Source code for openmc.deplete.cram

"""Chebyshev Rational Approximation Method module

Implements two different forms of CRAM for use in openmc.deplete.

import numbers
from itertools import repeat
from multiprocessing import Pool
import time

import numpy as np
import scipy.sparse as sp
import scipy.sparse.linalg as sla

from openmc.checkvalue import check_type, check_length
from .abc import DepSystemSolver

__all__ = [
    "deplete", "timed_deplete", "CRAM16", "CRAM48",
    "Cram16Solver", "Cram48Solver", "IPFCramSolver"]

[docs]def deplete(chain, x, rates, dt, matrix_func=None): """Deplete materials using given reaction rates for a specified time Parameters ---------- chain : openmc.deplete.Chain Depletion chain x : list of numpy.ndarray Atom number vectors for each material rates : openmc.deplete.ReactionRates Reaction rates (from transport operator) dt : float Time in [s] to deplete for maxtrix_func : Callable, optional Function to form the depletion matrix after calling ``matrix_func(chain, rates, fission_yields)``, where ``fission_yields = {parent: {product: yield_frac}}`` Expected to return the depletion matrix required by :func:`CRAM48`. Returns ------- x_result : list of numpy.ndarray Updated atom number vectors for each material """ fission_yields = chain.fission_yields if len(fission_yields) == 1: fission_yields = repeat(fission_yields[0]) elif len(fission_yields) != len(x): raise ValueError( "Number of material fission yield distributions {} is not equal " "to the number of compositions {}".format(len(fission_yields), len(x))) if matrix_func is None: matrices = map(chain.form_matrix, rates, fission_yields) else: matrices = map(matrix_func, repeat(chain), rates, fission_yields) # Use multiprocessing pool to distribute work with Pool() as pool: inputs = zip(matrices, x, repeat(dt)) x_result = list(pool.starmap(CRAM48, inputs)) return x_result
[docs]def timed_deplete(*args, **kwargs): """Wrapper over :func:`deplete` that also returns process time All arguments and keyword arguments are passed onto :func:`deplete` directly. Returns ------- proc_time: float Process time required to return from deplete results: list of numpy arrays Output from :func:`deplete` call """ start = time.time() results = deplete(*args, **kwargs) return time.time() - start, results
[docs]class IPFCramSolver(DepSystemSolver): r"""CRAM depletion solver that uses incomplete partial factorization Provides a :meth:`__call__` that utilizes an incomplete partial factorization (IPF) for the Chebyshev Rational Approximation Method (CRAM), as described in the following paper: M. Pusa, "`Higher-Order Chebyshev Rational Approximation Method and Application to Burnup Equations <>`_," Nucl. Sci. Eng., 182:3, 297-318. Parameters ---------- alpha : numpy.ndarray Complex residues of poles used in the factorization. Must be a vector with even number of items. theta : numpy.ndarray Complex poles. Must have an equal size as ``alpha``. alpha0 : float Limit of the approximation at infinity Attributes ---------- alpha : numpy.ndarray Complex residues of poles :attr:`theta` in the incomplete partial factorization. Denoted as :math:`\tilde{\alpha}` theta : numpy.ndarray Complex poles :math:`\theta` of the rational approximation alpha0 : float Limit of the approximation at infinity """ def __init__(self, alpha, theta, alpha0): check_type("alpha", alpha, np.ndarray, numbers.Complex) check_type("theta", theta, np.ndarray, numbers.Complex) check_length("theta", theta, alpha.size) check_type("alpha0", alpha0, numbers.Real) self.alpha = alpha self.theta = theta self.alpha0 = alpha0
[docs] def __call__(self, A, n0, dt): """Solve depletion equations using IPF CRAM Parameters ---------- A : scipy.sparse.csr_matrix Sparse transmutation matrix ``A[j, i]`` desribing rates at which isotope ``i`` transmutes to isotope ``j`` n0 : numpy.ndarray Initial compositions, typically given in number of atoms in some material or an atom density dt : float Time [s] of the specific interval to be solved Returns ------- numpy.ndarray Final compositions after ``dt`` """ A = sp.csr_matrix(A * dt, dtype=np.float64) y = np.asarray(n0, dtype=np.float64) ident = sp.eye(A.shape[0]) for alpha, theta in zip(self.alpha, self.theta): y += 2*np.real(alpha*sla.spsolve(A - theta*ident, y)) return y * self.alpha0
# Coefficients for IPF Cram 16 c16_alpha = np.array([ +5.464930576870210e+3 - 3.797983575308356e+4j, +9.045112476907548e+1 - 1.115537522430261e+3j, +2.344818070467641e+2 - 4.228020157070496e+2j, +9.453304067358312e+1 - 2.951294291446048e+2j, +7.283792954673409e+2 - 1.205646080220011e+5j, +3.648229059594851e+1 - 1.155509621409682e+2j, +2.547321630156819e+1 - 2.639500283021502e+1j, +2.394538338734709e+1 - 5.650522971778156e+0j], dtype=np.complex128) c16_theta = np.array([ +3.509103608414918 + 8.436198985884374j, +5.948152268951177 + 3.587457362018322j, -5.264971343442647 + 16.22022147316793j, +1.419375897185666 + 10.92536348449672j, +6.416177699099435 + 1.194122393370139j, +4.993174737717997 + 5.996881713603942j, -1.413928462488886 + 13.49772569889275j, -10.84391707869699 + 19.27744616718165j], dtype=np.complex128) c16_alpha0 = 2.124853710495224e-16 Cram16Solver = IPFCramSolver(c16_alpha, c16_theta, c16_alpha0) CRAM16 = Cram16Solver.__call__ del c16_alpha, c16_alpha0, c16_theta # Coefficients for 48th order IPF Cram theta_r = np.array([ -4.465731934165702e+1, -5.284616241568964e+0, -8.867715667624458e+0, +3.493013124279215e+0, +1.564102508858634e+1, +1.742097597385893e+1, -2.834466755180654e+1, +1.661569367939544e+1, +8.011836167974721e+0, -2.056267541998229e+0, +1.449208170441839e+1, +1.853807176907916e+1, +9.932562704505182e+0, -2.244223871767187e+1, +8.590014121680897e-1, -1.286192925744479e+1, +1.164596909542055e+1, +1.806076684783089e+1, +5.870672154659249e+0, -3.542938819659747e+1, +1.901323489060250e+1, +1.885508331552577e+1, -1.734689708174982e+1, +1.316284237125190e+1]) theta_i = np.array([ +6.233225190695437e+1, +4.057499381311059e+1, +4.325515754166724e+1, +3.281615453173585e+1, +1.558061616372237e+1, +1.076629305714420e+1, +5.492841024648724e+1, +1.316994930024688e+1, +2.780232111309410e+1, +3.794824788914354e+1, +1.799988210051809e+1, +5.974332563100539e+0, +2.532823409972962e+1, +5.179633600312162e+1, +3.536456194294350e+1, +4.600304902833652e+1, +2.287153304140217e+1, +8.368200580099821e+0, +3.029700159040121e+1, +5.834381701800013e+1, +1.194282058271408e+0, +3.583428564427879e+0, +4.883941101108207e+1, +2.042951874827759e+1]) c48_theta = np.array(theta_r + theta_i * 1j, dtype=np.complex128) alpha_r = np.array([ +6.387380733878774e+2, +1.909896179065730e+2, +4.236195226571914e+2, +4.645770595258726e+2, +7.765163276752433e+2, +1.907115136768522e+3, +2.909892685603256e+3, +1.944772206620450e+2, +1.382799786972332e+5, +5.628442079602433e+3, +2.151681283794220e+2, +1.324720240514420e+3, +1.617548476343347e+4, +1.112729040439685e+2, +1.074624783191125e+2, +8.835727765158191e+1, +9.354078136054179e+1, +9.418142823531573e+1, +1.040012390717851e+2, +6.861882624343235e+1, +8.766654491283722e+1, +1.056007619389650e+2, +7.738987569039419e+1, +1.041366366475571e+2]) alpha_i = np.array([ -6.743912502859256e+2, -3.973203432721332e+2, -2.041233768918671e+3, -1.652917287299683e+3, -1.783617639907328e+4, -5.887068595142284e+4, -9.953255345514560e+3, -1.427131226068449e+3, -3.256885197214938e+6, -2.924284515884309e+4, -1.121774011188224e+3, -6.370088443140973e+4, -1.008798413156542e+6, -8.837109731680418e+1, -1.457246116408180e+2, -6.388286188419360e+1, -2.195424319460237e+2, -6.719055740098035e+2, -1.693747595553868e+2, -1.177598523430493e+1, -4.596464999363902e+3, -1.738294585524067e+3, -4.311715386228984e+1, -2.777743732451969e+2]) c48_alpha = np.array(alpha_r + alpha_i * 1j, dtype=np.complex128) c48_alpha0 = 2.258038182743983e-47 Cram48Solver = IPFCramSolver(c48_alpha, c48_theta, c48_alpha0) del c48_alpha, c48_alpha0, c48_theta, alpha_r, alpha_i, theta_r, theta_i CRAM48 = Cram48Solver.__call__