openmc.deplete.CELIIntegrator¶
- class openmc.deplete.CELIIntegrator(operator, timesteps, power=None, power_density=None, source_rates=None, timestep_units='s', solver='cram48')[source]¶
Deplete using the CE/LI CFQ4 algorithm.
Implements the CE/LI Predictor-Corrector algorithm using the fourth order commutator-free integrator.
“CE/LI” stands for constant extrapolation on predictor and linear interpolation on corrector. This algorithm is mathematically defined as:
\[\begin{aligned} \mathbf{n}_{i+1}^p &= \exp \left ( h \mathbf{A}(\mathbf{n}_i ) \right ) \mathbf{n}_i \\ \mathbf{n}_{i+1} &= \exp \left( \frac{h}{12} \mathbf{A}(\mathbf{n}_i) + \frac{5h}{12} \mathbf{A}(\mathbf{n}_{i+1}^p) \right) \exp \left( \frac{5h}{12} \mathbf{A}(\mathbf{n}_i) + \frac{h}{12} \mathbf{A}(\mathbf{n}_{i+1}^p) \right) \mathbf{n}_i. \end{aligned} \]- Parameters
operator (openmc.deplete.abc.TransportOperator) – Operator to perform transport simulations
timesteps (iterable of float or iterable of tuple) – Array of timesteps. Note that values are not cumulative. The units are specified by the timestep_units argument when timesteps is an iterable of float. Alternatively, units can be specified for each step by passing an iterable of (value, unit) tuples.
power (float or iterable of float, optional) – Power of the reactor in [W]. A single value indicates that the power is constant over all timesteps. An iterable indicates potentially different power levels for each timestep. For a 2D problem, the power can be given in [W/cm] as long as the “volume” assigned to a depletion material is actually an area in [cm^2]. Either
power
,power_density
, orsource_rates
must be specified.power_density (float or iterable of float, optional) – Power density of the reactor in [W/gHM]. It is multiplied by initial heavy metal inventory to get total power if
power
is not specified.source_rates (float or iterable of float, optional) –
Source rate in [neutron/sec] or neutron flux in [neutron/s-cm^2] for each interval in
timesteps
New in version 0.12.1.
timestep_units ({'s', 'min', 'h', 'd', 'a', 'MWd/kg'}) – Units for values specified in the timesteps argument. ‘s’ means seconds, ‘min’ means minutes, ‘h’ means hours, ‘a’ means Julian years and ‘MWd/kg’ indicates that the values are given in burnup (MW-d of energy deposited per kilogram of initial heavy metal).
solver (str or callable, optional) –
If a string, must be the name of the solver responsible for solving the Bateman equations. Current options are:
cram16
- 16th order IPF CRAMcram48
- 48th order IPF CRAM [default]
If a function or other callable, must adhere to the requirements in
solver
.New in version 0.12.
- Variables
operator (openmc.deplete.abc.TransportOperator) – Operator to perform transport simulations
chain (openmc.deplete.Chain) – Depletion chain
timesteps (iterable of float) – Size of each depletion interval in [s]
source_rates (iterable of float) – Source rate in [W] or [neutron/sec] for each interval in
timesteps
solver (callable) –
Function that will solve the Bateman equations \(\frac{\partial}{\partial t}\vec{n} = A_i\vec{n}_i\) with a step size \(t_i\). Can be configured using the
solver
argument. User-supplied functions are expected to have the following signature:solver(A, n0, t) -> n1
whereA
is ascipy.sparse.csr_matrix
making up the depletion matrixn0
is a 1-Dnumpy.ndarray
of initial compositions for a given material in atoms/cm3t
is a float of the time step size in seconds, andn1
is anumpy.ndarray
of compositions at the next time step. Expected to be of the same shape asn0
New in version 0.12.
- __call__(bos_conc, rates, dt, source_rate, _i=None)[source]¶
Perform the integration across one time step
- Parameters
bos_conc (numpy.ndarray) – Initial concentrations for all nuclides in [atom]
rates (openmc.deplete.ReactionRates) – Reaction rates from operator
dt (float) – Time in [s] for the entire depletion interval
source_rate (float) – Power in [W] or source rate in [neutron/sec]
_i (int, optional) – Current iteration count. Not used
- Returns
proc_time (float) – Time spent in CRAM routines for all materials in [s]
conc_list (list of numpy.ndarray) – Concentrations at each of the intermediate points with the final concentration as the last element
op_results (list of openmc.deplete.OperatorResult) – Eigenvalue and reaction rates from intermediate transport simulation
- __iter__()¶
Return pair of time step in [s] and source rate in [W] or [neutron/sec]
- __len__()¶
Return integer number of depletion intervals
- integrate(final_step=True, output=True)¶
Perform the entire depletion process across all steps