# 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
<https://doi.org/10.13182/NSE15-26>_," 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__