openmc.deplete.EPCRK4Integrator

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

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.

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)[source]

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: Union[str, 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.