openmc.deplete.EPCRK4Integrator

class openmc.deplete.EPCRK4Integrator(operator: TransportOperator, timesteps: Union[Sequence[float], Sequence[tuple[float, str]]], power: Optional[Union[float, Sequence[float]]] = None, power_density: Optional[Union[float, Sequence[float]]] = None, source_rates: Optional[Union[float, Sequence[float]]] = None, timestep_units: str = 's', solver: str = 'cram48', continue_timesteps: bool = False)

Deplete using the EPC-RK4 algorithm.

Implements an extended predictor-corrector algorithm with traditional Runge-Kutta 4 method. This algorithm is mathematically defined as:

\[\begin{aligned} \mathbf{A}_1 &= h\mathbf{A}(\mathbf{n}_i) \\ \hat{\mathbf{n}}_1 &= \exp \left ( \frac{\mathbf{A}_1}{2} \right ) \mathbf{n}_i \\ \mathbf{A}_2 &= h\mathbf{A}(\hat{\mathbf{n}}_1) \\ \hat{\mathbf{n}}_2 &= \exp \left ( \frac{\mathbf{A}_2}{2} \right ) \mathbf{n}_i \\ \mathbf{A}_3 &= h \mathbf{A}(\hat{\mathbf{n}}_2) \\ \hat{\mathbf{n}}_3 &= \exp \left ( \mathbf{A}_3 \right ) \mathbf{n}_i \\ \mathbf{A}_4 &= h\mathbf{A}(\hat{\mathbf{n}}_3) \\ \mathbf{n}_{i+1} &= \exp \left ( \frac{\mathbf{A}_1}{6} + \frac{\mathbf{A}_2}{3} + \frac{\mathbf{A}_3}{3} + \frac{\mathbf{A}_4}{6} \right ) \mathbf{n}_i. \end{aligned} \]

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, or source_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 CRAM

  • cram48 - 48th order IPF CRAM [default]

  If a function or other callable, must adhere to the requirements in solver.

  New in version 0.12.

• continue_timesteps (bool, optional) –

  Whether or not to treat the current solve as a continuation of a previous simulation. Defaults to False. When False, the depletion steps provided are appended to any previous steps. If True, the timesteps provided to the Integrator must exacly match any that exist in the prev_results passed to the Operator. The power, power_density, or source_rates must match as well. The method of specifying power, power_density, or source_rates should be the same as the initial run.

  New in version 0.15.1.

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 where

  • A is a scipy.sparse.csc_matrix making up the depletion matrix

  • n0 is a 1-D numpy.ndarray of initial compositions for a given material in atoms/cm3

  • t is a float of the time step size in seconds, and

  • n1 is a numpy.ndarray of compositions at the next time step. Expected to be of the same shape as n0

• transfer_rates (openmc.deplete.TransferRates) –

  Instance of TransferRates class to perform continuous transfer during depletion

  New in version 0.14.0.

__call__(n, rates, dt, source_rate, _i=None)

Perform the integration across one time step

Parameters

• n (list of numpy.ndarray) – List of atom number arrays for each material. Each array in the list contains the number of [atom] of each nuclide.

• 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 depletion step index, unused.

Returns

• proc_time (float) – Time spent in CRAM routines for all materials in [s]

• n_list (list of 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 simulations

__iter__()

Return pair of time step in [s] and source rate in [W] or [neutron/sec]

__len__()

Return integer number of depletion intervals

add_transfer_rate(material: Union[str, int, Material], components: Sequence[str], transfer_rate: float, transfer_rate_units: str = '1/s', destination_material: Optional[Union[str, int, Material]] = None)

Add transfer rates to depletable material.

Parameters

• material (openmc.Material or str or int) – Depletable material

• components (list of str) – List of strings of elements and/or nuclides that share transfer rate. A transfer rate for a nuclide cannot be added to a material alongside a transfer rate for its element and vice versa.

• transfer_rate (float) – Rate at which elements are transferred. A positive or negative values set removal of feed rates, respectively.

• destination_material (openmc.Material or str or int, Optional) – Destination material to where nuclides get fed.

• transfer_rate_units ({'1/s', '1/min', '1/h', '1/d', '1/a'}) – Units for values specified in the transfer_rate argument. ‘s’ means seconds, ‘min’ means minutes, ‘h’ means hours, ‘a’ means Julian years.

integrate(final_step: bool = True, output: bool = True, path: str | os.PathLike = 'depletion_results.h5')

Perform the entire depletion process across all steps

Parameters

• final_step (bool, optional) –

  Indicate whether or not a transport solve should be run at the end of the last timestep.

  New in version 0.12.1.

• output (bool, optional) –

  Indicate whether to display information about progress

  New in version 0.13.1.

• path (PathLike) –

  Path to file to write. Defaults to ‘depletion_results.h5’.

  New in version 0.15.0.