diff --git a/CMakeLists.txt b/CMakeLists.txt
index da1d4bd2924c017dd82018265cd1f04d04a8d97e..627474c66f465e29bec62d833e44f9bda35c1a72 100644
--- a/CMakeLists.txt
+++ b/CMakeLists.txt
@@ -489,13 +489,6 @@ LIST(APPEND source_files ${hpp_for_lib} ${cpp_for_lib})
add_library(TurTLE ${source_files})
-# TODO: we should be using the bit below, but we need to fix everything else about TURTLE_LIBS before we can do that
-#target_link_libraries(TurTLE PUBLIC OpenMP::OpenMP_CXX)
-
-set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} ${OpenMP_CXX_FLAGS}")
-set(CMAKE_EXE_LINKER_FLAGS "${CMAKE_EXE_LINKER_FLAGS} ${OpenMP_EXE_LINKER_FLAGS}")
-list(APPEND TURTLE_LIBS "${OpenMP_CXX_LIB_NAMES}")
-
target_link_libraries(TurTLE ${TURTLE_LIBS})
target_compile_options(TurTLE PRIVATE $<$<CONFIG:DEBUG>:-O1>)
diff --git a/TurTLE/DNS.py b/TurTLE/DNS.py
index de8b7ff17bc251801093b8e04a6cc4f8f6f4a567..d2f7dba660f04bd6082ea87594b020892cbbe04d 100644
--- a/TurTLE/DNS.py
+++ b/TurTLE/DNS.py
@@ -344,13 +344,17 @@ class DNS(_code):
if k in pp_file.keys():
self.statistics[k] = pp_file[k][...]
# sanity check --- Parseval theorem check
- test_energy = (np.max(np.abs(
+ energy_error = np.max(np.abs(
self.statistics['renergy(t)'] -
- self.statistics['energy(t)']) / self.statistics['energy(t)']) < 1e-5)
- test_enstrophy = (np.max(np.abs(
+ self.statistics['energy(t)']) / self.statistics['energy(t)'])
+ test_energy = (energy_error < 1e-5)
+ enstrophy_error = np.max(np.abs(
self.statistics['renstrophy(t)'] -
- self.statistics['enstrophy(t)']) / self.statistics['enstrophy(t)']) < 1e-5)
+ self.statistics['enstrophy(t)']) / self.statistics['enstrophy(t)'])
+ test_enstrophy = (enstrophy_error < 1e-5)
if strict_Parseval_check:
+ #print('energy error', energy_error)
+ #print('enstrophy error', enstrophy_error)
assert(test_energy)
assert(test_enstrophy)
else:
@@ -383,7 +387,7 @@ class DNS(_code):
Further computation of statistics based on the contents of
``simname_cache.h5``.
Standard quantities are as follows
- (consistent with [Ishihara]_):
+ (consistent with [ishihara2007jfm]_):
.. math::
@@ -399,13 +403,6 @@ class DNS(_code):
Re = \\frac{U_{\\textrm{int}} L_{\\textrm{int}}}{\\nu}, \\hskip
.5cm
R_{\\lambda} = \\frac{U_{\\textrm{int}} \\lambda}{\\nu}
-
- .. [Ishihara] T. Ishihara et al,
- *Small-scale statistics in high-resolution direct numerical
- simulation of turbulence: Reynolds number dependence of
- one-point velocity gradient statistics*.
- J. Fluid Mech.,
- **592**, 335-366, 2007
"""
self.statistics['Uint(t)'] = np.sqrt(2*self.statistics['energy(t)'] / 3)
for key in ['energy',
@@ -820,7 +817,7 @@ class DNS(_code):
kM = opt.n * 0.5
if opt.dealias_type == 1:
kM *= 0.8
- # tweak forcing/viscosity based on forcint type
+ # tweak forcing/viscosity based on forcing type
if opt.forcing_type == 'linear':
# custom famplitude for 288 and 576
if opt.n == 288:
@@ -833,6 +830,11 @@ class DNS(_code):
opt.nu = (
opt.injection_rate *
(opt.kMeta / kM)**4)**(1./3)
+ if opt.injection_rate == 0:
+ opt.nu = (
+ (opt.kMeta / kM)**(4./3) *
+ (np.pi / 1.5)**(1./3) *
+ (2*self.parameters['energy'] / 3)**0.5)
elif opt.forcing_type == 'fixed_energy':
kf = 1. / (1./opt.fk0 +
1./opt.fk1)
@@ -1128,7 +1130,7 @@ class DNS(_code):
f = h5py.File(self.get_checkpoint_0_fname(), 'r')
try:
dset = f[checkpoint_field + '/complex/0']
- need_field = (dset.shape == (self.parameters['ny'],
+ need_field = (dset.shape != (self.parameters['ny'],
self.parameters['nz'],
self.parameters['nx']//2+1,
3))
@@ -1213,12 +1215,13 @@ class DNS(_code):
return None
def launch_jobs(
self,
- opt = None):
+ opt = None,
+ iter0 = 0):
if not os.path.exists(self.get_data_file_name()):
if not os.path.isdir(self.work_dir):
os.makedirs(self.work_dir)
self.generate_initial_condition(opt = opt)
- self.write_par()
+ self.write_par(iter0 = iter0)
if (('test' in self.dns_type) or
(self.dns_type in ['kraichnan_field'])):
self.check_current_vorticity_exists = False
diff --git a/TurTLE/PP.py b/TurTLE/PP.py
index 8382ab13619972c66ac5c2c456122115964e8dac..4699b52d8cd9f662f380bf0eff337e3595f9572c 100644
--- a/TurTLE/PP.py
+++ b/TurTLE/PP.py
@@ -238,7 +238,7 @@ class PP(_code):
Further computation of statistics based on the contents of
``simname_postprocess.h5``.
Standard quantities are as follows
- (consistent with [Ishihara]_):
+ (consistent with [ishihara2007jfm]_):
.. math::
@@ -254,13 +254,6 @@ class PP(_code):
Re = \\frac{U_{\\textrm{int}} L_{\\textrm{int}}}{\\nu}, \\hskip
.5cm
R_{\\lambda} = \\frac{U_{\\textrm{int}} \\lambda}{\\nu}
-
- .. [Ishihara] T. Ishihara et al,
- *Small-scale statistics in high-resolution direct numerical
- simulation of turbulence: Reynolds number dependence of
- one-point velocity gradient statistics*.
- J. Fluid Mech.,
- **592**, 335-366, 2007
"""
for key in ['energy', 'enstrophy']:
self.statistics[key + '(t)'] = (self.statistics['dk'] *
@@ -783,9 +776,9 @@ class PP(_code):
if 'field_dtype' in df.keys():
# we don't need to do anything, raw binary files are used
return None
- last_iteration = df['iteration'][...]
- cppf = df['parameters/checkpoints_per_file'][...]
- niter_out = df['parameters/niter_out'][...]
+ last_iteration = df['iteration'][()]
+ cppf = df['parameters/checkpoints_per_file'][()]
+ niter_out = df['parameters/niter_out'][()]
with h5py.File(os.path.join(self.work_dir, self.simname + '_fields.h5'), 'a') as ff:
ff.require_group('vorticity')
ff.require_group('vorticity/complex')
@@ -807,7 +800,13 @@ class PP(_code):
if self.dns_type == 'resize':
with h5py.File(os.path.join(self.work_dir, self.pp_parameters['new_simname'] + '.h5'), 'a') as ff:
for kk in df['parameters'].keys():
- ff['parameters/' + kk] = df['parameters/' + kk][...]
+ ff['parameters/' + kk] = df['parameters/' + kk][()]
+ ff['parameters/nx'][()] = self.pp_parameters['new_nx']
+ ff['parameters/ny'][()] = self.pp_parameters['new_ny']
+ ff['parameters/nz'][()] = self.pp_parameters['new_nz']
+ if 'iteration' not in ff.keys():
+ ff['iteration'] = last_iteration
+ ff['checkpoint'] = df['checkpoint'][()]
return None
def launch_jobs(
self,
diff --git a/TurTLE/test/particle_set/NSVEparticle_set.cpp b/TurTLE/test/particle_set/NSVEparticle_set.cpp
index ee89c2e36df943fd38dc04372dd2b459d5e21632..da986349dbe84e1db8e4945a4cf4e475ec894ea3 100644
--- a/TurTLE/test/particle_set/NSVEparticle_set.cpp
+++ b/TurTLE/test/particle_set/NSVEparticle_set.cpp
@@ -205,6 +205,7 @@ int NSVEparticle_set<rnumber>::read_parameters(void)
this->tracers0_neighbours = hdf5_tools::read_value<int>(parameter_file, "parameters/tracers0_neighbours");
this->tracers0_smoothness = hdf5_tools::read_value<int>(parameter_file, "parameters/tracers0_smoothness");
H5Fclose(parameter_file);
+ MPI_Barrier(this->comm);
return EXIT_SUCCESS;
}
diff --git a/TurTLE/tools.py b/TurTLE/tools.py
index 7ddbf32d2e6d0beee57d11db75c5b07dbd0e5f86..6d3869854a9070eaeca2c1bc8149e38637730df7 100644
--- a/TurTLE/tools.py
+++ b/TurTLE/tools.py
@@ -230,13 +230,8 @@ def get_fornberg_coeffs(
"""compute finite difference coefficients
Compute finite difference coefficients for a generic grid specified
- by ``x``, according to [Fornberg]_.
+ by ``x``, according to [fornberg1988mc]_.
The function is used by :func:`particle_finite_diff_test`.
-
- .. [Fornberg] B. Fornberg,
- *Generation of Finite Difference Formulas on Arbitrarily Spaced Grids*.
- Mathematics of Computation,
- **51:184**, 699-706, 1988
"""
N = len(a) - 1
d = []
diff --git a/cpp/field.cpp b/cpp/field.cpp
index 6e950ad450553dbfccfa3ba33f0ae0bab168c58b..7dd5889f0f57bbf270f55baa0507205694e9cc77 100644
--- a/cpp/field.cpp
+++ b/cpp/field.cpp
@@ -210,7 +210,7 @@ int field<rnumber, be, fc>::io(
}
if (read)
{
- DEBUG_MSG("field::io trying to read field from file %s\n", fname.c_str());
+ DEBUG_MSG("field::io trying to read field %s from file %s\n", dset_name.c_str(), fname.c_str());
assert(file_exists);
file_id = H5Fopen(fname.c_str(), H5F_ACC_RDONLY, plist_id);
}
diff --git a/cpp/full_code/NSE.cpp b/cpp/full_code/NSE.cpp
index 25a4c3956f98143bba4be477490499fa67533921..fac9a0b3c622d46bdbddd5e91fd9c7f40bed0140 100644
--- a/cpp/full_code/NSE.cpp
+++ b/cpp/full_code/NSE.cpp
@@ -460,7 +460,7 @@ int NSE<rnumber>::ShuOsher(void)
*this->tmp_velocity2 = rnumber(0);
this->add_nonlinear_term(this->tmp_velocity1, this->tmp_velocity2);
- this->add_forcing_term(this->velocity, this->tmp_velocity2);
+ this->add_forcing_term(this->tmp_velocity1, this->tmp_velocity2);
this->kk->CLOOP_K2(
[&](const ptrdiff_t cindex,
const ptrdiff_t xindex,
@@ -482,7 +482,7 @@ int NSE<rnumber>::ShuOsher(void)
*this->tmp_velocity1 = rnumber(0);
this->add_nonlinear_term(this->tmp_velocity2, this->tmp_velocity1);
- this->add_forcing_term(this->velocity, this->tmp_velocity1);
+ this->add_forcing_term(this->tmp_velocity2, this->tmp_velocity1);
this->kk->CLOOP_K2(
[&](const ptrdiff_t cindex,
const ptrdiff_t xindex,
diff --git a/cpp/full_code/NSVEparticles.cpp b/cpp/full_code/NSVEparticles.cpp
index f2c53a4eadb5770ffecb4bb440fbc5e899b662a6..3f24be4060ac03b45bdd55f13a81d46482327037 100644
--- a/cpp/full_code/NSVEparticles.cpp
+++ b/cpp/full_code/NSVEparticles.cpp
@@ -98,6 +98,7 @@ int NSVEparticles<rnumber>::initialize(void)
this->particles_output_writer_mpi->close_file();
}
+ DEBUG_MSG("fs->iteration = %d\n", this->fs->iteration);
DEBUG_MSG_WAIT(MPI_COMM_WORLD, "about to call particles_system_builder\n");
this->ps = particles_system_builder(
this->fs->cvelocity, // (field object)
diff --git a/cpp/kspace.cpp b/cpp/kspace.cpp
index 921e498de2330bc64b97fc76ccd3492955448ede..369db6236fb8eaad52c9e0c2a7fdca35fb5d5198 100644
--- a/cpp/kspace.cpp
+++ b/cpp/kspace.cpp
@@ -28,6 +28,7 @@
#include <cstdlib>
#include <algorithm>
#include <cassert>
+#include <stdexcept>
#include "kspace.hpp"
#include "scope_timer.hpp"
#include "shared_array.hpp"
@@ -201,8 +202,9 @@ int kspace<be, dt>::store(hid_t stat_file)
if (this->nshells != int(dims[0]))
{
DEBUG_MSG(
- "ERROR: computed nshells %d not equal to data file nshells %d\n",
- this->nshells, dims[0]);
+ "ERROR: computed nshells %d not equal to data file nshells %d for dealiasing %d\n",
+ this->nshells, dims[0], dt);
+ throw std::runtime_error("Computed nshells not equal to data file nshells. Something is probably wrong with the dealiasing option.\n");
}
H5Dwrite(
dset,
diff --git a/documentation/index.rst b/documentation/index.rst
index 0566f7b4dad43652deef3c17020987a4ae0b896b..7788a2f226901d8a66b728ff6aa45dda63e8ff33 100644
--- a/documentation/index.rst
+++ b/documentation/index.rst
@@ -18,6 +18,10 @@ Contents:
chapters/api
chapters/cpp_doxygen
+ sphinx_static/convergence
+
+ sphinx_static/bibliography
+
Indices and tables
==================
diff --git a/documentation/sphinx_static/bibliography.rst b/documentation/sphinx_static/bibliography.rst
new file mode 100644
index 0000000000000000000000000000000000000000..1b90b08284d75040e5732b0d84351946eb983036
--- /dev/null
+++ b/documentation/sphinx_static/bibliography.rst
@@ -0,0 +1,21 @@
+Bibliography
+============
+
+.. [fornberg1988mc] B. Fornberg,
+ *Generation of Finite Difference Formulas on Arbitrarily Spaced Grids*.
+ Mathematics of Computation,
+ **51:184**, 699-706, 1988
+
+.. [ishihara2007jfm] T. Ishihara et al,
+ *Small-scale statistics in high-resolution direct numerical
+ simulation of turbulence: Reynolds number dependence of
+ one-point velocity gradient statistics*.
+ J. Fluid Mech.,
+ **592**, 335-366, 2007
+
+.. [lalescu2010jcp] C. C. Lalescu, B. Teaca and D. Carati
+ *Implementation of high order spline interpolations for tracking test
+ particles in discretized fields*.
+ J. Comput. Phys.
+ **229**, 5862-5869, 2010
+
diff --git a/documentation/sphinx_static/convergence.rst b/documentation/sphinx_static/convergence.rst
new file mode 100644
index 0000000000000000000000000000000000000000..5d0b9574ed83d8efb1465766e800f98b13de628e
--- /dev/null
+++ b/documentation/sphinx_static/convergence.rst
@@ -0,0 +1,114 @@
+Tutorial: convergence test
+==========================
+
+An essential property of numerical methods is their accuracy. With
+standard approaches to differential equations, one may use
+generic convergence tests to characterize algorithms and/or
+implementations.
+
+As an example usage of TurTLE, we show here how to characterize
+convergence for the field solver, or the AdamsBashforth
+particle solver.
+The purpose of this tutorial is to introduce TurTLE users to nontrivial
+workflows that involve keeping track of a number of simulations and
+prescribing their initial conditions.
+
+The field solver test is a fairly simple pursuit, showing basic
+operations with the `turtle` launcher.
+
+The particle solver test is also minimal within the methodological
+constraints imposed by the use of the Adams Bashforth integration
+method, leading to a relatively increased complexity.
+The example shows how to account for a number of technical details: full
+control over multi-step method initialization and full control of fluid
+solver forcing parameters.
+
+
+Navier-Stokes solution
+----------------------
+
+TurTLE implements two distinct solvers of the Navier-Stokes equations:
+one for their vorticity formulation, and one for the more common velocity
+formulation.
+While there are inescapable differences in the two implementations, they
+both use the same time-stepping algorithm (a 3rd order RungeKutta
+method).
+
+We test convergence properties for a "statistically stationary initial
+condition", since the smoothness properties of the field will influence
+convergence properties in general.
+As an example of the variety of questions that we can address with
+TurTLE, we also perform a direct comparison of the NSVE (vorticity
+solver) and NSE (velocity solver) results.
+
+A basic Python script that performs this convergence test is provided
+under `examples/convergence/fields_temporal.py`, and it is discussed
+below.
+In brief, simulations are performed using the same initial conditions,
+but different time steps.
+Error estimates are computed as differences between the different
+solutions.
+The figure shows the L2 norm of these differences, normalized by
+the root-mean-squared velocity magnitude.
+Note that the actual value of this "relative error" is irrelevant
+(it depends on total integration time, spatial resolution, etc), but the
+plot shows that (1) solutions converge with the 3rd order of the
+timestep and (2) the systematic difference between NSE and NSVE behaves
+quite differently from the two individual error estimates.
+
+.. image:: err_vs_dt_field.svg
+
+Particle tracking errors
+------------------------
+
+There is a more complex mechanism that generates numerical errors for
+particle tracking in this setting.
+In particular, the accuracy of the field itself is relevant, as well as
+the accuracy of the interpolation method coupled to the ODE solver.
+For instance the default particle tracking solver is a 4th order
+Adams Bashforth method in TurTLE, which seems at first unnecessary since the
+fluid only provides a 3rd order solution.
+The default interpolation method is that of cubic splines, which should
+in principle limit ODE error convergence to 2nd order (because convergence
+relies on existence of relevant Taylor expansion terms, and interpolated
+fields only have a finite number of continuous derivatives).
+This discussion is briefly continued below, but we will not do it justice
+in this tutorial.
+
+We present below a Python script that performs a particle tracking
+convergence test, provided as
+`examples/convergence/particles_temporal.py`.
+The figure shows trajectory integration errors, in a fairly simple
+setting, for two interpolation methods: cubic splines (I4O3) and 5th
+order splines on a kernel of 8 points (I8O5).
+Furthermore, we show both the maximum and the mean errors for the same
+set of trajectories.
+While both mean errors seem to converge with the 3rd order of the
+timestep (consistent with the error of the velocity field), the maximum
+error obtained for cubic splines converges with the second order ---
+which is the expected behavior for particle tracking in a field that
+only has one continuous derivative.
+
+.. image:: err_vs_dt_particle.svg
+
+The results would merit a longer and more careful analysis outside the
+scope of this particular tutorial, so we limit ourselves to noting that
+(1) different ODEs will have different numerical properties (i.e.
+rotating particles, or inertial particles) and (2) interpolation errors
+may in general grow with the Reynolds number, since interpolation
+errors depend on values of gradients (however the distribution of
+extreme gradient values also changes with Reynolds number, so the
+overall behavior is complex).
+
+
+Python code for fields
+----------------------
+
+.. include:: convergence_fields.rst
+
+
+Python code for particles
+-------------------------
+
+.. include:: convergence_particles.rst
+
diff --git a/documentation/sphinx_static/convergence_fields.rst b/documentation/sphinx_static/convergence_fields.rst
new file mode 100644
index 0000000000000000000000000000000000000000..5830e4e7dff3e479d4e050678870cd68649d7e0d
--- /dev/null
+++ b/documentation/sphinx_static/convergence_fields.rst
@@ -0,0 +1,281 @@
+We present here a Python script that performs simple convergence tests
+with TurTLE. The script itself is provided as the file
+`examples/convergence/fields_temporal.py`.
+
+The preamble imports standard packages, TurTLE itself, and a few simple
+tools from `examples/convergence/fields_base.py` (not discussed here):
+
+.. code:: python
+
+ import numpy as np
+ import h5py
+ import matplotlib.pyplot as plt
+
+ import TurTLE
+ from TurTLE import DNS, PP
+
+ from fields_base import *
+
+As explained above, we should test convergence for a statistically
+stationary regime.
+The base simulation will use a real-space grid of :math:`128 \times 128 \times 128` points,
+and experience shows that reasonable quasistationarity is reached for this
+problem-size in a few hundred iterations (here `base_niterations`).
+For the convergence test itself, we use a relatively small number
+of iterations (i.e. `test_niterations`), otherwise deterministic chaos will dominate the
+difference between different solutions (rather than numerical integration
+errors).
+
+.. code:: python
+
+ base_niterations = 512
+ test_niterations = 32
+ grid_size = 128
+
+We distinguish between three steps of this script.
+First, we generate the statistically quasistationary regime.
+Second, we run the fluid solver.
+Finally, we "analyze" the results (i.e. we generate simple error plots).
+
+.. code:: python
+
+ def main():
+ generate_initial_conditions()
+
+ run_simulations_field(
+ err_vs_t = False)
+ plot_error_field(
+ err_vs_t = False)
+ return None
+
+The `err_vs_t` parameter may be used if one is interested in looking at
+the difference between the NSE and NSVE solutions as a function of time.
+It turns on a fine-grained output of the fields in
+`run_simulations_field` and it turns on the corresponding plot in
+`plot_error_field`.
+
+`generate_initial_conditions` performs a few calls to C++ executable
+codes generated with TurTLE. In brief, it
+
+ * generates a vorticity field in a quasistationary Navier-Stokes
+ regime (using the NSVE solver).
+ * generates the corresponding velocity field needed for the NSE solver
+
+We use the `launch` method of the `DNS` and `PP` Python
+classes --- the wrappers around the C++ `direct_numerical_simulation`
+and `postprocess` functionality.
+This means that the current Python code may be translated to a shell
+script directly, only by writing out the arguments (rather than placing
+them in a list of Python strings).
+The full function is given below:
+
+.. code:: python
+
+ def generate_initial_conditions():
+ # change these two values as needed.
+ # number of MPI processes to use
+ nprocesses = 8
+ # number of OpenMP threads per MPI process to use
+ nthreads_per_process = 1
+
+ # 1. Generate quasistationary state to use for initial conditions.
+ # create a dns object
+ c0 = DNS()
+ # launch the simulation
+ c0.launch([
+ 'NSVE',
+ '-n', '{0}'.format(grid_size),
+ '--np', '{0}'.format(nprocesses),
+ '--ntpp', '{0}'.format(nthreads_per_process),
+ '--precision', 'double',
+ '--src-simname', 'B32p1e4',
+ '--src-wd', TurTLE.data_dir,
+ '--src-iteration', '0',
+ '--simname', 'base',
+ '--niter_todo', '{0}'.format(base_niterations),
+ '--niter_out', '{0}'.format(base_niterations),
+ '--overwrite-src-parameters',
+ '--kMeta', '1.5',
+ '--fk0', '2',
+ '--fk1', '3',
+ '--niter_stat', '16',
+ '--checkpoints_per_file', '{0}'.format(64)])
+
+ # 3. Generate initial conditions for NSE runs.
+ # create postprocess object
+ cc = PP()
+ # launches code to compute velocity field at last iteration
+ cc.launch([
+ 'get_velocity',
+ '--np', '{0}'.format(nprocesses),
+ '--ntpp', '{0}'.format(nthreads_per_process),
+ '--simname', 'base',
+ '--precision', 'double',
+ '--iter0', '{0}'.format(base_niterations),
+ '--iter1', '{0}'.format(base_niterations)])
+ return None
+
+`run_simulations_field` runs six DNS (3 resolutions for each of NSE and
+NSVE solvers):
+
+.. code:: python
+
+ def run_simulations_field(err_vs_t = False):
+ # change these two values as needed.
+ # number of MPI processes to use
+ nprocesses = 8
+ # number of OpenMP threads per MPI process to use
+ nthreads_per_process = 1
+
+ if err_vs_t:
+ niter_out_factor = 1
+ else:
+ niter_out_factor = test_niterations
+
+ # 1. Run NSVE for the three resolutions.
+ for factor in [1, 2, 4]:
+ # create dns object
+ cc = DNS()
+ # launch simulation
+ cc.launch([
+ 'NSVE',
+ '-n', '{0}'.format(grid_size),
+ '--np', '{0}'.format(nprocesses),
+ '--ntpp', '{0}'.format(nthreads_per_process),
+ '--src-simname', 'base',
+ '--src-iteration', '{0}'.format(base_niterations),
+ '--simname', 'nsve_{0}x'.format(factor),
+ '--precision', 'double',
+ '--dtfactor', '{0}'.format(0.5 / factor),
+ '--niter_todo', '{0}'.format(test_niterations*factor),
+ '--niter_out', '{0}'.format(niter_out_factor*factor),
+ '--niter_stat', '{0}'.format(factor)])
+
+ # 2. Run NSE for the three resolutions.
+ for factor in [1, 2, 4]:
+ # create dns object
+ cc = DNS()
+ # launch simulation
+ cc.launch([
+ 'NSE',
+ '-n', '{0}'.format(grid_size),
+ '--np', '{0}'.format(nprocesses),
+ '--ntpp', '{0}'.format(nthreads_per_process),
+ '--src-simname', 'base',
+ '--src-iteration', '{0}'.format(base_niterations),
+ '--simname', 'nse_{0}x'.format(factor),
+ '--precision', 'double',
+ '--dtfactor', '{0}'.format(0.5 / factor),
+ '--niter_todo', '{0}'.format(test_niterations*factor),
+ '--niter_out', '{0}'.format(niter_out_factor*factor),
+ '--niter_out', '{0}'.format(test_niterations*factor),
+ '--niter_stat', '{0}'.format(factor)])
+ return None
+
+Once the runs are finished successfully, the code calls
+`plot_error_field`.
+Here we first read the output of the six different runs, and we compute
+their statistics, since the call to `compute_statistics` will also
+perform some sanity checks on the output.
+We then compute the error analysis for each of the two solvers, and also
+compare the two solutions.
+The `get_velocity` and `compute_vector_field_distance` methods are defined in
+`examples/convergence/fields_base.py`.
+
+.. code:: python
+
+ def plot_error_field(err_vs_t = False):
+ factor_list = [1, 2, 4]
+ c0_list = [DNS(simname = 'nsve_{0}x'.format(factor))
+ for factor in factor_list]
+ c1_list = [DNS(simname = 'nse_{0}x'.format(factor))
+ for factor in factor_list]
+ cl = [c0_list, c1_list]
+ # sanity checks
+ for cc in c0_list + c1_list:
+ cc.compute_statistics()
+
+ # errors for individual solvers
+ error_list = [[], [], []]
+ for ii in range(len(factor_list)-1):
+ factor = factor_list[ii]
+ for jj in [0, 1]:
+ vel1 = get_velocity(cl[jj][ii ], iteration = test_niterations*factor)
+ vel2 = get_velocity(cl[jj][ii+1], iteration = 2*test_niterations*factor)
+ dd = compute_vector_field_distance(vel1, vel2, figname = cl[jj][ii].simname + '_vs_' + cl[jj][ii+1].simname)
+ error_list[jj].append(dd['L2_rel'])
+
+ # comparisons of two solutions
+ for ii in range(len(factor_list)):
+ factor = factor_list[ii]
+ vel1 = get_velocity(cl[0][ii], iteration = test_niterations*factor)
+ vel2 = get_velocity(cl[1][ii], iteration = test_niterations*factor)
+ dd = compute_vector_field_distance(vel1, vel2)
+ error_list[2].append(dd['L2_rel'])
+
+ f = plt.figure(figsize = (4, 4))
+ a = f.add_subplot(111)
+ a.plot(factor_list[:len(error_list[0])],
+ error_list[0],
+ marker = '.',
+ dashes = (2, 3),
+ label = 'NSVE')
+ a.plot(factor_list[:len(error_list[1])],
+ error_list[1],
+ marker = '.',
+ dashes = (3, 5),
+ label = 'NSE')
+ a.plot(factor_list,
+ error_list[2],
+ marker = '.',
+ label = 'NSVE vs NSE')
+ fl = np.array(factor_list).astype(np.float)
+ for ee in [2, 3]:
+ a.plot(fl[:2], error_list[0][0] * fl[:2]**(-ee),
+ dashes = (ee, ee),
+ color = 'black',
+ label = '$\propto f^{{-{0}}}$'.format(ee),
+ zorder = -ee)
+ a.set_ylabel('relative error')
+ a.set_xlabel('resolution factor $f$')
+ a.set_xscale('log')
+ a.set_yscale('log')
+ a.legend(loc = 'best', fontsize = 6)
+ f.tight_layout()
+ f.savefig('err_vs_dt_field.pdf')
+ f.savefig('err_vs_dt_field.svg')
+ plt.close(f)
+
+ if err_vs_t:
+ t = range(test_niterations+1)
+ err_vs_t = [[], [], []]
+ # comparisons of two solvers
+ for ii in range(len(factor_list)):
+ factor = factor_list[ii]
+ for tt in t:
+ vel1 = get_velocity(cl[0][ii], iteration = tt*factor)
+ vel2 = get_velocity(cl[1][ii], iteration = tt*factor)
+ dd = compute_vector_field_distance(vel1, vel2)
+ err_vs_t[ii].append(dd['L2_rel'])
+ # distance between NSVE and NSE as a function of time
+ f = plt.figure(figsize = (4, 4))
+ a = f.add_subplot(111)
+ a.plot(t, err_vs_t[0], label = 'f = 1')
+ a.plot(t, err_vs_t[1], label = 'f = 2')
+ a.plot(t, err_vs_t[2], label = 'f = 4')
+ a.set_yscale('log')
+ a.legend(loc = 'best', fontsize = 6)
+ f.tight_layout()
+ f.savefig('err_vs_t_field.pdf')
+ f.savefig('err_vs_t_field.svg')
+ plt.close(f)
+ return None
+
+
+Finally, the script ends with a standard call to `main`:
+
+.. code:: python
+
+ if __name__ == '__main__':
+ main()
+
diff --git a/documentation/sphinx_static/convergence_particles.rst b/documentation/sphinx_static/convergence_particles.rst
new file mode 100644
index 0000000000000000000000000000000000000000..45fd4dab443a3dd3e51c911090ad0cfd1c01e316
--- /dev/null
+++ b/documentation/sphinx_static/convergence_particles.rst
@@ -0,0 +1,405 @@
+We present here a Python script that performs a particle tracking
+convergence test with TurTLE, in the current default configuration (a
+`particles_system` object coupled with an NSVE fluid solver).
+
+The preamble imports standard packages, and TurTLE itself:
+
+.. code:: python
+
+ import numpy as np
+ import h5py
+ import matplotlib.pyplot as plt
+
+ import TurTLE
+ from TurTLE import DNS
+
+We then set a few basic parameters, including a list of interpolation
+parameters.
+In brief, the spline interpolations used in TurTLE are parametrized by
+the size :math:`I` of the interpolation kernel and the order :math:`O` of
+the polynomial.
+In this example code we emphasize two equivalences:
+
+* :math:`I = 2 n + 2` where :math:`n` is the number of neighbouring
+ points (in one orientation) required to interpolate the field
+ within one cell.
+* :math:`O = 2 m + 1` where :math:`m` is the order of highest
+ continuous derivative of the interpolating polynomial.
+
+More details on the interpolation method can be found in
+[lalescu2010jcp]_.
+
+.. code:: python
+
+ base_niterations = 256
+
+ nparticles = 1000
+ particle_random_seed = 15
+ base_particle_dt = 0.5
+ test_niterations_particles = 128
+
+ neighbours_smoothness_list = [
+ (1, 1),
+ (3, 2)]
+
+We distinguish between three main parts of this script: generation of
+initial conditions with statistically stationary regime, computation of
+numerical solutions, analysis of results (plot commands):
+
+.. code:: python
+
+ def main():
+ generate_initial_conditions()
+
+ run_simulations_particles()
+
+ plot_error_particles()
+ plot_traj_particles()
+ return None
+
+`generate_initial_conditions` consists of a single launch of an NSVE
+run:
+
+.. code:: python
+
+ def generate_initial_conditions():
+ # change these two values as needed.
+ # number of MPI processes to use
+ nprocesses = 8
+ # number of OpenMP threads per MPI process to use
+ nthreads_per_process = 1
+
+ # 1. Generate quasistationary state to use for initial conditions.
+ # create a dns object
+ c0 = DNS()
+ # launch the simulation
+ c0.launch([
+ 'NSVE',
+ '-n', '32',
+ '--np', '{0}'.format(nprocesses),
+ '--ntpp', '{0}'.format(nthreads_per_process),
+ '--precision', 'double',
+ '--src-simname', 'B32p1e4',
+ '--src-wd', TurTLE.data_dir,
+ '--src-iteration', '0',
+ '--simname', 'base',
+ '--niter_todo', '{0}'.format(base_niterations),
+ '--niter_out', '{0}'.format(base_niterations),
+ '--overwrite-src-parameters',
+ '--kMeta', '1.',
+ '--niter_stat', '16',
+ '--checkpoints_per_file', '{0}'.format(16)])
+ return None
+
+We then define the function that runs the particle simulations,
+`run_simulations_particles`.
+Because we use an Adams-Bashforth method, we need to know the values of
+the particle velocity (i.e. the right-hand-side of the ODE) at previous
+steps.
+By default TurTLE accomplishes this by using the Euler integration scheme
+for "time step 0", then using a 2nd order Adams-Bashforth, etc, until
+the desired number of right-hand-side values is stored.
+Thus the code first integrates trajectories for 8 iterations, and the
+output will contain the additional information.
+This happens because such information is crucial for checkpointing ---
+i.e. stopping/restarting the simulation at arbitrary iterations without
+affecting the numerical solution.
+
+To put it simply, for the simulation "nsvep_base" the dataset
+"tracers0/rhs/0" contains only zeros, and TurTLE ignores them.
+However, the "tracers0/rhs/8" dataset for simulation "nsvep_1x" contains
+meaningful values (that **are** used because we request that `iter0` is
+different from 0).
+In order to generate consistent values of the "rhs" datasets, we first
+run a simulation at the highest resolution (smallest time-step), and
+then we extract the appropriate snapshots --- see also the second part
+of the trajectory plot function.
+
+Finally, the `launch` method is called in an unusual way for TurTLE.
+Firstly, we demand a DNS that starts at an iteration different from 0.
+Secondly, we manually copy the forcing parameters from the source
+simulation; since the source cannot be specified explicitly, the `DNS`
+class cannot search for the source parameters itself.
+Thirdly, I will note that one may in fact break this call by modifying
+the `iter0` and `checkpoints_per_file` values, since we have explicitly
+placed `iter0` in the `checkpoint` 0 file.
+While a full discussion of the `iter0` and `checkpoint` subtleties is
+beyond the scope of this tutorial, I will mention that it was a
+conscious choice to use a simple `iteration` to `checkpoint`
+correspondence, since we don't actually need anything smarter outside of
+extreme scenarios that need to be addressed individually anyway
+due to other complexities.
+
+.. code:: python
+
+ def run_simulations_particles():
+ # change these two values as needed.
+ # number of MPI processes to use
+ nprocesses = 8
+ # number of OpenMP threads per MPI process to use
+ nthreads_per_process = 1
+
+ # 1. Run NSVEparticles for a few iterations, to build up rhs values, consistent
+ # with the relevant velocity field.
+ factor = 1
+ for neighbours, smoothness in neighbours_smoothness_list:
+ interp_name = 'I{0}O{1}'.format(2*neighbours+2, 2*smoothness+1)
+ # create dns object
+ cc = DNS()
+ # launch simulation
+ cc.launch([
+ 'NSVEparticles',
+ '-n', '{0}'.format(factor*32),
+ '--np', '{0}'.format(nprocesses),
+ '--ntpp', '{0}'.format(nthreads_per_process),
+ '--src-simname', 'base',
+ '--src-iteration', '{0}'.format(base_niterations),
+ '--simname', 'nsvep_base_' + interp_name,
+ '--precision', 'double',
+ '--dtfactor', '{0}'.format(base_particle_dt/4),
+ '--kMeta', '{0}'.format(factor*1.0),
+ '--niter_todo', '{0}'.format(20),
+ '--niter_out', '{0}'.format(1),
+ '--niter_stat', '{0}'.format(1),
+ '--nparticles', '{0}'.format(nparticles),
+ '--niter_part', '{0}'.format(1),
+ '--tracers0_neighbours', '{0}'.format(neighbours),
+ '--tracers0_smoothness', '{0}'.format(smoothness),
+ '--tracers0_integration_steps', '4',
+ '--cpp_random_particles', '{0}'.format(particle_random_seed),
+ '--checkpoints_per_file', '{0}'.format(16)])
+
+ # 2. Prepare initial conditions
+ for neighbours, smoothness in neighbours_smoothness_list:
+ interp_name = 'I{0}O{1}'.format(2*neighbours+2, 2*smoothness+1)
+ for factor in [1, 2, 4]:
+ df = h5py.File('nsvep_{0}x_{1}_checkpoint_0.h5'.format(factor, interp_name), 'w')
+ # field
+ field_iteration = 16
+ df['vorticity/complex/{0}'.format(8*factor)] = h5py.ExternalLink(
+ 'nsvep_base_{0}_checkpoint_0.h5'.format(interp_name),
+ 'vorticity/complex/{0}'.format(field_iteration))
+ # particles
+ df['tracers0/state/{0}'.format(8*factor)] = h5py.ExternalLink(
+ 'nsvep_base_{0}_checkpoint_0.h5'.format(interp_name),
+ 'tracers0/state/16')
+ # copy rhs
+ source_file = h5py.File(
+ 'nsvep_base_{0}_checkpoint_0.h5'.format(interp_name), 'r')
+ rhs = source_file['tracers0/rhs/16'][()]
+ for tt in range(1, rhs.shape[0]):
+ ii = 16 - tt*4//factor + 1
+ #print(factor, tt, ii)
+ rhs[tt] = source_file['tracers0/rhs/{0}'.format(16 - tt*4//factor + 1)][1]
+ df['tracers0/rhs/{0}'.format(8*factor)] = rhs
+ df.close()
+
+ # 3. Run NSVEparticles
+ for factor in [1, 2, 4]:
+ # Test different interpolations.
+ # The loop iterates over number of neighbours and smoothness.
+ # The simulation names are defined based on "kernel size" and "order
+ # of polynomial".
+ # We do this to emphasize the one-to-one correspondence between
+ # neighbours and kernel size, and smoothness and order of polynomial.
+ for neighbours, smoothness in neighbours_smoothness_list:
+ interp_name = 'I{0}O{1}'.format(2*neighbours+2, 2*smoothness+1)
+ source_file = h5py.File('nsvep_base_{0}.h5'.format(interp_name), 'r')
+ # create dns object
+ cc = DNS()
+ # launch simulation
+ cc.launch([
+ 'NSVEparticles',
+ '-n', '{0}'.format(source_file['parameters/nx'][()]),
+ '--np', '{0}'.format(nprocesses),
+ '--ntpp', '{0}'.format(nthreads_per_process),
+ '--simname', 'nsvep_{0}x_{1}'.format(factor, interp_name),
+ '--precision', 'double',
+ '--dtfactor', '{0}'.format(base_particle_dt/factor),
+ '--niter_todo', '{0}'.format(test_niterations_particles*factor+8*factor),
+ '--niter_out', '{0}'.format(test_niterations_particles*factor+8*factor),
+ '--niter_stat', '{0}'.format(1),
+ ## ensure fluid physics is the same
+ '--dealias_type', '{0}'.format(source_file['parameters/dealias_type'][()]),
+ '--energy', '{0}'.format(source_file['parameters/energy'][()]),
+ '--famplitude', '{0}'.format(source_file['parameters/famplitude'][()]),
+ '--fk0', '{0}'.format(source_file['parameters/fk0'][()]),
+ '--fk1', '{0}'.format(source_file['parameters/fk1'][()]),
+ '--fmode', '{0}'.format(source_file['parameters/fmode'][()]),
+ '--friction_coefficient', '{0}'.format(source_file['parameters/friction_coefficient'][()]),
+ '--injection_rate', '{0}'.format(source_file['parameters/injection_rate'][()]),
+ '--nu', '{0}'.format(source_file['parameters/nu'][()]),
+ '--forcing_type', '{0}'.format(str(source_file['parameters/forcing_type'][()], 'ascii')),
+ ## particle params
+ '--nparticles', '{0}'.format(nparticles),
+ '--niter_part', '{0}'.format(1),
+ '--tracers0_neighbours', '{0}'.format(neighbours),
+ '--tracers0_smoothness', '{0}'.format(smoothness),
+ '--tracers0_integration_steps', '4',
+ '--cpp_random_particles', '0', # turn off C++ particle initialization
+ '--checkpoints_per_file', '{0}'.format(16)],
+ iter0 = 8*factor)
+ source_file.close()
+ return None
+
+To compute the trajectory integration error, we proceed as explained
+above, and compare each DNS with its corresponding double-the-resolution DNS:
+
+.. code:: python
+
+ def plot_error_particles():
+ f = plt.figure()
+ a = f.add_subplot(111)
+ factor_list = [1, 2]
+ factor_list = np.array(factor_list).astype(np.float)
+ for neighbours, smoothness in neighbours_smoothness_list:
+ interp_name = 'I{0}O{1}'.format(2*neighbours+2, 2*smoothness+1)
+ err_max_list = []
+ err_mean_list = []
+ for factor in factor_list:
+ factor = int(factor)
+ c1 = DNS(simname = 'nsvep_{0}x_'.format(int(factor)) + interp_name)
+ c2 = DNS(simname = 'nsvep_{0}x_'.format(int(factor)*2) + interp_name)
+ for cc in [c1, c2]:
+ cc.parameters['niter_part'] = cc.get_data_file()['parameters/niter_part'][()]
+ c1.compute_statistics(iter0 = 8*factor)
+ c2.compute_statistics(iter0 = 8*factor*2)
+ final_iter_x1 = c1.get_data_file()['iteration'][()]
+ final_iter_x2 = c2.get_data_file()['iteration'][()]
+ f1 = h5py.File(c1.simname + '_checkpoint_0.h5', 'r')
+ f2 = h5py.File(c2.simname + '_checkpoint_0.h5', 'r')
+ x1 = f1['tracers0/state/{0}'.format(final_iter_x1)][()]
+ x2 = f2['tracers0/state/{0}'.format(final_iter_x2)][()]
+ diff = np.sqrt(np.sum((x2-x1)**2, axis = -1))
+ err_max_list.append(np.max(diff))
+ err_mean_list.append(np.mean(diff))
+ f1.close()
+ f2.close()
+ err_max_list = np.array(err_max_list)
+ err_mean_list = np.array(err_mean_list)
+ a.plot(factor_list, err_max_list, marker = '.', label = 'max error ' + interp_name)
+ a.plot(factor_list, err_mean_list, marker = '.', label = 'mean error ' + interp_name)
+ for ee in [2, 3, 4]:
+ a.plot(factor_list,
+ (1e-4)*factor_list**(-ee),
+ color = 'black',
+ dashes = (ee, ee),
+ label = '$\propto f^{{-{0}}}$'.format(ee))
+ a.set_xscale('log')
+ a.set_yscale('log')
+ a.set_xlabel('resolution factor f')
+ a.set_ylabel('absolute trajectory error [DNS units]')
+ a.legend(loc = 'best', fontsize = 7)
+ f.tight_layout()
+ f.savefig('err_vs_dt_particle.pdf')
+ f.savefig('err_vs_dt_particle.svg')
+ plt.close(f)
+ return None
+
+For the beginner's convenience, we also provide a method to plot
+individual particle trajectories.
+Note that the second part of this function is in fact a sanity check for
+the initial conditions: the plot confirms that the data required for the
+multi-step method (explicitly placed in the checkpoint file) is
+consistent with the simulation parameters.
+
+.. code:: python
+
+ def plot_traj_particles():
+ ###########################################################################
+ f = plt.figure()
+ a = f.add_subplot(111)
+ for neighbours, smoothness in neighbours_smoothness_list:
+ interp_name = 'I{0}O{1}'.format(2*neighbours+2, 2*smoothness+1)
+ for factor in [1, 2, 4]:
+ cc = DNS(simname = 'nsvep_{0}x_'.format(factor) + interp_name)
+ cc.compute_statistics(iter0 = 8*factor)
+ tt = 3
+ xx = read_trajectory(cc, tt, iter0 = 8*factor)
+ a.plot(xx[:, 0],
+ xx[:, 1],
+ label = interp_name + '$f = {0}$'.format(factor),
+ marker = 'x',
+ dashes = (4/factor, 3, 4/factor),
+ alpha = 0.2)
+ #a.plot(xx[:, 0], xx[:, 1], dashes = (4/factor, 3, 4/factor), alpha = 0.2)
+ a.legend(loc = 'best', fontsize = 7)
+ a.set_xlabel('$x$ [code units]')
+ a.set_ylabel('$y$ [code units]')
+ f.tight_layout()
+ f.savefig('trajectories.pdf')
+ f.savefig('trajectories.svg')
+ plt.close(f)
+ ###########################################################################
+ ###########################################################################
+ traj_index = 3
+ for neighbours, smoothness in neighbours_smoothness_list:
+ interp_name = 'I{0}O{1}'.format(2*neighbours+2, 2*smoothness+1)
+ cc = DNS(simname = 'nsvep_base_' + interp_name)
+ cc.compute_statistics()
+ xx_base = read_trajectory(cc, traj_index, iter0 = 0, dset_name = 'velocity')
+ iter0 = 16
+ tt_base = np.array(range(iter0, iter0 + xx_base.shape[0]))*cc.parameters['dt']
+ f = plt.figure()
+ ax_counter = 1
+ for factor in [1, 2, 4]:
+ a = f.add_subplot(310 + ax_counter)
+ ax_counter += 1
+ a.plot(tt_base,
+ xx_base[:, 2],
+ label = 'base_' + interp_name,
+ marker = '+',
+ linewidth = 0.5,
+ alpha = 0.2)
+ cc = DNS(simname = 'nsvep_{0}x_'.format(factor) + interp_name)
+ cc.compute_statistics(iter0 = 8*factor)
+ xx = read_trajectory(cc, traj_index, iter0 = 8*factor, dset_name = 'velocity')
+ tt = np.array(range(8*factor, xx.shape[0]+8*factor))*cc.parameters['dt']
+ a.plot(tt,
+ xx[:, 2],
+ label = interp_name + '$f = {0}$'.format(factor),
+ marker = 'x',
+ dashes = (4/factor, 3, 4/factor),
+ alpha = 0.2)
+ df = h5py.File(cc.simname + '_checkpoint_0.h5', 'r')
+ xx = df['tracers0/rhs/{0}'.format(8*factor)][()]
+ for rhs_index in [1, 2, 3]:
+ a.scatter((8*factor - rhs_index)*cc.parameters['dt'],
+ xx[rhs_index, traj_index, 2],
+ marker = '*',
+ zorder = -10,
+ alpha = 0.5,
+ color = 'black')
+ df.close()
+ a.legend(loc = 'best', fontsize = 7)
+ a.set_xlabel('$t$ [code units]')
+ a.set_ylabel('$v_y$ [code units]')
+ f.tight_layout()
+ f.savefig('velocities_{0}.pdf'.format(interp_name))
+ plt.close(f)
+ ###########################################################################
+ return None
+
+ def read_trajectory(
+ cc,
+ traj_index,
+ iter0,
+ dset_name = 'position'):
+ cc.parameters['niter_part'] = cc.get_data_file()['parameters/niter_part'][()]
+ df = cc.get_particle_file()
+ xx = []
+ for ii in range(iter0, cc.get_data_file()['iteration'][()]+1, cc.parameters['niter_part']):
+ xx.append(df['tracers0/' + dset_name + '/{0}'.format(ii)][traj_index])
+ df.close()
+ return np.array(xx)
+
+.. image:: trajectories.svg
+
+Finally, the script ends with a standard call to `main`:
+
+.. code:: python
+
+ if __name__ == '__main__':
+ main()
+
+
diff --git a/documentation/sphinx_static/err_vs_dt_field.svg b/documentation/sphinx_static/err_vs_dt_field.svg
new file mode 100644
index 0000000000000000000000000000000000000000..96260bb1503d7d03103c838a9acd0d082c23ab1c
--- /dev/null
+++ b/documentation/sphinx_static/err_vs_dt_field.svg
@@ -0,0 +1,1157 @@
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diff --git a/examples/convergence/fields_base.py b/examples/convergence/fields_base.py
new file mode 100644
index 0000000000000000000000000000000000000000..d2225d3245c01f2b2ff24cad482dae7cc364d6e4
--- /dev/null
+++ b/examples/convergence/fields_base.py
@@ -0,0 +1,78 @@
+import numpy as np
+import h5py
+import matplotlib.pyplot as plt
+
+def main():
+ return None
+
+def get_velocity(
+ cc,
+ iteration = 0):
+ data_file = h5py.File(
+ cc.get_checkpoint_fname(iteration = iteration),
+ 'r')
+ if 'velocity' in data_file.keys():
+ if 'real' in data_file['velocity'].keys():
+ vel = data_file['velocity/real/{0}'.format(iteration)][()]
+ else:
+ cvel = data_file['velocity/complex/{0}'.format(iteration)][()]
+ vel = ift(cvel)
+ else:
+ assert('complex' in data_file['vorticity'].keys())
+ cvort = data_file['vorticity/complex/{0}'.format(iteration)][()]
+ cvel = compute_curl(cc, cvort, inverse_curl = True)
+ vel = ift(cvel)
+ data_file.close()
+ return vel
+
+def ift(cfield):
+ ff = np.fft.irfftn(cfield, axes = (0, 1, 2))
+ field = np.transpose(ff, (1, 0, 2, 3)).copy()
+ return field*(field.shape[0]*field.shape[1]*field.shape[2])
+
+def compute_curl(
+ cc,
+ cfield,
+ inverse_curl = False):
+ kx = cc.get_data_file()['kspace/kx'][()]
+ ky = cc.get_data_file()['kspace/ky'][()]
+ kz = cc.get_data_file()['kspace/kz'][()]
+ ff = cfield.copy()
+ ff[..., 0] = 1j*(ky[:, None, None]*cfield[..., 2] - kz[None, :, None]*cfield[..., 1])
+ ff[..., 1] = 1j*(kz[None, :, None]*cfield[..., 0] - kx[None, None, :]*cfield[..., 2])
+ ff[..., 2] = 1j*(kx[None, None, :]*cfield[..., 1] - ky[:, None, None]*cfield[..., 0])
+ if inverse_curl:
+ k2 = (kx[None, None, :]**2 +
+ ky[ :, None, None]**2 +
+ kz[None, :, None]**2)
+ k2[0, 0, 0] = 1.
+ ff /= k2[:, :, :, None]
+ return ff
+
+def compute_vector_field_distance(
+ f1, f2,
+ figname = None):
+ diff = f1 - f2
+ rms1 = np.mean(np.sum(f1**2, axis = 3))**0.5
+ rms2 = np.mean(np.sum(f2**2, axis = 3))**0.5
+ rms = np.sqrt(rms1 * rms2)
+ dd = np.sum(diff**2, axis = 3)
+ L2 = np.sqrt(np.mean(dd))
+ max_val = np.max(np.sqrt(dd))
+ if type(figname) != type(None):
+ fig = plt.figure(figsize = (8, 4))
+ a = fig.add_subplot(121)
+ a.imshow(f1[:, 5, :, 2], vmin = -1, vmax = 1, interpolation = 'none')
+ a = fig.add_subplot(122)
+ a.imshow(f2[:, 5, :, 2], vmin = -1, vmax = 1, interpolation = 'none')
+ fig.tight_layout()
+ fig.savefig('field_slice_' + figname + '.pdf')
+ plt.close(fig)
+ return {'L2' : L2,
+ 'L2_rel' : L2 / rms,
+ 'max' : max_val,
+ 'max_rel' : max_val / rms}
+
+if __name__ == '__main__':
+ main()
+
diff --git a/examples/convergence/fields_temporal.py b/examples/convergence/fields_temporal.py
new file mode 100644
index 0000000000000000000000000000000000000000..d5e271be336b70303c6e2845a438293930834a6e
--- /dev/null
+++ b/examples/convergence/fields_temporal.py
@@ -0,0 +1,208 @@
+import numpy as np
+import h5py
+import matplotlib.pyplot as plt
+
+import TurTLE
+from TurTLE import DNS, PP
+
+from fields_base import *
+
+base_niterations = 512
+test_niterations = 32
+grid_size = 128
+
+def main():
+ generate_initial_conditions()
+
+ run_simulations_field(
+ err_vs_t = False)
+ plot_error_field(
+ err_vs_t = False)
+ return None
+
+def generate_initial_conditions():
+ # change these two values as needed.
+ # number of MPI processes to use
+ nprocesses = 8
+ # number of OpenMP threads per MPI process to use
+ nthreads_per_process = 1
+
+# 1. Generate quasistationary state to use for initial conditions.
+ # create a dns object
+ c0 = DNS()
+ # launch the simulation
+ c0.launch([
+ 'NSVE',
+ '-n', '{0}'.format(grid_size),
+ '--np', '{0}'.format(nprocesses),
+ '--ntpp', '{0}'.format(nthreads_per_process),
+ '--precision', 'double',
+ '--src-simname', 'B32p1e4',
+ '--src-wd', TurTLE.data_dir,
+ '--src-iteration', '0',
+ '--simname', 'base',
+ '--niter_todo', '{0}'.format(base_niterations),
+ '--niter_out', '{0}'.format(base_niterations),
+ '--overwrite-src-parameters',
+ '--kMeta', '1.5',
+ '--fk0', '2',
+ '--fk1', '3',
+ '--niter_stat', '16',
+ '--checkpoints_per_file', '{0}'.format(64)])
+
+# 3. Generate initial conditions for NSE runs.
+ # create postprocess object
+ cc = PP()
+ # launches code to compute velocity field at last iteration
+ cc.launch([
+ 'get_velocity',
+ '--np', '{0}'.format(nprocesses),
+ '--ntpp', '{0}'.format(nthreads_per_process),
+ '--simname', 'base',
+ '--precision', 'double',
+ '--iter0', '{0}'.format(base_niterations),
+ '--iter1', '{0}'.format(base_niterations)])
+ return None
+
+def run_simulations_field(err_vs_t = False):
+ # change these two values as needed.
+ # number of MPI processes to use
+ nprocesses = 8
+ # number of OpenMP threads per MPI process to use
+ nthreads_per_process = 1
+
+ if err_vs_t:
+ niter_out_factor = 1
+ else:
+ niter_out_factor = test_niterations
+
+# 1. Run NSVE for the three resolutions.
+ for factor in [1, 2, 4]:
+ # create dns object
+ cc = DNS()
+ # launch simulation
+ cc.launch([
+ 'NSVE',
+ '-n', '{0}'.format(grid_size),
+ '--np', '{0}'.format(nprocesses),
+ '--ntpp', '{0}'.format(nthreads_per_process),
+ '--src-simname', 'base',
+ '--src-iteration', '{0}'.format(base_niterations),
+ '--simname', 'nsve_{0}x'.format(factor),
+ '--precision', 'double',
+ '--dtfactor', '{0}'.format(0.5 / factor),
+ '--niter_todo', '{0}'.format(test_niterations*factor),
+ '--niter_out', '{0}'.format(niter_out_factor*factor),
+ '--niter_stat', '{0}'.format(factor)])
+
+# 2. Run NSE for the three resolutions.
+ for factor in [1, 2, 4]:
+ # create dns object
+ cc = DNS()
+ # launch simulation
+ cc.launch([
+ 'NSE',
+ '-n', '{0}'.format(grid_size),
+ '--np', '{0}'.format(nprocesses),
+ '--ntpp', '{0}'.format(nthreads_per_process),
+ '--src-simname', 'base',
+ '--src-iteration', '{0}'.format(base_niterations),
+ '--simname', 'nse_{0}x'.format(factor),
+ '--precision', 'double',
+ '--dtfactor', '{0}'.format(0.5 / factor),
+ '--niter_todo', '{0}'.format(test_niterations*factor),
+ '--niter_out', '{0}'.format(niter_out_factor*factor),
+ '--niter_out', '{0}'.format(test_niterations*factor),
+ '--niter_stat', '{0}'.format(factor)])
+ return None
+
+def plot_error_field(err_vs_t = False):
+ factor_list = [1, 2, 4]
+ c0_list = [DNS(simname = 'nsve_{0}x'.format(factor))
+ for factor in factor_list]
+ c1_list = [DNS(simname = 'nse_{0}x'.format(factor))
+ for factor in factor_list]
+ cl = [c0_list, c1_list]
+ # sanity checks
+ for cc in c0_list + c1_list:
+ cc.compute_statistics()
+
+ # errors for individual solvers
+ error_list = [[], [], []]
+ for ii in range(len(factor_list)-1):
+ factor = factor_list[ii]
+ for jj in [0, 1]:
+ vel1 = get_velocity(cl[jj][ii ], iteration = test_niterations*factor)
+ vel2 = get_velocity(cl[jj][ii+1], iteration = 2*test_niterations*factor)
+ dd = compute_vector_field_distance(vel1, vel2, figname = cl[jj][ii].simname + '_vs_' + cl[jj][ii+1].simname)
+ error_list[jj].append(dd['L2_rel'])
+
+ # comparisons of two solutions
+ for ii in range(len(factor_list)):
+ factor = factor_list[ii]
+ vel1 = get_velocity(cl[0][ii], iteration = test_niterations*factor)
+ vel2 = get_velocity(cl[1][ii], iteration = test_niterations*factor)
+ dd = compute_vector_field_distance(vel1, vel2)
+ error_list[2].append(dd['L2_rel'])
+
+ f = plt.figure(figsize = (4, 4))
+ a = f.add_subplot(111)
+ a.plot(factor_list[:len(error_list[0])],
+ error_list[0],
+ marker = '.',
+ dashes = (2, 3),
+ label = 'NSVE')
+ a.plot(factor_list[:len(error_list[1])],
+ error_list[1],
+ marker = '.',
+ dashes = (3, 5),
+ label = 'NSE')
+ a.plot(factor_list,
+ error_list[2],
+ marker = '.',
+ label = 'NSVE vs NSE')
+ fl = np.array(factor_list).astype(np.float)
+ for ee in [2, 3]:
+ a.plot(fl[:2], error_list[0][0] * fl[:2]**(-ee),
+ dashes = (ee, ee),
+ color = 'black',
+ label = '$\propto f^{{-{0}}}$'.format(ee),
+ zorder = -ee)
+ a.set_ylabel('relative error')
+ a.set_xlabel('resolution factor $f$')
+ a.set_xscale('log')
+ a.set_yscale('log')
+ a.legend(loc = 'best', fontsize = 6)
+ f.tight_layout()
+ f.savefig('err_vs_dt_field.pdf')
+ f.savefig('err_vs_dt_field.svg')
+ plt.close(f)
+
+ if err_vs_t:
+ t = range(test_niterations+1)
+ err_vs_t = [[], [], []]
+ # comparisons of two solvers
+ for ii in range(len(factor_list)):
+ factor = factor_list[ii]
+ for tt in t:
+ vel1 = get_velocity(cl[0][ii], iteration = tt*factor)
+ vel2 = get_velocity(cl[1][ii], iteration = tt*factor)
+ dd = compute_vector_field_distance(vel1, vel2)
+ err_vs_t[ii].append(dd['L2_rel'])
+ # distance between NSVE and NSE as a function of time
+ f = plt.figure(figsize = (4, 4))
+ a = f.add_subplot(111)
+ a.plot(t, err_vs_t[0], label = 'f = 1')
+ a.plot(t, err_vs_t[1], label = 'f = 2')
+ a.plot(t, err_vs_t[2], label = 'f = 4')
+ a.set_yscale('log')
+ a.legend(loc = 'best', fontsize = 6)
+ f.tight_layout()
+ f.savefig('err_vs_t_field.pdf')
+ f.savefig('err_vs_t_field.svg')
+ plt.close(f)
+ return None
+
+if __name__ == '__main__':
+ main()
+
diff --git a/examples/convergence/particles_temporal.py b/examples/convergence/particles_temporal.py
new file mode 100644
index 0000000000000000000000000000000000000000..c7b0ac15573a97031bd9a72e706674d03f6d8a8f
--- /dev/null
+++ b/examples/convergence/particles_temporal.py
@@ -0,0 +1,304 @@
+import numpy as np
+import h5py
+import matplotlib.pyplot as plt
+
+import TurTLE
+from TurTLE import DNS
+
+base_niterations = 256
+
+nparticles = 1000
+particle_random_seed = 15
+base_particle_dt = 0.5
+test_niterations_particles = 128
+
+neighbours_smoothness_list = [
+ (1, 1),
+ (3, 2)]
+
+def main():
+ generate_initial_conditions()
+
+ run_simulations_particles()
+
+ plot_error_particles()
+ plot_traj_particles()
+ return None
+
+def generate_initial_conditions():
+ # change these two values as needed.
+ # number of MPI processes to use
+ nprocesses = 8
+ # number of OpenMP threads per MPI process to use
+ nthreads_per_process = 1
+
+# 1. Generate quasistationary state to use for initial conditions.
+ # create a dns object
+ c0 = DNS()
+ # launch the simulation
+ c0.launch([
+ 'NSVE',
+ '-n', '32',
+ '--np', '{0}'.format(nprocesses),
+ '--ntpp', '{0}'.format(nthreads_per_process),
+ '--precision', 'double',
+ '--src-simname', 'B32p1e4',
+ '--src-wd', TurTLE.data_dir,
+ '--src-iteration', '0',
+ '--simname', 'base',
+ '--niter_todo', '{0}'.format(base_niterations),
+ '--niter_out', '{0}'.format(base_niterations),
+ '--overwrite-src-parameters',
+ '--kMeta', '1.',
+ '--niter_stat', '16',
+ '--checkpoints_per_file', '{0}'.format(16)])
+ return None
+
+def run_simulations_particles():
+ # change these two values as needed.
+ # number of MPI processes to use
+ nprocesses = 8
+ # number of OpenMP threads per MPI process to use
+ nthreads_per_process = 1
+
+# 1. Run NSVEparticles for a few iterations, to build up rhs values, consistent
+# with the relevant velocity field.
+ factor = 1
+ for neighbours, smoothness in neighbours_smoothness_list:
+ interp_name = 'I{0}O{1}'.format(2*neighbours+2, 2*smoothness+1)
+ # create dns object
+ cc = DNS()
+ # launch simulation
+ cc.launch([
+ 'NSVEparticles',
+ '-n', '{0}'.format(factor*32),
+ '--np', '{0}'.format(nprocesses),
+ '--ntpp', '{0}'.format(nthreads_per_process),
+ '--src-simname', 'base',
+ '--src-iteration', '{0}'.format(base_niterations),
+ '--simname', 'nsvep_base_' + interp_name,
+ '--precision', 'double',
+ '--dtfactor', '{0}'.format(base_particle_dt/4),
+ '--kMeta', '{0}'.format(factor*1.0),
+ '--niter_todo', '{0}'.format(20),
+ '--niter_out', '{0}'.format(1),
+ '--niter_stat', '{0}'.format(1),
+ '--nparticles', '{0}'.format(nparticles),
+ '--niter_part', '{0}'.format(1),
+ '--tracers0_neighbours', '{0}'.format(neighbours),
+ '--tracers0_smoothness', '{0}'.format(smoothness),
+ '--tracers0_integration_steps', '4',
+ '--cpp_random_particles', '{0}'.format(particle_random_seed)])
+
+# 2. Prepare initial conditions
+ for neighbours, smoothness in neighbours_smoothness_list:
+ interp_name = 'I{0}O{1}'.format(2*neighbours+2, 2*smoothness+1)
+ for factor in [1, 2, 4]:
+ df = h5py.File('nsvep_{0}x_{1}_checkpoint_0.h5'.format(factor, interp_name), 'w')
+ # field
+ field_iteration = 16
+ df['vorticity/complex/{0}'.format(8*factor)] = h5py.ExternalLink(
+ 'nsvep_base_{0}_checkpoint_0.h5'.format(interp_name),
+ 'vorticity/complex/{0}'.format(field_iteration))
+ # particles
+ df['tracers0/state/{0}'.format(8*factor)] = h5py.ExternalLink(
+ 'nsvep_base_{0}_checkpoint_0.h5'.format(interp_name),
+ 'tracers0/state/16')
+ # copy rhs
+ source_file = h5py.File(
+ 'nsvep_base_{0}_checkpoint_0.h5'.format(interp_name), 'r')
+ rhs = source_file['tracers0/rhs/16'][()]
+ for tt in range(1, rhs.shape[0]):
+ ii = 16 - tt*4//factor + 1
+ #print(factor, tt, ii)
+ rhs[tt] = source_file['tracers0/rhs/{0}'.format(16 - tt*4//factor + 1)][1]
+ df['tracers0/rhs/{0}'.format(8*factor)] = rhs
+ df.close()
+
+# 3. Run NSVEparticles
+ for factor in [1, 2, 4]:
+ # Test different interpolations.
+ # The loop iterates over number of neighbours and smoothness.
+ # The simulation names are defined based on "kernel size" and "order
+ # of polynomial".
+ # We do this to emphasize the one-to-one correspondence between
+ # neighbours and kernel size, and smoothness and order of polynomial.
+ for neighbours, smoothness in neighbours_smoothness_list:
+ interp_name = 'I{0}O{1}'.format(2*neighbours+2, 2*smoothness+1)
+ source_file = h5py.File('nsvep_base_{0}.h5'.format(interp_name), 'r')
+ # create dns object
+ cc = DNS()
+ # launch simulation
+ cc.launch([
+ 'NSVEparticles',
+ '-n', '{0}'.format(source_file['parameters/nx'][()]),
+ '--np', '{0}'.format(nprocesses),
+ '--ntpp', '{0}'.format(nthreads_per_process),
+ '--simname', 'nsvep_{0}x_{1}'.format(factor, interp_name),
+ '--precision', 'double',
+ '--dtfactor', '{0}'.format(base_particle_dt/factor),
+ '--niter_todo', '{0}'.format(test_niterations_particles*factor+8*factor),
+ '--niter_out', '{0}'.format(test_niterations_particles*factor+8*factor),
+ '--niter_stat', '{0}'.format(1),
+ ## ensure fluid physics is the same
+ '--dealias_type', '{0}'.format(source_file['parameters/dealias_type'][()]),
+ '--energy', '{0}'.format(source_file['parameters/energy'][()]),
+ '--famplitude', '{0}'.format(source_file['parameters/famplitude'][()]),
+ '--fk0', '{0}'.format(source_file['parameters/fk0'][()]),
+ '--fk1', '{0}'.format(source_file['parameters/fk1'][()]),
+ '--fmode', '{0}'.format(source_file['parameters/fmode'][()]),
+ '--friction_coefficient', '{0}'.format(source_file['parameters/friction_coefficient'][()]),
+ '--injection_rate', '{0}'.format(source_file['parameters/injection_rate'][()]),
+ '--nu', '{0}'.format(source_file['parameters/nu'][()]),
+ '--forcing_type', '{0}'.format(str(source_file['parameters/forcing_type'][()], 'ascii')),
+ ## particle params
+ '--nparticles', '{0}'.format(nparticles),
+ '--niter_part', '{0}'.format(1),
+ '--tracers0_neighbours', '{0}'.format(neighbours),
+ '--tracers0_smoothness', '{0}'.format(smoothness),
+ '--tracers0_integration_steps', '4',
+ '--cpp_random_particles', '0'], # turn off C++ particle initialization
+ iter0 = 8*factor)
+ source_file.close()
+ return None
+
+def plot_error_particles():
+ f = plt.figure()
+ a = f.add_subplot(111)
+ factor_list = [1, 2]
+ factor_list = np.array(factor_list).astype(np.float)
+ for neighbours, smoothness in neighbours_smoothness_list:
+ interp_name = 'I{0}O{1}'.format(2*neighbours+2, 2*smoothness+1)
+ err_max_list = []
+ err_mean_list = []
+ for factor in factor_list:
+ factor = int(factor)
+ c1 = DNS(simname = 'nsvep_{0}x_'.format(int(factor)) + interp_name)
+ c2 = DNS(simname = 'nsvep_{0}x_'.format(int(factor)*2) + interp_name)
+ for cc in [c1, c2]:
+ cc.parameters['niter_part'] = cc.get_data_file()['parameters/niter_part'][()]
+ c1.compute_statistics(iter0 = 8*factor)
+ c2.compute_statistics(iter0 = 8*factor*2)
+ final_iter_x1 = c1.get_data_file()['iteration'][()]
+ final_iter_x2 = c2.get_data_file()['iteration'][()]
+ f1 = h5py.File(c1.simname + '_checkpoint_0.h5', 'r')
+ f2 = h5py.File(c2.simname + '_checkpoint_0.h5', 'r')
+ x1 = f1['tracers0/state/{0}'.format(final_iter_x1)][()]
+ x2 = f2['tracers0/state/{0}'.format(final_iter_x2)][()]
+ diff = np.sqrt(np.sum((x2-x1)**2, axis = -1))
+ err_max_list.append(np.max(diff))
+ err_mean_list.append(np.mean(diff))
+ f1.close()
+ f2.close()
+ err_max_list = np.array(err_max_list)
+ err_mean_list = np.array(err_mean_list)
+ a.plot(factor_list, err_max_list, marker = '.', label = 'max error ' + interp_name)
+ a.plot(factor_list, err_mean_list, marker = '.', label = 'mean error ' + interp_name)
+ for ee in [2, 3, 4]:
+ a.plot(factor_list,
+ (1e-4)*factor_list**(-ee),
+ color = 'black',
+ dashes = (ee, ee),
+ label = '$\propto f^{{-{0}}}$'.format(ee))
+ a.set_xscale('log')
+ a.set_yscale('log')
+ a.set_xlabel('resolution factor f')
+ a.set_ylabel('absolute trajectory error [DNS units]')
+ a.legend(loc = 'best', fontsize = 7)
+ f.tight_layout()
+ f.savefig('err_vs_dt_particle.pdf')
+ f.savefig('err_vs_dt_particle.svg')
+ plt.close(f)
+ return None
+
+def plot_traj_particles():
+ ###########################################################################
+ f = plt.figure()
+ a = f.add_subplot(111)
+ for neighbours, smoothness in neighbours_smoothness_list:
+ interp_name = 'I{0}O{1}'.format(2*neighbours+2, 2*smoothness+1)
+ for factor in [1, 2, 4]:
+ cc = DNS(simname = 'nsvep_{0}x_'.format(factor) + interp_name)
+ cc.compute_statistics(iter0 = 8*factor)
+ tt = 3
+ xx = read_trajectory(cc, tt, iter0 = 8*factor)
+ a.plot(xx[:, 0],
+ xx[:, 1],
+ label = interp_name + '$f = {0}$'.format(factor),
+ marker = 'x',
+ dashes = (4/factor, 3, 4/factor),
+ alpha = 0.2)
+ #a.plot(xx[:, 0], xx[:, 1], dashes = (4/factor, 3, 4/factor), alpha = 0.2)
+ a.legend(loc = 'best', fontsize = 7)
+ a.set_xlabel('$x$ [code units]')
+ a.set_ylabel('$y$ [code units]')
+ f.tight_layout()
+ f.savefig('trajectories.pdf')
+ f.savefig('trajectories.svg')
+ plt.close(f)
+ ###########################################################################
+ ###########################################################################
+ traj_index = 3
+ for neighbours, smoothness in neighbours_smoothness_list:
+ interp_name = 'I{0}O{1}'.format(2*neighbours+2, 2*smoothness+1)
+ cc = DNS(simname = 'nsvep_base_' + interp_name)
+ cc.compute_statistics()
+ xx_base = read_trajectory(cc, traj_index, iter0 = 0, dset_name = 'velocity')
+ iter0 = 16
+ tt_base = np.array(range(iter0, iter0 + xx_base.shape[0]))*cc.parameters['dt']
+ f = plt.figure()
+ ax_counter = 1
+ for factor in [1, 2, 4]:
+ a = f.add_subplot(310 + ax_counter)
+ ax_counter += 1
+ a.plot(tt_base,
+ xx_base[:, 2],
+ label = 'base_' + interp_name,
+ marker = '+',
+ linewidth = 0.5,
+ alpha = 0.2)
+ cc = DNS(simname = 'nsvep_{0}x_'.format(factor) + interp_name)
+ cc.compute_statistics(iter0 = 8*factor)
+ xx = read_trajectory(cc, traj_index, iter0 = 8*factor, dset_name = 'velocity')
+ tt = np.array(range(8*factor, xx.shape[0]+8*factor))*cc.parameters['dt']
+ a.plot(tt,
+ xx[:, 2],
+ label = interp_name + '$f = {0}$'.format(factor),
+ marker = 'x',
+ dashes = (4/factor, 3, 4/factor),
+ alpha = 0.2)
+ df = h5py.File(cc.simname + '_checkpoint_0.h5', 'r')
+ xx = df['tracers0/rhs/{0}'.format(8*factor)][()]
+ for rhs_index in [1, 2, 3]:
+ a.scatter((8*factor - rhs_index)*cc.parameters['dt'],
+ xx[rhs_index, traj_index, 2],
+ marker = '*',
+ zorder = -10,
+ alpha = 0.5,
+ color = 'black')
+ df.close()
+ a.legend(loc = 'best', fontsize = 7)
+ a.set_xlabel('$t$ [code units]')
+ a.set_ylabel('$v_y$ [code units]')
+ f.tight_layout()
+ f.savefig('velocities_{0}.pdf'.format(interp_name))
+ plt.close(f)
+ ###########################################################################
+ return None
+
+def read_trajectory(
+ cc,
+ traj_index,
+ iter0,
+ dset_name = 'position'):
+ cc.parameters['niter_part'] = cc.get_data_file()['parameters/niter_part'][()]
+ df = cc.get_particle_file()
+ xx = []
+ for ii in range(iter0, cc.get_data_file()['iteration'][()]+1, cc.parameters['niter_part']):
+ xx.append(df['tracers0/' + dset_name + '/{0}'.format(ii)][traj_index])
+ df.close()
+ return np.array(xx)
+
+if __name__ == '__main__':
+ main()
+