"""Chebyshev Rational Approximation Method module Implements two different forms of CRAM for use in openmc.deplete. """ import numbers 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__ = ["CRAM16", "CRAM48", "Cram16Solver", "Cram48Solver", "IPFCramSolver"] 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 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 = dt * sp.csc_matrix(A, dtype=np.float64) y = n0.copy() ident = sp.eye(A.shape[0], format='csc') 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__