902 *
constexpr LowStorageRungeKuttaScheme lsrk_scheme = stage_5_order_4;
906 * Eventually, we select a detail of the spatial discretization, namely the
907 * numerical flux (Riemann solver) at the faces between cells. For
this
908 * program, we have implemented a modified variant of the Lax--Friedrichs
909 * flux and the Harten--Lax--van Leer (HLL) flux.
912 *
enum EulerNumericalFlux
914 * lax_friedrichs_modified,
915 * harten_lax_vanleer,
917 *
constexpr EulerNumericalFlux numerical_flux_type = lax_friedrichs_modified;
924 * <a name=
"Equationdata"></a>
925 * <h3>Equation data</h3>
929 * We now define a
class with the exact solution for the test case 0 and one
930 * with a background flow field
for test
case 1 of the channel. Given that
931 * the Euler equations are a problem with @f$d+2@f$ equations in @f$d@f$ dimensions,
932 * we need to tell the
Function base
class about the correct number of
937 *
class ExactSolution :
public Function<dim>
940 * ExactSolution(
const double time)
945 *
const unsigned int component = 0)
const override;
952 * As far as the actual function implemented is concerned, the analytical
953 * test
case is an isentropic vortex
case (see
e.g. the book by Hesthaven
954 * and Warburton, Example 6.1 in Section 6.6 on page 209) which fulfills the
955 * Euler equations with zero force term on the right hand side. Given that
956 * definition, we
return either the density, the momentum, or the energy
957 * depending on which component is requested. Note that the original
958 * definition of the density involves the @f$\frac{1}{\
gamma -1}@f$-th power of
959 * some expression. Since `
std::pow()` has pretty slow implementations on
960 * some systems, we replace it by logarithm followed by exponentiation (of
961 * base 2), which is mathematically equivalent but usually much better
962 * optimized. This formula might lose accuracy in the last digits
963 *
for very small
numbers compared to `
std::pow()`, but we are happy with
964 * it anyway, since small
numbers map to data close to 1.
968 * For the channel test
case, we simply select a density of 1, a velocity of
969 * 0.4 in @f$x@f$ direction and zero in the other directions, and an energy that
970 * corresponds to a speed of sound of 1.3 measured against the background
971 * velocity field, computed from the relation @f$E = \frac{c^2}{\gamma (\gamma
972 * -1)} + \frac 12 \rho \|u\|^2@f$.
976 *
double ExactSolution<dim>::value(
const Point<dim> & x,
977 *
const unsigned int component)
const
979 *
const double t = this->
get_time();
985 *
Assert(dim == 2, ExcNotImplemented());
986 *
const double beta = 5;
990 *
const double radius_sqr =
991 * (x - x0).
norm_square() - 2. * (x[0] - x0[0]) * t + t * t;
992 *
const double factor =
994 *
const double density_log = std::log2(
995 *
std::abs(1. - (gamma - 1.) / gamma * 0.25 * factor * factor));
996 *
const double density = std::exp2(density_log * (1. / (gamma - 1.)));
997 *
const double u = 1. - factor * (x[1] - x0[1]);
998 *
const double v = factor * (x[0] - t - x0[0]);
1000 *
if (component == 0)
1002 *
else if (component == 1)
1003 *
return density * u;
1004 *
else if (component == 2)
1005 *
return density * v;
1008 *
const double pressure =
1009 * std::exp2(density_log * (gamma / (gamma - 1.)));
1010 *
return pressure / (
gamma - 1.) +
1011 * 0.5 * (density * u * u + density * v * v);
1017 *
if (component == 0)
1019 *
else if (component == 1)
1021 *
else if (component == dim + 1)
1022 *
return 3.097857142857143;
1028 *
Assert(
false, ExcNotImplemented());
1038 * <a name=
"LowstorageexplicitRungeKuttatimeintegrators"></a>
1039 * <h3>Low-storage
explicit Runge--Kutta time integrators</h3>
1043 * The next few lines implement a few low-storage variants of Runge--Kutta
1044 * methods. These methods have specific Butcher tableaux with coefficients
1045 * @f$b_i@f$ and @f$a_i@f$ as shown in the introduction. As usual in Runge--Kutta
1046 * method, we can deduce time steps, @f$c_i = \sum_{j=1}^{i-2} b_i + a_{i-1}@f$
1047 * from those coefficients. The main advantage of
this kind of scheme is the
1048 * fact that only two vectors are needed per stage, namely the accumulated
1049 * part of the solution @f$\mathbf{
w}@f$ (that will hold the solution
1050 * @f$\mathbf{
w}^{n+1}@f$ at the
new time @f$t^{n+1}@f$ after the last stage), the
1051 * update vector @f$\mathbf{r}_i@f$ that gets evaluated during the stages, plus
1052 * one vector @f$\mathbf{k}_i@f$ to hold the evaluation of the
operator. Such a
1053 * Runge--Kutta setup reduces the memory storage and memory access. As the
1054 * memory bandwidth is often the performance-limiting factor on modern
1055 * hardware when the evaluation of the differential
operator is
1056 * well-optimized, performance can be improved over standard time
1057 * integrators. This is
true also when taking into account that a
1058 * conventional Runge--Kutta scheme might allow
for slightly larger time
1059 * steps as more
free parameters allow
for better stability properties.
1063 * In
this tutorial programs, we concentrate on a few variants of
1064 * low-storage schemes defined in the article by Kennedy, Carpenter, and
1065 * Lewis (2000), as well as one variant described by Tselios and Simos
1066 * (2007). There is a large series of other schemes available, which could
1067 * be addressed by additional sets of coefficients or slightly different
1072 * We define a single
class for the four integrators, distinguished by the
1073 *
enum described above. To each scheme, we then fill the vectors
for the
1074 * @f$b_i@f$ and @f$a_i@f$ to the given variables in the
class.
1077 *
class LowStorageRungeKuttaIntegrator
1080 * LowStorageRungeKuttaIntegrator(const LowStorageRungeKuttaScheme scheme)
1085 * First comes the three-stage scheme of order three by Kennedy et al.
1086 * (2000). While its stability region is significantly smaller than
for
1087 * the other schemes, it only involves three stages, so it is very
1088 * competitive in terms of the work per stage.
1093 *
case stage_3_order_3:
1101 * The next scheme is a five-stage scheme of order four, again
1102 * defined in the paper by Kennedy et al. (2000).
1105 *
case stage_5_order_4:
1113 * The following scheme of seven stages and order four has been
1114 * explicitly derived
for acoustics problems. It is a balance of
1115 * accuracy
for imaginary
eigenvalues among fourth order schemes,
1116 * combined with a large stability region. Since DG schemes are
1117 * dissipative among the highest frequencies,
this does not
1118 * necessarily translate to the highest possible time step per
1119 * stage. In the context of the present tutorial program, the
1120 * numerical flux plays a crucial role in the dissipation and thus
1121 * also the maximal stable time step size. For the modified
1122 * Lax--Friedrichs flux,
this scheme is similar to the
1123 * `stage_5_order_4` scheme in terms of step size per stage
if only
1124 * stability is considered, but somewhat less efficient
for the HLL
1128 *
case stage_7_order_4:
1136 * The last scheme included here is the nine-stage scheme of order
1137 * five from Kennedy et al. (2000). It is the most accurate among
1138 * the schemes used here, but the higher order of accuracy
1139 * sacrifices some stability, so the step length normalized per
1140 * stage is less than
for the fourth order schemes.
1143 *
case stage_9_order_5:
1154 * rk_integrator(lsrk);
1155 * rk_integrator.get_coefficients(ai, bi, ci);
1158 *
unsigned int n_stages() const
1165 * The main function of the time integrator is to go through the stages,
1166 * evaluate the
operator, prepare the @f$\mathbf{r}_i@f$ vector
for the next
1167 * evaluation, and update the solution vector @f$\mathbf{
w}@f$. We hand off
1168 * the work to the `pde_operator` involved in order to be able to
merge
1169 * the vector operations of the Runge--Kutta setup with the evaluation of
1170 * the differential
operator for better performance, so all we
do here is
1171 * to delegate the vectors and coefficients.
1175 * We separately
call the
operator for the
first stage because we need
1176 * slightly modified arguments there: We evaluate the solution from
1177 * the old solution @f$\mathbf{
w}^n@f$ rather than a @f$\mathbf r_i@f$ vector, so
1178 * the
first argument is `solution`. We here let the stage vector
1179 * @f$\mathbf{r}_i@f$ also hold the temporary result of the evaluation, as it
1180 * is not used otherwise. For all subsequent stages, we use the vector
1181 * `vec_ki` as the
second vector argument to store the result of the
1182 *
operator evaluation. Finally, when we are at the last stage, we must
1183 * skip the computation of the vector @f$\mathbf{r}_{s+1}@f$ as there is no
1184 * coefficient @f$a_s@f$ available (nor will it be used).
1187 *
template <
typename VectorType,
typename Operator>
1188 *
void perform_time_step(
const Operator &pde_operator,
1189 *
const double current_time,
1190 *
const double time_step,
1191 * VectorType & solution,
1192 * VectorType & vec_ri,
1193 * VectorType & vec_ki)
const
1197 * pde_operator.perform_stage(current_time,
1198 * bi[0] * time_step,
1199 * ai[0] * time_step,
1205 *
for (
unsigned int stage = 1; stage < bi.size(); ++stage)
1207 *
const double c_i = ci[stage];
1208 * pde_operator.perform_stage(current_time + c_i * time_step,
1209 * bi[stage] * time_step,
1210 * (stage == bi.size() - 1 ?
1212 * ai[stage] * time_step),
1221 * std::vector<double> bi;
1222 * std::vector<double> ai;
1223 * std::vector<double> ci;
1231 * <a name=
"ImplementationofpointwiseoperationsoftheEulerequations"></a>
1232 * <h3>Implementation of
point-wise operations of the Euler equations</h3>
1236 * In the following
functions, we implement the various problem-specific
1237 * operators pertaining to the Euler equations. Each function acts on the
1238 * vector of conserved variables @f$[\rho, \rho\mathbf{u},
E]@f$ that we hold in
1239 * the solution vectors, and computes various derived quantities.
1243 * First out is the computation of the velocity, that we derive from the
1244 * momentum variable @f$\rho \mathbf{u}@f$ by division by @f$\rho@f$. One thing to
1245 * note here is that we decorate all those
functions with the keyword
1247 * compiler-specific keyword that tells the compiler to never create a
1248 * function
call for any of those
functions, and instead move the
1250 * href=
"https://en.wikipedia.org/wiki/Inline_function">
inline</a> to where
1251 * they are called. This is critical
for performance because we
call into some
1252 * of those
functions millions or billions of times: For example, we both use
1253 * the velocity
for the computation of the flux further down, but also
for the
1254 * computation of the pressure, and both of these places are evaluated at
1255 * every quadrature
point of every cell. Making sure these
functions are
1256 * inlined ensures not only that the processor does not have to execute a jump
1257 * instruction into the function (and the corresponding
return jump), but also
1258 * that the compiler can re-use intermediate information from one function
's
1259 * context in code that comes after the place where the function was called.
1260 * (We note that compilers are generally quite good at figuring out which
1261 * functions to inline by themselves. Here is a place where compilers may or
1262 * may not have figured it out by themselves but where we know for sure that
1263 * inlining is a win.)
1267 * Another trick we apply is a separate variable for the inverse density
1268 * @f$\frac{1}{\rho}@f$. This enables the compiler to only perform a single
1269 * division for the flux, despite the division being used at several
1270 * places. As divisions are around ten to twenty times as expensive as
1271 * multiplications or additions, avoiding redundant divisions is crucial for
1272 * performance. We note that taking the inverse first and later multiplying
1273 * with it is not equivalent to a division in floating point arithmetic due
1274 * to roundoff effects, so the compiler is not allowed to exchange one way by
1275 * the other with standard optimization flags. However, it is also not
1276 * particularly difficult to write the code in the right way.
1280 * To summarize, the chosen strategy of always inlining and careful
1281 * definition of expensive arithmetic operations allows us to write compact
1282 * code without passing all intermediate results around, despite making sure
1283 * that the code maps to excellent machine code.
1286 * template <int dim, typename Number>
1287 * inline DEAL_II_ALWAYS_INLINE
1288 * Tensor<1, dim, Number>
1289 * euler_velocity(const Tensor<1, dim + 2, Number> &conserved_variables)
1291 * const Number inverse_density = Number(1.) / conserved_variables[0];
1293 * Tensor<1, dim, Number> velocity;
1294 * for (unsigned int d = 0; d < dim; ++d)
1295 * velocity[d] = conserved_variables[1 + d] * inverse_density;
1302 * The next function computes the pressure from the vector of conserved
1303 * variables, using the formula @f$p = (\gamma - 1) \left(E - \frac 12 \rho
1304 * \mathbf{u}\cdot \mathbf{u}\right)@f$. As explained above, we use the
1305 * velocity from the `euler_velocity()` function. Note that we need to
1306 * specify the first template argument `dim` here because the compiler is
1307 * not able to deduce it from the arguments of the tensor, whereas the
1308 * second argument (number type) can be automatically deduced.
1311 * template <int dim, typename Number>
1312 * inline DEAL_II_ALWAYS_INLINE
1314 * euler_pressure(const Tensor<1, dim + 2, Number> &conserved_variables)
1316 * const Tensor<1, dim, Number> velocity =
1317 * euler_velocity<dim>(conserved_variables);
1319 * Number rho_u_dot_u = conserved_variables[1] * velocity[0];
1320 * for (unsigned int d = 1; d < dim; ++d)
1321 * rho_u_dot_u += conserved_variables[1 + d] * velocity[d];
1323 * return (gamma - 1.) * (conserved_variables[dim + 1] - 0.5 * rho_u_dot_u);
1328 * Here is the definition of the Euler flux function, i.e., the definition
1329 * of the actual equation. Given the velocity and pressure (that the
1330 * compiler optimization will make sure are done only once), this is
1331 * straight-forward given the equation stated in the introduction.
1334 * template <int dim, typename Number>
1335 * inline DEAL_II_ALWAYS_INLINE
1336 * Tensor<1, dim + 2, Tensor<1, dim, Number>>
1337 * euler_flux(const Tensor<1, dim + 2, Number> &conserved_variables)
1339 * const Tensor<1, dim, Number> velocity =
1340 * euler_velocity<dim>(conserved_variables);
1341 * const Number pressure = euler_pressure<dim>(conserved_variables);
1343 * Tensor<1, dim + 2, Tensor<1, dim, Number>> flux;
1344 * for (unsigned int d = 0; d < dim; ++d)
1346 * flux[0][d] = conserved_variables[1 + d];
1347 * for (unsigned int e = 0; e < dim; ++e)
1348 * flux[e + 1][d] = conserved_variables[e + 1] * velocity[d];
1349 * flux[d + 1][d] += pressure;
1350 * flux[dim + 1][d] =
1351 * velocity[d] * (conserved_variables[dim + 1] + pressure);
1359 * This next function is a helper to simplify the implementation of the
1360 * numerical flux, implementing the action of a tensor of tensors (with
1361 * non-standard outer dimension of size `dim + 2`, so the standard overloads
1362 * provided by deal.II's tensor classes
do not
apply here) with another
1363 * tensor of the same inner dimension, i.e., a matrix-vector product.
1366 * template <int n_components, int dim, typename Number>
1373 *
for (
unsigned int d = 0;
d < n_components; ++
d)
1374 * result[d] = matrix[d] * vector;
1380 * This function implements the numerical flux (Riemann solver). It gets the
1381 * state from the two sides of an
interface and the normal vector, oriented
1382 * from the side of the solution @f$\mathbf{
w}^-@f$ towards the solution
1383 * @f$\mathbf{
w}^+@f$. In finite
volume methods which rely on piece-wise
1384 *
constant data, the numerical flux is the central ingredient as it is the
1385 * only place where the physical information is entered. In DG methods, the
1386 * numerical flux is less central due to the polynomials within the elements
1387 * and the physical flux used there. As a result of higher-degree
1388 * interpolation with consistent
values from both sides in the limit of a
1389 * continuous solution, the numerical flux can be seen as a control of the
1390 * jump of the solution from both sides to weakly impose continuity. It is
1391 * important to realize that a numerical flux alone cannot stabilize a
1392 * high-order DG method in the presence of shocks, and thus any DG method
1393 * must be combined with further shock-capturing techniques to handle those
1394 * cases. In
this tutorial, we focus on wave-like solutions of the Euler
1395 * equations in the subsonic regime without strong discontinuities where our
1396 * basic scheme is sufficient.
1400 * Nonetheless, the numerical flux is decisive in terms of the numerical
1401 * dissipation of the overall scheme and influences the admissible time step
1402 * size with
explicit Runge--Kutta methods. We consider two choices, a
1403 * modified Lax--Friedrichs scheme and the widely used Harten--Lax--van Leer
1404 * (HLL) flux. For both variants, we
first need to get the velocities and
1405 * pressures from both sides of the interface and evaluate the physical
1410 * For the local Lax--Friedrichs flux, the definition is @f$\hat{\mathbf{F}}
1411 * =\frac{\mathbf{
F}(\mathbf{
w}^-)+\mathbf{
F}(\mathbf{
w}^+)}{2} +
1412 * \frac{\
lambda}{2}\left[\mathbf{
w}^--\mathbf{
w}^+\right]\otimes
1413 * \mathbf{n^-}@f$, where the factor @f$\lambda =
1414 * \max\left(\|\mathbf{u}^-\|+c^-, \|\mathbf{u}^+\|+c^+\right)@f$ gives the
1415 * maximal wave speed and @f$c = \sqrt{\
gamma p / \rho}@f$ is the speed of
1416 * sound. Here, we choose two modifications of that expression
for reasons
1417 * of computational efficiency, given the small impact of the flux on the
1418 * solution. For the above definition of the factor @f$\lambda@f$, we would need
1419 * to take four square roots, two
for the two velocity norms and two
for the
1420 * speed of sound on either side. The
first modification is hence to rather
1421 * use @f$\sqrt{\|\mathbf{u}\|^2+c^2}@f$ as an estimate of the maximal speed
1422 * (which is at most a factor of 2 away from the actual maximum, as shown in
1423 * the introduction). This allows us to pull the square root out of the
1424 * maximum and get away with a single square root computation. The
second
1425 * modification is to further relax on the parameter @f$\lambda@f$---the smaller
1426 * it is, the smaller the dissipation factor (which is multiplied by the
1427 * jump in @f$\mathbf{
w}@f$, which might result in a smaller or bigger
1428 * dissipation in the
end). This allows us to fit the spectrum into the
1429 * stability region of the
explicit Runge--Kutta integrator with bigger time
1430 * steps. However, we cannot make dissipation too small because otherwise
1431 * imaginary
eigenvalues grow larger. Finally, the current conservative
1432 * formulation is not energy-stable in the limit of @f$\lambda\to 0@f$ as it is
1433 * not skew-symmetric, and would need additional measures such as split-form
1434 * DG schemes in that
case.
1438 * For the HLL flux, we follow the formula from literature, introducing an
1439 * additional weighting of the two states from Lax--Friedrichs by a
1440 * parameter @f$s@f$. It is derived from the physical transport directions of
1441 * the Euler equations in terms of the current direction of velocity and
1442 * sound speed. For the velocity, we here choose a simple arithmetic average
1443 * which is sufficient
for DG scenarios and moderate jumps in material
1448 * Since the numerical flux is multiplied by the normal vector in the weak
1449 * form, we multiply by the result by the normal vector
for all terms in the
1450 * equation. In these multiplications, the `
operator*` defined above enables
1451 * a compact notation similar to the mathematical definition.
1455 * In
this and the following functions, we use variable suffixes `_m` and
1456 * `_p` to indicate quantities derived from @f$\mathbf{w}^-@f$ and @f$\mathbf{
w}^+@f$,
1457 * i.e.,
values "here" and
"there" relative to the current cell when looking
1458 * at a neighbor cell.
1461 *
template <
int dim,
typename Number>
1468 *
const auto velocity_m = euler_velocity<dim>(u_m);
1469 *
const auto velocity_p = euler_velocity<dim>(u_p);
1471 *
const auto pressure_m = euler_pressure<dim>(u_m);
1472 *
const auto pressure_p = euler_pressure<dim>(u_p);
1474 *
const auto flux_m = euler_flux<dim>(u_m);
1475 *
const auto flux_p = euler_flux<dim>(u_p);
1477 *
switch (numerical_flux_type)
1479 *
case lax_friedrichs_modified:
1483 * gamma * pressure_p * (1. / u_p[0]),
1484 * velocity_m.norm_square() +
1485 * gamma * pressure_m * (1. / u_m[0])));
1487 *
return 0.5 * (flux_m * normal + flux_p * normal) +
1488 * 0.5 * lambda * (u_m - u_p);
1491 *
case harten_lax_vanleer:
1493 *
const auto avg_velocity_normal =
1494 * 0.5 * ((velocity_m + velocity_p) * normal);
1497 * (pressure_p * (1. / u_p[0]) + pressure_m * (1. / u_m[0]))));
1498 *
const Number s_pos =
1499 *
std::max(Number(), avg_velocity_normal + avg_c);
1500 *
const Number s_neg =
1501 *
std::min(Number(), avg_velocity_normal - avg_c);
1502 *
const Number inverse_s = Number(1.) / (s_pos - s_neg);
1504 *
return inverse_s *
1505 * ((s_pos * (flux_m * normal) - s_neg * (flux_p * normal)) -
1506 * s_pos * s_neg * (u_m - u_p));
1511 *
Assert(
false, ExcNotImplemented());
1521 * This and the next function are helper
functions to provide compact
1522 * evaluation calls as multiple points get batched together via a
1523 *
VectorizedArray argument (see the @ref step_37
"step-37" tutorial
for details). This
1524 * function is used
for the subsonic outflow boundary conditions where we
1525 * need to
set the energy component to a prescribed
value. The next one
1526 * requests the solution on all components and is used
for inflow boundaries
1527 * where all components of the solution are
set.
1530 *
template <
int dim,
typename Number>
1534 *
const unsigned int component)
1537 *
for (
unsigned int v = 0; v < VectorizedArray<Number>::size(); ++v)
1540 *
for (
unsigned int d = 0;
d < dim; ++
d)
1541 * p[d] = p_vectorized[d][v];
1542 * result[v] = function.value(p, component);
1548 *
template <
int dim,
typename Number,
int n_components = dim + 2>
1555 *
for (
unsigned int v = 0; v < VectorizedArray<Number>::size(); ++v)
1558 *
for (
unsigned int d = 0;
d < dim; ++
d)
1559 * p[d] = p_vectorized[d][v];
1560 *
for (
unsigned int d = 0;
d < n_components; ++
d)
1561 * result[d][v] = function.value(p, d);
1571 * <a name=
"TheEulerOperationclass"></a>
1572 * <h3>The EulerOperation
class</h3>
1576 * This
class implements the evaluators for the Euler problem, in analogy to
1577 * the `LaplaceOperator`
class of @ref step_37
"step-37" or @ref step_59
"step-59". Since the present
1578 *
operator is non-linear and does not require a
matrix interface (to be
1579 * handed over to preconditioners), we skip the various `vmult`
functions
1580 * otherwise present in
matrix-
free operators and only implement an `
apply`
1581 * function as well as the combination of `
apply` with the required vector
1582 * updates
for the low-storage Runge--Kutta time integrator mentioned above
1583 * (called `perform_stage`). Furthermore, we have added three additional
1584 * functions involving matrix-free routines, namely one to compute an
1585 * estimate of the time step scaling (that is combined with the Courant
1586 * number
for the actual time step size) based on the velocity and speed of
1587 * sound in the elements, one
for the projection of solutions (specializing
1589 * against a possible analytical solution or norms against some background
1594 * The rest of the
class is similar to other
matrix-
free tutorials. As
1595 * discussed in the introduction, we provide a few
functions to allow a user
1596 * to pass in various forms of boundary conditions on different parts of the
1598 * possible body forces.
1601 *
template <
int dim,
int degree,
int n_po
ints_1d>
1602 *
class EulerOperator
1605 *
static constexpr unsigned int n_quadrature_points_1d = n_points_1d;
1615 *
void set_subsonic_outflow_boundary(
1621 *
void set_body_force(std::unique_ptr<
Function<dim>> body_force);
1623 *
void apply(
const double current_time,
1628 * perform_stage(
const Number cur_time,
1629 *
const Number factor_solution,
1630 *
const Number factor_ai,
1639 * std::array<double, 3> compute_errors(
1643 *
double compute_cell_transport_speed(
1654 * std::map<types::boundary_id, std::unique_ptr<Function<dim>>>
1655 * inflow_boundaries;
1656 * std::map<types::boundary_id, std::unique_ptr<Function<dim>>>
1657 * subsonic_outflow_boundaries;
1658 * std::set<types::boundary_id> wall_boundaries;
1659 * std::unique_ptr<Function<dim>> body_force;
1661 *
void local_apply_inverse_mass_matrix(
1665 *
const std::pair<unsigned int, unsigned int> & cell_range)
const;
1667 *
void local_apply_cell(
1671 *
const std::pair<unsigned int, unsigned int> & cell_range)
const;
1673 *
void local_apply_face(
1677 *
const std::pair<unsigned int, unsigned int> & face_range)
const;
1679 *
void local_apply_boundary_face(
1683 *
const std::pair<unsigned int, unsigned int> & face_range)
const;
1688 *
template <
int dim,
int degree,
int n_po
ints_1d>
1689 * EulerOperator<dim, degree, n_points_1d>::EulerOperator(
TimerOutput &timer)
1697 * For the initialization of the Euler
operator, we
set up the
MatrixFree
1698 * variable contained in the
class. This can be done given a mapping to
1699 * describe possible curved boundaries as well as a
DoFHandler object
1700 * describing the degrees of freedom. Since we use a discontinuous Galerkin
1701 * discretization in
this tutorial program where no constraints are imposed
1702 * strongly on the solution field, we
do not need to pass in an
1704 * construction. With respect to quadrature, we want to select two different
1705 * ways of computing the underlying integrals: The
first is a flexible one,
1706 * based on a
template parameter `n_points_1d` (that will be assigned the
1707 * `n_q_points_1d`
value specified at the top of
this file). More accurate
1708 * integration is necessary to avoid the aliasing problem due to the
1709 * variable coefficients in the Euler
operator. The
second less accurate
1710 * quadrature formula is a tight one based on `fe_degree+1` and needed
for
1711 * the inverse mass
matrix. While that formula provides an exact inverse
1712 * only on affine element shapes and not on deformed elements, it enables
1713 * the fast inversion of the mass matrix by tensor product techniques,
1714 * necessary to ensure optimal computational efficiency overall.
1717 * template <int dim, int degree, int n_points_1d>
1718 *
void EulerOperator<dim, degree, n_points_1d>::reinit(
1722 *
const std::vector<const DoFHandler<dim> *> dof_handlers = {&dof_handler};
1724 *
const std::vector<const AffineConstraints<double> *> constraints = {&dummy};
1725 *
const std::vector<Quadrature<1>> quadratures = {
QGauss<1>(n_q_points_1d),
1732 * additional_data.mapping_update_flags_inner_faces =
1735 * additional_data.mapping_update_flags_boundary_faces =
1738 * additional_data.tasks_parallel_scheme =
1742 * mapping, dof_handlers, constraints, quadratures, additional_data);
1747 *
template <
int dim,
int degree,
int n_po
ints_1d>
1748 *
void EulerOperator<dim, degree, n_points_1d>::initialize_vector(
1751 * data.initialize_dof_vector(vector);
1758 * The subsequent four member
functions are the ones that must be called from
1759 *
outside to specify the various
types of boundaries. For an inflow boundary,
1760 * we must specify all components in terms of density @f$\rho@f$, momentum @f$\rho
1761 * \mathbf{u}@f$ and energy @f$E@f$. Given
this information, we then store the
1762 * function alongside the respective boundary
id in a map member variable of
1763 *
this class. Likewise, we proceed
for the subsonic outflow boundaries (where
1764 * we request a function as well, which we use to retrieve the energy) and
for
1765 * wall (no-penetration) boundaries where we impose zero normal velocity (no
1766 * function necessary, so we only request the boundary
id). For the present
1767 * DG code where boundary conditions are solely applied as part of the weak
1768 * form (during time integration), the
call to
set the boundary conditions
1769 * can appear both before or after the `
reinit()`
call to
this class. This
1770 * is different from continuous finite element codes where the boundary
1772 * sent into
MatrixFree for initialization, thus requiring to be
set before
1773 * the initialization of the
matrix-
free data structures.
1777 * The checks added in each of the four function are used to
1778 * ensure that boundary conditions are mutually exclusive on the various
1779 * parts of the boundary, i.e., that a user does not accidentally designate a
1780 * boundary as both an inflow and say a subsonic outflow boundary.
1783 *
template <
int dim,
int degree,
int n_po
ints_1d>
1784 *
void EulerOperator<dim, degree, n_points_1d>::set_inflow_boundary(
1788 *
AssertThrow(subsonic_outflow_boundaries.find(boundary_id) ==
1789 * subsonic_outflow_boundaries.end() &&
1790 * wall_boundaries.find(boundary_id) == wall_boundaries.end(),
1791 * ExcMessage(
"You already set the boundary with id " +
1792 * std::to_string(
static_cast<int>(boundary_id)) +
1793 *
" to another type of boundary before now setting " +
1795 *
AssertThrow(inflow_function->n_components == dim + 2,
1796 * ExcMessage(
"Expected function with dim+2 components"));
1798 * inflow_boundaries[
boundary_id] = std::move(inflow_function);
1802 *
template <
int dim,
int degree,
int n_po
ints_1d>
1803 *
void EulerOperator<dim, degree, n_points_1d>::set_subsonic_outflow_boundary(
1807 *
AssertThrow(inflow_boundaries.find(boundary_id) ==
1808 * inflow_boundaries.end() &&
1809 * wall_boundaries.find(boundary_id) == wall_boundaries.end(),
1810 * ExcMessage(
"You already set the boundary with id " +
1811 * std::to_string(
static_cast<int>(boundary_id)) +
1812 *
" to another type of boundary before now setting " +
1813 *
"it as subsonic outflow"));
1814 *
AssertThrow(outflow_function->n_components == dim + 2,
1815 * ExcMessage(
"Expected function with dim+2 components"));
1817 * subsonic_outflow_boundaries[
boundary_id] = std::move(outflow_function);
1821 *
template <
int dim,
int degree,
int n_po
ints_1d>
1822 *
void EulerOperator<dim, degree, n_points_1d>::set_wall_boundary(
1825 *
AssertThrow(inflow_boundaries.find(boundary_id) ==
1826 * inflow_boundaries.end() &&
1827 * subsonic_outflow_boundaries.find(boundary_id) ==
1828 * subsonic_outflow_boundaries.end(),
1829 * ExcMessage(
"You already set the boundary with id " +
1830 * std::to_string(
static_cast<int>(boundary_id)) +
1831 *
" to another type of boundary before now setting " +
1832 *
"it as wall boundary"));
1834 * wall_boundaries.insert(boundary_id);
1838 *
template <
int dim,
int degree,
int n_po
ints_1d>
1839 *
void EulerOperator<dim, degree, n_points_1d>::set_body_force(
1844 * this->body_force = std::move(body_force);
1852 * <a name=
"Localevaluators"></a>
1853 * <h4>Local evaluators</h4>
1857 * Now we proceed to the local evaluators
for the Euler problem. The
1858 * evaluators are relatively simple and follow what has been presented in
1859 * @ref step_37
"step-37", @ref step_48
"step-48", or @ref step_59
"step-59". The
first notable difference is the fact
1860 * that we use an
FEEvaluation with a non-standard number of quadrature
1861 * points. Whereas we previously
always set the number of quadrature points
1862 * to
equal the polynomial degree plus one (ensuring exact integration on
1863 * affine element shapes), we now
set the number quadrature points as a
1864 * separate variable (
e.g. the polynomial degree plus two or three halves of
1865 * the polynomial degree) to more accurately handle nonlinear terms. Since
1866 * the evaluator is fed with the appropriate
loop lengths via the
template
1867 * argument and keeps the number of quadrature points in the whole cell in
1869 * the more accurate formula without further changes.
1873 * The
second difference is due to the fact that we are now evaluating a
1874 * multi-component system, as opposed to the
scalar systems considered
1875 * previously. The
matrix-
free framework provides several ways to handle the
1876 * multi-component
case. The variant shown here utilizes an
FEEvaluation
1877 *
object with multiple components embedded into it, specified by the fourth
1878 *
template argument `dim + 2`
for the components in the Euler system. As a
1880 * any more (that would return a
VectorizedArray type, collecting data from
1881 * several elements), but a
Tensor of `dim+2` components. The functionality
1882 * is otherwise similar to the scalar case; it is handled by a template
1884 * variant would have been to use several
FEEvaluation objects, a scalar one
1885 * for the density, a vector-valued one with `dim` components for the
1886 * momentum, and another scalar evaluator for the energy. To ensure that
1887 * those components point to the correct part of the solution, the
1888 * constructor of
FEEvaluation takes three optional integer arguments after
1890 * multi-
DoFHandler systems (taking the
first by default), the number of the
1891 * quadrature point in case there are multiple
Quadrature objects (see more
1892 * below), and as a third argument the component within a vector system. As
1893 * we have a single vector for all components, we would go with the third
1894 * argument, and set it to `0` for the density, `1` for the vector-valued
1895 * momentum, and `dim+1` for the energy slot.
FEEvaluation then picks the
1896 * appropriate subrange of the solution vector during
1898 *
FEEvaluation::distributed_local_to_global() or the more compact
1904 * When it comes to the evaluation of the body force vector, we distinguish
1905 * between two cases for efficiency reasons: In case we have a constant
1906 * function (derived from
Functions::ConstantFunction), we can precompute
1907 * the value outside the loop over quadrature points and simply use the
1908 * value everywhere. For a more general function, we instead need to call
1909 * the `evaluate_function()` method we provided above; this path is more
1910 * expensive because we need to access the memory associated with the
1911 * quadrature point data.
1915 * The rest follows the other tutorial programs. Since we have implemented
1916 * all physics for the Euler equations in the separate `euler_flux()`
1917 * function, all we have to do here is to call this function
1918 * given the current solution evaluated at quadrature points, returned by
1919 * `phi.get_value(q)`, and tell the
FEEvaluation object to queue the flux
1920 * for testing it by the gradients of the shape functions (which is a
Tensor
1921 * of outer `dim+2` components, each holding a tensor of `dim` components
1922 * for the @f$x,y,z@f$ component of the Euler flux). One final thing worth
1923 * mentioning is the order in which we queue the data for testing by the
1924 * value of the test function, `phi.submit_value()`, in case we are given an
1925 * external function: We must do this after calling `phi.get_value(q)`,
1926 * because `get_value()` (reading the solution) and `submit_value()`
1927 * (queuing the value for multiplication by the test function and summation
1928 * over quadrature points) access the same underlying data field. Here it
1929 * would be easy to achieve also without temporary variable `w_q` since
1930 * there is no mixing between values and gradients. For more complicated
1931 * setups, one has to
first copy out e.g. both the value and gradient at a
1932 * quadrature point and then queue results again by
1938 * argument of this function, which is a call-back from
MatrixFree::loop().
1939 * The interfaces imposes the present list of arguments, but since we are in
1940 * a member function where the
MatrixFree object is already available as the
1941 * `data` variable, we stick with that to avoid confusion.
1944 * template <
int dim,
int degree,
int n_points_1d>
1945 *
void EulerOperator<dim, degree, n_points_1d>::local_apply_cell(
1949 * const
std::pair<
unsigned int,
unsigned int> & cell_range) const
1957 *
if (constant_function)
1958 * constant_body_force = evaluate_function<dim, Number, dim>(
1961 *
for (
unsigned int cell = cell_range.first;
cell < cell_range.second; ++
cell)
1966 *
for (
unsigned int q = 0; q < phi.n_q_points; ++q)
1968 *
const auto w_q = phi.get_value(q);
1969 * phi.submit_gradient(euler_flux<dim>(w_q), q);
1970 *
if (body_force.get() !=
nullptr)
1973 * constant_function ? constant_body_force :
1974 * evaluate_function<dim, Number, dim>(
1975 * *body_force, phi.quadrature_point(q));
1978 *
for (
unsigned int d = 0;
d < dim; ++
d)
1979 * forcing[d + 1] = w_q[0] * force[d];
1980 *
for (
unsigned int d = 0;
d < dim; ++
d)
1981 * forcing[dim + 1] += force[d] * w_q[d + 1];
1983 * phi.submit_value(forcing, q);
1987 * phi.integrate_scatter(((body_force.get() !=
nullptr) ?
1999 * The next function concerns the computation of integrals on interior
2000 * faces, where we need evaluators from both cells adjacent to the face. We
2001 * associate the variable `phi_m` with the solution component @f$\mathbf{
w}^-@f$
2002 * and the variable `phi_p` with the solution component @f$\mathbf{
w}^+@f$. We
2004 *
second argument, with `
true`
for the interior side and `
false`
for the
2005 * exterior side, with interior and exterior denoting the orientation with
2006 * respect to the normal vector.
2012 * and the sum factorization parts. This combined operation not only saves a
2013 * line of code, but also contains an important optimization: Given that we
2014 * use a nodal basis in terms of the Lagrange polynomials in the points of
2015 * the Gauss-Lobatto quadrature formula, only @f$(p+1)^{
d-1}@f$ out of the
2016 * @f$(p+1)^
d@f$ basis
functions evaluate to non-zero on each face. Thus, the
2017 * evaluator only accesses the necessary data in the vector and skips the
2018 * parts which are multiplied by zero. If we had
first read the vector, we
2019 * would have needed to load all data from the vector, as the
call in
2020 * isolation would not know what data is required in subsequent
2022 * values and derivatives, indeed all @f$(p+1)^d@f$ vector entries for each
2023 * component are needed, as the normal derivative is nonzero for all basis
2028 * The arguments to the evaluators as well as the procedure is similar to
2029 * the cell evaluation. We again use the more accurate (over-)integration
2030 * scheme due to the nonlinear terms, specified as the third template
2031 * argument in the list. At the quadrature points, we then go to our
2032 * free-standing function for the numerical flux. It receives the solution
2033 * evaluated at quadrature points from both sides (i.e., @f$\mathbf{
w}^-@f$ and
2034 * @f$\mathbf{
w}^+@f$), as well as the normal vector onto the minus side. As
2035 * explained above, the numerical flux is already multiplied by the normal
2036 * vector from the minus side. We need to
switch the sign because the
2037 * boundary term comes with a minus sign in the weak form derived in the
2038 * introduction. The flux is then queued
for testing both on the minus sign
2039 * and on the plus sign, with switched sign as the normal vector from the
2040 * plus side is exactly opposed to the one from the minus side.
2043 * template <int dim, int degree, int n_points_1d>
2044 *
void EulerOperator<dim, degree, n_points_1d>::local_apply_face(
2048 *
const std::pair<unsigned int, unsigned int> & face_range)
const
2055 *
for (
unsigned int face = face_range.first; face < face_range.second; ++face)
2057 * phi_p.reinit(face);
2060 * phi_m.reinit(face);
2063 *
for (
unsigned int q = 0; q < phi_m.n_q_points; ++q)
2065 *
const auto numerical_flux =
2066 * euler_numerical_flux<dim>(phi_m.get_value(q),
2067 * phi_p.get_value(q),
2068 * phi_m.normal_vector(q));
2069 * phi_m.submit_value(-numerical_flux, q);
2070 * phi_p.submit_value(numerical_flux, q);
2082 * For faces located at the boundary, we need to impose the appropriate
2083 * boundary conditions. In
this tutorial program, we implement four cases as
2084 * mentioned above. (A fifth
case,
for supersonic outflow conditions is
2085 * discussed in the
"Results" section below.) The discontinuous Galerkin
2086 * method imposes boundary conditions not as constraints, but only
2087 * weakly. Thus, the various conditions are imposed by finding an appropriate
2088 * <i>exterior</i> quantity @f$\mathbf{
w}^+@f$ that is then handed to the
2089 * numerical flux function also used
for the interior faces. In essence,
2090 * we
"pretend" a state on the
outside of the domain in such a way that
2091 *
if that were reality, the solution of the PDE would satisfy the boundary
2092 * conditions we want.
2096 * For wall boundaries, we need to impose a no-normal-flux condition on the
2097 * momentum variable, whereas we use a Neumann condition
for the density and
2098 * energy with @f$\rho^+ = \rho^-@f$ and @f$E^+ =
E^-@f$. To achieve the no-normal
2099 * flux condition, we
set the exterior
values to the interior
values and
2100 * subtract two times the velocity in wall-normal direction, i.e., in the
2101 * direction of the normal vector.
2105 * For inflow boundaries, we simply
set the given Dirichlet data
2106 * @f$\mathbf{
w}_\mathrm{D}@f$ as a boundary
value. An alternative would have been
2107 * to use @f$\mathbf{
w}^+ = -\mathbf{
w}^- + 2 \mathbf{
w}_\mathrm{D}@f$, the
2108 * so-called mirror principle.
2112 * The imposition of outflow is essentially a Neumann condition, i.e.,
2113 * setting @f$\mathbf{
w}^+ = \mathbf{
w}^-@f$. For the
case of subsonic outflow,
2114 * we still need to impose a
value for the energy, which we derive from the
2115 * respective function. A special step is needed
for the
case of
2116 * <i>backflow</i>, i.e., the
case where there is a momentum flux into the
2117 * domain on the Neumann portion. According to the literature (a fact that can
2118 * be derived by appropriate energy arguments), we must
switch to another
2119 * variant of the flux on inflow parts, see Gravemeier, Comerford,
2120 * Yoshihara, Ismail, Wall,
"A novel formulation for Neumann inflow
2121 * conditions in biomechanics", Int. J. Numer. Meth. Biomed. Eng., vol. 28
2122 * (2012). Here, the momentum term needs to be added once again, which
2123 * corresponds to removing the flux contribution on the momentum
2124 * variables. We
do this in a post-processing step, and only
for the
case
2125 * when we both are at an outflow boundary and the dot product between the
2126 * normal vector and the momentum (or, equivalently, velocity) is
2127 *
negative. As we work on data of several quadrature points at once
for
2128 * SIMD vectorizations, we here need to explicitly
loop over the array
2129 * entries of the SIMD array.
2133 * In the implementation below, we
check for the various
types
2134 * of boundaries at the
level of quadrature points. Of course, we could also
2135 * have moved the decision out of the quadrature
point loop and treat entire
2136 * faces as of the same kind, which avoids some map/
set lookups in the inner
2137 *
loop over quadrature points. However, the loss of efficiency is hardly
2138 * noticeable, so we opt
for the simpler code here. Also note that the
final
2139 * `
else` clause will
catch the
case when some part of the boundary was not
2140 * assigned any boundary condition via `EulerOperator::set_..._boundary(...)`.
2143 *
template <
int dim,
int degree,
int n_po
ints_1d>
2144 *
void EulerOperator<dim, degree, n_points_1d>::local_apply_boundary_face(
2148 *
const std::pair<unsigned int, unsigned int> & face_range)
const
2152 *
for (
unsigned int face = face_range.first; face < face_range.second; ++face)
2157 *
for (
unsigned int q = 0; q < phi.n_q_points; ++q)
2159 *
const auto w_m = phi.get_value(q);
2160 *
const auto normal = phi.normal_vector(q);
2162 *
auto rho_u_dot_n = w_m[1] * normal[0];
2163 *
for (
unsigned int d = 1;
d < dim; ++
d)
2164 * rho_u_dot_n += w_m[1 + d] * normal[d];
2166 *
bool at_outflow =
false;
2169 *
const auto boundary_id = data.get_boundary_id(face);
2170 *
if (wall_boundaries.find(boundary_id) != wall_boundaries.end())
2173 *
for (
unsigned int d = 0;
d < dim; ++
d)
2174 * w_p[d + 1] = w_m[d + 1] - 2. * rho_u_dot_n * normal[d];
2175 * w_p[dim + 1] = w_m[dim + 1];
2177 *
else if (inflow_boundaries.find(boundary_id) !=
2178 * inflow_boundaries.end())
2180 * evaluate_function(*inflow_boundaries.find(boundary_id)->second,
2181 * phi.quadrature_point(q));
2182 *
else if (subsonic_outflow_boundaries.find(boundary_id) !=
2183 * subsonic_outflow_boundaries.end())
2186 * w_p[dim + 1] = evaluate_function(
2187 * *subsonic_outflow_boundaries.find(boundary_id)->second,
2188 * phi.quadrature_point(q),
2190 * at_outflow =
true;
2194 * ExcMessage(
"Unknown boundary id, did "
2195 *
"you set a boundary condition for "
2196 *
"this part of the domain boundary?"));
2198 *
auto flux = euler_numerical_flux<dim>(w_m, w_p, normal);
2201 *
for (
unsigned int v = 0; v < VectorizedArray<Number>::size(); ++v)
2203 *
if (rho_u_dot_n[v] < -1e-12)
2204 *
for (
unsigned int d = 0;
d < dim; ++
d)
2205 * flux[d + 1][v] = 0.;
2208 * phi.submit_value(-flux, q);
2219 * The next function implements the inverse mass
matrix operation. The
2220 * algorithms and rationale have been discussed extensively in the
2221 * introduction, so we here limit ourselves to the technicalities of the
2223 * operations as the forward evaluation of the mass
matrix, except with a
2224 * different interpolation
matrix, representing the inverse @f$S^{-1}@f$
2225 * factors. These represent a change of basis from the specified basis (in
2226 *
this case, the Lagrange basis in the points of the Gauss--Lobatto
2227 * quadrature formula) to the Lagrange basis in the points of the Gauss
2228 * quadrature formula. In the latter basis, we can
apply the inverse of the
2229 *
point-wise `JxW` factor, i.e., the quadrature weight times the
2230 *
determinant of the Jacobian of the mapping from reference to real
2231 * coordinates. Once
this is done, the basis is changed back to the nodal
2232 * Gauss-Lobatto basis again. All of these operations are done by the
2233 * `
apply()` function below. What we need to provide is the local fields to
2234 * operate on (which we extract from the global vector by an
FEEvaluation
2235 *
object) and write the results back to the destination vector of the mass
2240 * One thing to note is that we added two integer arguments (that are
2243 * only have one) and the
second being 1 to make the quadrature formula
2244 * selection. As we use the quadrature formula 0
for the over-integration of
2245 * nonlinear terms, we use the formula 1 with the
default @f$p+1@f$ (or
2246 * `fe_degree+1` in terms of the variable name) points
for the mass
2247 *
matrix. This leads to square contributions to the mass
matrix and ensures
2248 * exact integration, as explained in the introduction.
2251 *
template <
int dim,
int degree,
int n_po
ints_1d>
2252 *
void EulerOperator<dim, degree, n_points_1d>::local_apply_inverse_mass_matrix(
2256 *
const std::pair<unsigned int, unsigned int> & cell_range)
const
2262 *
for (
unsigned int cell = cell_range.first; cell < cell_range.second; ++cell)
2265 * phi.read_dof_values(src);
2267 * inverse.apply(phi.begin_dof_values(), phi.begin_dof_values());
2269 * phi.set_dof_values(dst);
2278 * <a name=
"Theapplyandrelatedfunctions"></a>
2279 * <h4>The
apply() and related functions</h4>
2283 * We now come to the function which implements the evaluation of the Euler
2284 * operator as a whole, i.e., @f$\mathcal M^{-1} \mathcal L(t, \mathbf{
w})@f$,
2285 * calling into the local evaluators presented above. The steps should be
2286 * clear from the previous code. One thing to note is that we need to adjust
2287 * the time in the functions we have associated with the various parts of
2288 * the boundary, in order to be consistent with the equation in
case the
2290 * perform the cell and face integrals, including the necessary ghost data
2291 * exchange in the `src` vector. The seventh argument to the function,
2292 * `
true`, specifies that we want to zero the `dst` vector as part of the
2293 *
loop, before we start accumulating integrals into it. This variant is
2294 * preferred over explicitly calling `dst = 0.;` before the
loop as the
2295 * zeroing operation is done on a subrange of the vector in parts that are
2296 * written by the integrals nearby. This enhances data locality and allows
2297 *
for caching, saving one roundtrip of vector data to main memory and
2298 * enhancing performance. The last two arguments to the
loop determine which
2299 * data is exchanged: Since we only access the
values of the shape
functions
2300 * one faces, typical of
first-order hyperbolic problems, and since we have
2301 * a nodal basis with nodes at the reference element surface, we only need
2302 * to exchange those parts. This again saves precious memory bandwidth.
2306 * Once the spatial
operator @f$\mathcal L@f$ is applied, we need to make a
2309 * is cheaper than the full loop as access only goes to the degrees of
2310 * freedom associated with the locally owned cells, which is simply the
2311 * locally owned degrees of freedom for DG discretizations. Thus, no ghost
2312 * exchange is needed here.
2316 * Around all these functions, we put timer scopes to record the
2317 * computational time for statistics about the contributions of the various
2321 * template <
int dim,
int degree,
int n_points_1d>
2322 *
void EulerOperator<dim, degree, n_points_1d>::apply(
2323 * const
double current_time,
2330 *
for (
auto &i : inflow_boundaries)
2331 * i.
second->set_time(current_time);
2332 *
for (
auto &i : subsonic_outflow_boundaries)
2333 * i.
second->set_time(current_time);
2335 * data.loop(&EulerOperator::local_apply_cell,
2336 * &EulerOperator::local_apply_face,
2337 * &EulerOperator::local_apply_boundary_face,
2349 * data.cell_loop(&EulerOperator::local_apply_inverse_mass_matrix,
2360 * Let us move to the function that does an entire stage of a Runge--Kutta
2361 * update. It calls EulerOperator::apply() followed by some updates
2362 * to the vectors, namely `next_ri = solution + factor_ai * k_i` and
2363 * `solution += factor_solution * k_i`. Rather than performing these
2364 * steps through the vector interfaces, we here present an alternative
2365 * strategy that is faster on cache-based architectures. As the memory
2366 * consumed by the vectors is often much larger than what fits into caches,
2367 * the data has to effectively come from the slow RAM memory. The situation
2368 * can be improved by loop fusion, i.e., performing both the updates to
2369 * `next_ki` and `solution` within a single sweep. In that case, we would
2370 * read the two vectors `rhs` and `solution` and write into `next_ki` and
2371 * `solution`, compared to at least 4 reads and two writes in the baseline
2372 * case. Here, we go one step further and perform the loop immediately when
2373 * the mass matrix inversion has finished on a part of the
2374 * vector.
MatrixFree::cell_loop() provides a mechanism to attach an
2375 * `
std::function` both before the loop over cells
first touches a vector
2376 * entry (which we do not use here, but is e.g. used for zeroing the vector)
2377 * and a
second `
std::function` to be called after the loop last touches
2378 * an entry. The callback is in form of a range over the given vector (in
2379 * terms of the local index numbering in the MPI universe) that can be
2380 * addressed by `local_element()` functions.
2384 * For this
second callback, we create a lambda that works on a range and
2385 * write the respective update on this range. Ideally, we would add the
2387 * compiler to SIMD parallelize this loop (which means in practice that we
2388 * ensure that there is no overlap, also called aliasing, between the index
2389 * ranges of the pointers we use inside the loops). It turns out that at the
2390 * time of this writing, GCC 7.2 fails to compile an OpenMP pragma inside a
2391 * lambda function, so we comment this pragma out below. If your compiler is
2392 * newer, you should be able to uncomment these lines again.
2396 * Note that we select a different code path for the last
2397 * Runge--Kutta stage when we do not need to update the `next_ri`
2398 * vector. This strategy gives a considerable speedup. Whereas the inverse
2399 * mass matrix and vector updates take more than 60% of the computational
2400 * time with default vector updates on a 40-core machine, the percentage is
2401 * around 35% with the more optimized variant. In other words, this is a
2402 * speedup of around a third.
2405 * template <
int dim,
int degree,
int n_points_1d>
2406 *
void EulerOperator<dim, degree, n_points_1d>::perform_stage(
2407 * const Number current_time,
2408 * const Number factor_solution,
2409 * const Number factor_ai,
2418 *
for (
auto &i : inflow_boundaries)
2419 * i.
second->set_time(current_time);
2420 *
for (
auto &i : subsonic_outflow_boundaries)
2421 * i.
second->set_time(current_time);
2423 * data.loop(&EulerOperator::local_apply_cell,
2424 * &EulerOperator::local_apply_face,
2425 * &EulerOperator::local_apply_boundary_face,
2438 * &EulerOperator::local_apply_inverse_mass_matrix,
2442 * std::function<
void(
const unsigned int,
const unsigned int)>(),
2443 * [&](
const unsigned int start_range,
const unsigned int end_range) {
2444 *
const Number ai = factor_ai;
2445 *
const Number bi = factor_solution;
2446 *
if (ai == Number())
2449 *
for (
unsigned int i = start_range; i < end_range; ++i)
2451 *
const Number k_i = next_ri.local_element(i);
2452 *
const Number sol_i = solution.local_element(i);
2453 * solution.local_element(i) = sol_i + bi * k_i;
2459 *
for (
unsigned int i = start_range; i < end_range; ++i)
2461 *
const Number k_i = next_ri.local_element(i);
2462 *
const Number sol_i = solution.local_element(i);
2463 * solution.local_element(i) = sol_i + bi * k_i;
2464 * next_ri.local_element(i) = sol_i + ai * k_i;
2475 * Having discussed the implementation of the
functions that deal with
2476 * advancing the solution by one time step, let us now move to
functions
2477 * that implement other, ancillary operations. Specifically, these are
2478 *
functions that compute projections, evaluate errors, and compute the speed
2479 * of information transport on a cell.
2485 * elements where there is no need to
set up and solve a linear system, as
2486 * each element has independent basis
functions. The reason why we show the
2487 * code here, besides a small speedup of
this non-critical operation, is that
2488 * it shows additional functionality provided by
2493 * The projection operation works as follows: If we denote the
matrix of
2494 * shape
functions evaluated at quadrature points by @f$S@f$, the projection on
2495 * cell @f$K@f$ is an operation of the form @f$\underbrace{S J^
K S^\mathrm
2496 * T}_{\mathcal M^
K} \mathbf{
w}^
K = S J^
K
2497 * \tilde{\mathbf{
w}}(\mathbf{x}_q)_{q=1:n_q}@f$, where @f$J^
K@f$ is the
diagonal
2499 * weight (JxW), @f$\mathcal M^
K@f$ is the cell-wise mass
matrix, and
2500 * @f$\tilde{\mathbf{
w}}(\mathbf{x}_q)_{q=1:n_q}@f$ is the evaluation of the
2501 * field to be projected onto quadrature points. (In reality the
matrix @f$S@f$
2502 * has additional structure through the tensor product, as explained in the
2503 * introduction.) This system can now equivalently be written as
2504 * @f$\mathbf{
w}^
K = \left(S J^K S^\mathrm T\right)^{-1} S J^
K
2505 * \tilde{\mathbf{
w}}(\mathbf{x}_q)_{q=1:n_q} = S^{-\mathrm T}
2506 * \left(J^K\right)^{-1} S^{-1} S J^
K
2507 * \tilde{\mathbf{
w}}(\mathbf{x}_q)_{q=1:n_q}@f$. Now, the term @f$S^{-1} S@f$ and
2508 * then @f$\left(J^K\right)^{-1} J^
K@f$ cancel, resulting in the
final
2509 * expression @f$\mathbf{
w}^
K = S^{-\mathrm T}
2510 * \tilde{\mathbf{
w}}(\mathbf{x}_q)_{q=1:n_q}@f$. This operation is
2513 * The name is derived from the fact that
this projection is simply
2514 * the multiplication by @f$S^{-\mathrm T}@f$, a basis change from the
2515 * nodal basis in the points of the Gaussian quadrature to the given finite
2517 * the result into the vector, overwriting previous content, rather than
2518 * accumulating the results as typical in integration tasks -- we can do
2519 * this because every vector entry has contributions from only a single
2520 * cell for discontinuous Galerkin discretizations.
2523 * template <
int dim,
int degree,
int n_points_1d>
2524 *
void EulerOperator<dim, degree, n_points_1d>::project(
2531 * solution.zero_out_ghost_values();
2535 *
for (
unsigned int q = 0; q < phi.n_q_points; ++q)
2536 * phi.submit_dof_value(evaluate_function(function,
2537 * phi.quadrature_point(q)),
2539 * inverse.transform_from_q_points_to_basis(dim + 2,
2540 * phi.begin_dof_values(),
2541 * phi.begin_dof_values());
2542 * phi.set_dof_values(solution);
2550 * The next function again repeats functionality also provided by the
2552 * the
explicit code to highlight how the vectorization across several cells
2553 * works and how to accumulate results via that interface: Recall that each
2554 * <i>lane</i> of the
vectorized array holds data from a different cell. By
2555 * the
loop over all cell batches that are owned by the current MPI process,
2557 * we would need to further go on and
sum across the entries in the SIMD
2558 * array. However, such a procedure is not stable as the SIMD array could in
2559 * fact not hold
valid data
for all its lanes. This happens when the number
2560 * of locally owned cells is not a multiple of the SIMD width. To avoid
2561 *
invalid data, we must explicitly skip those
invalid lanes when accessing
2562 * the data. While one could imagine that we could make it work by simply
2563 * setting the empty lanes to zero (and thus, not contribute to a sum), the
2564 * situation is more complicated than that: What
if we were to compute a
2565 * velocity out of the momentum? Then, we would need to divide by the
2566 * density, which is zero -- the result would consequently be NaN and
2567 * contaminate the result. This trap is avoided by accumulating the results
2568 * from the
valid SIMD range as we
loop through the cell batches,
using the
2570 * number of lanes with valid data. It equals
VectorizedArray::size() on
2571 * most cells, but can be less on the last cell batch if the number of cells
2572 * has a remainder compared to the SIMD width.
2575 * template <
int dim,
int degree,
int n_points_1d>
2576 *
std::array<
double, 3> EulerOperator<dim, degree, n_points_1d>::compute_errors(
2581 *
double errors_squared[3] = {};
2584 *
for (
unsigned int cell = 0; cell < data.n_cell_batches(); ++cell)
2589 *
for (
unsigned int q = 0; q < phi.n_q_points; ++q)
2591 *
const auto error =
2592 * evaluate_function(function, phi.quadrature_point(q)) -
2594 *
const auto JxW = phi.JxW(q);
2596 * local_errors_squared[0] += error[0] * error[0] * JxW;
2597 *
for (
unsigned int d = 0;
d < dim; ++
d)
2598 * local_errors_squared[1] += (error[d + 1] * error[d + 1]) * JxW;
2599 * local_errors_squared[2] += (error[dim + 1] * error[dim + 1]) * JxW;
2601 *
for (
unsigned int v = 0; v < data.n_active_entries_per_cell_batch(cell);
2603 *
for (
unsigned int d = 0;
d < 3; ++
d)
2604 * errors_squared[d] += local_errors_squared[d][v];
2609 * std::array<double, 3> errors;
2610 *
for (
unsigned int d = 0;
d < 3; ++
d)
2611 * errors[d] =
std::sqrt(errors_squared[d]);
2620 * This
final function of the EulerOperator
class is used to estimate the
2621 * transport speed, scaled by the mesh size, that is relevant
for setting
2622 * the time step size in the
explicit time integrator. In the Euler
2623 * equations, there are two speeds of transport, namely the convective
2624 * velocity @f$\mathbf{u}@f$ and the propagation of sound waves with sound
2625 * speed @f$c = \
sqrt{\
gamma p/\rho}@f$ relative to the medium moving at
2626 * velocity @f$\mathbf u@f$.
2630 * In the formula
for the time step size, we are interested not by
2631 * these absolute speeds, but by the amount of time it takes
for
2632 * information to cross a single cell. For information transported along with
2633 * the medium, @f$\mathbf u@f$ is scaled by the mesh size,
2634 * so an estimate of the maximal velocity can be obtained by computing
2635 * @f$\|J^{-\mathrm T} \mathbf{u}\|_\infty@f$, where @f$J@f$ is the Jacobian of the
2636 * transformation from real to the reference domain. Note that
2638 * Jacobian, representing the metric term from real to reference
2639 * coordinates, so we do not need to
transpose it again. We store this limit
2640 * in the variable `convective_limit` in the code below.
2644 * The sound propagation is isotropic, so we need to take mesh sizes in any
2645 * direction into account. The appropriate mesh size scaling is then given
2646 * by the minimal singular value of @f$J@f$ or, equivalently, the maximal
2647 * singular value of @f$J^{-1}@f$. Note that one could
approximate this quantity
2648 * by the minimal distance between
vertices of a cell when ignoring curved
2649 * cells. To get the maximal singular
value of the Jacobian, the
general
2650 * strategy would be some LAPACK function. Since all we need here is an
2651 * estimate, we can avoid the hassle of decomposing a tensor of
2653 * eigenvalue function without vectorization, and instead use a few
2654 * iterations (five in the code below) of the power method applied to
2655 * @f$J^{-1}J^{-\mathrm T}@f$. The speed of convergence of
this method depends
2656 * on the ratio of the largest to the next largest eigenvalue and the
2657 *
initial guess, which is the vector of all ones. This might suggest that
2658 * we get slow convergence on cells close to a cube shape where all
2659 * lengths are almost the same. However,
this slow convergence means that
2660 * the result will sit between the two largest singular
values, which both
2661 * are close to the maximal
value anyway. In all other cases, convergence
2662 * will be quick. Thus, we can merely hardcode 5 iterations here and be
2663 * confident that the result is good.
2666 *
template <
int dim,
int degree,
int n_po
ints_1d>
2667 *
double EulerOperator<dim, degree, n_points_1d>::compute_cell_transport_speed(
2671 * Number max_transport = 0;
2674 *
for (
unsigned int cell = 0; cell < data.n_cell_batches(); ++cell)
2679 *
for (
unsigned int q = 0; q < phi.n_q_points; ++q)
2681 *
const auto solution = phi.get_value(q);
2682 *
const auto velocity = euler_velocity<dim>(solution);
2683 *
const auto pressure = euler_pressure<dim>(solution);
2685 *
const auto inverse_jacobian = phi.inverse_jacobian(q);
2686 *
const auto convective_speed = inverse_jacobian * velocity;
2688 *
for (
unsigned int d = 0;
d < dim; ++
d)
2689 * convective_limit =
2692 *
const auto speed_of_sound =
2693 *
std::sqrt(gamma * pressure * (1. / solution[0]));
2696 *
for (
unsigned int d = 0;
d < dim; ++
d)
2697 * eigenvector[d] = 1.;
2698 *
for (
unsigned int i = 0; i < 5; ++i)
2700 * eigenvector =
transpose(inverse_jacobian) *
2701 * (inverse_jacobian * eigenvector);
2703 *
for (
unsigned int d = 0;
d < dim; ++
d)
2704 * eigenvector_norm =
2706 * eigenvector /= eigenvector_norm;
2708 *
const auto jac_times_ev = inverse_jacobian * eigenvector;
2709 *
const auto max_eigenvalue =
std::sqrt(
2710 * (jac_times_ev * jac_times_ev) / (eigenvector * eigenvector));
2713 * max_eigenvalue * speed_of_sound + convective_limit);
2718 * Similarly to the previous function, we must make sure to accumulate
2719 * speed only on the
valid cells of a cell batch.
2722 *
for (
unsigned int v = 0; v < data.n_active_entries_per_cell_batch(cell);
2724 *
for (
unsigned int d = 0;
d < 3; ++
d)
2725 * max_transport =
std::max(max_transport, local_max[v]);
2730 *
return max_transport;
2738 * <a name=
"TheEulerProblemclass"></a>
2739 * <h3>The EulerProblem
class</h3>
2743 * This
class combines the EulerOperator class with the time integrator and
2745 * actually
run the simulations of the Euler problem.
2749 * The member variables are a
triangulation, a finite element, a mapping (to
2750 * create high-order curved surfaces, see
e.g. @ref step_10
"step-10"), and a
DoFHandler to
2751 * describe the degrees of freedom. In addition, we keep an instance of the
2752 * EulerOperator described above around, which will
do all heavy lifting in
2753 * terms of integrals, and some parameters
for time integration like the
2754 * current time or the time step size.
2758 * Furthermore, we use a PostProcessor instance to write some additional
2759 * information to the output file, in similarity to what was done in
2760 * @ref step_33
"step-33". The
interface of the
DataPostprocessor class is intuitive,
2761 * requiring us to provide information about what needs to be evaluated
2762 * (typically only the
values of the solution, except
for the Schlieren plot
2763 * that we only enable in 2
d where it makes sense), and the names of what
2764 * gets evaluated. Note that it would also be possible to extract most
2765 * information by calculator tools within visualization programs such as
2766 * ParaView, but it is so much more convenient to
do it already when writing
2770 * template <int dim>
2771 *
class EulerProblem
2779 *
void make_grid_and_dofs();
2781 *
void output_results(
const unsigned int result_number);
2787 * #ifdef DEAL_II_WITH_P4EST
2799 * EulerOperator<dim, fe_degree, n_q_points_1d> euler_operator;
2801 *
double time, time_step;
2810 * std::vector<
Vector<double>> &computed_quantities)
const override;
2812 *
virtual std::vector<std::string>
get_names()
const override;
2814 *
virtual std::vector<
2821 *
const bool do_schlieren_plot;
2827 *
template <
int dim>
2828 * EulerProblem<dim>::Postprocessor::Postprocessor()
2829 * : do_schlieren_plot(dim == 2)
2836 * For the main evaluation of the field variables, we
first check that the
2837 * lengths of the arrays
equal the expected
values (the lengths `2*dim+4` or
2838 * `2*dim+5` are derived from the sizes of the names we specify in the
2839 * get_names() function below). Then we
loop over all evaluation points and
2840 * fill the respective information: First we fill the primal solution
2841 * variables of density @f$\rho@f$, momentum @f$\rho \mathbf{u}@f$ and energy @f$E@f$,
2842 * then we compute the derived velocity @f$\mathbf u@f$, the pressure @f$p@f$, the
2843 * speed of sound @f$c=\
sqrt{\
gamma p / \rho}@f$, as well as the Schlieren plot
2844 * showing @f$s = |\nabla \rho|^2@f$ in
case it is enabled. (See @ref step_69
"step-69" for
2845 * another example where we create a Schlieren plot.)
2848 *
template <
int dim>
2849 *
void EulerProblem<dim>::Postprocessor::evaluate_vector_field(
2853 *
const unsigned int n_evaluation_points = inputs.solution_values.size();
2855 *
if (do_schlieren_plot ==
true)
2856 *
Assert(inputs.solution_gradients.size() == n_evaluation_points,
2857 * ExcInternalError());
2859 *
Assert(computed_quantities.size() == n_evaluation_points,
2860 * ExcInternalError());
2861 *
Assert(inputs.solution_values[0].size() == dim + 2, ExcInternalError());
2862 *
Assert(computed_quantities[0].size() ==
2863 * dim + 2 + (do_schlieren_plot ==
true ? 1 : 0),
2864 * ExcInternalError());
2866 *
for (
unsigned int p = 0; p < n_evaluation_points; ++p)
2869 *
for (
unsigned int d = 0;
d < dim + 2; ++
d)
2870 * solution[d] = inputs.solution_values[p](d);
2872 *
const double density = solution[0];
2874 *
const double pressure = euler_pressure<dim>(solution);
2876 *
for (
unsigned int d = 0;
d < dim; ++
d)
2877 * computed_quantities[p](d) = velocity[
d];
2878 * computed_quantities[p](dim) = pressure;
2879 * computed_quantities[p](dim + 1) =
std::sqrt(gamma * pressure / density);
2881 *
if (do_schlieren_plot ==
true)
2882 * computed_quantities[p](dim + 2) =
2883 * inputs.solution_gradients[p][0] * inputs.solution_gradients[p][0];
2889 *
template <
int dim>
2890 * std::vector<std::string> EulerProblem<dim>::Postprocessor::get_names() const
2892 * std::vector<std::string> names;
2893 *
for (
unsigned int d = 0;
d < dim; ++
d)
2894 * names.emplace_back(
"velocity");
2895 * names.emplace_back(
"pressure");
2896 * names.emplace_back(
"speed_of_sound");
2898 *
if (do_schlieren_plot ==
true)
2899 * names.emplace_back(
"schlieren_plot");
2908 * For the interpretation of quantities, we have
scalar density, energy,
2909 * pressure, speed of sound, and the Schlieren plot, and vectors
for the
2910 * momentum and the velocity.
2913 *
template <
int dim>
2914 * std::vector<DataComponentInterpretation::DataComponentInterpretation>
2915 * EulerProblem<dim>::Postprocessor::get_data_component_interpretation() const
2917 * std::vector<DataComponentInterpretation::DataComponentInterpretation>
2919 *
for (
unsigned int d = 0;
d < dim; ++
d)
2920 * interpretation.push_back(
2925 *
if (do_schlieren_plot ==
true)
2926 * interpretation.push_back(
2929 *
return interpretation;
2936 * With respect to the necessary update flags, we only need the
values for
2937 * all quantities but the Schlieren plot, which is based on the density
2941 *
template <
int dim>
2942 *
UpdateFlags EulerProblem<dim>::Postprocessor::get_needed_update_flags() const
2944 *
if (do_schlieren_plot ==
true)
2954 * The constructor
for this class is unsurprising: We
set up a
parallel
2955 *
triangulation based on the `MPI_COMM_WORLD` communicator, a vector finite
2956 * element with `dim+2` components for density, momentum, and energy, a
2957 * high-order mapping of the same degree as the underlying finite element,
2958 * and initialize the time and time step to zero.
2961 * template <int dim>
2962 * EulerProblem<dim>::EulerProblem()
2963 * : pcout(std::cout, Utilities::MPI::this_mpi_process(MPI_COMM_WORLD) == 0)
2964 * #ifdef DEAL_II_WITH_P4EST
2967 * , fe(FE_DGQ<dim>(fe_degree), dim + 2)
2968 * , mapping(fe_degree)
2969 * , dof_handler(triangulation)
2970 * , timer(pcout, TimerOutput::never, TimerOutput::wall_times)
2971 * , euler_operator(timer)
2980 * As a mesh,
this tutorial program implements two options, depending on the
2981 * global variable `testcase`: For the analytical variant (`testcase==0`),
2982 * the domain is @f$(0, 10) \times (-5, 5)@f$, with Dirichlet boundary
2983 * conditions (inflow) all around the domain. For `testcase==1`, we
set the
2984 * domain to a
cylinder in a rectangular box, derived from the flow past
2985 *
cylinder testcase
for incompressible viscous flow by Schäfer and
2986 * Turek (1996). Here, we have a larger variety of boundaries. The inflow
2987 * part at the left of the channel is given the inflow type,
for which we
2988 * choose a
constant inflow profile, whereas we
set a subsonic outflow at
2989 * the right. For the boundary around the
cylinder (boundary
id equal to 2)
2990 * as well as the channel walls (boundary
id equal to 3) we use the wall
2991 * boundary type, which is no-normal flow. Furthermore,
for the 3
d cylinder
2992 * we also add a gravity force in vertical direction. Having the base mesh
2993 * in place (including the manifolds set by
2995 * specified number of global refinements, create the unknown numbering from
2997 * initialization of the EulerOperator.
3000 *
template <
int dim>
3001 *
void EulerProblem<dim>::make_grid_and_dofs()
3008 *
for (
unsigned int d = 1;
d < dim; ++
d)
3009 * lower_left[d] = -5;
3012 * upper_right[0] = 10;
3013 *
for (
unsigned int d = 1;
d < dim; ++
d)
3014 * upper_right[d] = 5;
3021 * euler_operator.set_inflow_boundary(
3022 * 0, std::make_unique<ExactSolution<dim>>(0));
3032 * euler_operator.set_inflow_boundary(
3033 * 0, std::make_unique<ExactSolution<dim>>(0));
3034 * euler_operator.set_subsonic_outflow_boundary(
3035 * 1, std::make_unique<ExactSolution<dim>>(0));
3037 * euler_operator.set_wall_boundary(2);
3038 * euler_operator.set_wall_boundary(3);
3041 * euler_operator.set_body_force(
3043 * std::vector<double>({0., 0., -0.2})));
3049 *
Assert(
false, ExcNotImplemented());
3054 * dof_handler.distribute_dofs(fe);
3056 * euler_operator.reinit(mapping, dof_handler);
3057 * euler_operator.initialize_vector(solution);
3061 * In the following, we output some statistics about the problem. Because we
3062 * often
end up with quite large
numbers of cells or degrees of freedom, we
3063 * would like to print them with a comma to separate each
set of three
3064 * digits. This can be done via
"locales", although the way
this works is
3065 * not particularly intuitive. @ref step_32
"step-32" explains
this in slightly more
3069 * std::locale s = pcout.get_stream().getloc();
3070 * pcout.get_stream().imbue(std::locale(
""));
3071 * pcout <<
"Number of degrees of freedom: " << dof_handler.n_dofs()
3072 * <<
" ( = " << (dim + 2) <<
" [vars] x "
3076 * pcout.get_stream().imbue(s);
3083 * For output, we
first let the Euler
operator compute the errors of the
3084 * numerical results. More precisely, we compute the error against the
3085 * analytical result
for the analytical solution
case, whereas we compute
3086 * the deviation against the background field with
constant density and
3087 * energy and
constant velocity in @f$x@f$ direction
for the
second test
case.
3091 * The next step is to create output. This is similar to what is done in
3092 * @ref step_33
"step-33": We let the postprocessor defined above control most of the
3093 * output, except
for the primal field that we write directly. For the
3094 * analytical solution test
case, we also perform another projection of the
3095 * analytical solution and print the difference between that field and the
3096 * numerical solution. Once we have defined all quantities to be written, we
3097 * build the patches
for output. Similarly to @ref step_65
"step-65", we create a
3098 * high-order VTK output by setting the appropriate flag, which enables us
3099 * to visualize fields of high polynomial degrees. Finally, we
call the
3101 * to the given file name. This function uses special MPI
parallel write
3102 * facilities, which are typically more optimized
for parallel file systems
3103 * than the standard library
's `std::ofstream` variants used in most other
3104 * tutorial programs. A particularly nice feature of the
3105 * `write_vtu_in_parallel()` function is the fact that it can combine output
3106 * from all MPI ranks into a single file, making it unnecessary to have a
3107 * central record of all such files (namely, the "pvtu" file).
3111 * For parallel programs, it is often instructive to look at the partitioning
3112 * of cells among processors. To this end, one can pass a vector of numbers
3113 * to DataOut::add_data_vector() that contains as many entries as the
3114 * current processor has active cells; these numbers should then be the
3115 * rank of the processor that owns each of these cells. Such a vector
3116 * could, for example, be obtained from
3117 * GridTools::get_subdomain_association(). On the other hand, on each MPI
3118 * process, DataOut will only read those entries that correspond to locally
3119 * owned cells, and these of course all have the same value: namely, the rank
3120 * of the current process. What is in the remaining entries of the vector
3121 * doesn't actually matter, and so we can just get away with a cheap trick: We
3123 * with the rank of the current MPI process. The key is that on each process,
3124 * only the entries corresponding to the locally owned cells will be read,
3125 * ignoring the (wrong)
values in other entries. The fact that every process
3126 * submits a vector in which the correct subset of entries is correct is all
3127 * that is necessary.
3131 * @note As of 2023, Visit 3.3.3 can still not deal with higher-order cells.
3132 * Rather, it simply reports that there is no data to show. To view the
3133 * results of
this program with Visit, you will want to comment out the
3134 * line that sets `flags.write_higher_order_cells =
true;`. On the other
3135 * hand, Paraview is able to understand VTU files with higher order cells
3139 *
template <
int dim>
3140 *
void EulerProblem<dim>::output_results(
const unsigned int result_number)
3142 *
const std::array<double, 3> errors =
3143 * euler_operator.compute_errors(ExactSolution<dim>(time), solution);
3144 *
const std::string quantity_name = testcase == 0 ?
"error" :
"norm";
3146 * pcout <<
"Time:" << std::setw(8) << std::setprecision(3) << time
3147 * <<
", dt: " << std::setw(8) << std::setprecision(2) << time_step
3148 * <<
", " << quantity_name <<
" rho: " << std::setprecision(4)
3149 * << std::setw(10) << errors[0] <<
", rho * u: " << std::setprecision(4)
3150 * << std::setw(10) << errors[1] <<
", energy:" << std::setprecision(4)
3151 * << std::setw(10) << errors[2] << std::endl;
3156 * Postprocessor postprocessor;
3161 * data_out.set_flags(flags);
3163 * data_out.attach_dof_handler(dof_handler);
3165 * std::vector<std::string> names;
3166 * names.emplace_back(
"density");
3167 *
for (
unsigned int d = 0;
d < dim; ++
d)
3168 * names.emplace_back(
"momentum");
3169 * names.emplace_back(
"energy");
3171 * std::vector<DataComponentInterpretation::DataComponentInterpretation>
3173 * interpretation.push_back(
3175 *
for (
unsigned int d = 0;
d < dim; ++
d)
3176 * interpretation.push_back(
3178 * interpretation.push_back(
3181 * data_out.add_data_vector(dof_handler, solution, names, interpretation);
3183 * data_out.add_data_vector(solution, postprocessor);
3186 *
if (testcase == 0 && dim == 2)
3188 * reference.
reinit(solution);
3189 * euler_operator.project(ExactSolution<dim>(time), reference);
3190 * reference.sadd(-1., 1, solution);
3191 * std::vector<std::string> names;
3192 * names.emplace_back(
"error_density");
3193 *
for (
unsigned int d = 0;
d < dim; ++
d)
3194 * names.emplace_back(
"error_momentum");
3195 * names.emplace_back(
"error_energy");
3197 * std::vector<DataComponentInterpretation::DataComponentInterpretation>
3199 * interpretation.push_back(
3201 *
for (
unsigned int d = 0;
d < dim; ++
d)
3202 * interpretation.push_back(
3204 * interpretation.push_back(
3207 * data_out.add_data_vector(dof_handler,
3215 * data_out.add_data_vector(mpi_owner,
"owner");
3217 * data_out.build_patches(mapping,
3221 *
const std::string filename =
3223 * data_out.write_vtu_in_parallel(filename, MPI_COMM_WORLD);
3231 * The EulerProblem::run() function puts all pieces together. It starts off
3232 * by calling the function that creates the mesh and sets up data structures,
3233 * and then initializing the time integrator and the two temporary vectors of
3234 * the low-storage integrator. We call these vectors `rk_register_1` and
3235 * `rk_register_2`, and use the
first vector to represent the quantity
3236 * @f$\mathbf{r}_i@f$ and the
second one
for @f$\mathbf{k}_i@f$ in the formulas
for
3237 * the Runge--Kutta scheme outlined in the introduction. Before we start the
3238 * time
loop, we compute the time step size by the
3239 * `EulerOperator::compute_cell_transport_speed()` function. For reasons of
3240 * comparison, we compare the result obtained there with the minimal mesh
3241 * size and print them to screen. For velocities and speeds of sound close
3242 * to unity as in
this tutorial program, the predicted effective mesh size
3243 * will be close, but they could vary
if scaling were different.
3246 *
template <
int dim>
3247 *
void EulerProblem<dim>::run()
3251 *
const unsigned int n_vect_bits = 8 *
sizeof(Number) * n_vect_number;
3253 * pcout <<
"Running with "
3255 * <<
" MPI processes" << std::endl;
3256 * pcout <<
"Vectorization over " << n_vect_number <<
' '
3257 * << (std::is_same<Number, double>::value ?
"doubles" :
"floats")
3258 * <<
" = " << n_vect_bits <<
" bits ("
3263 * make_grid_and_dofs();
3265 *
const LowStorageRungeKuttaIntegrator integrator(lsrk_scheme);
3269 * rk_register_1.
reinit(solution);
3270 * rk_register_2.reinit(solution);
3272 * euler_operator.project(ExactSolution<dim>(time), solution);
3274 *
double min_vertex_distance = std::numeric_limits<double>::max();
3275 *
for (
const auto &cell :
triangulation.active_cell_iterators())
3276 * if (cell->is_locally_owned())
3277 * min_vertex_distance =
3278 *
std::
min(min_vertex_distance, cell->minimum_vertex_distance());
3279 * min_vertex_distance =
3282 * time_step = courant_number * integrator.n_stages() /
3283 * euler_operator.compute_cell_transport_speed(solution);
3284 * pcout <<
"Time step size: " << time_step
3285 * <<
", minimal h: " << min_vertex_distance
3286 * <<
", initial transport scaling: "
3287 * << 1. / euler_operator.compute_cell_transport_speed(solution)
3291 * output_results(0);
3295 * Now we are ready to start the time
loop, which we
run until the time
3296 * has reached the desired
end time. Every 5 time steps, we compute a
new
3297 * estimate
for the time step -- since the solution is nonlinear, it is
3298 * most effective to adapt the
value during the course of the
3299 * simulation. In
case the Courant number was chosen too aggressively, the
3300 * simulation will typically blow up with time step NaN, so that is easy
3301 * to detect here. One thing to note is that roundoff errors might
3302 * propagate to the leading digits due to an interaction of slightly
3303 * different time step selections that in turn lead to slightly different
3304 * solutions. To decrease
this sensitivity, it is common practice to round
3305 * or truncate the time step size to a few digits,
e.g. 3 in
this case. In
3306 *
case the current time is near the prescribed
'tick' value for output
3307 * (
e.g. 0.02), we also write the output. After the end of the time loop,
3308 * we summarize the computation by printing some statistics, which is
3312 *
unsigned int timestep_number = 0;
3314 *
while (time < final_time - 1e-12)
3316 * ++timestep_number;
3317 *
if (timestep_number % 5 == 0)
3319 * courant_number * integrator.n_stages() /
3321 * euler_operator.compute_cell_transport_speed(solution), 3);
3325 * integrator.perform_time_step(euler_operator,
3333 * time += time_step;
3335 *
if (
static_cast<int>(time / output_tick) !=
3336 *
static_cast<int>((time - time_step) / output_tick) ||
3337 * time >= final_time - 1e-12)
3339 *
static_cast<unsigned int>(std::round(time / output_tick)));
3342 * timer.print_wall_time_statistics(MPI_COMM_WORLD);
3343 * pcout << std::endl;
3352 * The main() function is not surprising and follows what was done in all
3353 * previous MPI programs: As we run an MPI program, we need to call `MPI_Init()`
3355 *
Utilities::MPI::MPI_InitFinalize data structure. Note that we run the program
3356 * only with MPI, and set the thread count to 1.
3359 *
int main(
int argc,
char **argv)
3361 *
using namespace Euler_DG;
3362 *
using namespace dealii;
3370 * EulerProblem<dimension> euler_problem;
3371 * euler_problem.run();
3373 *
catch (std::exception &exc)
3375 * std::cerr << std::endl
3377 * <<
"----------------------------------------------------"
3379 * std::cerr <<
"Exception on processing: " << std::endl
3380 * << exc.what() << std::endl
3381 * <<
"Aborting!" << std::endl
3382 * <<
"----------------------------------------------------"
3389 * std::cerr << std::endl
3391 * <<
"----------------------------------------------------"
3393 * std::cerr <<
"Unknown exception!" << std::endl
3394 * <<
"Aborting!" << std::endl
3395 * <<
"----------------------------------------------------"
3403<a name=
"Results"></a><h1>Results</h1>
3406<a name=
"Programoutput"></a><h3>Program output</h3>
3409Running the program with the
default settings on a machine with 40 processes
3410produces the following output:
3412Running with 40 MPI processes
3413Vectorization over 8 doubles = 512 bits (AVX512)
3414Number of degrees of freedom: 147,456 ( = 4 [vars] x 1,024 [cells] x 36 [dofs/cell/var] )
3415Time step size: 0.00689325, minimal h: 0.3125,
initial transport scaling: 0.102759
3417Time: 0, dt: 0.0069, error rho: 2.76e-07, rho * u: 1.259e-06, energy: 2.987e-06
3418Time: 1.01, dt: 0.0069, error rho: 1.37e-06, rho * u: 2.252e-06, energy: 4.153e-06
3419Time: 2.01, dt: 0.0069, error rho: 1.561e-06, rho * u: 2.43e-06, energy: 4.493e-06
3420Time: 3.01, dt: 0.0069, error rho: 1.714e-06, rho * u: 2.591e-06, energy: 4.762e-06
3421Time: 4.01, dt: 0.0069, error rho: 1.843e-06, rho * u: 2.625e-06, energy: 4.985e-06
3422Time: 5.01, dt: 0.0069, error rho: 1.496e-06, rho * u: 1.961e-06, energy: 4.142e-06
3423Time: 6, dt: 0.0083, error rho: 1.007e-06, rho * u: 7.119e-07, energy: 2.972e-06
3424Time: 7, dt: 0.0095, error rho: 9.096e-07, rho * u: 3.786e-07, energy: 2.626e-06
3425Time: 8, dt: 0.0096, error rho: 8.439e-07, rho * u: 3.338e-07, energy: 2.43e-06
3426Time: 9, dt: 0.0096, error rho: 7.822e-07, rho * u: 2.984e-07, energy: 2.248e-06
3427Time: 10, dt: 0.0096, error rho: 7.231e-07, rho * u: 2.666e-07, energy: 2.074e-06
3429+-------------------------------------------+------------------+------------+------------------+
3430| Total wallclock time elapsed | 2.249s 30 | 2.249s | 2.249s 8 |
3432| Section | no. calls |
min time rank |
avg time |
max time rank |
3433+-------------------------------------------+------------------+------------+------------------+
3434| compute errors | 11 | 0.008066s 13 | 0.00952s | 0.01041s 20 |
3435| compute transport speed | 258 | 0.01012s 13 | 0.05392s | 0.08574s 25 |
3436| output | 11 | 0.9597s 13 | 0.9613s | 0.9623s 6 |
3437| rk time stepping total | 1283 | 0.9827s 25 | 1.015s | 1.06s 13 |
3438| rk_stage - integrals L_h | 6415 | 0.8803s 26 | 0.9198s | 0.9619s 14 |
3439| rk_stage - inv mass + vec upd | 6415 | 0.05677s 15 | 0.06487s | 0.07597s 13 |
3440+-------------------------------------------+------------------+------------+------------------+
3443The program output shows that all errors are small. This is due to the fact
3444that we use a relatively fine mesh of @f$32^2@f$ cells with polynomials of degree
34455 for a solution that is smooth. An interesting pattern shows for the time
3446step size: whereas it is 0.0069 up to time 5, it increases to 0.0096 for later
3447times. The step size increases once the vortex with some motion on top of the
3448speed of sound (and thus faster propagation) leaves the computational domain
3449between times 5 and 6.5. After that
point, the flow is simply uniform
3450in the same direction, and the maximum velocity of the gas is reduced
3451compared to the previous state where the uniform velocity was overlaid
3452by the vortex. Our time step formula recognizes this effect.
3454The final block of output shows detailed information about the timing
3455of individual parts of the programs; it breaks
this down by showing
3456the time taken by the fastest and the slowest processor, and the
3457average time --
this is often useful in very large computations to
3458find whether there are processors that are consistently overheated
3459(and consequently are throttling their clock speed) or consistently
3460slow
for other reasons.
3461The summary shows that 1283 time steps have been performed
3462in 1.02 seconds (looking at the average time among all MPI processes),
while
3463the output of 11 files has taken additional 0.96 seconds. Broken down per time
3464step and into the five Runge--Kutta stages, the compute time per evaluation is
34650.16 milliseconds. This high performance is typical of
matrix-
free evaluators
3466and a reason why
explicit time integration is very competitive against
3467implicit solvers, especially
for large-
scale simulations. The breakdown of
3468computational times at the
end of the program
run shows that the evaluation of
3469integrals in @f$\mathcal L_h@f$ contributes with around 0.92 seconds and the
3470application of the inverse mass
matrix with 0.06 seconds. Furthermore, the
3471estimation of the transport speed
for the time step size computation
3472contributes with another 0.05 seconds of compute time.
3474If we use three more levels of global refinement and 9.4 million DoFs in total,
3475the
final statistics are as follows (
for the modified Lax--Friedrichs flux,
3476@f$p=5@f$, and the same system of 40 cores of dual-socket Intel Xeon Gold 6230):
3478+-------------------------------------------+------------------+------------+------------------+
3479| Total wallclock time elapsed | 244.9s 12 | 244.9s | 244.9s 34 |
3481| Section | no. calls |
min time rank |
avg time |
max time rank |
3482+-------------------------------------------+------------------+------------+------------------+
3483| compute errors | 11 | 0.4239s 12 | 0.4318s | 0.4408s 9 |
3484| compute transport speed | 2053 | 3.962s 12 | 6.727s | 10.12s 7 |
3485| output | 11 | 30.35s 12 | 30.36s | 30.37s 9 |
3486| rk time stepping total | 10258 | 201.7s 7 | 205.1s | 207.8s 12 |
3487| rk_stage - integrals L_h | 51290 | 121.3s 6 | 126.6s | 136.3s 16 |
3488| rk_stage - inv mass + vec upd | 51290 | 66.19s 16 | 77.52s | 81.84s 10 |
3489+-------------------------------------------+------------------+------------+------------------+
3492Per time step, the solver now takes 0.02 seconds, about 25 times as long as
3493for the small problem with 147k unknowns. Given that the problem involves 64
3494times as many unknowns, the increase in computing time is not
3495surprising. Since we also do 8 times as many time steps, the compute time
3496should in theory increase by a factor of 512. The actual increase is 205 s /
34971.02 s = 202. This is because the small problem size cannot fully utilize the
349840 cores due to communication overhead. This becomes clear if we look into the
3499details of the operations done per time step. The evaluation of the
3500differential operator @f$\mathcal L_h@f$ with nearest neighbor communication goes
3501from 0.92 seconds to 127 seconds, i.
e., it increases with a factor of 138. On
3502the other hand, the cost for application of the inverse mass
matrix and the
3503vector updates, which do not need to communicate between the MPI processes at
3504all, has increased by a factor of 1195. The increase is more than the
3505theoretical factor of 512 because the operation is limited by the bandwidth
3506from RAM memory for the larger size while for the smaller size, all vectors
3507fit into the caches of the CPU. The
numbers show that the mass
matrix
3508evaluation and vector update part consume almost 40% of the time spent by the
3509Runge--Kutta stages -- despite using a low-storage Runge--Kutta integrator and
3510merging of vector operations! And despite using over-integration for the
3511@f$\mathcal L_h@f$ operator. For simpler differential operators and more expensive
3512time integrators, the proportion spent in the mass
matrix and vector update
3513part can also reach 70%. If we compute a throughput number in terms of DoFs
3514processed per
second and Runge--Kutta stage, we obtain @f[ \text{throughput} =
3515\frac{n_\mathrm{time steps} n_\mathrm{stages}
3516n_\mathrm{dofs}}{t_\mathrm{compute}} = \frac{10258 \cdot 5 \cdot
35179.4\,\text{MDoFs}}{205s} = 2360\, \text{MDoFs/s} @f] This throughput number is
3518very high, given that simply copying one vector to another one runs at
3519only around 10,000 MDoFs/s.
3521If we go to the next-larger size with 37.7 million DoFs, the overall
3522simulation time is 2196 seconds, with 1978 seconds spent in the time
3523stepping. The increase in
run time is a factor of 9.3
for the L_h
operator
3524(1179 versus 127 seconds) and a factor of 10.3
for the inverse mass matrix and
3525vector updates (797 vs 77.5 seconds). The reason
for this non-optimal increase
3526in
run time can be traced back to cache effects on the given hardware (with 40
3527MB of L2 cache and 55 MB of L3 cache): While not all of the relevant data fits
3528into caches for 9.4 million DoFs (one vector takes 75 MB and we have three
3529vectors plus some additional data in
MatrixFree), there is capacity for one and
3530a half vector nonetheless. Given that modern caches are more sophisticated than
3531the naive least-recently-used strategy (where we would have little re-use as
3532the data is used in a streaming-like fashion), we can assume that a sizeable
3533fraction of data can indeed be delivered from caches for the 9.4 million DoFs
3534case. For the larger case, even with optimal caching less than 10 percent of
3535data would fit into caches, with an associated loss in performance.
3538<a name=
"Convergenceratesfortheanalyticaltestcase"></a><h3>Convergence rates for the analytical test case</h3>
3541For the modified Lax--Friedrichs flux and measuring the error in the momentum
3542variable, we obtain the following convergence table (the rates are very
3543similar for the density and energy variables):
3545<table align=
"center" class=
"doxtable">
3548 <th colspan=
"3"><i>p</i>=2</th>
3549 <th colspan=
"3"><i>p</i>=3</th>
3550 <th colspan=
"3"><i>p</i>=5</th>
3565 <td align=
"right">16</td>
3572 <td align=
"right">2,304</td>
3573 <td align=
"center">1.373e-01</td>
3577 <td align=
"right">64</td>
3581 <td align=
"right">4,096</td>
3582 <td align=
"center">9.130e-02</td>
3584 <td align=
"right">9,216</td>
3585 <td align=
"center">8.899e-03</td>
3589 <td align=
"right">256</td>
3590 <td align=
"right">9,216</td>
3591 <td align=
"center">5.577e-02</td>
3593 <td align=
"right">16,384</td>
3594 <td align=
"center">7.381e-03</td>
3596 <td align=
"right">36,864</td>
3597 <td align=
"center">2.082e-04</td>
3601 <td align=
"right">1024</td>
3602 <td align=
"right">36,864</td>
3603 <td align=
"center">4.724e-03</td>
3605 <td align=
"right">65,536</td>
3606 <td align=
"center">3.072e-04</td>
3608 <td align=
"right">147,456</td>
3609 <td align=
"center">2.625e-06</td>
3613 <td align=
"right">4096</td>
3614 <td align=
"right">147,456</td>
3615 <td align=
"center">6.205e-04</td>
3617 <td align=
"right">262,144</td>
3618 <td align=
"center">1.880e-05</td>
3620 <td align=
"right">589,824</td>
3621 <td align=
"center">3.268e-08</td>
3625 <td align=
"right">16,384</td>
3626 <td align=
"right">589,824</td>
3627 <td align=
"center">8.279e-05</td>
3629 <td align=
"right">1,048,576</td>
3630 <td align=
"center">1.224e-06</td>
3632 <td align=
"right">2,359,296</td>
3633 <td align=
"center">9.252e-10</td>
3637 <td align=
"right">65,536</td>
3638 <td align=
"right">2,359,296</td>
3639 <td align=
"center">1.105e-05</td>
3641 <td align=
"right">4,194,304</td>
3642 <td align=
"center">7.871e-08</td>
3644 <td align=
"right">9,437,184</td>
3645 <td align=
"center">1.369e-10</td>
3649 <td align=
"right">262,144</td>
3650 <td align=
"right">9,437,184</td>
3651 <td align=
"center">1.615e-06</td>
3653 <td align=
"right">16,777,216</td>
3654 <td align=
"center">4.961e-09</td>
3656 <td align=
"right">37,748,736</td>
3657 <td align=
"center">7.091e-11</td>
3662If we
switch to the Harten-Lax-van Leer flux, the results are as follows:
3663<table align=
"center" class=
"doxtable">
3666 <th colspan=
"3"><i>p</i>=2</th>
3667 <th colspan=
"3"><i>p</i>=3</th>
3668 <th colspan=
"3"><i>p</i>=5</th>
3683 <td align=
"right">16</td>
3690 <td align=
"right">2,304</td>
3691 <td align=
"center">1.339e-01</td>
3695 <td align=
"right">64</td>
3699 <td align=
"right">4,096</td>
3700 <td align=
"center">9.037e-02</td>
3702 <td align=
"right">9,216</td>
3703 <td align=
"center">8.849e-03</td>
3707 <td align=
"right">256</td>
3708 <td align=
"right">9,216</td>
3709 <td align=
"center">4.204e-02</td>
3711 <td align=
"right">16,384</td>
3712 <td align=
"center">9.143e-03</td>
3714 <td align=
"right">36,864</td>
3715 <td align=
"center">2.501e-04</td>
3719 <td align=
"right">1024</td>
3720 <td align=
"right">36,864</td>
3721 <td align=
"center">4.913e-03</td>
3723 <td align=
"right">65,536</td>
3724 <td align=
"center">3.257e-04</td>
3726 <td align=
"right">147,456</td>
3727 <td align=
"center">3.260e-06</td>
3731 <td align=
"right">4096</td>
3732 <td align=
"right">147,456</td>
3733 <td align=
"center">7.862e-04</td>
3735 <td align=
"right">262,144</td>
3736 <td align=
"center">1.588e-05</td>
3738 <td align=
"right">589,824</td>
3739 <td align=
"center">2.953e-08</td>
3743 <td align=
"right">16,384</td>
3744 <td align=
"right">589,824</td>
3745 <td align=
"center">1.137e-04</td>
3747 <td align=
"right">1,048,576</td>
3748 <td align=
"center">9.400e-07</td>
3750 <td align=
"right">2,359,296</td>
3751 <td align=
"center">4.286e-10</td>
3755 <td align=
"right">65,536</td>
3756 <td align=
"right">2,359,296</td>
3757 <td align=
"center">1.476e-05</td>
3759 <td align=
"right">4,194,304</td>
3760 <td align=
"center">5.799e-08</td>
3762 <td align=
"right">9,437,184</td>
3763 <td align=
"center">2.789e-11</td>
3767 <td align=
"right">262,144</td>
3768 <td align=
"right">9,437,184</td>
3769 <td align=
"center">2.038e-06</td>
3771 <td align=
"right">16,777,216</td>
3772 <td align=
"center">3.609e-09</td>
3774 <td align=
"right">37,748,736</td>
3775 <td align=
"center">5.730e-11</td>
3780The tables show that we get optimal @f$\mathcal O\left(h^{p+1}\right)@f$
3781convergence rates
for both numerical fluxes. The errors are slightly smaller
3782for the Lax--Friedrichs flux
for @f$p=2@f$, but the picture is reversed
for
3783@f$p=3@f$; in any
case, the differences on
this testcase are relatively
3786For @f$p=5@f$, we reach the roundoff accuracy of @f$10^{-11}@f$ with both
3787fluxes on the finest grids. Also note that the errors are absolute with a
3788domain length of @f$10^2@f$, so relative errors are below @f$10^{-12}@f$. The HLL flux
3789is somewhat better
for the highest degree, which is due to a slight inaccuracy
3790of the Lax--Friedrichs flux: The Lax--Friedrichs flux sets a Dirichlet
3791condition on the solution that leaves the domain, which results in a small
3792artificial reflection, which is accentuated
for the Lax--Friedrichs
3793flux. Apart from that, we see that the influence of the numerical flux is
3794minor, as the polynomial part
inside elements is the main driver of the
3795accucary. The limited influence of the flux also has consequences when trying
3796to approach more challenging setups with the higher-order DG setup: Taking
for
3797example the parameters and grid of @ref step_33
"step-33", we get oscillations (which in turn
3798make density negative and make the solution explode) with both fluxes once the
3799high-mass part comes near the boundary, as opposed to the low-order finite
3800volume case (@f$p=0@f$). Thus, any
case that leads to shocks in the solution
3801necessitates some form of limiting or artificial dissipation. For another
3802alternative, see the @ref step_69
"step-69" tutorial program.
3805<a name=
"Resultsforflowinchannelaroundcylinderin2D"></a><h3>Results
for flow in channel around
cylinder in 2D</h3>
3808For the test
case of the flow around a
cylinder in a channel, we need to
3809change the
first code line to
3811 constexpr unsigned int testcase = 1;
3813This test
case starts with a background field of a
constant velocity
3815to go around an obstacle in the form of a
cylinder. Since we impose a
3816no-penetration condition on the
cylinder walls, the flow that
3817initially impinges head-on onto to
cylinder has to rearrange,
3818which creates a big sound wave. The following pictures show the pressure at
3819times 0.1, 0.25, 0.5, and 1.0 (top left to bottom right)
for the 2D
case with
38205 levels of global refinement,
using 102,400 cells with polynomial degree of
38215 and 14.7 million degrees of freedom over all 4 solution variables.
3822We clearly see the discontinuity that
3823propagates slowly in the upstream direction and more quickly in downstream
3824direction in the
first snapshot at time 0.1. At time 0.25, the sound wave has
3825reached the top and bottom walls and reflected back to the interior. From the
3826different distances of the reflected waves from lower and upper walls we can
3827see the slight asymmetry of the Schäfer-Turek test
case represented by
3829cylinder compared to below. At later times, the picture is more chaotic with
3830many sound waves all over the place.
3832<table align="
center" class="doxtable" style="width:85%">
3851The next picture shows an elevation plot of the pressure at time 1.0 looking
3852from the channel inlet towards the outlet at the same resolution -- here,
3853we can see the large number
3854of reflections. In the figure, two
types of waves are visible. The
3855larger-amplitude waves correspond to various reflections that happened as the
3856initial discontinuity hit the walls, whereas the small-amplitude waves of
3857size similar to the elements correspond to numerical artifacts. They have their
3858origin in the finite resolution of the scheme and appear as the discontinuity
3859travels through elements with high-order polynomials. This effect can be cured
3860by increasing resolution. Apart from this effect, the rich wave structure is
3861the result of the transport accuracy of the high-order DG method.
3865If we run the program with degree 2 and 6 levels of global refinements (410k
3866cells, 14.7M unknowns), we get the following evolution of the pressure
3867(elevation plot, colored by the value of the density):
3871 <iframe width="560" height="315" src="https:
3873 allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture"
3874 allowfullscreen></iframe>
3879With 2 levels of global refinement with 1,600 cells, the mesh and its
3880partitioning on 40 MPI processes looks as follows:
3884When we run the code with 4 levels of global refinements on 40 cores, we get
3885the following output:
3887Running with 40 MPI processes
3888Vectorization over 8 doubles = 512 bits (AVX512)
3889Number of degrees of freedom: 3,686,400 ( = 4 [vars] x 25,600 [cells] x 36 [dofs/cell/var] )
3890Time step size: 7.39876e-05, minimal h: 0.001875, initial transport scaling: 0.00110294
3892Time: 0, dt: 7.4e-05, norm rho: 4.17e-16, rho * u: 1.629e-16, energy: 1.381e-15
3893Time: 0.05, dt: 6.3e-05, norm rho: 0.02075, rho * u: 0.03801, energy: 0.08772
3894Time: 0.1, dt: 5.9e-05, norm rho: 0.02211, rho * u: 0.04515, energy: 0.08953
3895Time: 0.15, dt: 5.7e-05, norm rho: 0.02261, rho * u: 0.04592, energy: 0.08967
3896Time: 0.2, dt: 5.8e-05, norm rho: 0.02058, rho * u: 0.04361, energy: 0.08222
3897Time: 0.25, dt: 5.9e-05, norm rho: 0.01695, rho * u: 0.04203, energy: 0.06873
3898Time: 0.3, dt: 5.9e-05, norm rho: 0.01653, rho * u: 0.0401, energy: 0.06604
3899Time: 0.35, dt: 5.7e-05, norm rho: 0.01774, rho * u: 0.04264, energy: 0.0706
3903Time: 1.95, dt: 5.8e-05, norm rho: 0.01488, rho * u: 0.03923, energy: 0.05185
3904Time: 2, dt: 5.7e-05, norm rho: 0.01432, rho * u: 0.03969, energy: 0.04889
3906+-------------------------------------------+------------------+------------+------------------+
3907| Total wallclock time elapsed | 273.6s 13 | 273.6s | 273.6s 0 |
3909| Section | no. calls | min time rank | avg time | max time rank |
3910+-------------------------------------------+------------------+------------+------------------+
3911| compute errors | 41 | 0.01112s 35 | 0.0672s | 0.1337s 0 |
3912| compute transport speed | 6914 | 5.422s 35 | 15.96s | 29.99s 1 |
3913| output | 41 | 37.24s 35 | 37.3s | 37.37s 0 |
3914| rk time stepping total | 34564 | 205.4s 1 | 219.5s | 230.1s 35 |
3915| rk_stage - integrals L_h | 172820 | 153.6s 1 | 164.9s | 175.6s 27 |
3916| rk_stage - inv mass + vec upd | 172820 | 47.13s 13 | 53.09s | 64.05s 33 |
3917+-------------------------------------------+------------------+------------+------------------+
3920The norms shown here for the various quantities are the deviations
3921@f$\rho'@f$, @f$(\rho u)'@f$, and @f$E'@f$ against the background field (namely, the
3922initial condition). The distribution of run time is overall similar as in the
3923previous test case. The only slight difference is the larger proportion of
3924time spent in @f$\mathcal L_h@f$ as compared to the inverse mass matrix and vector
3925updates. This is because the geometry is deformed and the matrix-free
3926framework needs to load additional arrays for the geometry from memory that
3927are compressed in the affine mesh case.
3929Increasing the number of global refinements to 5, the output becomes:
3931Running with 40 MPI processes
3932Vectorization over 8 doubles = 512 bits (AVX512)
3933Number of degrees of freedom: 14,745,600 ( = 4 [vars] x 102,400 [cells] x 36 [dofs/cell/var] )
3937+-------------------------------------------+------------------+------------+------------------+
3938| Total wallclock time elapsed | 2693s 32 | 2693s | 2693s 23 |
3940| Section | no. calls | min time rank | avg time | max time rank |
3941+-------------------------------------------+------------------+------------+------------------+
3942| compute errors | 41 | 0.04537s 32 | 0.173s | 0.3489s 0 |
3943| compute transport speed | 13858 | 40.75s 32 | 85.99s | 149.8s 0 |
3944| output | 41 | 153.8s 32 | 153.9s | 154.1s 0 |
3945| rk time stepping total | 69284 | 2386s 0 | 2450s | 2496s 32 |
3946| rk_stage - integrals L_h | 346420 | 1365s 32 | 1574s | 1718s 19 |
3947| rk_stage - inv mass + vec upd | 346420 | 722.5s 10 | 870.7s | 1125s 32 |
3948+-------------------------------------------+------------------+------------+------------------+
3951The effect on performance is similar to the analytical test case -- in
3952theory, computation times should increase by a factor of 8, but we actually
3953see an increase by a factor of 11 for the time steps (219.5 seconds versus
39542450 seconds). This can be traced back to caches, with the small case mostly
3955fitting in caches. An interesting effect, typical of programs with a mix of
3956local communication (integrals @f$\mathcal L_h@f$) and global communication (computation of
3957transport speed) with some load imbalance, can be observed by looking at the
3958MPI ranks that encounter the minimal and maximal time of different phases,
3959respectively. Rank 0 reports the fastest throughput for the "rk time stepping
3960total" part. At the same time, it appears to be slowest for the "compute
3961transport speed" part, almost a factor of 2 slower than the
3962average and almost a factor of 4 compared to the faster rank.
3963Since the latter involves global communication, we can attribute the
3964slowness in this part to the fact that the local Runge--Kutta stages have
3965advanced more quickly on this rank and need to wait until the other processors
3966catch up. At this point, one can wonder about the reason for this imbalance:
3967The number of cells is almost the same on all MPI processes.
3968However, the matrix-free framework is faster on affine and Cartesian
3969cells located towards the outlet of the channel, to which the lower MPI ranks
3970are assigned. On the other hand, rank 32, which reports the highest run time
3971for the Runga--Kutta stages, owns the curved cells near the cylinder, for
3972which no data compression is possible. To improve throughput, we could assign
3973different weights to different cell
types when partitioning the
3975few time steps and try to rebalance then.
3977The throughput per Runge--Kutta stage can be computed to 2085 MDoFs/s for the
397814.7 million DoFs test case over the 346,000 Runge--Kutta stages, slightly slower
3979than the Cartesian mesh throughput of 2360 MDoFs/s reported above.
3981Finally, if we add one additional refinement, we record the following output:
3983Running with 40 MPI processes
3984Vectorization over 8 doubles = 512 bits (AVX512)
3985Number of degrees of freedom: 58,982,400 ( = 4 [vars] x 409,600 [cells] x 36 [dofs/cell/var] )
3989Time: 1.95, dt: 1.4e-05, norm rho: 0.01488, rho * u: 0.03923, energy: 0.05183
3990Time: 2, dt: 1.4e-05, norm rho: 0.01431, rho * u: 0.03969, energy: 0.04887
3992+-------------------------------------------+------------------+------------+------------------+
3993| Total wallclock time elapsed | 2.166e+04s 26 | 2.166e+04s | 2.166e+04s 24 |
3995| Section | no. calls | min time rank | avg time | max time rank |
3996+-------------------------------------------+------------------+------------+------------------+
3997| compute errors | 41 | 0.1758s 30 | 0.672s | 1.376s 1 |
3998| compute transport speed | 27748 | 321.3s 34 | 678.8s | 1202s 1 |
3999| output | 41 | 616.3s 32 | 616.4s | 616.4s 34 |
4000| rk time stepping total | 138733 | 1.983e+04s 1 | 2.036e+04s | 2.072e+04s 34 |
4001| rk_stage - integrals L_h | 693665 | 1.052e+04s 32 | 1.248e+04s | 1.387e+04s 19 |
4002| rk_stage - inv mass + vec upd | 693665 | 6404s 10 | 7868s | 1.018e+04s 32 |
4003+-------------------------------------------+------------------+------------+------------------+
4006The "rk time stepping total" part corresponds to a throughput of 2010 MDoFs/s. The
4007overall run time to perform 139k time steps is 20k seconds (5.7 hours) or 7
4008time steps per
second -- not so bad for having nearly 60 million
4009unknowns. More throughput can be achieved by adding more cores to
4013<a name="Resultsforflowinchannelaroundcylinderin3D"></a><h3>Results for flow in channel around cylinder in 3D</h3>
4016Switching the channel test case to 3D with 3 global refinements, the output is
4018Running with 40 MPI processes
4019Vectorization over 8 doubles = 512 bits (AVX512)
4020Number of degrees of freedom: 221,184,000 ( = 5 [vars] x 204,800 [cells] x 216 [dofs/cell/var] )
4024Time: 1.95, dt: 0.00011, norm rho: 0.01131, rho * u: 0.03056, energy: 0.04091
4025Time: 2, dt: 0.00011, norm rho: 0.0119, rho * u: 0.03142, energy: 0.04425
4027+-------------------------------------------+------------------+------------+------------------+
4028| Total wallclock time elapsed | 1.734e+04s 4 | 1.734e+04s | 1.734e+04s 38 |
4030| Section | no. calls | min time rank | avg time | max time rank |
4031+-------------------------------------------+------------------+------------+------------------+
4032| compute errors | 41 | 0.6551s 34 | 3.216s | 7.281s 0 |
4033| compute transport speed | 3546 | 160s 34 | 393.2s | 776.9s 0 |
4034| output | 41 | 1350s 34 | 1353s | 1357s 0 |
4035| rk time stepping total | 17723 | 1.519e+04s 0 | 1.558e+04s | 1.582e+04s 34 |
4036| rk_stage - integrals L_h | 88615 | 1.005e+04s 32 | 1.126e+04s | 1.23e+04s 11 |
4037| rk_stage - inv mass + vec upd | 88615 | 3056s 11 | 4322s | 5759s 32 |
4038+-------------------------------------------+------------------+------------+------------------+
4041The physics are similar to the 2D case, with a slight motion in the z
4042direction due to the gravitational force. The throughput per Runge--Kutta
4043stage in this case is
4045\text{throughput} = \frac{n_\mathrm{time steps} n_\mathrm{stages}
4046n_\mathrm{dofs}}{t_\mathrm{compute}} =
4047\frac{17723 \cdot 5 \cdot 221.2\,\text{M}}{15580s} = 1258\, \text{MDoFs/s}.
4050The throughput is lower than in 2D because the computation of the @f$\mathcal L_h@f$ term
4051is more expensive. This is due to over-integration with `degree+2` points and
4052the larger fraction of face integrals (worse volume-to-surface ratio) with
4053more expensive flux computations. If we only consider the inverse mass
matrix
4054and vector update part, we record a throughput of 4857 MDoFs/s
for the 2D
case
4055of the isentropic vortex with 37.7 million unknowns, whereas the 3D
case
4056runs with 4535 MDoFs/s. The performance is similar because both cases are in
4057fact limited by the memory bandwidth.
4059If we go to four levels of global refinement, we need to increase the number
4060of processes to fit everything in memory -- the computation needs around 350
4061GB of RAM memory in
this case. Also, the time it takes to complete 35k time
4062steps becomes more tolerable by adding additional resources. We therefore use
40636 nodes with 40 cores each, resulting in a computation with 240 MPI processes:
4065Running with 240 MPI processes
4066Vectorization over 8 doubles = 512 bits (AVX512)
4067Number of degrees of freedom: 1,769,472,000 ( = 5 [vars] x 1,638,400 [cells] x 216 [dofs/cell/var] )
4071Time: 1.95, dt: 5.6e-05,
norm rho: 0.01129, rho * u: 0.0306, energy: 0.04086
4072Time: 2, dt: 5.6e-05,
norm rho: 0.01189, rho * u: 0.03145, energy: 0.04417
4074+-------------------------------------------+------------------+------------+------------------+
4075| Total wallclock time elapsed | 5.396e+04s 151 | 5.396e+04s | 5.396e+04s 0 |
4077| Section | no. calls |
min time rank |
avg time |
max time rank |
4078+-------------------------------------------+------------------+------------+------------------+
4079| compute errors | 41 | 2.632s 178 | 7.221s | 16.56s 0 |
4080| compute transport speed | 7072 | 714s 193 | 1553s | 3351s 0 |
4081| output | 41 | 8065s 176 | 8070s | 8079s 0 |
4082| rk time stepping total | 35350 | 4.25e+04s 0 | 4.43e+04s | 4.515e+04s 193 |
4083| rk_stage - integrals L_h | 176750 | 2.936e+04s 134 | 3.222e+04s | 3.67e+04s 99 |
4084| rk_stage - inv mass + vec upd | 176750 | 7004s 99 | 1.207e+04s | 1.55e+04s 132 |
4085+-------------------------------------------+------------------+------------+------------------+
4087This simulation had nearly 2 billion unknowns -- quite a large
4088computation indeed, and still only needed around 1.5 seconds per time
4092<a name=
"Possibilitiesforextensions"></a><h3>Possibilities for extensions</h3>
4095The code presented here straight-forwardly extends to adaptive meshes, given
4096appropriate indicators for setting the refinement flags. Large-
scale
4097adaptivity of a similar solver in the context of the acoustic wave equation
4098has been achieved by the <a href=
"https://github.com/kronbichler/exwave">exwave
4099project</a>. However, in the present context, the benefits of adaptivity are often
4100limited to early times and effects close to the origin of sound waves, as the
4101waves eventually reflect and diffract. This leads to steep
gradients all over
4102the place, similar to turbulent flow, and a more or less globally
4105Another topic that we did not discuss in the results section is a comparison
4106of different time integration schemes. The program provides four variants of
4107low-storage Runga--Kutta integrators that each have slightly different
4108accuracy and stability behavior. Among the schemes implemented here, the
4109higher-order ones provide additional accuracy but come with slightly lower
4110efficiency in terms of step size per stage before they violate the CFL
4111condition. An interesting extension would be to compare the low-storage
4112variants proposed here with standard Runge--Kutta integrators or to use vector
4113operations that are
run separate from the mass
matrix operation and compare
4117<a name=
"Moreadvancednumericalfluxfunctionsandskewsymmetricformulations"></a><h4>More advanced numerical flux
functions and skew-
symmetric formulations</h4>
4120As mentioned in the introduction, the modified Lax--Friedrichs flux and the
4121HLL flux employed in this program are only two variants of a large body of
4122numerical fluxes available in the literature on the Euler equations. One
4123example is the HLLC flux (Harten-Lax-van Leer-Contact) flux which adds the
4124effect of rarefaction waves missing in the HLL flux, or the Roe flux. As
4125mentioned in the introduction, the effect of numerical fluxes on high-order DG
4126schemes is debatable (unlike for the case of low-order discretizations).
4128A related improvement to increase the stability of the solver is to also
4129consider the spatial integral terms. A shortcoming in the rather naive
4130implementation used above is the fact that the energy conservation of the
4131original Euler equations (in the absence of shocks) only holds up to a
4132discretization error. If the solution is under-resolved, the discretization
4133error can give rise to an increase in the numerical energy and eventually
4134render the discretization unstable. This is because of the inexact numerical
4135integration of the terms in the Euler equations, which both contain rational
4136nonlinearities and higher-degree content from curved cells. A way out of this
4137dilemma are so-called skew-
symmetric formulations, see @cite Gassner2013 for a
4138simple variant. Skew symmetry means that switching the role of the solution
4139@f$\mathbf{
w}@f$ and test
functions @f$\mathbf{v}@f$ in the weak form produces the
4140exact
negative of the original quantity, apart from some boundary terms. In
4141the discrete setting, the challenge is to keep this skew symmetry also when
4142the integrals are only computed approximately (in the continuous case,
4143skew-symmetry is a consequence of integration by parts). Skew-
symmetric
4144numerical schemes balance spatial derivatives in the conservative form
4145@f$(\nabla \mathbf v, \mathbf{
F}(\mathbf
w))_{
K}@f$ with contributions in the
4146convective form @f$(\mathbf v, \tilde{\mathbf{
F}}(\mathbf
w)\nabla
4147\mathbf{
w})_{
K}@f$ for some @f$\tilde{\mathbf{
F}}@f$. The precise terms depend on
4148the equation and the integration formula, and can in some cases by understood
4149by special skew-
symmetric finite difference schemes.
4151To get started, interested readers could take a look at
4153skew-
symmetric DG formulation is implemented with deal.II for a simple advection
4156<a name=
"Equippingthecodeforsupersoniccalculations"></a><h4>Equipping the code for supersonic calculations</h4>
4159As mentioned in the introduction, the solution to the Euler equations develops
4160shocks as the Mach number increases, which require additional mechanisms to
4161stabilize the scheme,
e.g. in the form of limiters. The main challenge besides
4162actually implementing the limiter or artificial viscosity approach would be to
4163load-balance the computations, as the additional computations involved for
4164limiting the oscillations in troubled cells would make them more expensive than the
4165plain DG cells without limiting. Furthermore, additional numerical fluxes that
4166better cope with the discontinuities would also be an option.
4168One ingredient also necessary for supersonic flows are appropriate boundary
4169conditions. As opposed to the subsonic outflow boundaries discussed in the
4170introduction and implemented in the program, all characteristics are outgoing
4171for supersonic outflow boundaries, so we do not want to prescribe any external
4174\mathbf{
w}^+ = \mathbf{
w}^- = \
begin{pmatrix} \rho^-\\
4175(\rho \mathbf u)^- \\
E^-\
end{pmatrix} \quad
4179In the code, we would simply add the additional statement
4181 else if (supersonic_outflow_boundaries.find(
boundary_id) !=
4182 supersonic_outflow_boundaries.
end())
4188in the `local_apply_boundary_face()` function.
4190<a name=
"ExtensiontothelinearizedEulerequations"></a><h4>Extension to the linearized Euler equations</h4>
4193When the interest with an Euler solution is mostly in the propagation of sound
4194waves, it often makes sense to linearize the Euler equations around a
4195background state, i.e., a given density, velocity and energy (or pressure)
4196field, and only compute the change against these fields. This is the setting
4197of the wide field of aeroacoustics. Even though the resolution requirements
4198are sometimes considerably reduced, implementation gets somewhat more
4199complicated as the linearization gives rise to additional terms. From a code
4200perspective, in the
operator evaluation we also need to equip the code with
4201the state to linearize against. This information can be provided either by
4202analytical
functions (that are evaluated in terms of the position of the
4203quadrature points) or by a vector similar to the solution. Based on that
4204vector, we would create an additional
FEEvaluation object to read from it and
4205provide the
values of the field at quadrature points. If the background
4206velocity is zero and the density is
constant, the linearized Euler equations
4207further simplify and can equivalently be written in the form of the
4208acoustic wave equation.
4210A challenge in the context of sound propagation is often the definition of
4211boundary conditions, as the computational domain needs to be of finite size,
4212whereas the actual simulation often spans an infinite (or at least much
4213larger) physical domain. Conventional Dirichlet or Neumann boundary conditions
4214give rise to reflections of the sound waves that eventually propagate back to
4215the region of interest and spoil the solution. Therefore, various variants of
4216non-reflecting boundary conditions or sponge layers, often in the form of
4218href=
"https://en.wikipedia.org/wiki/Perfectly_matched_layer">perfectly
4219matched layers</a> -- where the solution is damped without reflection
4223<a name=
"ExtensiontothecompressibleNavierStokesequations"></a><h4>Extension to the compressible Navier-Stokes equations</h4>
4226The solver presented in
this tutorial program can also be extended to the
4227compressible Navier--Stokes equations by adding viscous terms, as described in
4228@cite FehnWallKronbichler2019. To keep as much of the performance obtained
4229here despite the additional cost of elliptic terms,
e.g. via an interior
4231the @ref step_59
"step-59" tutorial program.
4234<a name=
"Usingcellcentricloopsandsharedmemory"></a><h4>Using cell-centric loops and shared memory</h4>
4237In
this tutorial, we used face-centric loops. Here, cell and face integrals
4238are treated in separate loops, resulting in multiple writing accesses into the
4239result vector, which is relatively expensive on modern hardware since writing
4240operations generally result also in an implicit read operation. Element-centric
4241loops, on the other hand, are processing a cell and in direct succession
4242processing all its 2
d faces. Although
this kind of
loop implies that fluxes have
4243to be computed twice (
for each side of an interior face), the fact that the
4244result vector has to accessed only once might - and the fact that the resulting
4245algorithm is
free of race-conditions and as such perfectly suitable
for
4246shared memory - already give a performance
boost. If you are interested in these
4247advanced topics, you can take a look at @ref step_76
"step-76" where we take the present
4248tutorial and modify it so that we can use these features.
4251<a name=
"PlainProg"></a>
4252<h1> The plain program</h1>
4253@include
"step-67.cc"
void write_vtu_in_parallel(const std::string &filename, const MPI_Comm comm) const
void add_data_vector(const VectorType &data, const std::vector< std::string > &names, const DataVectorType type=type_automatic, const std::vector< DataComponentInterpretation::DataComponentInterpretation > &data_component_interpretation={})
virtual UpdateFlags get_needed_update_flags() const =0
virtual void evaluate_vector_field(const DataPostprocessorInputs::Vector< dim > &input_data, std::vector< Vector< double > > &computed_quantities) const
virtual std::vector< std::string > get_names() const =0
virtual std::vector< DataComponentInterpretation::DataComponentInterpretation > get_data_component_interpretation() const
void set_dof_values(VectorType &dst, const unsigned int first_index=0, const std::bitset< VectorizedArrayType::size()> &mask=std::bitset< VectorizedArrayType::size()>().flip()) const
value_type get_value(const unsigned int q_point) const
const ShapeInfoType * data
Tensor< 2, dim, VectorizedArrayType > inverse_jacobian(const unsigned int q_point) const
const unsigned int n_q_points
void gather_evaluate(const VectorType &input_vector, const EvaluationFlags::EvaluationFlags evaluation_flag)
void evaluate(const EvaluationFlags::EvaluationFlags evaluation_flag)
virtual RangeNumberType value(const Point< dim > &p, const unsigned int component=0) const
void reinit(const size_type size, const bool omit_zeroing_entries=false)
unsigned int depth_console(const unsigned int n)
Abstract base class for mapping classes.
void transform_from_q_points_to_basis(const unsigned int n_actual_components, const VectorizedArrayType *in_array, VectorizedArrayType *out_array) const
void loop(const std::function< void(const MatrixFree< dim, Number, VectorizedArrayType > &, OutVector &, const InVector &, const std::pair< unsigned int, unsigned int > &)> &cell_operation, const std::function< void(const MatrixFree< dim, Number, VectorizedArrayType > &, OutVector &, const InVector &, const std::pair< unsigned int, unsigned int > &)> &face_operation, const std::function< void(const MatrixFree< dim, Number, VectorizedArrayType > &, OutVector &, const InVector &, const std::pair< unsigned int, unsigned int > &)> &boundary_operation, OutVector &dst, const InVector &src, const bool zero_dst_vector=false, const DataAccessOnFaces dst_vector_face_access=DataAccessOnFaces::unspecified, const DataAccessOnFaces src_vector_face_access=DataAccessOnFaces::unspecified) const
unsigned int n_active_entries_per_cell_batch(const unsigned int cell_batch_index) 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
void reinit(const MappingType &mapping, const DoFHandler< dim > &dof_handler, const AffineConstraints< number2 > &constraint, const QuadratureType &quad, const AdditionalData &additional_data=AdditionalData())
constexpr numbers::NumberTraits< Number >::real_type norm_square() const
void print_wall_time_statistics(const MPI_Comm mpi_comm, const double print_quantile=0.) const
static constexpr std::size_t size()
#define DEAL_II_ALWAYS_INLINE
#define DEAL_II_OPENMP_SIMD_PRAGMA
__global__ void set(Number *val, const Number s, const size_type N)
#define Assert(cond, exc)
#define AssertDimension(dim1, dim2)
#define AssertThrow(cond, exc)
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())
@ update_values
Shape function values.
@ update_normal_vectors
Normal vectors.
@ update_JxW_values
Transformed quadrature weights.
@ update_gradients
Shape function gradients.
@ update_quadrature_points
Transformed quadrature points.
CGAL::Exact_predicates_exact_constructions_kernel_with_sqrt K
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)
DataComponentInterpretation
@ component_is_part_of_vector
void approximate(SynchronousIterators< std::tuple< typename DoFHandler< dim, spacedim >::active_cell_iterator, Vector< float >::iterator > > const &cell, const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof_handler, const InputVector &solution, const unsigned int component)
The namespace for the EvaluationFlags enum.
void hyper_rectangle(Triangulation< dim, spacedim > &tria, const Point< dim > &p1, const Point< dim > &p2, const bool colorize=false)
void cylinder(Triangulation< dim > &tria, const double radius=1., const double half_length=1.)
void channel_with_cylinder(Triangulation< dim > &tria, const double shell_region_width=0.03, const unsigned int n_shells=2, const double skewness=2.0, const bool colorize=false)
@ valid
Iterator points to a valid object.
@ matrix
Contents is actually a matrix.
@ symmetric
Matrix is symmetric.
@ diagonal
Matrix is diagonal.
@ general
No special properties.
double norm(const FEValuesBase< dim > &fe, const ArrayView< const std::vector< Tensor< 1, dim > > > &Du)
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
SymmetricTensor< 2, dim, Number > E(const Tensor< 2, dim, Number > &F)
Tensor< 2, dim, Number > w(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
SymmetricTensor< 2, dim, Number > e(const Tensor< 2, dim, Number > &F)
SymmetricTensor< 2, dim, Number > d(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
Tensor< 2, dim, Number > F(const Tensor< 2, dim, Number > &Grad_u)
void call(const std::function< RT()> &function, internal::return_value< RT > &ret_val)
@ LOW_STORAGE_RK_STAGE9_ORDER5
@ LOW_STORAGE_RK_STAGE3_ORDER3
@ LOW_STORAGE_RK_STAGE7_ORDER4
@ LOW_STORAGE_RK_STAGE5_ORDER4
VectorType::value_type * end(VectorType &V)
VectorType::value_type * begin(VectorType &V)
T sum(const T &t, const MPI_Comm mpi_communicator)
unsigned int n_mpi_processes(const MPI_Comm mpi_communicator)
T max(const T &t, const MPI_Comm mpi_communicator)
T min(const T &t, const MPI_Comm mpi_communicator)
unsigned int this_mpi_process(const MPI_Comm mpi_communicator)
const std::string get_current_vectorization_level()
constexpr T pow(const T base, const int iexp)
Number truncate_to_n_digits(const Number number, const unsigned int n_digits)
std::string int_to_string(const unsigned int value, const unsigned int digits=numbers::invalid_unsigned_int)
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)
bool check(const ConstraintKinds kind_in, const unsigned int dim)
long double gamma(const unsigned int n)
DEAL_II_HOST constexpr TableIndices< 2 > merge(const TableIndices< 2 > &previous_indices, const unsigned int new_index, const unsigned int position)
unsigned int n_cells(const internal::TriangulationImplementation::NumberCache< 1 > &c)
int(&) functions(const void *v1, const void *v2)
void reinit(MatrixBlock< MatrixType > &v, const BlockSparsityPattern &p)
static constexpr double PI
::VectorizedArray< Number, width > exp(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > min(const ::VectorizedArray< Number, width > &, const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > max(const ::VectorizedArray< Number, width > &, const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > sqrt(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > pow(const ::VectorizedArray< Number, width > &, const Number p)
::VectorizedArray< Number, width > abs(const ::VectorizedArray< Number, width > &)
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
****code * * MPI_Finalize()
bool write_higher_order_cells
UpdateFlags mapping_update_flags
DEAL_II_HOST constexpr Number determinant(const SymmetricTensor< 2, dim, Number > &)
std::array< Number, 1 > eigenvalues(const SymmetricTensor< 2, 1, Number > &T)
const TriangulationDescription::Settings settings
1.9.7