A new approach to the stabilization and convergence acceleration in coupled Monte CarloeCFD calculations: The Newton method via Monte Carlo perturbation theory

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A new approach to the stabilization and convergence acceleration in coupled Monte CarloeCFD calculations: The Newton method via Monte Carlo perturbation theory. This paper proposes the adoption of Monte Carlo perturbation theory to approximate the Jacobian matrix of coupled neutronics/thermal-hydraulics problems. The projected Jacobian is obtained from the eigenvalue decomposition of the fission matrix, and it is adopted to solve the coupled problem via the Newton method.
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A new approach to the stabilization and convergence acceleration in coupled Monte CarloeCFD calculations: The Newton method via Monte Carlo perturbation theory. This paper proposes the adoption of Monte Carlo perturbation theory to approximate the Jacobian matrix of coupled neutronics/thermal-hydraulics problems. The projected Jacobian is obtained from the eigenvalue decomposition of the fission matrix, and it is adopted to solve the coupled problem via the Newton method..

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Nuclear Engineering and Technology 49 (2017) 1181e1188 Contents lists available at ScienceDirect Nuclear Engineering and Technology journal homepage: www.elsevier.com/locate/net Original Article A new approach to the stabilization and convergence acceleration in coupled Monte CarloeCFD calculations: The Newton method via Monte Carlo perturbation theory Manuele Aufiero*, Massimiliano Fratoni Department of Nuclear Engineering, University of California, Berkeley, CA 94720-1730, USA a r t i c l e i n f o a b s t r a c t Article history: Received 30 May 2017 Received in revised form 26 July 2017 Accepted 7 August 2017 Available online 12 August 2017 Keywords: Jacobian M&C2017 Monte Carlo Multiphysics Newton Perturbation Theory 1. Introduction This paper proposes the adoption of Monte Carlo perturbation theory to approximate the Jacobian matrix of coupled neutronics/thermal-hydraulics problems. The projected Jacobian is obtained from the eigenvalue decomposition of the fission matrix, and it is adopted to solve the coupled problem via the Newton method. This avoids numerical differentiations commonly adopted in Jacobian-free Newton eKrylov methods that tend to become expensive and inaccurate in the presence of Monte Carlo statis-tical errors in the residual. The proposed approach is presented and preliminarily demonstrated for a simple two-dimensional pressurized water reactor case study. © 2017 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 2. The coupled neutronics/thermal-hydraulics nonlinear problem Multiphysics modeling of fission reactors represents a field of growing interest in the nuclear community [1,2]. Although coupled neutronics/thermal-hydraulics reactor simulations usually employ deterministic codes, more recently, several studies have proposed the adoption of continuous energy Monte Carlo codes for the neutronics solution of multiphysics problems [3e5]. For the purpose of the present work, it is useful to describe the coupled neutronics/thermal-hydraulics problem as a system of two equations. The first equation represents the generic neutron transport eigenvalue problem: The use of Monte Carlo in coupled simulations is motivated by the desire to obtain more accurate results and more flexible implementations, with respect to legacy deterministic codes. On ½L S4 ¼ 1 F4 (1) eff the other hand, stochastic neutron transport usually involves higher computational requirements when compared to determin-istic approaches, and poses barriers to the adoption of common techniques for the solution of nonlinear problems. This work presents a new approach to stabilize and accelerate the convergence of steady-state coupled Monte Carlo/thermal-hydraulics simulations, by combining the Newton method and Monte Carlo perturbation theory. The method is demonstrated in a simplified pressurized water reactor (PWR) multiphysics simulation. * Corresponding author. E-mail address: manuele.aufiero@berkeley.edu (M. Aufiero). where keff is the fundamental eigenvalue, L, S and F are the loss, scattering and fission production operators, and f represents the neutron flux solution of the eigenvalue problem. The second equation is represented here as a generic nonlinear equation in which the thermal-hydraulics (TH) solution T depends on the fission source distribution 4: T ¼ Qð4Þ (2) In the considered cases, T represents the material temperature and density distributions, and the main feedback of T on the neu-tronics solution is driven by the Doppler effect and moderator expansion effect. The generic coupling terms can be introduced in http://dx.doi.org/10.1016/j.net.2017.08.005 1738-5733/© 2017 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/). 1182 M. Aufiero, M. Fratoni / Nuclear Engineering and Technology 49 (2017) 1181e1188 Eq. (1) by allowing the L, S and F operators to be dependent on the generic TH solution T. The fixed-point iteration method is very simple and does not require major modifications to the code used to solve the neu- ½LðTÞ SðTÞ4 ¼ 1 FðTÞ4 eff tronics and thermal-hydraulics problems. Unfortunately, this (3) approach is prone to numerical instabilities and a low speed of convergence. For simplicity, it is assumed that the fissionpower distribution 4 is the only term of Eq. (3) required for solving the coupled problem. Thus, the neutronics equation can be simplified to: A coupled Serpent/OpenFOAM simulation of a PWR core [8] is used in order to test the fixed-point iteration method. The CFD solution is obtained with a coarse-mesh/porous-media approach, in which power densities and the coolant temperature are ho- 4 ¼ FðTÞ (4) Eq. (4) depends on T only. Replacing T with Eq. (2): 4 ¼ F½Qð4Þ (5) mogenized over a scale of several centimeters. Fig. 1 illustrates the case study. Instabilities in the convergence of the fixed-point iteration commonly arise when dealing with coupled neutronics/thermal- hydraulics calculations, for example, in light water reactors. Fig. 2 it is shown that the fission power distribution 4 depends on the material temperatures and densities (T), which depend on the power distribution itself. Eq. (5) can be written as shows the radial power distribution and coolant density distribu-tion in the PWR case study, for two consecutive iterations. In this case, an unbalance arises in the power distribution, most likely due to the randomness of the Monte Carlo sampling. An unbalance in the fuel temperature and coolant density follows in the next TH 4 ¼ Gð4Þ (6) calculation. Due to the strong negative Doppler and moderator feedbacks, the following Monte Carlo solution results in the so that the coupled neutronics/thermal-hydraulics problem re-duces to finding 4, the solution to G. In practical applications of multiphysics reactor analysis, the power distribution 4 is scored or discretized into N volumes within the reactor core. In this case, 4 is a vector of N components: opposite power unbalance (see Fig. 2). Fig. 3 shows the onset of numerical oscillations in the fission rate distribution during the first 20 iterations, at five different points in the two-dimensional (2D) PWR test case. With reference to the geometry description in Fig. 1, the selected points are: 4 ¼ ð41;42…4NÞ (7) and G(4) is a function G : ℝN/ℝN: Gð4Þ ¼ ðG1;G2…GNÞ (8) 3. Monte Carlo/CFD coupling: fixed-point iteration Point a: central assembly (12, M); Point b: left reactor side (12, G); Point c: right reactor side (12, T); Point d: upper reactor side (17, M); Point e: lower reactor side (7, M). In the initial iteration, the power distribution is uniform in the In the present work, the solution to the nonlinear equation T ¼ Q(4) is obtained via CFD, adopting the multiphysics Cþþ toolkit OpenFOAM [6]. The fission power distribution 4 ¼ F(T) is obtained via the Monte Carlo code Serpent [7]. One of the most common methods of solving the nonlinear problem 4 ¼ G(4) is to apply the operator splitting approach along with the fixed-point iteration method. This approach iterates be-tween the neutronics and the thermal-hydraulics codes, using the output of the previous run as the input to each simulation. At each radial points (Points bee) and slightly higher in the central point (Point a). After the first few iterations, Fig. 3 shows linearly growing oscillation amplitudes up to a saturation point due to nonlinearity effects. These oscillations will be damped or amplified according to the peculiarities of the system (dimensions, power level, magni-tude of the TH feedback on neutronics, etc.). As a first order approximation, if the initial iteration is close to the fixed-point solution, the condition for the stability of the fixed-point iteration 4(nþ1) ¼ G(4(n)) can be expressed as: coupled iteration n the following equations are solved: Tðnþ1Þ ¼ Q 4ðnÞ rðJGÞ<1 (14) (9) where r(JG) is the spectral radius of the Jacobian matrix JG of G: 4ðnþ1Þ ¼ F Tðnþ1Þ (10) rðJGÞ ¼ maxfjl1j;jl2j…jlNjg (15) or: 4ðnþ1Þ ¼ G 4ðnÞ (11) and fjl1j;jl2j…jlNjg are the eigenvalues of JG. The J element of the Jacobian matrix JG is the derivative of the ith value Gi of the vector function G, with respect to the jth value 4j of the vector input 4: That is, at each iteration, the new value for the fission power distribution 4(nþ1) is the output obtained from the coupled simu-lation, using the previous value of 4(n) as the input. The residual of any iteration n can be defined as: rðnÞ ¼ 4ðnÞ G 4ðnÞ (12) therefore, using Eq. (11): 4ðnþ1Þ 4ðnÞ ¼ rðnÞ (13) 2 dG1 dG1 6 d41 d42 6 dG2 dG2 JG ¼ 6 d41 d42 6 « « 6dGN dGN d41 d42 / / 1 / dG1 3 d4N 7 dG2 7 d4N 7 (16) « 7 dGN 7 d4N M. Aufiero, M. Fratoni / Nuclear Engineering and Technology 49 (2017) 1181e1188 1183 Fig. 1. Geometry of the considered PWR case study. Thus, if the power distribution 4 is scored in N different volumes, JG is an N N matrix. Typically, the presence of strong negative power feedback en- sures that most diagonal elements of JG are negative, and the ei-genvalues l1 l2, l3,… are negative, which ensures the stability of the reactor. To overcome the problem of numerical instabilities in the fixed-point iteration, under-relaxation is commonly employed to enforce the convergence of the coupled simulation. This is obtained by multiplying the residuals r(n) by a scalar a with 0 < a< 1 before the update of one of the coupled variables: Unfortunately, the spectral radius r(JG) is not known when dealing with a generic multiphysics problem with a fixed-point iteration. Thus, a is usually selected by the user based on its experience. Dufek and Gudowski [10] proposed an optimal iterative pro-cedure adopting decreasing under-relaxation factor and an increasing neutron population for the Monte Carlo simulation. Unfortunately, any approach based on under-relaxed fixed-point iteration, features averyslowconvergence rate (i.e., requires a large number of coupled iterations). Even in the case of variable popu- lation size, this might lead to large computational requirements, 4ðnþ1Þ 4ðnÞ ¼ a,rðnÞ especially if the thermal/hydraulic solution is obtained via expen- (17) sive CFD calculations. In Fig. 4, the fission power distribution in selected points is represented for the first few under-relaxed iterations, when assuming a¼0.65. Iteration #0 in the under-relaxed simulation has been selected as iteration #38 from the previous unstable case, highlighting how the selection of a suitable a parameter is effec-tively able to dump severe numerical oscillations. The optimal under-relaxation factor a can be obtained as [9]: a ¼ 2 þ2ðJGÞ (18) 4. The Newton method One of the classical methods of stabilizing and accelerating the convergence of a nonlinear problem is the Newton method. At each Newton iteration, 4(nþ1) is obtained by solving the following linear system: JðnÞ4ðnþ1Þ 4ðnÞ ¼ rðnÞ (19) 1184 M. Aufiero, M. Fratoni / Nuclear Engineering and Technology 49 (2017) 1181e1188 Fig. 2. Radial power distribution (top) and coolant density distribution (bottom) for coupled iterations #37 (left) and #38 (right). Fig. 3. Instabilities in the power distribution at four points in the 2D case study (nu-merical instabilities in the coupled Picard iterations). where r(n) are the residuals obtained at the nth iteration as [4(n)G(4(n))], and JðnÞ is the Jacobian matrix of G evaluated in 4ðnÞ. The Newton method can lead to quadratic convergence in most multiphysics problems that are of interest in the field of compu-tational physics. Unfortunately, its use as described by Eq. (19) is impractical in common applications, due to the difficulty of obtaining the full Jacobian JG in large multiphysics problems. For this reason, Newton methods adopting an approximated Jacobian have gained popularity in the last few decades. One of the Fig. 4. Effect of the under-relaxation in the 2D case study (effect of under-relaxation on the coupled Picard iterations). most popular approaches is the family of Jacobian-free New-toneKrylov (JFNK) methods [11]. These methods do not require the compilation of the full JG. At each Newton iteration, the system in Eq. (19) is solved by approximately applying the Krylov subspace method for the Jacobian. Rather than calculating the full JG, JFNK approaches only require the calculation of a Jacobian-vector product, through the evaluation of the nonlinear function G. For example, an approximation of the Jacobian-product JGð4ðnÞÞ ei can be obtained as [11]: M. Aufiero, M. Fratoni / Nuclear Engineering and Technology 49 (2017) 1181e1188 1185 JG4ðnÞeixG4ðnÞ þ ε,ei G4ðnÞ (20) first few eigentriplets, in which case, the eigenvalues are always real. Provided that the forward and adjoint eigenmodes respect the