758 *
return 1. / (0.05 + 2. * p.square());
764 *
double Coefficient<dim>::value(
const Point<dim> & p,
765 *
const unsigned int component)
const
767 *
return value<double>(p, component);
774 * <a name=
"Matrixfreeimplementation"></a>
775 * <h3>Matrix-
free implementation</h3>
779 * The following
class, called <code>LaplaceOperator</code>, implements the
780 * differential
operator. For all practical purposes, it is a
matrix, i.e.,
781 * you can ask it
for its size (member functions <code>m(), n()</code>) and
782 * you can apply it to a vector (the <code>vmult()</code> function). The
783 * difference to a real matrix of course lies in the fact that this class
784 * does not actually store the <i>elements</i> of the matrix, but only knows
785 * how to compute the action of the operator when applied to a vector.
789 * The infrastructure describing the matrix size, the initialization from a
790 *
MatrixFree object, and the various interfaces to matrix-vector products
791 * through vmult() and Tvmult() methods, is provided by the class
792 * MatrixFreeOperator::Base from which this class derives. The
793 * LaplaceOperator class defined here only has to provide a few interfaces,
794 * namely the actual action of the operator through the apply_add() method
795 * that gets used in the vmult() functions, and a method to compute the
796 * diagonal entries of the underlying matrix. We need the diagonal for the
797 * definition of the multigrid smoother. Since we consider a problem with
798 * variable coefficient, we further implement a method that can fill the
799 * coefficient values.
803 * Note that the file <code>include/deal.II/matrix_free/operators.h</code>
804 * already contains an implementation of the Laplacian through the class
806 * operator is re-implemented in this tutorial program, explaining the
807 * ingredients and
concepts used there.
811 * This program makes use of the data cache for finite element operator
812 * application that is integrated in deal.II. This data cache class is
813 * called
MatrixFree. It contains mapping information (Jacobians) and index
814 * relations between local and global degrees of freedom. It also contains
815 * constraints like the ones from hanging nodes or Dirichlet boundary
816 * conditions. Moreover, it can issue a loop over all cells in %
parallel,
817 * making sure that only cells are worked on that do not share any degree of
818 * freedom (this makes the loop thread-safe when writing into destination
819 * vectors). This is a more advanced strategy compared to the
WorkStream
820 * class described in the @ref threads module. Of course, to not destroy
821 * thread-safety, we have to be careful when writing into class-global
826 * The class implementing the Laplace operator has three template arguments,
827 * one for the dimension (as many deal.II classes carry), one for the degree
828 * of the finite element (which we need to enable efficient computations
829 * through the
FEEvaluation class), and one for the underlying scalar
830 * type. We want to use <code>
double</code>
numbers (i.e.,
double precision,
831 * 64-bit floating point) for the final matrix, but floats (single
832 * precision, 32-bit floating point
numbers) for the multigrid
level
833 * matrices (as that is only a preconditioner, and floats can be processed
834 * twice as fast). The class
FEEvaluation also takes a template argument for
835 * the number of quadrature points in one dimension. In the code below, we
836 * hard-code it to <code>fe_degree+1</code>. If we wanted to change it
837 * independently of the polynomial degree, we would need to add a template
842 * As a sidenote, if we implemented several different operations on the same
843 * grid and degrees of freedom (like a @ref GlossMassMatrix "mass matrix" and a Laplace matrix), we
844 * would define two classes like the current one for each of the operators
846 * refer to the same
MatrixFree data cache from the general problem
848 * only provide a minimal set of functions. This concept allows for writing
849 * complex application codes with many matrix-free operations.
854 * requires care: Here, we use the deal.II table class which is prepared to
855 * hold the data with correct alignment. However, storing e.g. an
857 * vectorization: A certain alignment of the data with the memory address
858 * boundaries is required (essentially, a
VectorizedArray that is 32 bytes
859 *
long in case of AVX needs to start at a memory address that is divisible
860 * by 32). The table class (as well as the
AlignedVector class it is based
861 * on) makes sure that this alignment is respected, whereas
std::vector does
862 * not in general, which may lead to segmentation faults at strange places
863 * for some systems or suboptimal performance for other systems.
866 * template <
int dim,
int fe_degree, typename number>
867 * class LaplaceOperator
872 *
using value_type = number;
876 *
void clear()
override;
878 *
void evaluate_coefficient(
const Coefficient<dim> &coefficient_function);
883 *
virtual void apply_add(
891 *
const std::pair<unsigned int, unsigned int> &cell_range)
const;
893 *
void local_compute_diagonal(
896 *
const unsigned int & dummy,
897 *
const std::pair<unsigned int, unsigned int> &cell_range)
const;
906 * This is the constructor of the @p LaplaceOperator
class. All it does is
907 * to
call the
default constructor of the base
class
909 *
class that asserts that this class is not accessed after going out of scope
910 *
e.g. in a preconditioner.
913 *
template <
int dim,
int fe_degree,
typename number>
914 * LaplaceOperator<dim, fe_degree, number>::LaplaceOperator()
921 *
template <
int dim,
int fe_degree,
typename number>
922 *
void LaplaceOperator<dim, fe_degree, number>::clear()
924 * coefficient.
reinit(0, 0);
934 * <a name=
"Computationofcoefficient"></a>
935 * <h4>Computation of coefficient</h4>
939 * To initialize the coefficient, we directly give it the Coefficient
class
940 * defined above and then select the method
941 * <code>coefficient_function.value</code> with
vectorized number (which the
942 * compiler can deduce from the point data type). The use of the
943 *
FEEvaluation class (and its
template arguments) will be explained below.
946 * template <int dim, int fe_degree, typename number>
947 *
void LaplaceOperator<dim, fe_degree, number>::evaluate_coefficient(
948 *
const Coefficient<dim> &coefficient_function)
950 *
const unsigned int n_cells = this->data->n_cell_batches();
953 * coefficient.reinit(n_cells, phi.n_q_points);
954 *
for (
unsigned int cell = 0; cell <
n_cells; ++cell)
957 *
for (
unsigned int q = 0; q < phi.n_q_points; ++q)
958 * coefficient(cell, q) =
959 * coefficient_function.value(phi.quadrature_point(q));
968 * <a name=
"LocalevaluationofLaplaceoperator"></a>
969 * <h4>Local evaluation of Laplace
operator</h4>
973 * Here comes the main function of
this class, the evaluation of the
974 *
matrix-vector product (or, in general, a finite element
operator
975 * evaluation). This is done in a function that takes exactly four
976 * arguments, the
MatrixFree object, the destination and source vectors, and
977 * a range of cells that are to be worked on. The method
978 * <code>cell_loop</code> in the
MatrixFree class will internally
call this
979 * function with some range of cells that is obtained by checking which
980 * cells are possible to work on simultaneously so that write operations
do
981 * not cause any race condition. Note that the cell range used in the
loop
982 * is not directly the number of (active) cells in the current mesh, but
983 * rather a collection of batches of cells. In other word,
"cell" may be
985 * cells together. This means that in the
loop over quadrature points we are
986 * actually seeing a
group of quadrature points of several cells as one
987 * block. This is done to enable a higher degree of vectorization. The
988 * number of such
"cells" or
"cell batches" is stored in
MatrixFree and can
990 * cell iterators, in
this class all cells are laid out in a plain array
991 * with no direct knowledge of
level or neighborship relations, which makes
992 * it possible to
index the cells by
unsigned integers.
996 * The implementation of the Laplace
operator is quite simple: First, we
997 * need to create an
object FEEvaluation that contains the computational
998 * kernels and has data fields to store temporary results (
e.g. gradients
999 * evaluated on all quadrature points on a collection of a few cells). Note
1000 * that temporary results
do not use a lot of memory, and since we specify
1001 *
template arguments with the element order, the data is stored on the
1002 * stack (without expensive memory allocation). Usually, one only needs to
1003 *
set two
template arguments, the dimension as a
first argument and the
1004 * degree of the finite element as the
second argument (
this is
equal to the
1005 * number of degrees of freedom per dimension minus one
for FE_Q
1006 * elements). However, here we also want to be able to use
float numbers for
1007 * the multigrid preconditioner, which is the last (fifth)
template
1008 * argument. Therefore, we cannot rely on the
default template arguments and
1009 * must also fill the third and fourth field, consequently. The third
1010 * argument specifies the number of quadrature points per direction and has
1011 * a
default value equal to the degree of the element plus one. The fourth
1012 * argument sets the number of components (one can also evaluate
1013 * vector-valued functions in systems of PDEs, but the
default is a scalar
1014 * element), and
finally the last argument sets the number type.
1018 * Next, we
loop over the given cell range and then we
continue with the
1019 * actual implementation: <ol> <li>Tell the
FEEvaluation object the (macro)
1020 * cell we want to work on. <li>Read in the
values of the source vectors
1021 * (@p read_dof_values), including the resolution of constraints. This
1022 * stores @f$u_\mathrm{cell}@f$ as described in the introduction. <li>Compute
1023 * the unit-cell
gradient (the evaluation of finite element
1025 *
gradient computations, it uses a unified
interface to all kinds of
1026 * derivatives of order between zero and two. We only want
gradients, no
1027 *
values and no
second derivatives, so we
set the function arguments to
1029 * (
first slot). There is also a third slot
for the Hessian which is
1030 *
false by
default, so it needs not be given. Note that the
FEEvaluation
1031 *
class internally evaluates shape functions in an efficient way where one
1032 * dimension is worked on at a time (
using the tensor product form of shape
1033 * functions and quadrature points as mentioned in the introduction). This
1034 * gives complexity
equal to @f$\mathcal O(d^2 (p+1)^{
d+1})@f$
for polynomial
1035 * degree @f$p@f$ in @f$d@f$ dimensions, compared to the naive approach with loops
1036 * over all local degrees of freedom and quadrature points that is used in
1037 *
FEValues and costs @f$\mathcal O(d (p+1)^{2
d})@f$. <li>Next comes the
1038 * application of the Jacobian transformation, the multiplication by the
1039 * variable coefficient and the quadrature weight.
FEEvaluation has an
1040 * access function @p get_gradient that applies the Jacobian and returns the
1041 * gradient in real space. Then, we just need to multiply by the (scalar)
1042 * coefficient, and let the function @p submit_gradient
apply the
second
1043 * Jacobian (
for the test function) and the quadrature weight and Jacobian
1045 * data field as where it is read from in @p get_gradient. Therefore, you
1046 * need to make sure to not read from the same quadrature
point again after
1047 * having called @p submit_gradient on that particular quadrature
point. In
1048 *
general, it is a good idea to
copy the result of @p get_gradient when it
1049 * is used more often than once. <li>Next follows the summation over
1050 * quadrature points
for all test
functions that corresponds to the actual
1051 * integration step. For the Laplace
operator, we just multiply by the
1052 *
gradient, so we
call the integrate function with the respective argument
1053 *
set. If you have an equation where you test by both the
values of the
1055 * to
true. Calling
first the integrate function
for values and then
1057 *
call will internally overwrite the results from the
first call. Note that
1058 * there is no function argument
for the
second derivative
for integrate
1059 * step. <li>Eventually, the local contributions in the vector
1060 * @f$v_\mathrm{cell}@f$ as mentioned in the introduction need to be added into
1061 * the result vector (and constraints are applied). This is done with a
call
1062 * to @p distribute_local_to_global, the same name as the corresponding
1064 * in the
FEEvaluation object, as are the indices between local and global
1065 * degrees of freedom). </ol>
1068 *
template <
int dim,
int fe_degree,
typename number>
1069 *
void LaplaceOperator<dim, fe_degree, number>::local_apply(
1073 *
const std::pair<unsigned int, unsigned int> & cell_range)
const
1077 *
for (
unsigned int cell = cell_range.first; cell < cell_range.second; ++cell)
1083 * phi.read_dof_values(src);
1085 *
for (
unsigned int q = 0; q < phi.n_q_points; ++q)
1086 * phi.submit_gradient(coefficient(cell, q) * phi.get_gradient(q), q);
1088 * phi.distribute_local_to_global(dst);
1096 * This function implements the
loop over all cells
for the
1097 * Base::apply_add() interface. This is done with the @p cell_loop of the
1098 *
MatrixFree class, which takes the operator() of this class with arguments
1099 *
MatrixFree, OutVector, InVector, cell_range. When working with MPI
1100 * parallelization (but no threading) as is done in this tutorial program,
1101 * the cell loop corresponds to the following three lines of code:
1105 * <div class=CodeFragmentInTutorialComment>
1107 * src.update_ghost_values();
1108 * local_apply(*this->data, dst, src,
std::make_pair(0U,
1109 * data.n_cell_batches()));
1116 * Here, the two calls update_ghost_values() and compress() perform the data
1117 * exchange on processor boundaries for MPI, once for the source vector
1118 * where we need to read from entries owned by remote processors, and once
1119 * for the destination vector where we have accumulated parts of the
1120 * residuals that need to be added to the respective entry of the owner
1121 * processor. However,
MatrixFree::cell_loop does not only abstract away
1122 * those two calls, but also performs some additional optimizations. On the
1123 * one hand, it will split the update_ghost_values() and compress() calls in
1124 * a way to allow for overlapping communication and computation. The
1125 * local_apply function is then called with three cell ranges representing
1126 * partitions of the cell range from 0 to
MatrixFree::n_cell_batches(). On
1127 * the other hand, cell_loop also supports thread parallelism in which case
1128 * the cell ranges are split into smaller chunks and scheduled in an
1129 * advanced way that avoids access to the same vector entry by several
1130 * threads. That feature is explained in @ref step_48 "step-48".
1134 * Note that after the cell loop, the constrained degrees of freedom need to
1135 * be touched once more for sensible vmult() operators: Since the assembly
1136 * loop automatically resolves constraints (just as the
1138 * compute any contribution for constrained degrees of freedom, leaving the
1139 * respective entries zero. This would represent a matrix that had empty
1140 * rows and columns for constrained degrees of freedom. However, iterative
1141 * solvers like CG only work for non-singular matrices. The easiest way to
1142 * do that is to set the sub-block of the matrix that corresponds to
1143 * constrained DoFs to an
identity matrix, in which case application of the
1144 * matrix would simply copy the elements of the right hand side vector into
1145 * the left hand side. Fortunately, the vmult() implementations
1147 * apply_add() function, so we do not need to take further action here.
1152 * with MPI, there is one aspect to be careful about — the indexing
1153 * used for accessing the vector. For performance reasons,
MatrixFree and
1154 *
FEEvaluation are designed to access vectors in MPI-local index space also
1155 * when working with multiple processors. Working in local index space means
1156 * that no index translation needs to be performed at the place the vector
1157 * access happens, apart from the unavoidable indirect addressing. However,
1158 * local index spaces are ambiguous: While it is standard convention to
1159 * access the locally owned range of a vector with indices between 0 and the
1160 * local size, the numbering is not so clear for the ghosted entries and
1161 * somewhat arbitrary. For the matrix-vector product, only the indices
1162 * appearing on locally owned cells (plus those referenced via hanging node
1163 * constraints) are necessary. However, in deal.II we often set all the
1164 * degrees of freedom on ghosted elements as ghosted vector entries, called
1166 * @ref GlossLocallyRelevantDof "locally relevant DoFs described in the glossary".
1167 * In that case, the MPI-local index of a ghosted vector entry can in
1168 * general be different in the two possible ghost sets, despite referring
1169 * to the same global index. To avoid problems,
FEEvaluation checks that
1170 * the partitioning of the vector used for the matrix-vector product does
1171 * indeed match with the partitioning of the indices in
MatrixFree by a
1175 * mechanism to fit the ghost set to the correct layout. This happens in the
1176 * ghost region of the vector, so keep in mind that the ghost region might
1177 * be modified in both the destination and source vector after a call to a
1178 * vmult() method. This is legitimate because the ghost region of a
1179 * distributed deal.II vector is a mutable section and filled on
1180 * demand. Vectors used in matrix-vector products must not be ghosted upon
1181 * entry of vmult() functions, so no information gets lost.
1184 * template <
int dim,
int fe_degree, typename number>
1185 *
void LaplaceOperator<dim, fe_degree, number>::apply_add(
1189 * this->data->cell_loop(&LaplaceOperator::local_apply,
this, dst, src);
1196 * The following function implements the computation of the
diagonal of the
1198 * turns out to be more complicated than evaluating the
1199 *
operator. Fundamentally, we could obtain a
matrix representation of the
1200 *
operator by applying the
operator on <i>all</i> unit vectors. Of course,
1201 * that would be very inefficient since we would need to perform <i>n</i>
1202 *
operator evaluations to retrieve the whole
matrix. Furthermore,
this
1203 * approach would completely ignore the
matrix sparsity. On an individual
1204 * cell, however,
this is the way to go and actually not that inefficient as
1205 * there usually is a coupling between all degrees of freedom
inside the
1211 * layout. This vector is encapsulated in a member called
1212 * inverse_diagonal_entries of type
DiagonalMatrix in the base
class
1214 * need to initialize and then get the vector representing the
diagonal
1215 * entries in the
matrix. As to the actual
diagonal computation, we again
1216 * use the cell_loop infrastructure of
MatrixFree to invoke a local worker
1217 * routine called local_compute_diagonal(). Since we will only write into a
1218 * vector but not have any source vector, we put a dummy argument of type
1219 * <tt>
unsigned int</tt> in place of the source vector to confirm with the
1220 * cell_loop interface. After the
loop, we need to
set the vector entries
1221 * subject to Dirichlet boundary conditions to one (either those on the
1223 * the indices at the interface between different grid levels in adaptive
1224 * multigrid). This is done through the function
1226 * with the setting in the matrix-vector product provided by the Base
1227 * operator. Finally, we need to
invert the diagonal entries which is the
1228 * form required by the Chebyshev smoother based on the Jacobi iteration. In
1229 * the loop, we assert that all entries are non-zero, because they should
1230 * either have obtained a positive contribution from integrals or be
1231 * constrained and treated by @p set_constrained_entries_to_one() following
1235 * template <
int dim,
int fe_degree, typename number>
1236 *
void LaplaceOperator<dim, fe_degree, number>::compute_diagonal()
1242 * this->data->initialize_dof_vector(inverse_diagonal);
1243 *
unsigned int dummy = 0;
1244 * this->data->cell_loop(&LaplaceOperator::local_compute_diagonal,
1251 *
for (
unsigned int i = 0; i < inverse_diagonal.locally_owned_size(); ++i)
1253 *
Assert(inverse_diagonal.local_element(i) > 0.,
1254 * ExcMessage(
"No diagonal entry in a positive definite operator "
1255 *
"should be zero"));
1256 * inverse_diagonal.local_element(i) =
1257 * 1. / inverse_diagonal.local_element(i);
1266 * columns in the local
matrix and putting the entry 1 in the <i>i</i>th
1267 * slot and a zero entry in all other slots, i.e., we
apply the cell-wise
1268 * differential
operator on one unit vector at a time. The inner part
1270 * FEEvalution::integrate, is exactly the same as in the local_apply
1271 * function. Afterwards, we pick out the <i>i</i>th entry of the local
1272 * result and put it to a temporary storage (as we overwrite all entries in
1274 * iteration). Finally, the temporary storage is written to the destination
1276 *
FEEvaluation::submit_dof_value() to read and write to the data field that
1277 *
FEEvaluation uses for the integration on the one hand and writes into the
1278 * global vector on the other hand.
1282 * Given that we are only interested in the matrix diagonal, we simply throw
1283 * away all other entries of the local matrix that have been computed along
1284 * the way. While it might seem wasteful to compute the complete cell matrix
1285 * and then throw away everything but the diagonal, the integration are so
1286 * efficient that the computation does not take too much time. Note that the
1287 * complexity of operator evaluation per element is @f$\mathcal
1288 * O((p+1)^{
d+1})@f$
for polynomial degree @f$k@f$, so computing the whole matrix
1289 * costs us @f$\mathcal O((p+1)^{2
d+1})@f$ operations, not too far away from
1290 * @f$\mathcal O((p+1)^{2
d})@f$ complexity
for computing the diagonal with
1292 * vectorization and other optimizations, the diagonal computation with
this
1293 * function is actually the fastest (simple) variant. (It would be possible
1294 * to compute the
diagonal with
sum factorization techniques in @f$\mathcal
1295 * O((p+1)^{
d+1})@f$ operations involving specifically adapted
1296 * kernels—but since such kernels are only useful in that particular
1297 * context and the
diagonal computation is typically not on the critical
1298 * path, they have not been implemented in deal.II.)
1302 * Note that the code that calls distribute_local_to_global on the vector to
1303 * accumulate the
diagonal entries into the global
matrix has some
1304 * limitations. For operators with hanging node constraints that distribute
1305 * an integral contribution of a constrained DoF to several other entries
1306 *
inside the distribute_local_to_global
call, the vector
interface used
1307 * here does not exactly compute the
diagonal entries, but lumps some
1310 * result is correct up to discretization accuracy as explained in <a
1311 * href=
"http://dx.doi.org/10.4208/cicp.101214.021015a">Kormann (2016),
1312 * section 5.3</a>, but not mathematically
equal. In
this tutorial program,
1313 * no harm can happen because the
diagonal is only used
for the multigrid
1314 *
level matrices where no hanging node constraints appear.
1317 *
template <
int dim,
int fe_degree,
typename number>
1318 *
void LaplaceOperator<dim, fe_degree, number>::local_compute_diagonal(
1321 *
const unsigned int &,
1322 *
const std::pair<unsigned int, unsigned int> &cell_range)
const
1328 *
for (
unsigned int cell = cell_range.first; cell < cell_range.second; ++cell)
1334 *
for (
unsigned int i = 0; i < phi.dofs_per_cell; ++i)
1336 *
for (
unsigned int j = 0; j < phi.dofs_per_cell; ++j)
1338 * phi.submit_dof_value(make_vectorized_array<number>(1.), i);
1341 *
for (
unsigned int q = 0; q < phi.n_q_points; ++q)
1342 * phi.submit_gradient(coefficient(cell, q) * phi.get_gradient(q),
1345 *
diagonal[i] = phi.get_dof_value(i);
1347 *
for (
unsigned int i = 0; i < phi.dofs_per_cell; ++i)
1348 * phi.submit_dof_value(diagonal[i], i);
1349 * phi.distribute_local_to_global(dst);
1358 * <a name=
"LaplaceProblemclass"></a>
1359 * <h3>LaplaceProblem
class</h3>
1363 * This
class is based on the one in @ref step_16
"step-16". However, we replaced the
1365 * that we can also skip the sparsity patterns. Notice that we define the
1366 * LaplaceOperator
class with the degree of finite element as template
1367 * argument (the value is defined at the top of the file), and that we use
1372 * The
class also has a member variable to keep track of all the detailed
1373 * timings
for setting up the entire chain of data before we actually go
1374 * about solving the problem. In addition, there is an output stream (that
1375 * is disabled by
default) that can be used to output details
for the
1376 * individual setup operations instead of the summary only that is printed
1381 * Since
this program is designed to be used with MPI, we also provide the
1382 * usual @p pcout output stream that only prints the information of the
1383 * processor with MPI rank 0. The grid used
for this programs can either be
1384 * a distributed
triangulation based on p4est (in
case deal.II is configured
1385 * to use p4est), otherwise it is a
serial grid that only runs without MPI.
1388 *
template <
int dim>
1389 *
class LaplaceProblem
1396 *
void setup_system();
1397 *
void assemble_rhs();
1399 *
void output_results(
const unsigned int cycle)
const;
1401 * #ifdef DEAL_II_WITH_P4EST
1413 *
using SystemMatrixType =
1414 * LaplaceOperator<dim, degree_finite_element, double>;
1415 * SystemMatrixType system_matrix;
1418 *
using LevelMatrixType = LaplaceOperator<dim, degree_finite_element, float>;
1424 *
double setup_time;
1433 * When we initialize the finite element, we of course have to use the
1434 * degree specified at the top of the file as well (otherwise, an exception
1435 * will be thrown at some point, since the computational kernel defined in
1436 * the templated LaplaceOperator
class and the information from the finite
1437 * element read out by
MatrixFree will not match). The constructor of the
1439 * conform to the 2:1 cell balance over
vertices, which is needed
for the
1440 * convergence of the geometric multigrid routines. For the distributed
1441 * grid, we also need to specifically enable the multigrid hierarchy.
1444 *
template <
int dim>
1445 * LaplaceProblem<dim>::LaplaceProblem()
1446 * #ifdef DEAL_II_WITH_P4EST
1450 * dim>::construct_multigrid_hierarchy)
1454 * , fe(degree_finite_element)
1460 * The LaplaceProblem
class holds an additional output stream that
1461 * collects detailed timings about the setup phase. This stream, called
1462 * time_details, is disabled by
default through the @p
false argument
1463 * specified here. For detailed timings, removing the @p
false argument
1464 * prints all the details.
1467 * , time_details(std::cout,
1477 * <a name=
"LaplaceProblemsetup_system"></a>
1478 * <h4>LaplaceProblem::setup_system</h4>
1482 * The setup stage is in analogy to @ref step_16
"step-16" with relevant changes due to the
1484 * including the degrees of freedom
for the multigrid levels, and to
1485 * initialize constraints from hanging nodes and homogeneous Dirichlet
1486 * conditions. Since we intend to use
this programs in %
parallel with MPI,
1487 * we need to make sure that the constraints get to know the locally
1488 * relevant degrees of freedom, otherwise the storage would explode when
1489 *
using more than a few hundred millions of degrees of freedom, see
1490 * @ref step_40
"step-40".
1494 * Once we have created the multigrid dof_handler and the constraints, we
1496 * each
level of the multigrid scheme. The main action is to
set up the
1497 * <code>
MatrixFree </code> instance
for the problem. The base
class of the
1499 * initialized with a shared pointer to
MatrixFree object. This way, we can
1500 * simply create it here and then pass it on to the system
matrix and
level
1501 * matrices, respectively. For setting up
MatrixFree, we need to activate
1502 * the update flag in the AdditionalData field of
MatrixFree that enables
1503 * the storage of quadrature
point coordinates in real space (by
default, it
1504 * only caches data
for gradients (inverse transposed Jacobians) and JxW
1505 * values). Note that
if we
call the
reinit function without specifying the
1506 *
level (i.e., giving <code>
level = numbers::invalid_unsigned_int</code>),
1507 *
MatrixFree constructs a
loop over the active cells. In
this tutorial, we
1508 *
do not use threads in addition to MPI, which is why we explicitly disable
1511 * and vectors are initialized as explained above.
1514 *
template <
int dim>
1515 *
void LaplaceProblem<dim>::setup_system()
1520 * system_matrix.clear();
1521 * mg_matrices.clear_elements();
1523 * dof_handler.distribute_dofs(fe);
1524 * dof_handler.distribute_mg_dofs();
1526 * pcout <<
"Number of degrees of freedom: " << dof_handler.n_dofs()
1529 *
const IndexSet locally_relevant_dofs =
1532 * constraints.clear();
1533 * constraints.reinit(locally_relevant_dofs);
1537 * constraints.close();
1538 * setup_time += time.wall_time();
1539 * time_details <<
"Distribute DoFs & B.C. (CPU/wall) " << time.cpu_time()
1540 * <<
"s/" << time.wall_time() <<
's' << std::endl;
1547 * additional_data.mapping_update_flags =
1549 * std::shared_ptr<MatrixFree<dim, double>> system_mf_storage(
1551 * system_mf_storage->reinit(mapping,
1556 * system_matrix.initialize(system_mf_storage);
1559 * system_matrix.evaluate_coefficient(Coefficient<dim>());
1561 * system_matrix.initialize_dof_vector(solution);
1562 * system_matrix.initialize_dof_vector(system_rhs);
1564 * setup_time += time.wall_time();
1565 * time_details <<
"Setup matrix-free system (CPU/wall) " << time.cpu_time()
1566 * <<
"s/" << time.wall_time() <<
's' << std::endl;
1571 * Next, initialize the matrices
for the multigrid method on all the
1573 * the indices subject to boundary conditions as well as the indices on
1574 * edges between different refinement levels as described in the @ref step_16
"step-16"
1575 * tutorial program. We then go through the levels of the mesh and
1576 * construct the constraints and matrices on each
level. These follow
1577 * closely the construction of the system
matrix on the original mesh,
1578 * except the slight difference in naming when accessing information on
1579 * the levels rather than the active cells.
1582 *
const unsigned int nlevels =
triangulation.n_global_levels();
1583 * mg_matrices.resize(0, nlevels - 1);
1585 *
const std::set<types::boundary_id> dirichlet_boundary_ids = {0};
1586 * mg_constrained_dofs.initialize(dof_handler);
1587 * mg_constrained_dofs.make_zero_boundary_constraints(dof_handler,
1588 * dirichlet_boundary_ids);
1595 * level_constraints.
reinit(relevant_dofs);
1596 * level_constraints.add_lines(
1597 * mg_constrained_dofs.get_boundary_indices(
level));
1598 * level_constraints.close();
1603 * additional_data.mapping_update_flags =
1605 * additional_data.mg_level =
level;
1606 * std::shared_ptr<MatrixFree<dim, float>> mg_mf_storage_level(
1608 * mg_mf_storage_level->reinit(mapping,
1610 * level_constraints,
1614 * mg_matrices[
level].initialize(mg_mf_storage_level,
1615 * mg_constrained_dofs,
1617 * mg_matrices[
level].evaluate_coefficient(Coefficient<dim>());
1619 * setup_time += time.wall_time();
1620 * time_details <<
"Setup matrix-free levels (CPU/wall) " << time.cpu_time()
1621 * <<
"s/" << time.wall_time() <<
's' << std::endl;
1629 * <a name=
"LaplaceProblemassemble_rhs"></a>
1630 * <h4>LaplaceProblem::assemble_rhs</h4>
1634 * The
assemble function is very simple since all we have to
do is to
1636 * cached in the
MatrixFree class, which we query from
1639 * alternative), we must not forget to
call compress() at the end of the
1640 * assembly to send all the contributions of the right hand side to the
1641 * owner of the respective degree of freedom.
1644 * template <
int dim>
1645 *
void LaplaceProblem<dim>::assemble_rhs()
1651 * *system_matrix.get_matrix_free());
1652 *
for (
unsigned int cell = 0;
1653 * cell < system_matrix.get_matrix_free()->n_cell_batches();
1657 *
for (
unsigned int q = 0; q < phi.n_q_points; ++q)
1658 * phi.submit_value(make_vectorized_array<double>(1.0), q);
1660 * phi.distribute_local_to_global(system_rhs);
1664 * setup_time += time.wall_time();
1665 * time_details <<
"Assemble right hand side (CPU/wall) " << time.cpu_time()
1666 * <<
"s/" << time.wall_time() <<
's' << std::endl;
1674 * <a name=
"LaplaceProblemsolve"></a>
1675 * <h4>LaplaceProblem::solve</h4>
1679 * The solution process is similar as in @ref step_16
"step-16". We start with the setup of
1682 * interpolation between the grid levels with the same fast
sum
1683 * factorization kernels that get also used in
FEEvaluation.
1686 *
template <
int dim>
1687 *
void LaplaceProblem<dim>::solve()
1691 * mg_transfer.build(dof_handler);
1692 * setup_time += time.wall_time();
1693 * time_details <<
"MG build transfer time (CPU/wall) " << time.cpu_time()
1694 * <<
"s/" << time.wall_time() <<
"s\n";
1699 * As a smoother,
this tutorial program uses a Chebyshev iteration instead
1700 * of SOR in @ref step_16
"step-16". (SOR would be very difficult to implement because we
1701 *
do not have the
matrix elements available explicitly, and it is
1702 * difficult to make it work efficiently in %
parallel.) The smoother is
1703 * initialized with our
level matrices and the mandatory additional data
1704 *
for the Chebyshev smoother. We use a relatively high degree here (5),
1705 * since
matrix-vector products are comparably cheap. We choose to smooth
1707 * in the smoother where @f$\hat{\
lambda}_{\
max}@f$ is an estimate of the
1708 * largest eigenvalue (the factor 1.2 is applied inside
1710 * Chebyshev initialization performs a few steps of a CG algorithm
1711 * without preconditioner. Since the highest eigenvalue is usually the
1712 * easiest one to find and a rough estimate is enough, we choose 10
1713 * iterations. Finally, we also
set the inner preconditioner type in the
1714 * Chebyshev method which is a Jacobi iteration. This is represented by
1716 * by our LaplaceOperator
class.
1720 * On
level zero, we initialize the smoother differently because we want
1722 * allows the user to
switch to solver mode where the number of iterations
1723 * is internally chosen to the correct
value. In the additional data
1724 * object,
this setting is activated by choosing the polynomial degree to
1727 *
matrix. The number of steps in the Chebyshev smoother are chosen such
1728 * that the Chebyshev convergence estimates guarantee to
reduce the
1729 * residual by the number specified in the variable @p
1730 * smoothing_range. Note that
for solving, @p smoothing_range is a
1731 * relative tolerance and chosen smaller than one, in
this case, we select
1732 * three orders of magnitude, whereas it is a number larger than 1 when
1737 * From a computational
point of view, the Chebyshev iteration is a very
1738 * attractive coarse grid solver as
long as the coarse size is
1739 * moderate. This is because the Chebyshev method performs only
1740 *
matrix-vector products and vector updates, which typically parallelize
1741 * better to the largest cluster size with more than a few tens of
1742 * thousands of cores than inner product involved in other iterative
1743 * methods. The former involves only local communication between neighbors
1744 * in the (coarse) mesh, whereas the latter
requires global communication
1745 * over all processors.
1748 *
using SmootherType =
1761 * smoother_data[
level].smoothing_range = 15.;
1762 * smoother_data[
level].degree = 5;
1763 * smoother_data[
level].eig_cg_n_iterations = 10;
1767 * smoother_data[0].smoothing_range = 1
e-3;
1769 * smoother_data[0].eig_cg_n_iterations = mg_matrices[0].m();
1771 * mg_matrices[
level].compute_diagonal();
1772 * smoother_data[
level].preconditioner =
1773 * mg_matrices[
level].get_matrix_diagonal_inverse();
1775 * mg_smoother.initialize(mg_matrices, smoother_data);
1783 * The next step is to
set up the
interface matrices that are needed for the
1784 *
case with hanging nodes. The adaptive multigrid realization in deal.II
1785 * implements an approach called local smoothing. This means that the
1786 * smoothing on the finest
level only covers the local part of the mesh
1787 * defined by the fixed (finest) grid
level and ignores parts of the
1788 * computational domain where the terminal cells are coarser than
this
1789 *
level. As the method progresses to coarser levels, more and more of the
1790 * global mesh will be covered. At some coarser
level, the whole mesh will
1791 * be covered. Since all
level matrices in the multigrid method cover a
1792 * single
level in the mesh, no hanging nodes appear on the
level matrices.
1793 * At the
interface between multigrid levels, homogeneous Dirichlet boundary
1794 * conditions are
set while smoothing. When the residual is transferred to
1795 * the next coarser
level, however, the coupling over the multigrid
1796 *
interface needs to be taken into account. This is done by the so-called
1797 * interface (or edge) matrices that compute the part of the residual that
1799 * homogeneous Dirichlet conditions. We refer to the @ref mg_paper
1800 *
"Multigrid paper by Janssen and Kanschat" for more details.
1804 * For the implementation of those
interface matrices, there is already a
1808 * vmult() and @p Tvmult() operations (that were originally written for
1809 * matrices, hence expecting those names). Note that vmult_interface_down
1810 * is used during the restriction phase of the multigrid V-cycle, whereas
1811 * vmult_interface_up is used during the prolongation phase.
1815 * Once the interface matrix is created, we set up the remaining
Multigrid
1816 * preconditioner infrastructure in complete analogy to @ref step_16 "step-16" to obtain
1817 * a @p preconditioner
object that can be applied to a matrix.
1824 * mg_interface_matrices;
1825 * mg_interface_matrices.resize(0,
triangulation.n_global_levels() - 1);
1828 * mg_interface_matrices[
level].initialize(mg_matrices[
level]);
1830 * mg_interface_matrices);
1833 * mg_matrix, mg_coarse, mg_transfer, mg_smoother, mg_smoother);
1834 *
mg.set_edge_matrices(mg_interface, mg_interface);
1839 * preconditioner(dof_handler,
mg, mg_transfer);
1843 * The setup of the multigrid routines is quite easy and one cannot see
1844 * any difference in the solve process as compared to @ref step_16 "step-16". All the
1845 * magic is hidden behind the implementation of the LaplaceOperator::vmult
1846 * operation. Note that we print out the solve time and the accumulated
1847 * setup time through standard out, i.e., in any case, whereas detailed
1848 * times for the setup operations are only printed in case the flag for
1849 * detail_times in the constructor is changed.
1855 *
SolverControl solver_control(100, 1e-12 * system_rhs.l2_norm());
1857 * setup_time += time.wall_time();
1858 * time_details << "MG build smoother time (CPU/wall) " << time.cpu_time()
1859 * << "s/" << time.wall_time() << "s\n";
1860 * pcout << "Total setup time (wall) " << setup_time << "s\n";
1864 * constraints.set_zero(solution);
1865 * cg.solve(system_matrix, solution, system_rhs, preconditioner);
1867 * constraints.distribute(solution);
1869 * pcout << "Time solve (" << solver_control.last_step() << " iterations)"
1870 * << (solver_control.last_step() < 10 ? " " : " ") << "(CPU/wall) "
1871 * << time.cpu_time() << "s/" << time.wall_time() << "s\n";
1879 * <a name="LaplaceProblemoutput_results"></a>
1880 * <h4>LaplaceProblem::output_results</h4>
1884 * Here is the data output, which is a simplified version of @ref step_5 "step-5". We use
1885 * the standard VTU (= compressed VTK) output for each grid produced in the
1886 * refinement process. In addition, we use a compression algorithm that is
1887 * optimized for speed rather than disk usage. The default setting (which
1888 * optimizes for disk usage) makes saving the output take about 4 times as
1889 *
long as running the linear solver, while setting
1891 * best_speed lowers this to only one fourth the time
1892 * of the linear solve.
1896 * We disable the output when the mesh gets too large. A variant of this
1897 * program has been run on hundreds of thousands MPI ranks with as many as
1898 * 100 billion grid cells, which is not directly accessible to classical
1899 * visualization tools.
1902 * template <
int dim>
1903 *
void LaplaceProblem<dim>::output_results(const
unsigned int cycle) const
1911 * solution.update_ghost_values();
1913 * data_out.add_data_vector(solution,
"solution");
1914 * data_out.build_patches(mapping);
1918 * data_out.set_flags(flags);
1919 * data_out.write_vtu_with_pvtu_record(
1920 *
"./",
"solution", cycle, MPI_COMM_WORLD, 3);
1922 * time_details <<
"Time write output (CPU/wall) " << time.cpu_time()
1923 * <<
"s/" << time.wall_time() <<
"s\n";
1931 * <a name=
"LaplaceProblemrun"></a>
1932 * <h4>LaplaceProblem::run</h4>
1936 * The function that runs the program is very similar to the one in
1937 * @ref step_16
"step-16". We
do few refinement steps in 3
d compared to 2
d, but that
's
1942 * Before we run the program, we output some information about the detected
1943 * vectorization level as discussed in the introduction.
1946 * template <int dim>
1947 * void LaplaceProblem<dim>::run()
1950 * const unsigned int n_vect_doubles = VectorizedArray<double>::size();
1951 * const unsigned int n_vect_bits = 8 * sizeof(double) * n_vect_doubles;
1953 * pcout << "Vectorization over " << n_vect_doubles
1954 * << " doubles = " << n_vect_bits << " bits ("
1955 * << Utilities::System::get_current_vectorization_level() << ')
'
1959 * for (unsigned int cycle = 0; cycle < 9 - dim; ++cycle)
1961 * pcout << "Cycle " << cycle << std::endl;
1965 * GridGenerator::hyper_cube(triangulation, 0., 1.);
1966 * triangulation.refine_global(3 - dim);
1968 * triangulation.refine_global(1);
1972 * output_results(cycle);
1973 * pcout << std::endl;
1976 * } // namespace Step37
1983 * <a name="Thecodemaincodefunction"></a>
1984 * <h3>The <code>main</code> function</h3>
1988 * Apart from the fact that we set up the MPI framework according to @ref step_40 "step-40",
1989 * there are no surprises in the main function.
1992 * int main(int argc, char *argv[])
1996 * using namespace Step37;
1998 * Utilities::MPI::MPI_InitFinalize mpi_init(argc, argv, 1);
2000 * LaplaceProblem<dimension> laplace_problem;
2001 * laplace_problem.run();
2003 * catch (std::exception &exc)
2005 * std::cerr << std::endl
2007 * << "----------------------------------------------------"
2009 * std::cerr << "Exception on processing: " << std::endl
2010 * << exc.what() << std::endl
2011 * << "Aborting!" << std::endl
2012 * << "----------------------------------------------------"
2018 * std::cerr << std::endl
2020 * << "----------------------------------------------------"
2022 * std::cerr << "Unknown exception!" << std::endl
2023 * << "Aborting!" << std::endl
2024 * << "----------------------------------------------------"
2032<a name="Results"></a><h1>Results</h1>
2035<a name="Programoutput"></a><h3>Program output</h3>
2038Since this example solves the same problem as @ref step_5 "step-5" (except for
2039a different coefficient), there is little to say about the
2040solution. We show a picture anyway, illustrating the size of the
2041solution through both isocontours and volume rendering:
2043<img src="https://www.dealii.org/images/steps/developer/step-37.solution.png" alt="">
2045Of more interest is to evaluate some aspects of the multigrid solver.
2046When we run this program in 2D for quadratic (@f$Q_2@f$) elements, we get the
2047following output (when run on one core in release mode):
2049Vectorization over 2 doubles = 128 bits (SSE2)
2051Number of degrees of freedom: 81
2052Total setup time (wall) 0.00159788s
2053Time solve (6 iterations) (CPU/wall) 0.000951s/0.000951052s
2056Number of degrees of freedom: 289
2057Total setup time (wall) 0.00114608s
2058Time solve (6 iterations) (CPU/wall) 0.000935s/0.000934839s
2061Number of degrees of freedom: 1089
2062Total setup time (wall) 0.00244665s
2063Time solve (6 iterations) (CPU/wall) 0.00207s/0.002069s
2066Number of degrees of freedom: 4225
2067Total setup time (wall) 0.00678205s
2068Time solve (6 iterations) (CPU/wall) 0.005616s/0.00561595s
2071Number of degrees of freedom: 16641
2072Total setup time (wall) 0.0241671s
2073Time solve (6 iterations) (CPU/wall) 0.019543s/0.0195441s
2076Number of degrees of freedom: 66049
2077Total setup time (wall) 0.0967851s
2078Time solve (6 iterations) (CPU/wall) 0.07457s/0.0745709s
2081Number of degrees of freedom: 263169
2082Total setup time (wall) 0.346374s
2083Time solve (6 iterations) (CPU/wall) 0.260042s/0.265033s
2086As in @ref step_16 "step-16", we see that the number of CG iterations remains constant with
2087increasing number of degrees of freedom. A constant number of iterations
2088(together with optimal computational properties) means that the computing time
2089approximately quadruples as the problem size quadruples from one cycle to the
2090next. The code is also very efficient in terms of storage. Around 2-4 million
2091degrees of freedom fit into 1 GB of memory, see also the MPI results below. An
2092interesting fact is that solving one linear system is cheaper than the setup,
2093despite not building a matrix (approximately half of which is spent in the
2094DoFHandler::distribute_dofs() and DoFHandler::distribute_mg_dofs()
2095calls). This shows the high efficiency of this approach, but also that the
2096deal.II data structures are quite expensive to set up and the setup cost must
2097be amortized over several system solves.
2099Not much changes if we run the program in three spatial dimensions. Since we
2100use uniform mesh refinement, we get eight times as many elements and
2101approximately eight times as many degrees of freedom with each cycle:
2104Vectorization over 2 doubles = 128 bits (SSE2)
2106Number of degrees of freedom: 125
2107Total setup time (wall) 0.00231099s
2108Time solve (6 iterations) (CPU/wall) 0.000692s/0.000922918s
2111Number of degrees of freedom: 729
2112Total setup time (wall) 0.00289083s
2113Time solve (6 iterations) (CPU/wall) 0.001534s/0.0024128s
2116Number of degrees of freedom: 4913
2117Total setup time (wall) 0.0143182s
2118Time solve (6 iterations) (CPU/wall) 0.010785s/0.0107841s
2121Number of degrees of freedom: 35937
2122Total setup time (wall) 0.087064s
2123Time solve (6 iterations) (CPU/wall) 0.063522s/0.06545s
2126Number of degrees of freedom: 274625
2127Total setup time (wall) 0.596306s
2128Time solve (6 iterations) (CPU/wall) 0.427757s/0.431765s
2131Number of degrees of freedom: 2146689
2132Total setup time (wall) 4.96491s
2133Time solve (6 iterations) (CPU/wall) 3.53126s/3.56142s
2136Since it is so easy, we look at what happens if we increase the polynomial
2137degree. When selecting the degree as four in 3D, i.e., on @f$\mathcal Q_4@f$
2138elements, by changing the line <code>const unsigned int
2139degree_finite_element=4;</code> at the top of the program, we get the
2140following program output:
2143Vectorization over 2 doubles = 128 bits (SSE2)
2145Number of degrees of freedom: 729
2146Total setup time (wall) 0.00633097s
2147Time solve (6 iterations) (CPU/wall) 0.002829s/0.00379395s
2150Number of degrees of freedom: 4913
2151Total setup time (wall) 0.0174279s
2152Time solve (6 iterations) (CPU/wall) 0.012255s/0.012254s
2155Number of degrees of freedom: 35937
2156Total setup time (wall) 0.082655s
2157Time solve (6 iterations) (CPU/wall) 0.052362s/0.0523629s
2160Number of degrees of freedom: 274625
2161Total setup time (wall) 0.507943s
2162Time solve (6 iterations) (CPU/wall) 0.341811s/0.345788s
2165Number of degrees of freedom: 2146689
2166Total setup time (wall) 3.46251s
2167Time solve (7 iterations) (CPU/wall) 3.29638s/3.3265s
2170Number of degrees of freedom: 16974593
2171Total setup time (wall) 27.8989s
2172Time solve (7 iterations) (CPU/wall) 26.3705s/27.1077s
2175Since @f$\mathcal Q_4@f$ elements on a certain mesh correspond to @f$\mathcal Q_2@f$
2176elements on half the mesh size, we can compare the run time at cycle 4 with
2177fourth degree polynomials with cycle 5 using quadratic polynomials, both at
21782.1 million degrees of freedom. The surprising effect is that the solver for
2179@f$\mathcal Q_4@f$ element is actually slightly faster than for the quadratic
2180case, despite using one more linear iteration. The effect that higher-degree
2181polynomials are similarly fast or even faster than lower degree ones is one of
2182the main strengths of matrix-free operator evaluation through sum
2183factorization, see the <a
2184href="http://dx.doi.org/10.1016/j.compfluid.2012.04.012">matrix-free
2185paper</a>. This is fundamentally different to matrix-based methods that get
2186more expensive per unknown as the polynomial degree increases and the coupling
2189In addition, also the setup gets a bit cheaper for higher order, which is
2190because fewer elements need to be set up.
2192Finally, let us look at the timings with degree 8, which corresponds to
2193another round of mesh refinement in the lower order methods:
2196Vectorization over 2 doubles = 128 bits (SSE2)
2198Number of degrees of freedom: 4913
2199Total setup time (wall) 0.0842004s
2200Time solve (8 iterations) (CPU/wall) 0.019296s/0.0192959s
2203Number of degrees of freedom: 35937
2204Total setup time (wall) 0.327048s
2205Time solve (8 iterations) (CPU/wall) 0.07517s/0.075999s
2208Number of degrees of freedom: 274625
2209Total setup time (wall) 2.12335s
2210Time solve (8 iterations) (CPU/wall) 0.448739s/0.453698s
2213Number of degrees of freedom: 2146689
2214Total setup time (wall) 16.1743s
2215Time solve (8 iterations) (CPU/wall) 3.95003s/3.97717s
2218Number of degrees of freedom: 16974593
2219Total setup time (wall) 130.8s
2220Time solve (8 iterations) (CPU/wall) 31.0316s/31.767s
2223Here, the initialization seems considerably slower than before, which is
2224mainly due to the computation of the diagonal of the matrix, which actually
2225computes a 729 x 729 matrix on each cell and throws away everything but the
2226diagonal. The solver times, however, are again very close to the quartic case,
2227showing that the linear increase with the polynomial degree that is
2228theoretically expected is almost completely offset by better computational
2229characteristics and the fact that higher order methods have a smaller share of
2230degrees of freedom living on several cells that add to the evaluation
2233<a name="Comparisonwithasparsematrix"></a><h3>Comparison with a sparse matrix</h3>
2236In order to understand the capabilities of the matrix-free implementation, we
2237compare the performance of the 3d example above with a sparse matrix
2238implementation based on TrilinosWrappers::SparseMatrix by measuring both the
2239computation times for the initialization of the problem (distribute DoFs,
2240setup and assemble matrices, setup multigrid structures) and the actual
2241solution for the matrix-free variant and the variant based on sparse
2242matrices. We base the preconditioner on float numbers and the actual matrix
2243and vectors on double numbers, as shown above. Tests are run on an Intel Core
2244i7-5500U notebook processor (two cores and <a
2245href="http://en.wikipedia.org/wiki/Advanced_Vector_Extensions">AVX</a>
2246support, i.e., four operations on doubles can be done with one CPU
2247instruction, which is heavily used in FEEvaluation), optimized mode, and two
2250<table align="center" class="doxtable">
2253 <th colspan="2">Sparse matrix</th>
2254 <th colspan="2">Matrix-free implementation</th>
2258 <th>Setup + assemble</th>
2259 <th> Solve </th>
2260 <th>Setup + assemble</th>
2261 <th> Solve </th>
2264 <td align="right">125</td>
2265 <td align="center">0.0042s</td>
2266 <td align="center">0.0012s</td>
2267 <td align="center">0.0022s</td>
2268 <td align="center">0.00095s</td>
2271 <td align="right">729</td>
2272 <td align="center">0.012s</td>
2273 <td align="center">0.0040s</td>
2274 <td align="center">0.0027s</td>
2275 <td align="center">0.0021s</td>
2278 <td align="right">4,913</td>
2279 <td align="center">0.082s</td>
2280 <td align="center">0.012s</td>
2281 <td align="center">0.011s</td>
2282 <td align="center">0.0057s</td>
2285 <td align="right">35,937</td>
2286 <td align="center">0.73s</td>
2287 <td align="center">0.13s</td>
2288 <td align="center">0.048s</td>
2289 <td align="center">0.040s</td>
2292 <td align="right">274,625</td>
2293 <td align="center">5.43s</td>
2294 <td align="center">1.01s</td>
2295 <td align="center">0.33s</td>
2296 <td align="center">0.25s</td>
2299 <td align="right">2,146,689</td>
2300 <td align="center">43.8s</td>
2301 <td align="center">8.24s</td>
2302 <td align="center">2.42s</td>
2303 <td align="center">2.06s</td>
2307The table clearly shows that the matrix-free implementation is more than twice
2308as fast for the solver, and more than six times as fast when it comes to
2309initialization costs. As the problem size is made a factor 8 larger, we note
2310that the times usually go up by a factor eight, too (as the solver iterations
2311are constant at six). The main deviation is in the sparse matrix between 5k
2312and 36k degrees of freedom, where the time increases by a factor 12. This is
2313the threshold where the (L3) cache in the processor can no longer hold all
2314data necessary for the matrix-vector products and all matrix elements must be
2315fetched from main memory.
2317Of course, this picture does not necessarily translate to all cases, as there
2318are problems where knowledge of matrix entries enables much better solvers (as
2319happens when the coefficient is varying more strongly than in the above
2320example). Moreover, it also depends on the computer system. The present system
2321has good memory performance, so sparse matrices perform comparably
2322well. Nonetheless, the matrix-free implementation gives a nice speedup already
2323for the <i>Q</i><sub>2</sub> elements used in this example. This becomes
2324particularly apparent for time-dependent or nonlinear problems where sparse
2325matrices would need to be reassembled over and over again, which becomes much
2326easier with this class. And of course, thanks to the better complexity of the
2327products, the method gains increasingly larger advantages when the order of the
2328elements increases (the matrix-free implementation has costs
23294<i>d</i><sup>2</sup><i>p</i> per degree of freedom, compared to
23302<i>p<sup>d</sup></i> for the sparse matrix, so it will win anyway for order 4
2333<a name="ResultsforlargescaleparallelcomputationsonSuperMUC"></a><h3> Results for large-scale parallel computations on SuperMUC</h3>
2336As explained in the introduction and the in-code comments, this program can be
2337run in parallel with MPI. It turns out that geometric multigrid schemes work
2338really well and can scale to very large machines. To the authors' knowledge,
2339the geometric multigrid results shown here are the largest computations done
2340with deal.II as of late 2016, run on up to 147,456 cores of the <a
2341href=
"https://www.lrz.de/services/compute/supermuc/systemdescription/">complete
2342SuperMUC Phase 1</a>. The ingredients
for scalability beyond 1000 cores are
2343that no data structure that depends on the global problem size is held in its
2344entirety on a single processor and that the communication is not too frequent
2345in order not to run into latency issues of the network. For PDEs solved with
2346iterative solvers, the communication latency is often the limiting factor,
2347rather than the throughput of the network. For the example of the SuperMUC
2348system, the point-to-point latency between two processors is between 1e-6 and
23491e-5 seconds, depending on the proximity in the MPI network. The matrix-vector
2350products with @p LaplaceOperator from
this class involves several
2351point-to-point communication steps, interleaved with computations on each
2352core. The resulting latency of a matrix-vector product is around 1e-4
2353seconds. Global communication,
for example an @p MPI_Allreduce operation that
2354accumulates the sum of a single number per rank over all ranks in the MPI
2355network, has a latency of 1e-4 seconds. The multigrid V-cycle used in
this
2356program is also a form of global communication. Think about the coarse grid
2357solve that happens on a single processor: It accumulates the contributions
2358from all processors before it starts. When completed, the coarse grid solution
2359is transferred to finer levels, where more and more processors help in
2360smoothing until the fine grid. Essentially, this is a tree-like pattern over
2361the processors in the network and controlled by the mesh. As opposed to the
2362@p MPI_Allreduce operations where the tree in the
reduction is optimized to the
2363actual links in the MPI network, the multigrid V-cycle does this according to
2364the partitioning of the mesh. Thus, we cannot expect the same
2365optimality. Furthermore, the multigrid cycle is not simply a walk up and down
2366the refinement tree, but also communication on each
level when doing the
2367smoothing. In other words, the global communication in multigrid is more
2368challenging and related to the mesh that provides less optimization
2369opportunities. The measured latency of the V-cycle is between 6
e-3 and 2
e-2
2370seconds, i.
e., the same as 60 to 200 MPI_Allreduce operations.
2372The following figure shows a scaling experiments on @f$\mathcal Q_3@f$
2373elements. Along the lines, the problem size is held
constant as the number of
2374cores is increasing. When doubling the number of cores, one expects a halving
2375of the computational time, indicated by the dotted gray lines. The results
2376show that the implementation shows almost ideal behavior until an absolute
2377time of around 0.1 seconds is reached. The solver tolerances have been
set
2378such that the solver performs five iterations. This way of plotting data is
2379the <
b>strong scaling</
b> of the algorithm. As we go to very large core
2380counts, the curves flatten out a bit earlier, which is because of the
2381communication network in SuperMUC where communication between processors
2382farther away is slightly slower.
2384<img src=
"https://www.dealii.org/images/steps/developer/step-37.scaling_strong.png" alt=
"">
2386In addition, the plot also contains results for <
b>weak scaling</
b> that lists
2387how the algorithm behaves as both the number of processor cores and elements
2388is increased at the same pace. In this situation, we expect that the compute
2389time remains
constant. Algorithmically, the number of CG iterations is
2390constant at 5, so we are good from that
end. The lines in the plot are
2391arranged such that the top left
point in each data series represents the same
2392size per processor, namely 131,072 elements (or approximately 3.5 million
2393degrees of freedom per core). The gray lines indicating ideal strong scaling
2394are by the same factor of 8 apart. The results show again that the scaling is
2395almost ideal. The
parallel efficiency when going from 288 cores to 147,456
2396cores is at around 75% for a local problem size of 750,000 degrees of freedom
2397per core which takes 1.0s on 288 cores, 1.03s on 2304 cores, 1.19s on 18k
2398cores, and 1.35s on 147k cores. The algorithms also reach a very high
2399utilization of the processor. The largest computation on 147k cores reaches
2400around 1.7 PFLOPs/s on SuperMUC out of an arithmetic peak of 3.2 PFLOPs/s. For
2401an iterative PDE solver, this is a very high number and significantly more is
2402often only reached for dense linear algebra. Sparse linear algebra is limited
2403to a tenth of this
value.
2405As mentioned in the introduction, the
matrix-
free method reduces the memory
2406consumption of the data structures. Besides the higher performance due to less
2407memory transfer, the algorithms also allow for very large problems to fit into
2408memory. The figure below shows the computational time as we increase the
2409problem size until an upper limit where the computation exhausts memory. We do
2410this for 1k cores, 8k cores, and 65k cores and see that the problem size can
2411be varied over almost two orders of magnitude with ideal scaling. The largest
2412computation shown in this picture involves 292 billion (@f$2.92 \cdot 10^{11}@f$)
2413degrees of freedom. On a DG computation of 147k cores, the above algorithms
2414were also run involving up to 549 billion (2^39) DoFs.
2416<img src=
"https://www.dealii.org/images/steps/developer/step-37.scaling_size.png" alt=
"">
2418Finally, we note that
while performing the tests on the large-
scale system
2419shown above, improvements of the multigrid algorithms in deal.II have been
2420developed. The original version contained the sub-optimal code based on
2422all vector entries are zero) were done on each smoothing
2423operation on each
level, which only became apparent on 65k cores and
2424more. However, the following picture shows that the improvement already pay
2425off on a smaller
scale, here shown on computations on up to 14,336 cores
for
2426@f$\mathcal Q_5@f$ elements:
2428<img src=
"https://www.dealii.org/images/steps/developer/step-37.scaling_oldnew.png" alt=
"">
2431<a name=
"Adaptivity"></a><h3> Adaptivity</h3>
2434As explained in the code, the algorithm presented here is prepared to
run on
2435adaptively refined meshes. If only part of the mesh is refined, the multigrid
2436cycle will
run with local smoothing and impose Dirichlet conditions along the
2437interfaces which differ in refinement
level for smoothing through the
2439distributed over levels, relating the owner of the
level cells to the owner of
2440the
first descendant active cell, there can be an imbalance between different
2441processors in MPI, which limits scalability by a factor of around two to five.
2443<a name=
"Possibilitiesforextensions"></a><h3> Possibilities
for extensions</h3>
2446<a name=
"Kellyerrorestimator"></a><h4> Kelly error estimator </h4>
2449As mentioned above the code is ready
for locally adaptive h-refinement.
2450For the Poisson equation one can employ the Kelly error indicator,
2452with the ghost indices of
parallel vectors.
2453In order to evaluate the jump terms in the error indicator, each MPI process
2454needs to know locally relevant DoFs.
2456some locally relevant DoFs.
2457The ghost indices made available in the vector are a tight set of only those indices
2458that are touched in the cell integrals (including constraint resolution).
2459This choice has performance reasons, because sending all locally relevant degrees
2460of freedom would be too expensive compared to the matrix-vector product.
2461Consequently the solution vector as-is is
2463The trick is to change the ghost part of the partition, for example using a
2468const
IndexSet locally_relevant_dofs =
DoFTools::extract_locally_relevant_dofs(dof_handler);
2470solution.reinit(dof_handler.locally_owned_dofs(),
2471 locally_relevant_dofs,
2473solution.copy_locally_owned_data_from(copy_vec);
2474constraints.distribute(solution);
2475solution.update_ghost_values();
2478<a name="Sharedmemoryparallelization"></a><h4> Shared-memory parallelization</h4>
2481This program is parallelized with MPI only. As an alternative, the
MatrixFree
2482loop can also be issued in
hybrid mode, for example by using MPI parallelizing
2483over the nodes of a cluster and with threads through Intel TBB within the
2484shared memory region of one node. To use this, one would need to both set the
2485number of threads in the MPI_InitFinalize data structure in the main function,
2486and set the
MatrixFree::AdditionalData::tasks_parallel_scheme to
2487partition_color to actually do the loop in
parallel. This use case is
2488discussed in @ref step_48 "step-48".
2490<a name="InhomogeneousDirichletboundaryconditions"></a><h4> Inhomogeneous Dirichlet boundary conditions </h4>
2493The presented program assumes homogeneous Dirichlet boundary conditions. When
2494going to non-homogeneous conditions, the situation is a bit more intricate. To
2495understand how to implement such a setting, let us
first recall how these
2496arise in the mathematical formulation and how they are implemented in a
2497matrix-based variant. In essence, an inhomogeneous Dirichlet condition sets
2498some of the nodal values in the solution to given values rather than
2499determining them through the variational principles,
2501u_h(\mathbf{x}) = \sum_{i\in \mathcal N} \varphi_i(\mathbf{x}) u_i =
2502\sum_{i\in \mathcal N \setminus \mathcal N_D} \varphi_i(\mathbf{x}) u_i +
2503\sum_{i\in \mathcal N_D} \varphi_i(\mathbf{x}) g_i,
2505where @f$u_i@f$ denotes the nodal
values of the solution and @f$\mathcal N@f$ denotes
2506the
set of all nodes. The
set @f$\mathcal N_D\subset \mathcal N@f$ is the subset
2507of the nodes that are subject to Dirichlet boundary conditions where the
2508solution is forced to
equal @f$u_i = g_i = g(\mathbf{x}_i)@f$ as the interpolation
2509of boundary values on the Dirichlet-constrained node points @f$i\in \mathcal
2510N_D@f$. We then insert
this solution
2511representation into the weak form,
e.g. the Laplacian shown above, and move
2512the known quantities to the right hand side:
2514(\nabla \varphi_i, \nabla u_h)_\Omega &=& (\varphi_i, f)_\Omega \quad \Rightarrow \\
2515\sum_{j\in \mathcal N \setminus \mathcal N_D}(\nabla \varphi_i,\nabla \varphi_j)_\Omega \, u_j &=&
2516(\varphi_i, f)_\Omega
2517-\sum_{j\in \mathcal N_D} (\nabla \varphi_i,\nabla\varphi_j)_\Omega\, g_j.
2519In this formula, the equations are tested for all basis
functions @f$\varphi_i@f$
2520with @f$i\in N \setminus \mathcal N_D@f$ that are not related to the nodes
2521constrained by Dirichlet conditions.
2523In the implementation in deal.II, the integrals @f$(\nabla \varphi_i,\nabla \varphi_j)_\Omega@f$
2524on the right hand side are already contained in the local
matrix contributions
2525we
assemble on each cell. When using
2527@ref step_6
"step-6" and @ref step_7
"step-7" tutorial programs, we can account for the contribution of
2528inhomogeneous constraints <i>j</i> by multiplying the columns <i>j</i> and
2529rows <i>i</i> of the local
matrix according to the integrals @f$(\varphi_i,
2530\varphi_j)_\Omega@f$ by the inhomogeneities and subtracting the resulting from
2531the position <i>i</i> in the global right-hand-side vector, see also the @ref
2532constraints module. In essence, we use some of the integrals that get
2533eliminated from the left hand side of the equation to finalize the right hand
2534side contribution. Similar mathematics are also involved when
first writing
2535all entries into a left hand side
matrix and then eliminating
matrix rows and
2538In principle, the components that belong to the constrained degrees of freedom
2539could be eliminated from the linear system because they do not carry any
2540information. In practice, in deal.II we
always keep the size of the linear
2541system the same to avoid handling two different numbering systems and avoid
2542confusion about the two different
index sets. In order to ensure that the
2543linear system does not get singular when not adding anything to constrained
2544rows, we then add dummy entries to the
matrix diagonal that are otherwise
2545unrelated to the real entries.
2547In a
matrix-
free method, we need to take a different approach, since the @p
2548LaplaceOperator class represents the
matrix-vector product of a
2549<
b>homogeneous</
b> operator (the left-hand side of the last formula). It does
2552MatrixFree::cell_loop()
call will only resolve the homogeneous part of the
2553constraints as long as it represents a <
b>linear</
b> operator.
2555In our
matrix-
free code, the right hand side computation where the
2556contribution of inhomogeneous conditions ends up is completely decoupled from
2557the
matrix operator and handled by a different function above. Thus, we need
2558to explicitly generate the data that enters the right hand side rather than
2559using a byproduct of the
matrix assembly. Since we already know how to
apply
2560the operator on a vector, we could try to use those facilities for a vector
2568 BoundaryValueFunction<dim>(),
2570 for (
const std::pair<const types::global_dof_index, double> &pair : boundary_values)
2571 if (solution.locally_owned_elements().is_element(pair.
first))
2574or, equivalently,
if we already had filled the inhomogeneous constraints into
2581We could then pass the vector @p solution to the @p
2583system_rhs vector that gets filled in the @p LaplaceProblem::assemble_rhs()
2584function. However, this idea does not work because the
2585FEEvaluation::read_dof_values() call used inside the vmult() functions assumes
2586homogeneous values on all constraints (otherwise the operator would not be a
2587linear operator but an affine one). To also retrieve the values of the
2588inhomogeneities, we could select one of two following strategies.
2590<a name="UseFEEvaluationread_dof_values_plaintoavoidresolvingconstraints"></a><h5> Use
FEEvaluation::read_dof_values_plain() to avoid resolving constraints </h5>
2593The class
FEEvaluation has a facility that addresses precisely this
2594requirement: For non-homogeneous Dirichlet values, we do want to skip the
2595implicit imposition of homogeneous (Dirichlet) constraints upon reading the
2596data from the vector @p solution. For example, we could extend the @p
2597LaplaceProblem::assemble_rhs() function to deal with inhomogeneous Dirichlet
2598values as follows, assuming the Dirichlet values have been interpolated into
2599the
object @p constraints:
2602void LaplaceProblem<dim>::assemble_rhs()
2605 constraints.distribute(solution);
2606 solution.update_ghost_values();
2611 for (
unsigned int cell = 0;
2612 cell < system_matrix.get_matrix_free()->n_cell_batches();
2616 phi.read_dof_values_plain(solution);
2618 for (
unsigned int q = 0; q < phi.n_q_points; ++q)
2620 phi.submit_gradient(-coefficient(cell, q) * phi.get_gradient(q), q);
2621 phi.submit_value(make_vectorized_array<double>(1.0), q);
2624 phi.distribute_local_to_global(system_rhs);
2631tentative solution vector by
FEEvaluation::read_dof_values_plain() that
2632ignores all constraints. Due to this setup, we must make sure that other
2633constraints, e.g. by hanging nodes, are correctly distributed to the input
2634vector already as they are not resolved as in
2635FEEvaluation::read_dof_values_plain(). Inside the loop, we then evaluate the
2636Laplacian and repeat the
second derivative call with
2637FEEvaluation::submit_gradient() from the @p LaplaceOperator class, but with the
2638sign switched since we wanted to subtract the contribution of Dirichlet
2639conditions on the right hand side vector according to the formula above. When
2640we invoke the
FEEvaluation::integrate() call, we then set both arguments
2641regarding the value slot and
first derivative slot to true to account for both
2642terms added in the loop over quadrature points. Once the right hand side is
2643assembled, we then go on to solving the linear system for the homogeneous
2644problem, say involving a variable @p solution_update. After solving, we can
2645add @p solution_update to the @p solution vector that contains the final
2646(inhomogeneous) solution.
2648Note that the negative sign for the Laplacian alongside with a positive sign
2649for the forcing that we needed to build the right hand side is a more general
2650concept: We have implemented nothing else than Newton's method for nonlinear
2651equations, but applied to a linear system. We have used an initial guess for
2652the variable @p solution in terms of the Dirichlet boundary conditions and
2653computed a residual @f$r = f - Au_0@f$. The linear system was then solved as
2654@f$\Delta u = A^{-1} (f-Au)@f$ and we
finally computed @f$u = u_0 + \Delta u@f$. For a
2655linear system, we obviously reach the exact solution after a single
2656iteration. If we wanted to extend the code to a nonlinear problem, we would
2657rename the @p assemble_rhs() function into a more descriptive name like @p
2658assemble_residual() that computes the (weak) form of the residual, whereas the
2659@p LaplaceOperator::apply_add() function would get the linearization of the
2660residual with respect to the solution variable.
2662<a name="UseLaplaceOperatorwithasecondAffineConstraintsobjectwithoutDirichletconditions"></a><h5> Use LaplaceOperator with a
second AffineConstraints object without Dirichlet conditions </h5>
2665A
second alternative to get the right hand side that re-uses the @p
2666LaplaceOperator::apply_add() function is to instead add a
second LaplaceOperator
2667that skips Dirichlet constraints. To do this, we initialize a
second MatrixFree
2668object which does not have any boundary value constraints. This @p matrix_free
2669object is then passed to a @p LaplaceOperator class instance @p
2670inhomogeneous_operator that is only used to create the right hand side:
2673void LaplaceProblem<dim>::assemble_rhs()
2677 no_constraints.
close();
2678 LaplaceOperator<dim, degree_finite_element, double> inhomogeneous_operator;
2683 std::shared_ptr<MatrixFree<dim, double>> matrix_free(
2685 matrix_free->reinit(dof_handler,
2689 inhomogeneous_operator.initialize(matrix_free);
2692 constraints.distribute(solution);
2693 inhomogeneous_operator.evaluate_coefficient(Coefficient<dim>());
2694 inhomogeneous_operator.vmult(system_rhs, solution);
2698 *inhomogeneous_operator.get_matrix_free());
2699 for (
unsigned int cell = 0;
2700 cell < inhomogeneous_operator.get_matrix_free()->n_cell_batches();
2704 for (
unsigned int q = 0; q < phi.n_q_points; ++q)
2705 phi.submit_value(make_vectorized_array<double>(1.0), q);
2707 phi.distribute_local_to_global(system_rhs);
2713A more sophisticated implementation of
this technique could reuse the original
2716object. Doing
this would require making substantial modifications to the
2718comes with the library can do this. See the discussion on blocks in
2721<a name="Furtherperformanceimprovements"></a><h4> Further performance improvements </h4>
2724While the performance achieved in this tutorial program is already very good,
2725there is functionality in deal.II to further improve the performance. On the
2726one hand, increasing the polynomial degree to three or four will further
2727improve the time per unknown. (Even higher degrees typically get slower again,
2728because the multigrid iteration counts increase slightly with the chosen
2729simple smoother. One could then use hybrid multigrid algorithms to use
2730polynomial coarsening through MGTransferGlobalCoarsening, to reduce the impact
2731of the coarser level on the communication latency.) A more significant
2732improvement can be obtained by data-locality optimizations. The class
2734preconditioner as in the present class, can overlap the vector operations with
2735the
matrix-vector product. As the former are typically constrained by memory
2736bandwidth, reducing the number of loads helps to achieve this goal. The two
2737ingredients to achieve this are
2739<li> to provide LaplaceOperator class of this tutorial program with a `vmult`
2740function that takes two `std::function` objects, which can be passed on to
2742will then pick up this interface and schedule its vector operations), and </li>
2743<li> to compute a numbering that optimizes for data locality, as provided by
2748<a name="PlainProg"></a>
2749<h1> The plain program</h1>
2750@include
"step-37.cc"
void reinit(const IndexSet &local_constraints=IndexSet())
void distribute(VectorType &vec) const
void attach_dof_handler(const DoFHandler< dim, spacedim > &)
value_type get_dof_value(const unsigned int dof) const
void read_dof_values(const VectorType &src, const unsigned int first_index=0, const std::bitset< VectorizedArrayType::size()> &mask=std::bitset< VectorizedArrayType::size()>().flip())
void evaluate(const EvaluationFlags::EvaluationFlags evaluation_flag)
void reinit(const size_type size, const bool omit_zeroing_entries=false)
void initialize(const MGSmootherBase< VectorType > &coarse_smooth)
void resize(const unsigned int new_minlevel, const unsigned int new_maxlevel, Args &&...args)
void set_constrained_entries_to_one(VectorType &dst) const
void vmult_add(VectorType &dst, const VectorType &src) const
void vmult_interface_down(VectorType &dst, const VectorType &src) const
std::shared_ptr< DiagonalMatrix< VectorType > > inverse_diagonal_entries
unsigned int n_cell_batches() const
void initialize_dof_vector(VectorType &vec, const unsigned int dof_handler_index=0) const
void cell_loop(const std::function< void(const MatrixFree< dim, Number, VectorizedArrayType > &, OutVector &, const InVector &, const std::pair< unsigned int, unsigned int > &)> &cell_operation, OutVector &dst, const InVector &src, const bool zero_dst_vector=false) const
__global__ void reduction(Number *result, const Number *v, const size_type N)
__global__ void set(Number *val, const Number s, const size_type N)
#define Assert(cond, exc)
#define AssertDimension(dim1, dim2)
void loop(ITERATOR begin, std_cxx20::type_identity_t< ITERATOR > 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())
void make_hanging_node_constraints(const DoFHandler< dim, spacedim > &dof_handler, AffineConstraints< number > &constraints)
@ update_JxW_values
Transformed quadrature weights.
@ update_gradients
Shape function gradients.
@ update_quadrature_points
Transformed quadrature points.
void apply(const Kokkos::TeamPolicy< MemorySpace::Default::kokkos_space::execution_space >::member_type &team_member, const Kokkos::View< Number *, MemorySpace::Default::kokkos_space > shape_data, const ViewTypeIn in, ViewTypeOut out)
void matrix_free_data_locality(DoFHandler< dim, spacedim > &dof_handler, const MatrixFree< dim, Number, VectorizedArrayType > &matrix_free)
@ matrix
Contents is actually a matrix.
@ diagonal
Matrix is diagonal.
@ general
No special properties.
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
SymmetricTensor< 2, dim, Number > e(const Tensor< 2, dim, Number > &F)
SymmetricTensor< 2, dim, Number > 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)
VectorType::value_type * begin(VectorType &V)
std::vector< unsigned int > serial(const std::vector< unsigned int > &targets, const std::function< RequestType(const unsigned int)> &create_request, const std::function< AnswerType(const unsigned int, const RequestType &)> &answer_request, const std::function< void(const unsigned int, const AnswerType &)> &process_answer, const MPI_Comm comm)
T sum(const T &t, const MPI_Comm mpi_communicator)
unsigned int this_mpi_process(const MPI_Comm mpi_communicator)
T reduce(const T &local_value, const MPI_Comm comm, const std::function< T(const T &, const T &)> &combiner, const unsigned int root_process=0)
std::string compress(const std::string &input)
void run(const Iterator &begin, const std_cxx20::type_identity_t< Iterator > &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)
unsigned int n_cells(const internal::TriangulationImplementation::NumberCache< 1 > &c)
void copy(const T *begin, const T *end, U *dest)
int(&) functions(const void *v1, const void *v2)
void assemble(const MeshWorker::DoFInfoBox< dim, DOFINFO > &dinfo, A *assembler)
void reinit(MatrixBlock< MatrixType > &v, const BlockSparsityPattern &p)
static const unsigned int invalid_unsigned_int
unsigned int global_dof_index
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
DataOutBase::CompressionLevel compression_level
TasksParallelScheme tasks_parallel_scheme
UpdateFlags mapping_update_flags
DEAL_II_HOST constexpr Number determinant(const SymmetricTensor< 2, dim, Number > &)
DEAL_II_HOST constexpr SymmetricTensor< 2, dim, Number > invert(const SymmetricTensor< 2, dim, Number > &)
std::array< Number, 1 > eigenvalues(const SymmetricTensor< 2, 1, Number > &T)
1.9.7