iterator i2) {
3003 *
double tau_max_on_subrange = std::numeric_limits<double>::infinity();
3005 *
for (
const auto i :
3009 *
double d_sum = 0.;
3011 *
for (
auto jt = sparsity.begin(i); jt != sparsity.end(i); ++jt)
3013 *
const auto j = jt->column();
3018 * d_sum -= get_entry(dij_matrix, jt);
3023 * We store the negative
sum of the d_ij entries at the
3027 * dij_matrix.diag_element(i) = d_sum;
3030 * and compute the maximal local time-step size
3034 *
const double mass = lumped_mass_matrix.diag_element(i);
3035 *
const double tau = cfl_update * mass / (-2. * d_sum);
3036 * tau_max_on_subrange =
std::min(tau_max_on_subrange, tau);
3041 * <code>tau_max_on_subrange</code> contains the largest possible
3042 * time-step size computed
for the (thread local) subrange. At
this
3043 *
point we have to synchronize the
value over all threads. This is
3044 * were we use the <a
3045 * href=
"http://www.cplusplus.com/reference/atomic/atomic/"><code>std::atomic<double></code></a>
3046 * <i>compare exchange</i> update mechanism:
3049 *
double current_tau_max = tau_max.load();
3050 *
while (current_tau_max > tau_max_on_subrange &&
3051 * !tau_max.compare_exchange_weak(current_tau_max,
3052 * tau_max_on_subrange))
3057 * indices_relevant.end(),
3063 * After all threads have finished we can simply synchronize the
3064 *
value over all MPI processes:
3074 * This is a good
point to verify that the computed
3075 * <code>tau_max</code> is indeed a
valid floating
point number.
3079 * !std::isnan(tau_max.load()) && !std::isinf(tau_max.load()) &&
3080 * tau_max.load() > 0.,
3082 *
"I'm sorry, Dave. I'm afraid I can't do that. - We crashed."));
3087 * <
b>Step 3</
b>: Perform update.
3091 * At
this point, we have computed all viscosity coefficients @f$d_{ij}@f$
3092 * and we know the maximal admissible time-step size
3093 * @f$\tau_{\text{
max}}@f$. This means we can now compute the update:
3098 * \mathbf{
U}_i^{n+1} = \mathbf{
U}_i^{n} - \frac{\tau_{\text{
max}} }{m_i}
3099 * \sum_{j \in \mathcal{I}(i)} (\mathbb{f}(\mathbf{
U}_j^{n}) -
3100 * \mathbb{f}(\mathbf{
U}_i^{n})) \cdot \mathbf{c}_{ij} - d_{ij}
3101 * (\mathbf{
U}_j^{n} - \mathbf{
U}_i^{n})
3106 * This update formula is slightly different from what was discussed in
3107 * the introduction (in the pseudo-code). However, it can be shown that
3108 * both equations are algebraically equivalent (they will produce the
3109 * same numerical values). We favor
this second formula since it has
3110 * natural cancellation properties that might help avoid numerical
3119 *
"time_stepping - 3 perform update");
3121 *
const auto on_subranges =
3125 *
unsigned int>::iterator i2) {
3130 *
const auto U_i =
gather(
U, i);
3132 *
const auto f_i = ProblemDescription<dim>::flux(U_i);
3133 *
const double m_i = lumped_mass_matrix.diag_element(i);
3135 *
auto U_i_new = U_i;
3137 *
for (
auto jt = sparsity.begin(i); jt != sparsity.end(i); ++jt)
3139 *
const auto j = jt->column();
3141 *
const auto U_j =
gather(
U, j);
3142 *
const auto f_j = ProblemDescription<dim>::flux(U_j);
3144 *
const auto c_ij = gather_get_entry(cij_matrix, jt);
3145 *
const auto d_ij = get_entry(dij_matrix, jt);
3147 *
for (
unsigned int k = 0; k < problem_dimension; ++k)
3151 * (-(f_j[k] - f_i[k]) * c_ij + d_ij * (U_j[k] - U_i[k]));
3155 * scatter(temporary_vector, U_i_new, i);
3160 * indices_owned.end(),
3167 * <
b>Step 4</
b>: Fix up boundary states.
3171 * As a last step in the Forward Euler method, we have to fix up all
3172 * boundary states. As discussed in the intro we
3173 * -
advance in time satisfying no boundary condition at all,
3174 * - at the
end of the time step enforce boundary conditions strongly
3175 * in a post-processing step.
3179 * Here, we compute the correction
3181 * \mathbf{m}_i \dealcoloneq \mathbf{m}_i - (\boldsymbol{\nu}_i \cdot
3182 * \mathbf{m}_i) \boldsymbol{\nu}_i,
3184 * which removes the normal component of @f$\mathbf{m}@f$.
3192 *
"time_stepping - 4 fix boundary states");
3194 *
for (
auto it : boundary_normal_map)
3196 *
const auto i = it.first;
3200 * We only iterate over the locally owned subset:
3203 *
if (i >= n_locally_owned)
3206 *
const auto &normal = std::get<0>(it.second);
3207 *
const auto &
id = std::get<1>(it.second);
3208 *
const auto &position = std::get<2>(it.second);
3210 *
auto U_i =
gather(temporary_vector, i);
3214 * On
free slip boundaries we remove the normal component of the
3218 *
if (
id == Boundaries::free_slip)
3220 *
auto m = ProblemDescription<dim>::momentum(U_i);
3221 * m -= (m * normal) * normal;
3222 *
for (
unsigned int k = 0; k < dim; ++k)
3223 * U_i[k + 1] = m[k];
3228 * On Dirichlet boundaries we enforce
initial conditions
3232 *
else if (
id == Boundaries::dirichlet)
3234 * U_i = initial_values->initial_state(position, t + tau_max);
3237 * scatter(temporary_vector, U_i, i);
3243 * <
b>Step 5</
b>: We now update the ghost layer over all MPI ranks,
3244 *
swap the temporary vector with the solution vector <code>
U</code>
3245 * (that will get returned by reference) and
return the chosen
3246 * time-step size @f$\tau_{\text{
max}}@f$:
3252 *
for (
auto &it : temporary_vector)
3253 * it.update_ghost_values();
3255 *
U.swap(temporary_vector);
3263 * <a name=
"Schlierenpostprocessing"></a>
3264 * <h4>Schlieren postprocessing</h4>
3268 * At various intervals we will output the current state <code>
U</code>
3269 * of the solution together with a so-called Schlieren plot.
3270 * The constructor of the <code>SchlierenPostprocessor</code>
class again
3271 * contains no surprises. We simply supply
default values to and
register
3274 * is an ad-hoc positive amplification factor in order to enhance the
3275 * contrast in the visualization. Its actual
value is a matter of
3277 * - schlieren_index: is an integer indicating which component of the
3278 * state @f$[\rho, \mathbf{m},
E]@f$ are we going to use in order to generate
3279 * the visualization.
3285 *
template <
int dim>
3286 * SchlierenPostprocessor<dim>::SchlierenPostprocessor(
3289 *
const OfflineData<dim> &offline_data,
3290 *
const std::string & subsection )
3292 * , mpi_communicator(mpi_communicator)
3293 * , computing_timer(computing_timer)
3294 * , offline_data(&offline_data)
3296 * schlieren_beta = 10.;
3297 * add_parameter(
"schlieren beta",
3299 *
"Beta factor used in Schlieren-type postprocessor");
3301 * schlieren_index = 0;
3302 * add_parameter(
"schlieren index",
3304 *
"Use the corresponding component of the state vector for the "
3305 *
"schlieren plot");
3310 * Again, the <code>prepare()</code>
function initializes two temporary
3311 * the vectors (<code>r</code> and <code>schlieren</code>).
3317 *
template <
int dim>
3318 *
void SchlierenPostprocessor<dim>::prepare()
3321 *
"schlieren_postprocessor - prepare scratch space");
3323 * r.reinit(offline_data->n_locally_relevant);
3324 * schlieren.reinit(offline_data->partitioner);
3329 * We now discuss the implementation of the
class member
3330 * <code>SchlierenPostprocessor<dim>::compute_schlieren()</code>, which
3331 * basically takes a component of the state vector <code>
U</code> and
3332 * computes the Schlieren indicator
for such component (the formula of the
3333 * Schlieren indicator can be found just before the declaration of the
class
3334 * <code>SchlierenPostprocessor</code>). We start by noting
3335 * that
this formula requires the
"nodal gradients" @f$\nabla r_j@f$.
3336 * However, nodal values of gradients are not defined
for @f$\mathcal{
C}^0@f$
3337 * finite element
functions. More generally, pointwise values of
3338 * gradients are not defined
for @f$W^{1,p}(\Omega)@f$
functions. The
3339 * simplest technique we can use to recover gradients at nodes is
3340 * weighted-averaging i.e.
3344 * \f[ \nabla r_j \dealcoloneq \frac{1}{\int_{S_i} \omega_i(\mathbf{x}) \,
3345 * \mathrm{
d}\mathbf{x}}
3346 * \int_{S_i} r_h(\mathbf{x}) \omega_i(\mathbf{x}) \, \mathrm{
d}\mathbf{x}
3347 * \ \ \ \ \ \mathbf{(*)} \f]
3351 * where @f$S_i@f$ is the support of the shape
function @f$\phi_i@f$, and
3352 * @f$\omega_i(\mathbf{x})@f$ is the weight. The weight could be any
3353 * positive
function such as
3354 * @f$\omega_i(\mathbf{x}) \equiv 1@f$ (that would allow us to recover the usual
3355 * notion of
mean value). But as usual, the goal is to reuse the off-line
3356 * data as much as possible. In
this sense, the most natural
3357 * choice of weight is @f$\omega_i = \phi_i@f$. Inserting
this choice of
3358 * weight and the expansion @f$r_h(\mathbf{x}) = \sum_{j \in \mathcal{
V}}
3359 * r_j \phi_j(\mathbf{x})@f$ into @f$\mathbf{(*)}@f$ we get :
3363 * \f[ \nabla r_j \dealcoloneq \frac{1}{m_i} \sum_{j \in \mathcal{I}(i)} r_j
3364 * \mathbf{c}_{ij} \ \ \ \ \ \mathbf{(**)} \, . \f]
3368 * Using
this last formula we can recover averaged nodal gradients without
3369 * resorting to any form of quadrature. This idea aligns quite well with
3370 * the whole spirit of edge-based schemes (or algebraic schemes) where
3371 * we want to operate on matrices and vectors as directly as
3372 * it could be possible avoiding by all means assembly of bilinear
3373 * forms, cell-loops, quadrature, or any other
3374 * intermediate construct/operation between the input arguments (the state
3375 * from the previous time-step) and the actual matrices and vectors
3376 * required to compute the update.
3380 * The
second thing to note is that we have to compute global minimum and
3381 * maximum @f$\max_j |\nabla r_j|@f$ and @f$\min_j |\nabla r_j|@f$. Following the
3382 * same ideas used to compute the time step size in the
class member
3383 * <code>%
TimeStepping%<dim%>::%step()</code> we define @f$\max_j |\nabla r_j|@f$
3384 * and @f$\min_j |\nabla r_j|@f$ as atomic doubles in order to resolve any
3385 * conflicts between threads. As usual, we use
3388 * among all MPI processes.
3392 * Finally, it is not possible to compute the Schlieren indicator in a single
3393 *
loop over all nodes. The entire operation requires two loops over nodes:
3397 * - The
first loop computes @f$|\nabla r_i|@f$
for all @f$i \in \mathcal{
V}@f$ in
3398 * the mesh, and the bounds @f$\max_j |\nabla r_j|@f$ and
3399 * @f$\min_j |\nabla r_j|@f$.
3400 * - The
second loop finally computes the Schlieren indicator
using the
3405 * \f[ \text{schlieren}[i] =
e^{\beta \frac{ |\nabla r_i|
3406 * - \min_j |\nabla r_j| }{\max_j |\nabla r_j| - \min_j |\nabla r_j| } }
3411 * This means that we will have to define two workers
3412 * <code>on_subranges</code>
for each one of these stages.
3418 *
template <
int dim>
3419 *
void SchlierenPostprocessor<dim>::compute_schlieren(
const vector_type &
U)
3422 * computing_timer,
"schlieren_postprocessor - compute schlieren plot");
3424 *
const auto &sparsity = offline_data->sparsity_pattern;
3425 *
const auto &lumped_mass_matrix = offline_data->lumped_mass_matrix;
3426 *
const auto &cij_matrix = offline_data->cij_matrix;
3427 *
const auto &boundary_normal_map = offline_data->boundary_normal_map;
3428 *
const auto &n_locally_owned = offline_data->n_locally_owned;
3430 *
const auto indices =
3436 * We define the r_i_max and r_i_min in the current MPI process as
3437 * atomic doubles in order to avoid race conditions between threads:
3440 * std::atomic<double> r_i_max{0.};
3441 * std::atomic<double> r_i_min{std::numeric_limits<double>::infinity()};
3445 * First
loop: compute the averaged gradient at each node and the
3446 * global maxima and minima of the gradients.
3450 *
const auto on_subranges =
3454 *
unsigned int>::iterator i2) {
3455 *
double r_i_max_on_subrange = 0.;
3456 *
double r_i_min_on_subrange = std::numeric_limits<double>::infinity();
3464 *
for (
auto jt = sparsity.begin(i); jt != sparsity.end(i); ++jt)
3466 *
const auto j = jt->column();
3471 *
const auto U_js =
U[schlieren_index].local_element(j);
3472 *
const auto c_ij = gather_get_entry(cij_matrix, jt);
3473 * r_i += c_ij * U_js;
3478 * We fix up the gradient r_i at
free slip boundaries similarly to
3479 * how we fixed up boundary states in the forward Euler step.
3480 * This avoids sharp, artificial gradients in the Schlieren
3481 * plot at
free slip boundaries and is a purely cosmetic choice.
3487 *
const auto bnm_it = boundary_normal_map.find(i);
3488 *
if (bnm_it != boundary_normal_map.end())
3490 *
const auto &normal = std::get<0>(bnm_it->second);
3491 *
const auto &
id = std::get<1>(bnm_it->second);
3493 *
if (
id == Boundaries::free_slip)
3494 * r_i -= 1. * (r_i * normal) * normal;
3501 * We remind the reader that we are not interested in the nodal
3502 * gradients per se. We only want their norms in order to
3503 * compute the Schlieren indicator (weighted with the lumped
3504 * mass
matrix @f$m_i@f$):
3507 * const
double m_i = lumped_mass_matrix.diag_element(i);
3508 * r[i] = r_i.
norm() / m_i;
3509 * r_i_max_on_subrange =
std::max(r_i_max_on_subrange, r[i]);
3510 * r_i_min_on_subrange =
std::min(r_i_min_on_subrange, r[i]);
3515 * We compare the current_r_i_max and current_r_i_min (in the
3516 * current subrange) with r_i_max and r_i_min (
for the current MPI
3517 * process) and update them
if necessary:
3523 *
double current_r_i_max = r_i_max.load();
3524 *
while (current_r_i_max < r_i_max_on_subrange &&
3525 * !r_i_max.compare_exchange_weak(current_r_i_max,
3526 * r_i_max_on_subrange))
3529 *
double current_r_i_min = r_i_min.load();
3530 *
while (current_r_i_min > r_i_min_on_subrange &&
3531 * !r_i_min.compare_exchange_weak(current_r_i_min,
3532 * r_i_min_on_subrange))
3544 * And synchronize <code>r_i_max</code> and <code>r_i_min</code> over
3545 * all MPI processes.
3556 * Second
loop: we now have the vector <code>r</code> and the scalars
3557 * <code>r_i_max</code> and <code>r_i_min</code> at our disposal. We
3558 * are thus in a position to actually compute the Schlieren indicator.
3565 *
const auto on_subranges =
3569 *
unsigned int>::iterator i2) {
3574 * schlieren.local_element(i) =
3575 * 1. -
std::exp(-schlieren_beta * (r[i] - r_i_min) /
3576 * (r_i_max - r_i_min));
3588 * And
finally, exchange ghost elements.
3591 * schlieren.update_ghost_values();
3597 * <a name=
"Themainloop"></a>
3598 * <h4>The main
loop</h4>
3602 * With all classes implemented it is time to create an instance of
3603 * <code>Discretization<dim></code>, <code>OfflineData<dim></code>,
3604 * <code>InitialValues<dim></code>, <code>TimeStepping<dim></code>, and
3605 * <code>SchlierenPostprocessor<dim></code>, and
run the forward Euler
3610 * In the constructor of <code>MainLoop<dim></code> we now initialize an
3611 * instance of all classes, and declare a number of parameters
3612 * controlling output. Most notable, we declare a
boolean parameter
3613 * <code>resume</code> that will control whether the program attempts to
3614 * restart from an interrupted computation, or not.
3620 *
template <
int dim>
3621 * MainLoop<dim>::MainLoop(
const MPI_Comm mpi_communicator)
3623 * , mpi_communicator(mpi_communicator)
3624 * , computing_timer(mpi_communicator,
3629 * , discretization(mpi_communicator, computing_timer,
"B - Discretization")
3630 * , offline_data(mpi_communicator,
3633 *
"C - OfflineData")
3634 * , initial_values(
"D - InitialValues")
3635 * , time_stepping(mpi_communicator,
3639 *
"E - TimeStepping")
3640 * , schlieren_postprocessor(mpi_communicator,
3643 *
"F - SchlierenPostprocessor")
3645 * base_name =
"test";
3646 * add_parameter(
"basename", base_name,
"Base name for all output files");
3649 * add_parameter(
"final time", t_final,
"Final time");
3651 * output_granularity = 0.02;
3652 * add_parameter(
"output granularity",
3653 * output_granularity,
3654 *
"time interval for output");
3657 * asynchronous_writeback =
true;
3659 * asynchronous_writeback =
false;
3661 * add_parameter(
"asynchronous writeback",
3662 * asynchronous_writeback,
3663 *
"Write out solution in a background thread performing IO");
3666 * add_parameter(
"resume", resume,
"Resume an interrupted computation.");
3671 * We start by implementing a helper function <code>print_head()</code>
3672 * in an anonymous
namespace that is used to output messages in the
3673 * terminal with some nice formatting.
3682 *
const std::string & header,
3683 *
const std::string & secondary =
"")
3685 *
const auto header_size = header.size();
3686 *
const auto padded_header = std::string((34 - header_size) / 2,
' ') +
3688 * std::string((35 - header_size) / 2,
' ');
3690 *
const auto secondary_size = secondary.size();
3691 *
const auto padded_secondary =
3692 * std::string((34 - secondary_size) / 2,
' ') + secondary +
3693 * std::string((35 - secondary_size) / 2,
' ');
3696 * pcout << std::endl;
3697 * pcout <<
" ####################################################" << std::endl;
3698 * pcout <<
" ######### #########" << std::endl;
3699 * pcout <<
" #########" << padded_header <<
"#########" << std::endl;
3700 * pcout <<
" #########" << padded_secondary <<
"#########" << std::endl;
3701 * pcout <<
" ######### #########" << std::endl;
3702 * pcout <<
" ####################################################" << std::endl;
3703 * pcout << std::endl;
3710 * With <code>print_head</code> in place it is now time to implement the
3718 *
template <
int dim>
3723 * We start by reading in parameters and initializing all objects. We
3725 * all parameters from the parameter file (whose name is given as a
3728 * declarations
for all
class instances that are derived from
3729 * ParameterAceptor. The
call to initialize enters the subsection
for
3730 * each each derived
class, and sets all variables that were added
3737 * pcout <<
"Reading parameters and allocating objects... " << std::flush;
3740 * pcout <<
"done" << std::endl;
3745 * scratch space, and initialize the
DataOut<dim> object:
3752 * print_head(pcout,
"create triangulation");
3753 * discretization.setup();
3755 * pcout <<
"Number of active cells: "
3756 * << discretization.triangulation.n_global_active_cells()
3759 * print_head(pcout,
"compute offline data");
3760 * offline_data.setup();
3761 * offline_data.assemble();
3763 * pcout <<
"Number of degrees of freedom: "
3764 * << offline_data.dof_handler.n_dofs() << std::endl;
3766 * print_head(pcout,
"set up time step");
3767 * time_stepping.prepare();
3768 * schlieren_postprocessor.prepare();
3773 * We will store the current time and state in the variable
3774 * <code>t</code> and vector <code>
U</code>:
3781 *
unsigned int output_cycle = 0;
3783 * print_head(pcout,
"interpolate initial values");
3784 * vector_type
U = interpolate_initial_values();
3789 * <a name=
"Resume"></a>
3794 * By
default the boolean <code>resume</code> is
set to
false, i.e. the
3795 * following code snippet is not
run. However,
if
3796 * <code>resume==
true</code> we indicate that we have indeed an
3797 * interrupted computation and the program shall restart by reading in
3798 * an old state consisting of <code>t</code>,
3799 * <code>output_cycle</code>, and <code>
U</code> from a checkpoint
3800 * file. These checkpoint files will be created in the
3801 * <code>output()</code> routine discussed below.
3809 * print_head(pcout,
"restore interrupted computation");
3811 *
const unsigned int i =
3812 * discretization.triangulation.locally_owned_subdomain();
3814 *
const std::string name = base_name +
"-checkpoint-" +
3816 * std::ifstream file(name, std::ios::binary);
3820 * We use a <code>boost::archive</code> to store and read in the
3821 * contents the checkpointed state.
3827 * boost::archive::binary_iarchive ia(file);
3828 * ia >> t >> output_cycle;
3830 *
for (
auto &it1 :
U)
3834 * <code>it1</code> iterates over all components of the state
3835 * vector <code>
U</code>. We read in every entry of the
3836 * component in sequence and update the ghost layer afterwards:
3839 *
for (
auto &it2 : it1)
3841 * it1.update_ghost_values();
3847 * With either the
initial state
set up, or an interrupted state
3848 * restored it is time to enter the main
loop:
3854 * output(
U, base_name, t, output_cycle++);
3856 * print_head(pcout,
"enter main loop");
3858 *
for (
unsigned int cycle = 1; t < t_final; ++cycle)
3862 * We
first print an informative status message
3868 * std::ostringstream head;
3869 * std::ostringstream secondary;
3872 * << std::fixed << std::setprecision(1) << t / t_final * 100
3874 * secondary <<
"at time t = " << std::setprecision(8) << std::fixed << t;
3876 * print_head(pcout, head.str(), secondary.str());
3880 * and then perform a single forward Euler step. Note that the
3881 * state vector <code>
U</code> is updated in place and that
3882 * <code>time_stepping.make_one_step()</code> returns the chosen step
3889 * t += time_stepping.make_one_step(
U, t);
3893 * Post processing, generating output and writing out the current
3894 * state is a CPU and IO intensive task that we cannot afford to
do
3895 * every time step - in particular with
explicit time stepping. We
3896 * thus only schedule output by calling the <code>output()</code>
3897 *
function if we are past a threshold
set by
3898 * <code>output_granularity</code>.
3904 *
if (t > output_cycle * output_granularity)
3906 * output(
U, base_name, t, output_cycle,
true);
3913 * We wait
for any remaining background output thread to finish before
3914 * printing a summary and exiting.
3917 *
if (background_thread_state.valid())
3918 * background_thread_state.wait();
3920 * computing_timer.print_summary();
3921 * pcout << timer_output.str() << std::endl;
3926 * The <code>interpolate_initial_values</code> takes an
initial time
"t"
3927 * as input argument and populates a state vector <code>
U</code> with the
3928 * help of the <code>InitialValues<dim>::initial_state</code>
object.
3934 *
template <
int dim>
3935 *
typename MainLoop<dim>::vector_type
3936 * MainLoop<dim>::interpolate_initial_values(
const double t)
3938 * pcout <<
"MainLoop<dim>::interpolate_initial_values(t = " << t <<
")"
3941 *
"main_loop - setup scratch space");
3945 *
for (
auto &it :
U)
3946 * it.reinit(offline_data.partitioner);
3948 * constexpr
auto problem_dimension =
3949 * ProblemDescription<dim>::problem_dimension;
3953 * The
function signature of
3954 * <code>InitialValues<dim>::initial_state</code> is not quite right
3956 * creating a
lambda function that
for a given position <code>x</code>
3957 * returns just the
value of the <code>i</code>th component. This
3958 *
lambda in turn is converted to a ::Function with the help of
3965 *
for (
unsigned int i = 0; i < problem_dimension; ++i)
3969 * return initial_values.initial_state(x, t)[i];
3973 *
for (
auto &it :
U)
3974 * it.update_ghost_values();
3982 * <a name=
"Outputandcheckpointing"></a>
3983 * <h5>Output and checkpointing</h5>
3987 * Writing out the
final vtk files is quite an IO intensive task that can
3988 * stall the main
loop for a
while. In order to avoid
this we use an <a
3989 * href=
"https://en.wikipedia.org/wiki/Asynchronous_I/O">asynchronous
3990 * IO</a> strategy by creating a background thread that will perform IO
3991 *
while the main
loop is allowed to
continue. In order
for this to work
3992 * we have to be mindful of two things:
3993 * - Before running the <code>output_worker</code> thread, we have to create
3994 * a
copy of the state vector <code>
U</code>. We store it in the
3995 * vector <code>output_vector</code>.
3996 * - We have to avoid any MPI communication in the background thread,
3997 * otherwise the program might deadlock. This implies that we have to
3998 *
run the postprocessing outside of the worker thread.
4004 *
template <
int dim>
4005 *
void MainLoop<dim>::output(
const typename MainLoop<dim>::vector_type &
U,
4006 *
const std::string & name,
4008 *
const unsigned int cycle,
4009 *
const bool checkpoint)
4011 * pcout <<
"MainLoop<dim>::output(t = " << t
4012 * <<
", checkpoint = " << checkpoint <<
")" << std::endl;
4016 * If the asynchronous writeback option is
set we launch a background
4017 * thread performing all the slow IO to disc. In that
case we have to
4018 * make sure that the background thread actually finished running. If
4019 * not, we have to wait to
for it to finish. We launch said background
4021 * href=
"https://en.cppreference.com/w/cpp/thread/async"><code>std::async()</code></a>
4023 * href=
"https://en.cppreference.com/w/cpp/thread/future"><code>std::future</code></a>
4024 *
object. This <code>std::future</code>
object contains the
return
4025 *
value of the
function, which is in our
case simply
4026 * <code>
void</code>.
4032 *
if (background_thread_state.valid())
4035 * background_thread_state.wait();
4038 * constexpr
auto problem_dimension =
4039 * ProblemDescription<dim>::problem_dimension;
4043 * At
this point we make a
copy of the state vector,
run the schlieren
4045 * output code is standard: We create a
DataOut instance, attach all
4046 * data vectors we want to output and
call
4047 *
DataOut<dim>::build_patches(). There is
one twist, however. In order
4048 * to perform asynchronous IO on a background thread we create the
4049 *
DataOut<dim>
object as a shared pointer that we pass on to the
4050 * worker thread to ensure that once we exit this function and the
4051 * worker thread finishes the
DataOut<dim>
object gets destroyed again.
4057 * for (
unsigned int i = 0; i < problem_dimension; ++i)
4059 * output_vector[i] =
U[i];
4060 * output_vector[i].update_ghost_values();
4063 * schlieren_postprocessor.compute_schlieren(output_vector);
4065 *
auto data_out = std::make_shared<DataOut<dim>>();
4067 * data_out->attach_dof_handler(offline_data.dof_handler);
4069 *
const auto &component_names = ProblemDescription<dim>::component_names;
4071 *
for (
unsigned int i = 0; i < problem_dimension; ++i)
4072 * data_out->add_data_vector(output_vector[i], component_names[i]);
4074 * data_out->add_data_vector(schlieren_postprocessor.schlieren,
4075 *
"schlieren_plot");
4077 * data_out->build_patches(discretization.mapping,
4078 * discretization.finite_element.degree - 1);
4082 * Next we create a
lambda function for the background thread. We <a
4083 * href=
"https://en.cppreference.com/w/cpp/language/lambda">capture</a>
4084 * the <code>
this</code> pointer as well as most of the arguments of
4085 * the output
function by
value so that we have access to them inside
4089 *
const auto output_worker = [
this, name, t, cycle, checkpoint, data_out]() {
4094 * We checkpoint the current state by doing the precise inverse
4095 * operation to what we discussed
for the <a href=
"Resume">resume
4102 *
const unsigned int i =
4103 * discretization.triangulation.locally_owned_subdomain();
4104 * std::string filename =
4107 * std::ofstream file(filename, std::ios::binary | std::ios::trunc);
4109 * boost::archive::binary_oarchive oa(file);
4111 *
for (
const auto &it1 : output_vector)
4112 *
for (
const auto &it2 : it1)
4120 * data_out->set_flags(flags);
4122 * data_out->write_vtu_with_pvtu_record(
4123 *
"", name +
"-solution", cycle, mpi_communicator, 6);
4128 * If the asynchronous writeback option is
set we launch a
new
4129 * background thread with the help of
4131 * href=
"https://en.cppreference.com/w/cpp/thread/async"><code>std::async</code></a>
4132 *
function. The
function returns a <a
4133 * href=
"https://en.cppreference.com/w/cpp/thread/future"><code>std::future</code></a>
4134 *
object that we can use to query the status of the background thread.
4135 * At
this point we can
return from the <code>output()</code>
function
4136 * and resume with the time stepping in the main
loop - the thread will
4137 *
run in the background.
4140 *
if (asynchronous_writeback)
4143 * background_thread_state = std::async(std::launch::async, output_worker);
4148 *
"\"asynchronous_writeback\" was set to true but deal.II was built "
4149 *
"without thread support (\"DEAL_II_WITH_THREADS=false\")."));
4162 * And
finally, the main
function.
4168 *
int main(
int argc,
char *argv[])
4172 * constexpr
int dim = 2;
4174 *
using namespace dealii;
4175 *
using namespace Step69;
4179 *
MPI_Comm mpi_communicator(MPI_COMM_WORLD);
4180 * MainLoop<dim> main_loop(mpi_communicator);
4184 *
catch (std::exception &exc)
4186 * std::cerr << std::endl
4188 * <<
"----------------------------------------------------"
4190 * std::cerr <<
"Exception on processing: " << std::endl
4191 * << exc.what() << std::endl
4192 * <<
"Aborting!" << std::endl
4193 * <<
"----------------------------------------------------"
4199 * std::cerr << std::endl
4201 * <<
"----------------------------------------------------"
4203 * std::cerr <<
"Unknown exception!" << std::endl
4204 * <<
"Aborting!" << std::endl
4205 * <<
"----------------------------------------------------"
4211 <a name=
"Results"></a>
4212 <a name=
"Results"></a><h1>Results</h1>
4215 Running the program with
default parameters in release mode takes about 1
4216 minute on a 4 core machine (with hyperthreading):
4218 # mpirun -np 4 ./step-69 | tee output
4219 Reading parameters and allocating objects... done
4221 ####################################################
4225 ####################################################
4227 Number of active cells: 36864
4229 ####################################################
4231 ######### compute offline data #########
4233 ####################################################
4235 Number of degrees of freedom: 37376
4237 ####################################################
4239 #########
set up time step #########
4241 ####################################################
4243 ####################################################
4248 ####################################################
4250 TimeLoop<dim>::interpolate_initial_values(t = 0)
4251 TimeLoop<dim>::output(t = 0, checkpoint = 0)
4253 ####################################################
4255 ######### enter main
loop #########
4258 ####################################################
4260 ####################################################
4262 ######### Cycle 000001 (0.0%) #########
4263 ######### at time t = 0.00000000 #########
4265 ####################################################
4269 ####################################################
4271 ######### Cycle 007553 (100.0%) #########
4272 ######### at time t = 3.99984036 #########
4274 ####################################################
4276 TimeLoop<dim>::output(t = 4.00038, checkpoint = 1)
4278 +------------------------------------------------------------------------+------------+------------+
4279 | Total CPU time elapsed since start | 357s | |
4281 | Section | no. calls | CPU time | % of total |
4282 +------------------------------------------------------------+-----------+------------+------------+
4283 | discretization - setup | 1 | 0.113s | 0% |
4284 | offline_data -
assemble lumped mass
matrix, and c_ij | 1 | 0.167s | 0% |
4285 | offline_data - compute |c_ij|, and n_ij | 1 | 0.00255s | 0% |
4286 | offline_data - create sparsity pattern and
set up matrices | 1 | 0.0224s | 0% |
4287 | offline_data - distribute dofs | 1 | 0.0617s | 0% |
4288 | offline_data - fix slip boundary c_ij | 1 | 0.0329s | 0% |
4289 | schlieren_postprocessor - compute schlieren plot | 201 | 0.811s | 0.23% |
4290 | schlieren_postprocessor - prepare scratch space | 1 | 7.6
e-05s | 0% |
4291 | time_loop - setup scratch space | 1 | 0.127s | 0% |
4292 | time_loop - stalled output | 200 | 0.000685s | 0% |
4293 | time_step - 1 compute d_ij | 7553 | 240s | 67% |
4294 | time_step - 2 compute d_ii, and tau_max | 7553 | 11.5s | 3.2% |
4295 | time_step - 3 perform update | 7553 | 101s | 28% |
4296 | time_step - 4 fix boundary states | 7553 | 0.724s | 0.2% |
4297 | time_step - prepare scratch space | 1 | 0.00245s | 0% |
4298 +------------------------------------------------------------+-----------+------------+------------+
4301 One thing that becomes evident is the fact that the program spends two
4302 thirds of the execution time computing the graph viscosity d_ij and about a
4303 third of the execution time in performing the update, where computing the
4304 flux @f$f(
U)@f$ is the expensive operation. The preset default resolution is
4305 about 37k gridpoints, which amounts to about 148k spatial degrees of
4306 freedom in 2D. An animated schlieren plot of the solution looks as follows:
4308 <img src=
"https://www.dealii.org/images/steps/developer/step-69.coarse.gif" alt=
"" height=
"300">
4310 It is evident that 37k gridpoints for the
first-order method is nowhere
4311 near the resolution needed to resolve any flow features. For comparison,
4312 here is a
"reference" computation with a
second-order method and about 9.5M
4315 <img src=
"https://www.dealii.org/images/steps/developer/step-69.2nd-order.t400.jpg" alt=
"" height=
"300">
4317 So, we give the
first-order method a
second chance and
run it with about
4318 2.4M gridpoints on a small compute server:
4321 # mpirun -np 16 ./step-69 | tee output
4325 ####################################################
4327 ######### Cycle 070216 (100.0%) #########
4328 ######### at time t = 3.99999231 #########
4330 ####################################################
4332 TimeLoop<dim>::output(t = 4.00006, checkpoint = 1)
4336 +------------------------------------------------------------------------+------------+------------+
4337 | Total wallclock time elapsed since start | 6.75
e+03s | |
4339 | Section | no. calls | wall time | % of total |
4340 +------------------------------------------------------------+-----------+------------+------------+
4341 | discretization - setup | 1 | 1.97s | 0% |
4342 | offline_data -
assemble lumped mass
matrix, and c_ij | 1 | 1.19s | 0% |
4343 | offline_data - compute |c_ij|, and n_ij | 1 | 0.0172s | 0% |
4344 | offline_data - create sparsity pattern and
set up matrices | 1 | 0.413s | 0% |
4345 | offline_data - distribute dofs | 1 | 1.05s | 0% |
4346 | offline_data - fix slip boundary c_ij | 1 | 0.252s | 0% |
4347 | schlieren_postprocessor - compute schlieren plot | 201 | 1.82s | 0% |
4348 | schlieren_postprocessor - prepare scratch space | 1 | 0.000497s | 0% |
4349 | time_loop - setup scratch space | 1 | 1.45s | 0% |
4350 | time_loop - stalled output | 200 | 0.00342s | 0% |
4351 | time_step - 1 compute d_ij | 70216 | 4.38
e+03s | 65% |
4352 | time_step - 2 compute d_ii, and tau_max | 70216 | 419s | 6.2% |
4353 | time_step - 3 perform update | 70216 | 1.87
e+03s | 28% |
4354 | time_step - 4 fix boundary states | 70216 | 24s | 0.36% |
4355 | time_step - prepare scratch space | 1 | 0.0227s | 0% |
4356 +------------------------------------------------------------+-----------+------------+------------+
4359 And with the following result:
4361 <img src=
"https://www.dealii.org/images/steps/developer/step-69.fine.gif" alt=
"" height=
"300">
4363 That
's substantially better, although of course at the price of having run
4364 the code for roughly 2 hours on 16 cores.
4367 <a name="PlainProg"></a>
4368 <h1> The plain program</h1>
4369 @include "step-69.cc"