Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation (Woodhead Publishing Series in Energy) 0128182210, 9780128182215

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Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Woodhead Publishing Series in Energy

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation Edited by Liangzhi Cao Professor, School of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an, People’s Republic of China Hongchun Wu Professor, School of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an, People’s Republic of China

An imprint of Elsevier

Woodhead Publishing is an imprint of Elsevier The Officers’ Mess Business Centre, Royston Road, Duxford, CB22 4QH, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, OX5 1GB, United Kingdom Copyright © 2021 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/ permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-818221-5 (print) ISBN: 978-0-12-818222-2 (online) For information on all Woodhead publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Candice G. Janco Acquisitions Editor: Maria Convey Editorial Project Manager: Chiara Giglio Production Project Manager: Vijayaraj Purushothaman Cover Designer: Greg Harris Typeset by SPi Global, India

Contributors

Liangzhi Cao Xi’an Jiaotong University, Xi’an, People’s Republic of China Chao Fang Xi’an Jiaotong University, Xi’an, People’s Republic of China Qingming He Xi’an Jiaotong University, Xi’an, People’s Republic of China Yunzhao Li Xi’an Jiaotong University, Xi’an, People’s Republic of China Zhouyu Liu Xi’an Jiaotong University, Xi’an, People’s Republic of China Hongchun Wu Xi’an Jiaotong University, Xi’an, People’s Republic of China Haochun Zhang Harbin Institute of Technology, Harbin, People’s Republic of China Yining Zhang Harbin Institute of Technology, Harbin, People’s Republic of China Youqi Zheng Xi’an Jiaotong University, Xi’an, People’s Republic of China

Foreword

The design, analysis, optimization, and licensing of nuclear reactor plants require accurate computational tools for predicting the complex interactions of neutrons with nuclear fuel and structural materials within reactor cores. In recent years, the increasing complexity of nuclear fuel designs and the development of advanced reactor concepts have combined to challenge the capabilities of traditional analysis tools. Across the globe, researchers are pushing the frontiers in solving the fundamental Boltzmann neutron transport equation (NTE) by utilizing massively parallel computers and advanced methods that minimize approximations in geometrical representations and energy resolution. While stochastic (Monte Carlo) methods are systematically maturing in their ability to produce high-fidelity solutions to full-core reactor models for pseudo-steady-state problems, the intractability of Monte Carlo for solving timedependent reactor neutron transport problems has left the multigroup deterministic methods as the most promising of techniques for accurately solving the broadest class of challenging reactor neutron transport problems. Deterministic methods for solving the reactor NTE have been in existence for at least 50 years, and dozens of fundamentally different computational approaches have been developed during this time. Until very recently, all reactor applications relied on lower dimensionality models (e.g., 1-D and 2-D), compromised geometrical representations (e.g., Cartesian meshes), and low-resolution angular representations (e.g., diffusion theory). Today, researchers strive to solve directly the heterogeneous, three-dimensional, neutron transport equation for precise geometrical representations of nuclear reactor cores. As a consequence, each of the traditional approaches for solving the NTE has been reexamined for its suitability for extension to these extremely challenging reactor problems. Students of nuclear engineering often struggle to understand the important differences between these methods, and students must individually assemble numerous proceedings of topical conferences and various textbooks in order to study the myriad of techniques that have been developed to date—before they can commence development or extension of modern tools for solving the most challenging of reactor neutron transport problems. One of the larger efforts to develop new methods and tools for solving the NTE has occurred in the universities of the Peoples Republic of China. In particular, the Nuclear Engineering Computational Physics Laboratory at Xi’an Jiaotong University, under Prof. Liangzhi Cao and Prof. Hongchun Wu, has produced a succession of students that have systematically studied and developed codes/tools for each of the traditional methods of solving the NTE—with a particular focus on general geometry extensions and solutions on modern computational platforms. Profs. Cao and Wu have coalesced their student contributions to produce this comprehensive textbook of

x

Foreword

techniques for solving the NTE. Students will find this textbook particularly useful because it compiles, into a single document, the wealth of techniques for solving the NTE—saving students a tremendous amount of effort in acquiring the materials needed to understand the differences and applicability of these diverse techniques. This textbook systematically progresses through many of the classic solution methods in a sequence of chapters covering: Collision Probabilities, Transmission Probabilities, Current-Coupled Collision Probabilities, Method of Characteristics, Spherical Harmonics/Finite Elements, Discrete Ordinates, Mesh-Free Diffusion Theory, as well as an introduction to Wavelets, Variational Nodal, and Hybrid Monte Carlo/MOC methods. In each chapter, systematic mathematical derivations and formulations are presented for each method, and numerous numerical examples are used to provide the reader with an understanding of the required discretization/accuracy of each method. Many of the examples are for complex geometrical representations on unstructured meshes—which are seldom treated in other textbooks. Students and nuclear professionals will find this textbook to be a valuable asset when starting research into modern methods for solving the NTE and their applications for the analysis of complex nuclear fuels and reactor designs that are rapidly evolving in the world of advanced nuclear reactor engineering.

Kord Smith Professor of the Practice of Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, United States

Preface

Neutron transport equation, derived from the Boltzmann equation, is the basic equation to describe neutron behaviors during their transport in different media. Numerical solution to the neutron transport equation is of great importance in the nuclear areas thus attracting more and more attentions in the community of nuclear engineering as well as nuclear technology applications. During the past decades, many effective numerical methods have been proposed and developed to solve the neutron transport equation. They can be classified into two categories: Monte Carlo methods and deterministic methods. Because the basic idea of Monte Carlo methods is to tally the number of particles by tracking the neutron in the real geometric structures, it offers the Monte Carlo methods high geometric adaptability by nature, although the computational cost is high. For the deterministic methods, however, the angular and spatial variables are always discretized by meshes, so the geometric adaptability relies highly on the discretization method. Traditional deterministic methods are mainly developed based on the structured meshes, which means they can only be used for regular geometric problems, or the real engineering problems have to be converted to regular geometry with approximations. In order to reduce those approximations and improve the accuracy of the solution, deterministic methods should be developed based on unstructured meshes. In recent years, the demand for high-fidelity numerical solution to neutron transport equation is increasing due to the rapid development of the advanced nuclear reactor concepts as well as the high-performance computational technologies. However, even though there are many journal and conference papers discussing the recent progresses in the deterministic neutron transport calculation methods, it is still very difficult to find a comprehensive book focusing on this topic up to now. There are a few books discussing the FEM solutions to NTE and some other books on traditional SN and PN methods, but not focusing on the unstructured meshes. Actually, the method of characteristic (MOC), for example, has been well developed for more than 20 years and has been widely employed in the industrial codes like CASMO5, APOLLO3, etc. Even for the traditional collision probability method (CPM) and transmission probability method (TPM), recent study has extended them to triangular unstructured meshes. Those progresses have not yet been well summarized. So, the primary motivation of this book is to put together those newly developed methods for the solution to NTE with unstructured meshes. The Nuclear Engineering Computational Physics (NECP) Lab at Xi’an Jiaotong University, created and led by the authors since 2004, has been focusing on the development of deterministic numerical methods for unstructured-mesh neutron transport calculation from the very beginning of its establishment. Over the past 16 years, dozens of graduate students have worked on this topic by developing different

xii

Preface

methods and testing them with numerical simulation. Many in-house codes have been written to verify those methods against some benchmark problems. Among them, we selected some relatively mature ones to composite this book. The first chapter is devoted to the derivation of the transport equation and its different forms including first-order and second-order differential forms and integral forms. The adjoint neutron transport equation is also introduced in Chapter 1. Then, in Chapters 2–4, some traditional methods, CPM, TPM, and CCCP, based on integral transport equation will be expanded into unstructured meshes. Chapter 5 introduces the most widely used MOC method for complex geometry problems. Chapters 6 and 7 are devoted to two classical deterministic methods, PN and SN method, with unstructured meshes mainly coupled with FEM. In Chapter 8, the mesh-free method which has been applied to neutron diffusion and transport calculation in recent years will be briefly introduced. Finally, Chapter 9 includes some other nonclassical methods which are still under development, such as the wavelet method, variational nodal method, and the Monte Carlo deterministic hybrid method. In most of the abovementioned methods, the theoretical model will be derived first and some numerical results will follow. The authors wish to express their appreciation to many contributors to this book. First of all, they acknowledge former and current graduate students of NECP lab who developed those methods and codes and contributed greatly to the contents of this book. They are Haoliang Lu (Chapter 7), Pingping Liu (Chapter 3), Guoming Liu (Chapter 3), Qichang Chen (Chapter 5), Qi Zheng (Chapter 9). The authors also thank Dr. Wei Shen at CANDU Owner Group who offered thoughtful suggestions and comments to the manuscript. During the preparation of this book, many current graduate students provided a lot of helps to them. They are Chao Fang, Qi Zheng, Yifan Zhang, Jianxin Miao, Xiaoyang Zou, etc. At last, but not the least, the authors thank Ms. Maria Convey, Ms. Michelle Fisher, and other colleagues from Elsevier for their kind support and constant help to make this book happen. Liangzhi Cao Hongchun Wu Xi’an Jiaotong University, Xi’an, People’s Republic of China

Neutron transport equation Liangzhi Cao Xi’an Jiaotong University, Xi’an, People’s Republic of China

1.1

1

Introduction

Neutrons play a very important role in many nuclear facilities, for example, nuclear fission and fusion reactors. The theory to describe and simulate the transport behavior of neutrons is called neutron transport theory. The roots of transport theory go back more than a century to the Boltzmann equation, first formulated for the study of the kinetic theory of gases [1]. With the rapid development of high-performance computer technology, the numerical simulation of neutron transport behavior is becoming more and more important in engineering design and analysis. In recent years, the demand for high-fidelity numerical solution to neutron transport equation (NTE) has increased due to the rapid development of the advanced nuclear reactor concepts as well as the high-performance computater technologies [2, 3]. This chapter introduces the basic concept of neutron transport equation and its alternative forms. Some fundamentals for the numerical solution of neutron transport calculation are also briefly introduced in this chapter.

1.2

Definition of the ordinates and basic elements

In the transport theory, neutrons are considered as point particles, which means the neutron motion state can be represented by the determined position and velocity. As shown in Fig. 1.1, the position of neutrons in space can be expressed by r. The velocity vector ν of the particles is written in terms of its solid angle as ν ¼ υΩ

(1.1)

where υ ¼ jνj is the magnitude of velocity, and its relation to the kinetic energy of neutrons E is E ¼ mυ2/2, where m is the mass of neutrons and Ω is the unit vector of the direction of motion. Its modulus is equal to 1. Its direction is expressed by polar coordinate system through polar angle θ and azimuth angle φ. Therefore, at any moment, the state of neutron motion can be described by six independent variables such as position vector r(x,y,z), energy E, and direction of motion Ω(θ, φ). For different coordinate systems, the expressions of r and Ω are different.

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation. https://doi.org/10.1016/B978-0-12-818221-5.00003-9 Copyright © 2021 Elsevier Ltd. All rights reserved.

2

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

ez

Fig. 1.1 Definition of vector r and Ω. z

Wz q

W Wy

Wx

dW ey

j

ex r

O

y

x

1.2.1 Coordinate system (1) Cartisian coordinate (x, y, z) Cartisian coordinate is the most commonly used coordinate system. Fig. 1.2 shows the three-dimensional cartisian coordinate and the representation of Ω in it. Coordinates of space points are determined by (x, y, z), volume element dV ¼ dxdydz, and Ω ¼ Ωx ex + Ωy ey + Ωz ez

(1.2)

Fig. 1.2 Cartisian coordinate.

z

ez x W q

j (x,y,z)

h

ey

m ex x

y

Neutron transport equation

3

Among which ex, ey, and ez are the unit vectors in the directions of three coordinate axes. And Ωx ¼ Ω  ex ¼ cosθ ¼ μ

(1.3)

Ωy ¼ Ω  ey ¼ sin θ cos φ ¼

pffiffiffiffiffiffiffiffiffiffiffiffi 1  μ2 cos φ ¼ η

(1.4)

pffiffiffiffiffiffiffiffiffiffiffiffi Ωz ¼ Ω  ez ¼ sinθ sinφ ¼ 1  μ2 sin φ ¼ ξ

(1.5)

dΩ ¼ sinθdθdφ ¼ dμdφ

(1.6)

It should be noted that the polar angle θ is the angle between Ω and ex (Fig. 1.2) which is different from that shown in Fig. 1.1. (2) Cylindrical coordinate (r, ψ, z) The cylindrical coordinate and the representation of Ω in it are shown in Fig. 1.3. Coordinates of space points are determined by (r, ψ, z), volume elementdV ¼ rdrdψdz, and Ω ¼ Ωr er + Ωθ eθ + Ωz ez

(1.7)

Among which ez and er are axial and radial unit vectors, respectively. And eθ is unit vectors perpendicular to the plane (ez, er) Ωr ¼ Ω  er ¼ sinθ cosφ ¼

qffiffiffiffiffiffiffiffiffiffiffiffi 1  ξ2 cos φ ¼ μ

(1.8)

Ωθ ¼ Ω  eθ ¼ sinθ sinφ ¼

qffiffiffiffiffiffiffiffiffiffiffiffi 1  ξ2 sinφ ¼ η

(1.9)

Ωz ¼ Ω  ez ¼ cosθ ¼ ξ

(1.10)

dΩ ¼ dξdφ

(1.11)

ez

z

Fig. 1.3 Cylindrical coordinate. W

x

r

h

q j

(r,y,z)

er

z

y x

eq

eq

y

4

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Fig. 1.4 Spherical coordinate.

z

W

q′

w

w q

r

y j

x

It is apparent that the direction of er and eθ changes when the space coordinate r or ψ changes. (3) Spherical coordinate (r, θ, φ) Fig. 1.4 shows the spherical coordinate and the representation of Ω in it, coordinates of space points are determined by (r, ψ, z). The direction of neutron motion is determined by θ0 between Ω and r (or μ ¼ cos θ0 ) and angle ω, which is between the plane (r, Ω) and the plane (r, z). In fact, three-dimensional spherical coordinate system is seldom used to solve neutron transport equation. One-dimensional spherically symmetric coordinate system is often used. Because of its symmetry, the space points are determined only by coordinate r, dV ¼ 4πr2dr. The direction of motion Ω is determined by Ωr ¼ μ ¼ cos θ0 , and is independent of ω.

1.2.2 Neutron density, angular flux, and current (1) Neutron density In order to fully describe the distribution of neutron in the medium, the distribution of space coordinate r, energy E, and the direction of motion Ω with time t of neutron must be given. Thus, we introduce the neutron density distribution function—the neutron angular density n(r, E, Ω, t). It is defined as the number of neutrons in unit volume at point r with energy E in dE and direction Ω in dΩ at time t. Therefore, 3 Number of neutrons in the volume of dr at time t 7 6 and at coordinate r, 7 nðr, E, Ω, tÞdrdEdΩ ¼ 6 5 4 energy between E and E + dE, and in the unit solid angle dΩ with the direction of Ω 2

(1.12)

Neutron transport equation

5

After integrating the neutron angular density with all the three-dimensional angular directions, the neutron density n(r, E, t) independent of the angle is obtained, i.e., the total neutron density ð nðr, E, tÞ ¼

nðr, E, Ω, tÞdΩ

(1.13)

4π

where n(r, E, t)drdE is the total number of neutrons in volume r and energy between E and E + dE at time t and coordinate r (including all directions of motion). (2) Neutron angular flux The neutron angular flux ϕ(r, E, Ω, t) is defined as the product of the neutron angular density and the neutron velocity ϕðr, E, Ω, tÞ ¼ υnðr, E, Ω, tÞ

(1.14)

It indicates total travel length per unit time, in unit volume at point r with energy E in dE and direction Ω in dΩ at time t. Similarly, from formula (1.13), the flux of neutrons ϕ(r, E, t) independent of direction can be defined as ð ϕðr, E, tÞ ¼ υnðr, E, tÞ ¼

ϕðr, E, Ω, tÞdΩ

(1.15)

4π

Therefore, the neutron flux ϕ(r, E, t) can be regarded as the sum of the intensities of infinite differential neutron beams ϕ(r, E, Ω, t) in all directions. Sometimes we call ϕ(r, E, t) which is independent of Ω as the total neutron flux or scalar flux. (3) Neutron current We use n to describe the unit normal vector perpendicular to the dS surface at r (Fig. 1.5). Therefore, in the steady state, the number of neutrons moving in the direction of Ω with energy equal to E passing through dS in unit time is given by n(r, E, Ω)υ j Ω  nj dS. We specify that the positive side of n is “+” and the negative side is “.” We also assume that the total number of neutrons passing from the “” side in all directions to the “+” side is J+n dS per second, and vice versa J n dS, then Fig. 1.5 Neutron current. Jn n

J dS

r

W

6

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

ð Jn+ dS ¼ dS ð ¼ dS Jn dS ¼ dS

ðΩ, nÞ>0

jΩ  njnðr, E, ΩÞυdΩ (1.16)

ðΩ, nÞ>0

jΩ  njϕðr, E, ΩÞdΩ

ð ðΩ, nÞ 0 (or < 0) in the formula means that only those Ω (i.e., hemispheres) with (Ω  n) > 0 (or < 0) are integral. J+n and J n along the positive and negative directions at r are called the partial neutron current. If the neutron angular flux is isotropic, then J+n ¼ J n , that is to say the net neutron number passing through dS per second or net neutron flow per second is equal to zero. In general, if J+n 6¼ J n , then the net neutron number passing through dS per second or net neutron flow per second is equal to   Jn ðr, EÞdS ¼ Jn+  Jn dS ¼ dS

ð ðΩ  nÞϕðr, E, ΩÞdΩ

(1.18)

4π

The above formula can be rewritten as ð Jn ðr, EÞdS ¼ dSn 

Ωϕðr, E, ΩÞdΩ ¼ n  Jðr, EÞdS

(1.19)

4π

Among them ð Ωϕðr, E, ΩÞdΩ

Jðr, EÞ ¼

(1.20)

4π

So Jn ðr, EÞ ¼ Jðr, EÞ  n

(1.21)

Vector J(r, E) is defined as neutron current. Its projection (or component) in the direction n Jn(r, E) is equal to the number of net neutron (net flux) passing through the unit area perpendicular to n per unit time. If Jn > 0, then J+n > J n , the direction of the net neutron flow is same as that of the normal vector n. Conversely, if Jn < 0, the direction of the net neutron flow is opposite to that of the normal vector n. From formula (1.19) we can see that the net neutron current passing through dS per second is not only related to distribution of ϕ(r, E, Ω), but also to n the direction of the area element dS. Obviously, if the direction of n is the same as that of J, the value of net neutron current is the largest.

1.3

First-order transport equation

1.3.1 Derivation of the NTE Typically, the neutron density in the reactor is significantly smaller than the atom density of the medium. For example, even in the thermal spectrum reactor with neutron

Neutron transport equation

7

flux ϕ  1020 n/(m2 s), the magnitude of the neutron density is lower than 1017 n/(m3), but the magnitude of the atom density of a typical solid is about 1028/(m3). Therefore, the motion of neutrons in the medium is influenced dominantly by the collision of the neutrons and nuclei in the medium, while the collision among neutrons is negligible. Due to the motion of neutrons and the collision of neutrons with the nuclei, the neutron at some position with some energy moving in some direction will appear at another position with another energy moving in another direction. This process is called neutron transport process, and the corresponding theory studying the neutron transport process is called the neutron transport theory [2]. For a single neutron, it moves in the medium randomly with a rambling zigzag trajectory until it is absorbed or escapes from the reactor, which is a random process. However, what we are interested in is not the behavior or position of individual neutron, but the macroscopic statistical spatial distribution of the neutron density. Like the gas molecular dynamics, a kind of macroscopic theory to handle the behavior of large quantities of neutrons is adopted to derive the neutron transport equation called Boltzmann transport equation, which is similar to the gas molecular transport equation. The neutron transport equation is derived as follows. A basic principle employed in the study of neutron transport process is the balance of neutron or the conservation of the neutron number, i.e., the variation rate of the neutron density is equal to the generation rate minus the loss rate from the streaming and collision ∂n ¼QLR ∂t

(1.22)

where ∂n ∂t is the variation rate of neutron density, Q is the neutron generation rate, L is the neutron loss rate from streaming, and R is the neutron loss rate from collision. When the system stays at the equilibrium state or steady state, the variation rate is equal to zero. At time t, the neutron balance at the phase space (r, E, Ω) is discussed as follows. For a point M, its position is defined by the vector r0. Along the moving direction Ω, define a position r where a neutron arrives by traveling from M with a distance of s. Consider a small volume increment with length ds and cross-sectional area dA, and its volume is dV ¼ dsdA, which is shown in Fig. 1.6. Then the neutron balance problem in the volume increment dV centered about r, and traveling in a cone of directions dΩ about direction Ω, with energy between E and E + dE is analyzed, and the expression for each term in Eq. (1.22) is derived. (1) Loss rate from streaming According to the definition of angular flux, the number of neutrons with energy between E and E + dE enters the volume increment dV through dA per unit time in a cone of directions dΩ about direction Ω is ϕðr0 + sΩ, E, Ω, tÞdAdEdΩ While the number of neutrons that exit from the volume increment is

(1.23)

8

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

W

ds s

dV M

dA

r0

r

O

Fig. 1.6 Loss of neutrons from streaming.

ϕðr 0 + ðs + dsÞΩ, E, Ω, tÞdAdEdΩ

(1.24)

Hence the number of neutrons leaking from the volume increment is L ¼ ½ϕðr 0 + ðs + dsÞΩ, E, Ω, tÞ  ϕðr0 + sΩ, E, Ω, tÞdAdEdΩ ¼

(1.25)

dϕðr 0 + sΩ, E, Ω, tÞ dVdEdΩ ds

Note that r ¼ r0 + sΩ, and the total derivative

dϕ ds

can be represented as

dϕ ∂ϕ dx ∂ϕ dy ∂ϕ dz ¼ + + ds ∂x ds ∂y ds ∂z ds ∂ϕ ∂ϕ ∂ϕ ¼ Ωx + Ωy + Ωz ¼ Ω  rϕ ∂x ∂y ∂z

(1.26)

Here the operator r can be interpreted as a vector in form, i.e., r¼i

∂ ∂ ∂ +j +k ∂x ∂y ∂z

rϕ ¼ grad ϕ ¼

∂ϕ ∂ϕ ∂ϕ i+ j+ k ∂x ∂y ∂z

(1.27)

(1.28)

Hence the neutron loss rate from streaming is derived as L ¼ Ω  rϕdVdEdΩ

(1.29)

(2) Loss rate from collision The loss number of neutrons per unit time from the element dVdEdΩ of the phase space per unit time can be attributed to two phenomena: (I) the neutron is absorbed in dV; (II) the neutron is scattered by the nucleus, and elastic or inelastic scattering will alter the neutron

Neutron transport equation

9

energy E or the motion direction Ω, which causes the loss of neutrons from the element of the phase space (r,E,Ω). Hence, the loss number of neutrons from the element dVdEdΩ due to the collision is R ¼ ðΣ s + Σ a Þϕðr, E, Ω, tÞdVdEdΩ ¼ Σ t ϕdVdEdΩ

(1.30)

Here, Σ s , Σ a , and Σ t represent the macroscopic scattering, absorption, and total cross sections, respectively. (3) Generation rate There are three kinds of sources contributing to the generation of neutrons in the element dVdEdΩ. The first one is the scattering source. The neutrons with energy E0 moving in direction Ω0 can be transformed to the neutrons with energy E moving in direction Ω by scattering. According to the definition of the scattering function, the number of neutrons in dV with energy from E0 to E0 + dE0 traveling in the direction Ω0 scattered into energy E and direction Ω is Σ s ðr, E0 Þf ðr, E0 ! E, Ω0 ! ΩÞϕðr, E0 , Ω0 , tÞdVdE0 dΩ0

(1.31)

Here, f is the scattering function, which is defined as the probability that neutrons with energy E0 traveling in direction Ω0 scatter into energy E and direction Ω. By integrating the foregoing formulation over E0 and Ω0 , the number of neutrons with various energy and directions scattered into the element dEdΩ over the volume increment dV per unit time is obtained as Qs ¼ dVdEdΩ

ð∞

0

dE 0

ð Ω’

Σ s ðr, E0 Þf ðr, E0 ! E, Ω0 ! ΩÞϕðr, E0 , Ω0 , tÞdΩ0

(1.32)

The second one is the fission source, i.e., the neutrons generated by the nuclear fission reaction in the reactor. ν(E0 ) is the average number of neutrons with energy E0 generated per fission reaction, and χ(E) is the fission spectrum. Assuming that all the fission neutrons are generated promptly and isotopically, then the neutron generation rate due to the fission reaction is   ð ð χ ðEÞ ∞ 0 Qf ¼ dVdEdΩ dE νðE0 ÞΣ f ðr, E0 Þϕðr, E0 , Ω0 , tÞdΩ0 4π 0 Ω0

(1.33)

The third one is the external neutron source. The neutron source strength is S(r, E, Ω, t), and its contribution is S ¼ Sðr, E, Ω, tÞdVdEdΩ

(1.34)

Therefore, the total generation rate Q is Qðr, E, Ω, tÞ ¼ Qs + Qf + S

(1.35)

Finally, the variation rate of neutron density is ∂ 1∂ nðr, E, Ω, tÞ ¼ ϕðr, E, Ω, tÞ ∂t v ∂t

(1.36)

10

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Combining Eqs. (1.25), (1.36), (1.22) and eliminating dVdEdΩ, the neutron balance equation at time t at phase space (r, E, Ω) is obtained as 1 ∂ϕ + Ω  rϕ + Σ t ðr, EÞϕ v ∂t ð∞ ð ¼ Σ s ðr, E0 Þf ðr, E0 ! E, Ω0 ! ΩÞϕðr, E0 , Ω0 , tÞdE0 dΩ0 0

(1.37)

Ω0

+ Qf ðr, E, Ω, tÞ + Sðr, E, Ω, tÞ where ϕ ¼ ϕ(r, E, Ω, t). This is the neutron transport equation under unsteady state or the Boltzmann equation, and it forms the basis of the reactor physics and the neutron transport theory. At the steady state, ∂ n/∂ t ¼ 0, and the steady-state neutron transport equation is Ω  rϕ + Σ t ðr, EÞϕ ¼

ð∞ ð 0

Ω0

Σ s ðr, E0 Þf ðr, E0 ! E, Ω0 ! ΩÞϕðr, E0 , Ω0 , tÞdE0 dΩ0

+ Qf ðr, E, Ω, tÞ + Sðr, E, Ω, tÞ (1.38)

1.3.2 Different forms of divergence operator From the foregoing formulation, the neutron transport equation is a linear integrodifferential equation. It contains six variables including r(x, y, z), E, and Ω(θ, φ) at the steady state. This kind of equation is difficult to be solved directly from a mathematical point of view, and it is impossible to obtain the solution of some complex problems. Therefore, in the reactor physics community, the numerical solution is usually obtained by establishing some simplified model and approximate numerical computational method. Eq. (1.38) is a steady-state neutron transport equation. Under specific conditions, different coordinate systems are usually adopted for different problems. The divergence operator Ω  r ϕ ¼ Ω  grad ϕ in Eq. (1.38) has different forms in different coordinates. According to the field theory, Ω  grad ϕ is the directional derivative of the function ϕ in the direction Ω. Consider a ray s with direction Ω, and the increment along the direction Ω is ds (Fig. 1.7), and Ω  rϕ ¼ Ω  grad ϕ ¼

dϕ ds

(1.39)

Then the expressions of Ω  r ϕ and the neutron transport equation are derived in several common coordinates in the reactor analysis as follows. (1) Cartesian coordinates The expression of Ω  r ϕ in three-dimensional Cartesian coordinates has already been obtained during the process of deriving the neutron transport equation, which can be represented as follows according to the definition of Ω in Fig. 1.1:

Neutron transport equation

11

s ds

W

r

O

Fig. 1.7 Directional derivative.

∂ϕ ∂ϕ ∂ϕ + Ωy + Ωz ∂x ∂y ∂z   pffiffiffiffiffiffiffiffiffiffiffiffi ∂ϕ ∂ϕ ∂ϕ +μ Ω  rϕ ¼ 1  μ2 cos ϕ + sinϕ ∂x ∂y ∂z

Ω  rϕ ¼ Ωx

(1.40)

where μ ¼ cos θ, or be represented as follows according to the definition of Ω in Fig. 1.2: Ω  rϕ ¼ μ

∂ϕ ∂ϕ ∂ϕ +η +ξ ∂x ∂y ∂z

(1.41)

For the two-dimensional (x,y) case symmetric about the z axis in the reactor design Ω  rϕ ¼ μ

∂ϕ ∂ϕ +η ∂x ∂y

(1.42)

One-dimensional slab problem is the simplest case in the Cartesian coordinate (Fig. 1.8), where the neutron flux is just a function of one-dimensional coordinate z and the angle θ (or μ ¼ cos θ) between Ω and the z axis, hence Ω  rϕ ¼ μ

∂ϕ ∂z

(1.43)

O

W q j

z

Fig. 1.8 One-dimensional plane coordinate system.

12

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

where, ϕðz, E, μÞdμ ¼ dμ

ð 2π

ϕðz, E, ΩÞdϕ

(1.44)

0

or ϕðz, E, μÞ ¼ 2πϕðz, E, ΩÞ

(1.45)

In a similar way, we define S(z, E, μ) as follows: Sðz, E, μÞ ¼ 2πSðz, E, ΩÞ

(1.46)

Assuming that the scattering function f(E0 ! E, Ω0 ! Ω) is just an angle between the direction Ω0 before the scattering and the direction Ω after the scattering (i.e., scattering angle θ0), let μ0 ¼ cos(Ω0 , Ω) ¼ cos θ0, and then similarly f ðE0 ! E; μ0 Þ ¼ 2πf ðE0 ! E, Ω0 ! ΩÞ

(1.47)

Considering Eqs.(1.43)–(1.47), the steady-state neutron transport equation in the onedimensional slab geometry is written as ∂ϕðz, E, μÞ + Σ t ϕðz, E, μÞ ∂z ð ð ð 1 ∞ 0 1 2π dE Σ s f ðE0 ! E, μ0 Þϕðz, E0 , μ0 Þdφ0 dμ0 + Sðz, E, μÞ ¼ 2π 0 1 0

μ

(1.48)

(2) Cylindrical coordinates Cylindrical geometry is often encountered in the reactor physics analysis, such as the reactor core, fuel rod, etc. It is convenient to adopt the cylindrical coordinates. In the cylindrical coordinates Ω  rϕ ¼

dϕ ∂ϕ dr ∂ϕ dψ ∂ϕ dz ∂ϕ dξ ∂ϕ dφ ¼ + + + + ds ∂r ds ∂ψ ds ∂z ds ∂ξ ds ∂φ ds

(1.49)

Referring to Fig. 1.3, we have dr/ds ¼ μ, dψ/ds ¼ η/r, dz/ds ¼ ξ, dξ/ds ¼ 0, and dφ/ ds ¼  η/r. Hence, Ω  rϕ ¼

   qffiffiffiffiffiffiffiffiffiffiffiffi ∂ϕ sin φ ∂ϕ ∂ϕ ∂ϕ  +ξ 1  ξ2 cosφ + ∂r r ∂ψ ∂φ ∂z

¼μ

∂ϕ η ∂ϕ η ∂ϕ ∂ϕ +  +ξ ∂r r ∂ψ r ∂φ ∂z

As to the two-dimensional (r,z) case, Ω  rϕ ¼ μ

∂ϕ ∂ψ ¼ 0

(1.50)

in the above equation, hence

∂ϕ η ∂ϕ ∂ϕ  +ξ ∂r r ∂φ ∂z

Similarly, for the two-dimensional (r, ψ) case, ∂ϕ ∂z ¼ 0 in Eq. (1.50).

(1.51)

13

W

Neutron transport equation

q′ –dq

–rdq

dl

dr

q

–dq o

r

Fig. 1.9 The diagram of the motion of neutrons in the spherical symmetrical case. (3) Spherical coordinates Only the one-dimensional spherical symmetrical case is discussed here because it is the most commonly used case. The neutron flux is a function of the position coordinate r and the direction angle θ (the angle between the direction Ω and r), let μ ¼ cos θ, then we have Ω  rϕðr, E, μÞ ¼

∂ϕ ∂ϕ dr ∂ϕ dμ ¼ + ∂s ∂r ds ∂μ ds

(1.52)

According to Fig. 1.9, we have dr ¼ cos θ ¼ μ ds

(1.53)

  dμ dμ dθ sinθ 1  μ2 ¼ ¼  sinθ  ¼ r ds dθ ds r

(1.54)

Hence, Ω  rϕðr, E, μÞ ¼ μ

∂ϕ 1  μ2 ∂ϕ + r ∂μ ∂r

(1.55)

The expressions of Ω  r ϕ in various coordinates with different dimensions are listed in Table 1.1.

1.3.3 Definite conditions The neutron transport equation is an integrodifferential equation. From the derivation process, it only represents the mathematical form of the neutron balance, i.e., the equation that the neutron density distribution function observes during the motion process of neutrons in the medium. But it does not provide a complete description of a specific state of the physical problem. For the nuclear reactors with the same material

Table 1.1 The expression of Ω  r ϕ in various coordinates.

Cartesian coordinates

Cylindrical coordinates

Spherical coordinates

Spatial variable

Directional variable

z

θ

x,y

θ, φ

x,y,z

θ, φ

r

θ, φ

r,z

θ, φ

r, ψ

θ, φ

r, ψ, z

θ, φ

r

θ0

r, θ, φ

θ0 , ω

Ω —ϕ

dV

∂ϕ ∂x ∂ϕ ∂ϕ μ +η ∂x ∂y

dx

μ

μ

∂ϕ ∂ϕ ∂ϕ +η +ξ ∂x ∂y ∂z

dxdy

2πrdr

∂ϕ η ∂ϕ ∂ϕ  +ξ ∂r r ∂φ ∂z   ∂ϕ η ∂ϕ ∂ϕ  μ + ∂r r ∂ψ ∂φ

2πrdrdz

μ

∂ϕ η ∂ϕ η ∂ϕ ∂ϕ +  +ξ ∂r r ∂ψ r ∂φ ∂z

∂ϕ 1  μ2 ∂ϕ μ + ∂r prffiffiffiffiffiffiffiffiffiffiffiffi ∂μ  1  μ2 sinω ∂ϕ ∂ϕ μ + r ∂r sinθ ∂φ ∂ϕ + cosω  sinω cotθ  ∂θ 2 ∂ϕ 1  μ ∂ϕ + r ∂μ ∂ω

dΩ Ð 2π 11dμ Ð 2π Ð1 1dμ 0 dφ

Referring to Fig. 1.2 μ ¼ Ωx ¼ cosθ pffiffiffiffiffiffiffiffiffiffiffiffi η ¼ Ωy ¼ 1  μ2 cos φ pffiffiffiffiffiffiffiffiffiffiffiffi ξ ¼ Ωz ¼ 1  μ2 sinφ

dxdydz

∂ϕ η ∂ϕ μ  ∂r r ∂φ μ

Ð

Ð

Ð1

2π 1dξ 0 dφ

Referring to Fig. 1.3 pffiffiffiffiffiffiffiffiffiffiffiffi μ ¼ Ωr ¼ 1  ξ2 cos φ pffiffiffiffiffiffiffiffiffiffiffiffi η ¼ Ωθ ¼ 1  ξ2 sin φ ξ ¼ Ωx ¼ cos θ

rdrdψ rdψdz 4πr2dr r 2 sinθdr dθdφ

2π Ð1

Ð1

1dμ

Ð

2π 1dμ 0 dω

Referring to Fig. 1.4 μ ¼ Ωr ¼ cos θ0 ω is the angle between the plane determined by r and Ω and the plane determined by r and z

Neutron transport equation

15

composition, they satisfy the same equation no matter how their initial state, shape, and boundary conditions are defined. Hence for a specific physical problem, the boundary and initial conditions should be given. From a mathematical point of view, arbitrary integral constants are contained in the ordinary solution of the neutron transport equation. In order to determine these constants, some appropriate definite conditions (i.e., boundary and initial conditions) need to be given. The number of the definite conditions should be proper to guarantee the unique solution of the given problem. Here, the boundary conditions which are often used when solving the neutron transport equation are discussed as follows: (1) In the domain where the neutron transport equation is defined, the neutron flux should be finite and nonnegative real numbers. (2) For an interface boundary between two different media, direct contact of the two media is assumed without inserting source or other medium. According to the continuity condition, the boundary condition should be: ϕ(r, E, Ω) is a continuous function of r on the interface boundary along Ω. If the interface boundary is inserted by third medium or source, the effect of the neutrons passing through the medium should be considered. The boundary condition should be modified in this case. (3) Free surface Free surface means the surface or the portion of the surface without external neutrons streaming into the medium. Assume that the domain where the neutron transport happens consists of several convex and piecewise smooth curved surfaces Γ. “Convex” surface means that any line starting from the surface will not intersect the surface again, see Fig. 1.10. Therefore, any neutron leaking from the surface will not return to the domain. Hence, the convex surface interfacing with vacuum is a free surface, and its boundary condition can be written as ϕðr, E, ΩÞ ¼ 0, rΓ,

if n  Ω < 0

(1.56)

where n is the outgoing normal unit vector at point r on the boundary Г . (4) Reflective boundary Reflective boundary appears on the symmetric surface of the symmetric system. On the reflective boundary, the incoming angular neutron flux at such direction is equal to the outgoing angular neutron flux on the corresponding reflecting direction (such as mirror-like

Fig. 1.10 Convex and concave surfaces.

16

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

reflecting direction). For instance, in the Cartesian coordinates, let some Y-Z plane be the reflective boundary, then the following condition is applied on the boundary: ϕðr s , E, μ, η, ξÞ ¼ ϕðr s , E,  μ, η, ξÞs

(1.57)

Reflective boundary condition also appears on the outer radial surface of the cylindrical cells. (5) White boundary White boundary is usually adopted on the outer surface of the cylindrical cell or bundles. On the white boundary, the outgoing angular neutron flux is isotropic and constant, and its value is equal to the average value of the incoming angular neutron flux. For example, the white boundary on the outer surface of the 1D cylinder with radius R is ð 2π ð 1 ϕðR, E, μ, φÞ ¼

0

0

μ0 ϕðR, E, μ0 , ϕ0 Þdμ0 dφ0 ð 2π ð 1 μdμdφ 0

μð1, 0Þ

(1.58)

0

Hence the net neutron current through the white boundary is zero. (6) Albedo boundary Albedo boundary is similar to the above two boundaries, but the incoming angular neutron flux or current is not equal to the outgoing angular neutron flux or current. The ratio between the incoming and outgoing ones is equal to a constant α, α < 1. Traditionally, α is called the albedo coefficient. Corresponding to the boundary conditions (4) and (5), albedo boundary condition can be represented as ϕðr s , E, μ, η, ξÞ ¼ αϕðr s , E,  μ, η, ξÞ

(1.59)

where rs is one point on the boundary. If α ¼ 1, then it is a reflective boundary condition. The albedo boundary condition can also be represented as ð 2π ð 1 ϕðR, E, μ, φÞ ¼ α

0

0

μ0 ϕðR, E, μ0 , ϕ0 Þdμ0 dφ , μð1, 0Þ ð 2π ð 1 μdμdφ 0

(1.60)

0

Similarly, if α ¼ 1, then it is a white boundary condition. In the unsteady-state case, the initial condition should be given along with the boundary conditions, such as the angular neutron flux distribution ϕ(R, E, μ, 0) at t ¼ 0.

1.4

Second-order transport equation

The first-order differential-integral equation obtained in Section 1.3 is the basis of the neutron transport theory, and it has been widely used in the study of many problems, but its disadvantage is that its operators are nonself-adjoint. Therefore, in the numerical solution of many problems, such as finite element method or variational method, the asymmetric matrix equation is obtained due to the nonself-adjoint of operators,

Neutron transport equation

17

which brings a lot of trouble to the solution. This section describes two methods for transforming the first-order differential-integral equation into the second-order neutron transport equation. After transformation, the first-order transport equation is transformed into second-order even symmetric transport equation and second-order self-conjugate SAAF (self-adjoint angular flux) transport equation, respectively. The advantage of the second-order transport equation is that the operator is selfadjoint, so the symmetric matrix equation can be obtained from it, and a series of standard numerical methods can be used to solve the equation more effectively.

1.4.1 Second-order even-parity neutron transport equation For the sake of convenience, we discuss the case of single energy, then the transport equation can be written as Ω  rϕðr, ΩÞ + Σ t ðr Þϕðr, ΩÞ ð ¼ Σ s ðr, Ω0 ! ΩÞϕðr, Ω0 ÞdΩ0 + Qðr, ΩÞ

(1.61)

Ω0

the boundary condition is ϕðr, ΩÞ ¼ 0, n  Ω < 0, rΓ

(1.62)

where Γ is the free boundary. To derive even symmetric transport equation, an operator was defined, which acts on an arbitrary function ϕ(r, Ω), and make Uϕ(r, Ω) ¼ ϕ(r,  Ω), while the even neutron angular flux density and the odd neutron angular flux density are defined as 1 1 ψ + ðr, ΩÞ ¼ ½ϕðr, ΩÞ + Uϕðr, ΩÞ ¼ ½ϕðr, ΩÞ + ϕðr,  ΩÞ 2 2

(1.63)

1 1 ψ  ðr, ΩÞ ¼ ½ϕðr, ΩÞ  Uϕðr, ΩÞ ¼ ½ϕðr, ΩÞ  ϕðr,  ΩÞ 2 2

(1.64)

where ψ +(r, Ω) is an even function of Ω, ψ +(r, Ω) ¼ Uψ +(r, Ω), ψ (r, Ω) is an odd function of Ω, ψ (r, Ω) ¼  Uψ (r, Ω). Neutron scalar flux density is defined as the integral of the even neutron angular flux density ð ϕð r Þ ¼

Ω

ψ + ðr, ΩÞdΩ

(1.65)

The neutron current can be obtained by using the odd neutron angular flux density as ð J ðr Þ ¼

Ω

Ωψ  ðr, ΩÞdΩ

(1.66)

18

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Firstly, the operator U is applied to the transport equation, and the transport equation along the angle  Ω can be obtained Ω  rϕðr,  ΩÞ + Σ t ðr Þϕðr,  ΩÞ ð ¼ Σ s ðr, Ω0 ! ΩÞϕðr, E0 , Ω0 ÞdΩ0 + Qðr,  ΩÞ

(1.67)

Ω0

By adding Eqs. (1.61) and (1.67), and subtracting Eq. (1.61) from Eq. (1.67), we get the following equations, respectively: Ω  rψ  ðr, ΩÞ + Σ t ðr Þψ + ðr, ΩÞ ð ¼ Σ s+ ðr, Ω0 ! ΩÞϕðr, Ω0 ÞdΩ0 + Q + ðr, ΩÞ

(1.68)

Ω  rψ + ðr, ΩÞ + Σ t ðr Þψ  ðr, ΩÞ ð 0 0 0  ¼ Σ s ðr, Ω ! ΩÞϕðr, Ω ÞdΩ + Q ðr, ΩÞ

(1.69)

Ω0

Ω0

In the equations above 1 0 0 0 Σ s ðr, Ω ! ΩÞ ¼ ½Σ s ðr, Ω ! ΩÞ  Σ s ðr, Ω ! ΩÞ 2

(1.70)

1 Q ðr, ΩÞ ¼ ½Q + ðr, ΩÞ  Q ðr, ΩÞ 2

(1.71)

If the scattering and source term are assumed to be isotropic, + 0 i.e., Σ s+ ðr, Ω0 ! ΩÞ ¼ Σ s ðr, Ω0 ! ΩÞ, Σ  s ðr, Ω ! ΩÞ ¼ 0, Q (r, Ω) ¼ Q(r, Ω),  Q (r, Ω) ¼ 0, then from Eq. (1.69) we get + ψ  ðr, ΩÞ ¼ Σ 1 t ðr ÞΩ  rψ ðr, ΩÞ

(1.72)

Substituting Eq. (1.72) into Eq. (1.68), the transport equation in the form of secondorder even-parity can be obtained as

+ + Ω  r Σ 1 t ðr ÞΩ  rψ ðr, ΩÞ + Σ t ðr Þψ ðr, ΩÞ ð ¼ Σ s ðr, Ω0 ! ΩÞψ + ðr, Ω0 ÞdΩ0 + Q + ðr, ΩÞ

(1.73)

Ω0

For the boundary conditions of ψ +(r, Ω), on vacuum boundary, there are + ð Ω  nÞ < 0 ϕðr s , ΩÞ ¼ ψ + ðr, ΩÞ  Σ 1 t ðr ÞΩ  rψ ðr, ΩÞ ¼ 0, 1 + ϕðr s ,  ΩÞ ¼ ψ ðr, ΩÞ + Σ t ðr ÞΩ  rψ + ðr, ΩÞ ¼ 0, ðΩ  nÞ > 0

on symmetric boundary, there are

 (1.74)

Neutron transport equation

ψ + ðr, ΩÞ ¼ ψ + ðr, Ω0 Þ

19

(1.75)

Eq. (1.73), together with boundary condition (1.74) or condition (1.75), is the secondorder even symmetric transport equation. Once ψ + is solved from the equation, the neutron flux density ϕ(r) and the neutron current vector J(r) can be obtained. It can be proved that Eq. (1.73) is self-adjoint. Therefore, variational method or finite element method can be used to solve the equation, and symmetrical matrix equation can be obtained.

1.4.2 Second-order SAAF transport equation Compared with the second-order even-parity transport equation, the second-order self-adjoint angular flux (SAAF) transport equation has some unique properties: it can be directly solved to obtain the full-angle flux; it can accurately express the reflective boundary condition; and it can be employed to solve problems containing void regions. The derivation of the SAAF equation is described below. For convenience, we discuss the single-energy case, the steady-state neutron transport equation can be written in the form of Eq. (1.61). In the equation, the scattering term can be expressed as an operator form: ð Ω0

Σ s ðr, Ω0 ! ΩÞϕðr, Ω0 ÞdΩ0 ¼ Sϕðr, ΩÞ

(1.76)

Then Eq. (1.61) can be rewritten as Ω  rϕ + Σ t ϕ ¼ Sϕ + q

(1.77)

From Eq. (1.77) we get ϕ ¼ ðΣ t  SÞ1 Ω  rϕ + ðΣ t  SÞ1 q

(1.78)

Substituting Eq. (1.78) into the first term of Eq. (1.77), we get Ω  rðΣ t  SÞ1 Ω  rϕ + ðΣ t  SÞϕ ¼ q  Ω  rðΣ t  SÞ1 q

(1.79)

This is the SAAF equation. The boundary conditions of the SAAF equation are similar to those of the first-order transport equation, i.e., at the boundary Γ, the angular flux of incident neutrons satisfies the boundary conditions: ϕðrb , ΩÞ ¼ f ðΩÞ, r b Γ, Ω  n < 0

(1.80)

20

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

where f(Ω) is the incident angular flux density on the boundary; for the vacuum (free) boundary, f ¼ 0. However, since the SAAF equation is a second-order equation, the required number of boundary conditions is double that for the first-order one. It is generally required that the angular flux of outgoing neutrons satisfy the first-order transport equation on the boundary. ϕðr b , ΩÞ + ðΣ t  SÞ1 Ω  rϕðr b , ΩÞ ¼ ðΣ t  SÞ1 qðr b , ΩÞ,Ω  n > 0

(1.81)

Eq. (1.79), together with boundary conditions (1.80) and (1.81), constitutes the second-order SAAF transport equation.

1.5

Integral form of transport equation

Neutron transport equation (1.38) is a first-order partial differential-integral equation. According to the principle of conservation of neutron number, an integral form of transport equation can be obtained from another point of view. It can be proved that these two forms of transport equations are mathematically equivalent. It only needs to prove that the transport equation in integral form can be derived from the steady-state differential-integral equation (1.82) directly. Ω  rϕ + Σ t ðr, EÞϕ ¼

ð∞ ð 0

Σ s ðr, E0 Þf ðr, E0 ! E, Ω0 ! ΩÞϕðr, E0 , Ω0 ÞdE0 dΩ0

4π

+ Qf ðr, E, ΩÞ + Sðr, E, ΩÞ (1.82) First note that in Eq. (1.82), Ω  r ϕ is the derivative along the direction Ω. Consider a certain point P(r  lΩ) in Ω (shown in Fig. 1.11). For point P, replace r with (r lΩ) in Eq.(1.82), and the steady-state Boltzmann equation can be rewritten as d  ϕðr  lΩ, E, ΩÞ + Σ t ϕðr  lΩ, E, ΩÞ ¼ Qðr  lΩ, E, ΩÞ dl

l0 P

W r–l

l

W

r

Fig. 1.11 The trajectories of neutron along Ω direction.

(1.83)

Neutron transport equation

21

in the equation, Q(r  lΩ, E, Ω) can be obtained by using Eq. (1.35) without considering the time variable. If the source term Q(r  lΩ, E, Ω) is assumed to be known, then Eq. (1.83) would be a first-order linear ordinary differential equation about ϕ(r  lΩ, E, Ω). With the boundary condition known as ϕ(r  l0Ω, E, Ω), it is easy to find its solution as  ð l0  0 0 ϕðr  lΩ, E, ΩÞ ¼ ϕðr  l0 Ω, E, ΩÞexp  Σ t ðr  l Ω, EÞdl l

ð l0 +

" ð0 # l 0 00 00 Qðr  l Ω, E, ΩÞ exp  Σ t ðr  l Ω, EÞdl dl0

l

(1.84)

l

where l0 is the value of l on the boundary Γ. Setting l ¼ 0 can yield  ð l0  ϕðr, E, ΩÞ ¼ ϕðr  l0 Ω, E, ΩÞ exp  Σ t ðr  l0 Ω, EÞdl0 0

ð l0 +

" ð0 # l Qðr  l0 Ω, E, ΩÞ exp  Σ t ðr  l00 Ω, EÞdl00 dl0

0

(1.85)

0

If the medium is infinite, the upper limit l0 can be replaced by ∞ while the incoming angular flux can be set as 0. It leads to ϕðr, E, ΩÞ ¼

ð∞

 ðl  0 0 Qðr  l Ω, E, ΩÞexp  Σ t ðr  l Ω, EÞdl dl 0

0

(1.86)

0

This is the integral form transport equation. It is mathematically equivalent to Eq. (1.38). In the case of isotropic source and isotropic scattering, the source term Q can be simplified as Qðr 0 , E, ΩÞ ¼

1 Qðr 0 , EÞ 4π

(1.87)

Substituting Eq. (1.87) into Eq. (1.84) and integrating the equation on Ω yield ð

Qðr 0 , EÞ

ϕðr, EÞ ¼

exp ½τðE, r 0 ! r Þ

dv 4π jr 0  r j2    ð  r  rs r  r s exp ½τðE, rs ! r Þ  ns ϕ r s , E, + ds jr  r s j jr  r s j2 S jr  r s j V

(1.88)

where τðE, r 0 ! r Þ ¼

ð jrr0 j 0

Σ t ðE0 , l0 Þdl0

(1.89)

22

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

In Eq. (1.89), the integral of l0 is along the path of r0 ! r, τ(E, r0 ! r), which is called straight “optical distance” from point r0 to point r, that is, the distance measured by mean free path λt as units. When Σ t is a constant, τ is equal to j r0 ! rj/λt. Like Boltzmann equation, Eq. (1.88) has the physical meaning of neutron balance. In fact, Q(r0 , E) in Eq. (1.87) is the number of neutrons with energy E produced in unit time at point r0 , and neutrons with (4π j r  r0 j2)1 move toward r. exp[ τ(E, r0 ! r)]0 is the weakening factor. Thus, the integrand in Eq. (1.85) is the contribution of the neutrons produced at point r0 to point r, while Eq. (1.88) means that the total neutron flux is the sum of contributions of all neutron fluxes produced at point r0 of the whole space to point r. From the previous discussion, we can see that both the transport equation (1.38) and Eq. (1.85) have been derived from the principle of neutron conservation. The difference is that Boltzmann equation (1.38) is based on the neutron conservation principle in a specific phase element, while the integral transport equation is based on the neutron conservation principle in the whole system space. Generally, the transport equation in integral form is less used, because the approximate or numerical solution of differential equation is much easier to be obtained than that of integral equation. However, for some reactor physics problems, the integral form of equation is more convenient, and more accurate results can be easily obtained only by less computational cost. For example, integral form of transport equation is widely used in the study of the calculations of neutron absorption in heterogeneous media, the calculation of energy spectra of fuel elements or assemblies based on the collision probability method or transmission probability method.

1.6

Brief introduction to the numerical solution to NTE

1.6.1 Eigenvalue problem and source iteration method There are two different cases for the steady-state neutron transport equation. The first one is in a nonbreeding medium (or subcritical) system, solving steady-state distribution of the neutron flux due to the external neutron source. This type of problem is often encountered in shielding calculations. At this time, the neutron transport equation Ω  rϕ + Σ t ðr, EÞϕ ¼

ð∞ ð 0

Σ s ðr, E0 Þf ðr, E0 ! E, Ω0 ! ΩÞϕðr, E0 , Ω0 ÞdE0 dΩ0

4π

+ Qf ðr, E, ΩÞ + Sðr, E, ΩÞ (1.90) is a nonhomogeneous equation under steady-state conditions. Under the known external source distribution and given boundary conditions, the unique solution can be obtained.

Neutron transport equation

23

The other case is to discuss the critical problem of the fission system in the absence of an external neutron source. In this case, the neutron transport equation (1.90) can be written as an operator Lϕ ¼ Fϕ

(1.91)

where ϕ ¼ ϕ(r, E, Ω), operators L and F are expressed as Lϕ ¼ ðΩð rð + Σ t Þϕ 

∞

0

Fϕ ¼

χ ðEÞ 4π

Ω0

ð∞ 0

Σ s f ðr, E0 ! E, Ω0 ! ΩÞϕðr, E0 , Ω0 ÞdE0 dΩ0

dE0

ð Ω0

vΣ f ðr, E0 Þϕðr, E0 , Ω0 ÞdΩ0

(1.92)

(1.93)

Eq. (1.91) is a linear homogeneous equation for ϕ. It is mathematically known that it has a nonzero solution only under certain conditions, and its solution can be attributed to the following eigenvalue problems: Lϕ ¼ λFϕ

(1.94)

where λ is called the eigenvalue of the equation, and the corresponding solution ϕ is called the eigenfunction. It is mathematically possible to prove the existence of these sets of eigenvalues {λn}, λ1 < λ2 < ⋯. In the critical or steady-state situation, only the first eigenvalue is meaningful, and the neutron flux distribution in the system is determined by its corresponding eigenfunction. The aim of the critical calculation of the reactor is to find the first eigenvalue and the corresponding eigenfunction that make Eq. (1.94) have a nonzero solution. The above situation can be explained with the following physical fact. For a given system with parameters Σ s , vΣ f , etc. (i.e., given material composition and characteristics), the system can only reach a critical state under a certain geometric size. Correspondingly, certain geometric properties require certain material properties. Thus, for any given system, Eq. (1.91) generally has no solution. However, this can be physically adjusted by introducing a parameter to achieve a critical status. That is, the number of neutrons released per fission is divided by a parameter k. By changing the value of k one can make the system critical. If the original problem is subcritical, dividing it by a positive k value 1 can also make the system critical by changing the k value. Therefore, the problem of the solution of Eq. (1.91) can be reduced to the solution of the following equation, that is, 1 Lϕ ¼ Fϕ k

(1.95)

24

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Compared with Eq. (1.94), it is obvious that k ¼ 1λ in the formula. As can be seen from the physical concept introduced by k it is the effective multiplication factor of the system. Eq. (1.95) is the basic equation for us to study the critical problem of reactor systems in the future. It is very difficult to directly find the eigenvalue k from Eq. (1.95). In general, a more effective method is to apply the source iteration method. It is especially suitable for numerical calculations. The calculation steps are as follows: first we arbitrarily give a distribution source or ϕ(r, E, Ω) ð0Þ

Q ðr Þ ¼

ð∞ dE 0

0

ð Ω0

vðE0 ÞΣ f ϕðr, E0 , Ω0 ÞdΩ0

(1.96)

For example, let Q(r) ¼ Q(0)(r), substitute Q(0)(r) into the right-hand side of Eq. (1.91) and solve the neutron flux ϕ(1)(r, E, Ω), and substitute Q(0)(r) into Eq. (1.95) to find the new distribution source Q(1)(r). Then use Q(1)(r) to resolve Eq. (1.91) to find a new solution ϕ(2)(r, E, Ω)⋯⋯, and so on. For example, for the Nth time iteration LϕðnÞ ¼ Fϕðn1Þ ¼

χ ðEÞ ðn1Þ Q ðr Þ 4π

(1.97)

This iterative process continues, and it is expected that Q(n)(r) and Φ(n) will increase in successive iterations for supercritical systems, and will decrease for subcritical systems, and will tend to be constant for critical systems. But in either case, the ratio of ϕ or Q(r) for two consecutive iterations after enough iterations will approach a constant independent of r, Ω, and E. From the physical meaning of k, it should be the neutron multiplication factor k, that is, QðnÞ ðr Þ n!∞ Qðn1Þ ðr Þ

k ¼ lim

(1.98)

In the actual calculation, the iterative process is terminated when the difference between the k values obtained after the last two iterations is within the predefined acceptance criteria. Here, we have physically demonstrated the convergence of the iterative process. Convergence of the iterative process can also be proved mathematically, but it is beyond the scope of this book. The source iteration method, also known as the power iteration method, is a commonly used and effective method for solving the eigenvalues and neutron flux distributions of Eqs. (1.91) in reactor physical numerical calculations, and is used in the later chapters of this book.

1.6.2 Overview of approximate solution for neutron transport equation The neutron transport equation is a differential-integral equation containing seven independent variables including spatial coordinate r(x, y, z), energy E, neutron flying direction Ω(θ, φ), and time t. Even in the case of steady state, due to the

Neutron transport equation

25

complexity of geometry and structure in practical problems, considering the large number of details of the cross section of the nuclei of various materials (fissionable and nonfissionable) with energy, it is impossible to accurately solve this equation, except for very simple or simplified cases. Therefore, in the actual calculation problem, this equation is usually solved by some approximation methods. Among all the approximation methods, the numerical discrete method is the most important and effective method because it can quickly find the numerical solution of the required precision using a computer. Therefore, the numerical method and its corresponding computer software have become an indispensable method and means for the current reactor physical design. As with other physical or engineering problems, the methods for studying neutron transport problems can be classified into two categories. The method in the first category is called the deterministic method. In this method, a mathematical model established according to the physical properties of the problem can be represented by one or a set of determined mathematical physics equations. For example, based on the neutron transport equation (1.38) established earlier, mathematical methods are used to find its exact or approximate solution. The method in the second category is called the stochastic or Monte Carlo method, also known as the nondeterministic method, which is a numerical method based on the statistics (or probability) theory. In the Monte Carlo method, a series of random numbers are used to simulate the movement of neutrons in the medium, the history of each neutron is tracked, and the obtained information is analyzed. The Monte Carlo method has the ability to calculate any complex geometry domain and the neutron cross section with complex energy variations and to obtain accurate results. However, it requires huge computer time and memory, which are major obstacles to its widespread use. Therefore, the deterministic method is mainly applied in the physical design of reactors. However, in some reactor physics fields, such as shielding calculations, the Monte Carlo method has proven to be very useful [3]. The discrete and approximate methods of the variables in the numerical solution of the neutron transport equation are discussed as following. (1) Discretization of energy variables In the neutron transport problem, the neutron energy can vary continuously from 0 eV to tens or even hundreds of MeV. We note that in the transport equation, the energy appears as an integral variable only in the source term on the right-hand side of the equation, and on the left-hand side is a simple parametric variable. Therefore, the multigroup method is the most common and simplest discrete method. In the multigroup method, the neutron flux density distribution range (E0, 0) is divided into intervals, for example, G discrete energy intervals ΔEg : (E0, E1), ..., (Eg1, Eg), ..., (EG1, EG), each of which is referred to as an energy group (see Fig. 1.12). Integrating the neutron transport equation (1.38) (the steady state) at each energy interval ΔEg, the following equations are obtained: Ω  rϕg ðr, ΩÞ + Σ t, g ðrÞϕg ðr, ΩÞ G ð X Σ g0 g ðr, Ω0 ! ΩÞϕg0 ðr, Ω0 ÞdΩ0 + Qg ðr, ΩÞ g ¼ 1, ⋯,G ¼ g0 ¼1

(1.99)

26

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

ΔEg = Eg – 1 –Eg – 1

EG = 0

Eg

EG – 1

E2

Eg – 1

The G group

E1

E0

The first group

The g group

Fig. 1.12 Grouping diagram.

where ð ϕg ðr, ΩÞ≡

ΔEg

ϕðr, E, ΩÞdE

(1.100)

ð ΔEg

Σ t, g ðr, ΩÞ≡

Σ t ðr, EÞϕðr, E, ΩÞdE (1.101)

ϕg ðr, ΩÞ ð

0

Σ g0 g ðr, Ω ! ΩÞ≡

ð dE ΔEg

ΔEg0

Σ s ðr, E0 ! E, Ω0 ! ΩÞϕðr, E0 , Ω0 ÞdE0 ϕg ðr, Ω0 Þ

(1.102)

ð Qg ðr, ΩÞ≡

ΔEg

Qðr, E, ΩÞdE

(1.103)

Here ϕg(r, Ω) is called the group flux, which is the sum of the flux in the energy interval ΔEg. In this way, by means of multigroup approximation, the original equation (1.38) with continuous independent variable E is discretized into the solution of G-group equation (1.99) with ϕg(r, Ω), and the number of energy groups, G, is determined by the nature and accuracy of the problem studied. The nature of the problem and the accuracy requirements determine the number of energy groups used in the calculation, which can be from 1 to 103. Σ s, g ðr, ΩÞ and Σ g0 g ðr, Ω0 ! ΩÞ are called group constants, and these group constants must be determined before solving the multigroup equations (1.99). However, from definition of the group constant Eqs. (1.101), (1.102), we find that to calculate the group constant, we must first know the neutron flux distribution ϕ(r, E, Ω), and it is exactly what we want. Therefore, strictly speaking, this is a nonlinear problem. In the actual calculation, an approximation method is usually used to solve the problem. That is, an approximation of the neutron flux density is obtained by some approximation methods or assumptions, and then it is substituted into the group constant expression to calculate the group constant. Finally, the obtained group constants are applied to solve the multigroup equations (1.99). Therefore, the accuracy of the multigroup method depends not only on the number of energy groups, but also on the approximate calculation of the group constants, especially in the few group calculations with large energy intervals.

Neutron transport equation

27

(2) Approximation of angular variable (a) spherical harmonic expansion approximation Among all the approximation methods used for the angle variable Ω in solving the neutron transport equation, the spherical harmonic approximation is the most widely used and the most famous method. The basic idea is to use a set of complete orthogonal spherical harmonic functions as expansion functions for some functions with variable Ω in the equation, for example, the neutron flux ϕ(r, E, Ω), the scattering function, etc., and take the N-orders of the expansion series (called PN approximation) ϕðr, E, ΩÞ ¼

N n X 2n + 1 X n¼0

4π

φn, m ðr, EÞYn, m ðΩ, θÞ

(1.104)

m¼n

Here Yn, m(θ, φ) ≡ Yn, m(Ω) is a spherical harmonic function, and φn, m(r, E) is a set of pending functions. In this way, the problem is transformed into a problem for solving a set of undetermined functions. This can be achieved by substituting Eq. (1.104) into transport equation, using the orthogonal properties of the spherical harmonic function and using the weight function method transform the original equation into a set of differential equations containing the coefficientφn, m(r, E), and then solve this system of equations, from which we can determine each of the coefficients in the expansion. In the spherical harmonic approximation, the simplest and most common case is N ¼ 1, called P1 approximation or diffusion approximation, which is the basis of reactor physical calculations, especially for large reactor calculations. (b) Discrete ordinate method Another common approximation method is the numerical method of directly discretizing the variable Ω. That is, solving the transport equation only for a selected number of discrete directions Ωm . Now, the angular variable Ωm is only one of parameters in the equation. After finding ϕðr, E, Ωm Þ from the transport equation, the integral of direction Ω is approximated by numerical integral, for example, ð ϕðr, EÞ ¼

Ω

ϕðr, E, ΩÞdΩ ¼

M X

ωm ϕðr, E, Ωm Þ

(1.105)

m¼1

where ωm is the quadrature factor. The number of discrete directions depends on the accuracy of the calculation. This is the so-called discrete ordinate method, which is commonly referred as the SN method. Here, the subscript N indicates the number of discrete points of the vector Ω in a certain direction. When N is relatively large, for example, N > 8, the SN method can achieve high accuracy. It is currently an effective numerical method for solving neutron transport equations. In the reactor shielding calculation, the SN method is widely used because the neutron angular flux distribution is highly anisotropy. In some fuel cell and assembly calculations, the SN method is also used because the neutron flux distribution is highly heterogeneous. (c) Other methods For some problems with small size but strong nonuniformity, such as the calculation of the fuel cell and fuel assembly, we usually use the integral transport theory method. It starts from the integral form of the neutron transport equation, when the source term and neutron scattering term are isotropic, the equation is integrated over Ω. In this way, the

28

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

variable Ω is eliminated and an integral transport equation (1.67) for ϕ(r, E) independent of the direction Ω of neutron motion is obtained. It is much more accurate than P1 approximation or diffusion theory, and its calculation is much simpler than SN method. On this basis, the well-known collision probability method (CPM) and transmisssion probability method (TPM) are developed, which are widely used in reactor physics calculation. The method of characteristics (MOC) is also widely used. The MOC simultaneously processes angular variable (Ω) and spatial variable (r) in the neutron transport equation, starting from the differential form of the neutron transport equation. By integrating along the trajectory of the neutron transport (called the characteristic line), the neutron transport equation is transformed into a set of characteristic line equations, and the whole neutron transport equation of computational geometry is solved by solving a large number of parallel characteristic line equations. (3) Spatial variable discretization Among the methods for approximating spatial variables, traditional discrete numerical methods, especially finite difference methods, are the most common and effective methods. Finite difference method is widely used for numerical solutions of neutron transport equations and neutron diffusion equations. It has a relatively complete theoretical basis, and currently, a couple of computer codes based on finite differential method are applied to reactor physical design in the field of nuclear engineering. The main disadvantage of the finite differential method is that the mesh must be small in order to guarantee a certain degree of computational accuracy. Therefore, it takes up a lot of storage and computation time, especially for multidimensional problems. For example, for 3D numerical analysis of pressurized water reactors, tens of millions of points and 10 h of CPU calculation time are required. Therefore, for engineering problems that require repeated solutions (such as fuel management, etc.), 3D finite differential method has little engineering values. To overcome these shortcomings, many coarse mesh or nodal methods were developed in the mid-1970s. The neutron flux density in the node is expanded by a higher order polynomial to achieve higher accuracy over a wider node. Currently, nodal methods, such as the node expansion method (NEM) and the node Green function method (NGFM), have been successfully developed and widely used in the calculation of the neutron diffusion problem of the nuclear reactors. Since the 1950s, the finite element method has been widely used in the numerical solution of engineering problems. Essentially, it is a discretization method based on the variational principle, which uses a piecewise polynomial to approximate the solution of the problem. In 1970s the finite element method began to be applied to the numerical solution of the neutron diffusion equation and the neutron transport equation and good results were achieved. In the approximation of spatial variables, it is often possible to perform dimensionality reduction based on the specific circumstances of the problem. For example, in the fuel cell calculation, it is common to reduce the original two-dimensional or three-dimensional problem to the one-dimensional cylinder problem. In addition, mathematical methods such as flux synthesis can be used to transform a three-dimensional problem into a twodimensional or one-dimensional problems. Table 1.2 lists some of the approximate methods commonly used for variables r, Ω, and E. Obviously, these methods can be reasonably selected and matched according to the specific situation of the problem and the requirements of the calculation. For example, the commonly used two-group diffusion approximate finite difference method, the two-group diffusion approximate finite element method, the multigroup discrete coordinate (SN) finite difference method, etc.

Neutron transport equation

29

Table 1.2 Approximate solution of neutron transport equation. Independent variable Approximate method

1.7

E

r

Ω

Group approximation Single group Double group Multigroup

Finite difference method Coarse grid method Nodal expansion method Analytical nodal method Green function nodal method Finite element method Dimensionality reduction method Comprehensive flux method

Spherical harmonic approximation method Diffusion approximation PN approximation Discrete coordinate method(SN) Integral transport method Collision probability method Surface flow method Characteristic line method

The adjoint transport equation

1.7.1 Adjoint operator First, a brief introduction is given to the definition of the adjoint operator. Let the functions ϕ(p) and ϕ∗(p) be defined in the domain G(p), where p is the point on the phase space r  E  Ω, then the inner product of the defined functions ϕ(p) and ϕ∗(p) is ð ðϕ, ϕ∗ Þ ¼

ϕϕ∗ dp

(1.106)

G

An operator L is provided on the function ϕ(p), which is any function defined on G of the function set {ϕ} that satisfies the continuous condition and the boundary condition. At the same time, another operator L∗ is set, which acts on any function ϕ∗ in the continuous function set {ϕ∗} (the boundary conditions satisfied by ϕ and ϕ∗ can be different), if Lϕ and L∗ ϕ∗ satisfy the following condition: ðϕ∗ , LϕÞ ¼ ðϕ, L∗ ϕ∗ Þ

(1.107)

then we call L∗ an adjoint operator of L. If L ¼ L∗, then L is called the self-adjoint operator. Suppose there is an equation Lϕ ¼ 0

(1.108)

30

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

If the adjoint operator of L is L∗, the solution ϕ∗ of the equation L ∗ ϕ∗ ¼ 0

(1.109)

and the solution ϕ of Eq. (1.108) clearly satisfy (1.107). We call Eq. (1.109) the adjoint equation (1.108), and its solution ϕ∗ is called the adjoint function of ϕ. For the neutron transport equation of the reactor, the operator L and its adjoint operator L∗ can usually be expressed as L ¼ M + λF

(1.110)

L∗ ¼ M∗ + λF∗

(1.111)

where λ is the eigenvalue, the operators M and M∗, F and F∗ are also adjoint to each other, which means that they all satisfy Eq. (1.107).

1.7.2 Adjoint neutron transport equation In a steady state, without external neutron source, the neutron transport equation in the reactor can be written in the following operator form: Lϕ ¼ λFϕ

(1.112)

where λ ¼ 1/k is the eigenvalue, operators L and F are as follows: Lϕ ¼ ðΩð rð + Σ t Þϕ 

∞

0

Fϕ ¼

χ ð EÞ 4π

Ω

ð∞

0

Σ s ðr, E0 Þf ðE0 ! E, Ω0 ! ΩÞϕðr, E0 , Ω0 ÞdE0 dΩ0

dE0

ð 0

Ω

0

νΣ f ðr, E0 Þϕðr, E0 , Ω0 ÞdΩ0

(1.113)

(1.114)

Now let us find the adjoint operators L∗ and F∗ of L and F. Observe the first item in Eq. (1.113) ð ð ð ðϕ∗ , Ω  rϕÞ ¼ dE dΩ ϕ  Ω  rϕdr ð  ∗ ∗ ¼ dE dΩ r  ðΩϕϕ Þ  ϕΩ  rϕ dr ð

ð

(1.115)

According to the Gauss formula ð

ð div ðΩϕϕ∗ ÞdV ¼ V

ðΩ  nÞϕϕ∗ dS S

(1.116)

Neutron transport equation

31

Considering the boundary condition of ϕ(r, E, Ω) on the outer surface, if ϕ∗(r, E, Ω) satisfies the following boundary conditions ϕ∗ ðr, E, ΩÞ ¼ 0,rΓ and ðΩ  nÞ > 0

(1.117)

then the integral of Eq. (1.116) will be equal to zero. Then ðϕ∗ , ΩrϕÞ ¼ ðϕ,  Ωrϕ∗ Þ

(1.118)

For the second item, it is obvious that there are ðϕ∗ , Σ t ϕÞ ¼ ðϕ, Σ t ϕ∗ Þ

(1.119)

For the third term, applying the exchange integral order one can prove that

ÐÐ ϕ∗ , Σ s ðr, E0 Þf ðE0 ! E, Ω0 ! ΩÞϕðr, E0 , Ω0 ÞdE0 dΩ0 ð ð ð ðð ∗ ¼ dr dE dΩdϕ ðr, E, ΩÞ Σ s ðr, E0 Þf ðE0 ! E, Ω0 ! ΩÞϕðr, E0 , Ω0 ÞdE0 dΩ0 ð ð ð ðð ¼ dr dE0 dΩ0 ϕðr, E0 , Ω0 ÞΣ s ðr, E0 Þ f ðE0 ! E, Ω0 ! ΩÞϕ∗ ðr, E, ΩÞdEdΩ ðð ð ð ð ¼ dr dE dΩϕðr, E, ΩÞΣ s ðr, EÞ f ðE ! E0 , Ω ! Ω0 Þϕ∗ ðr, E0 , Ω0 ÞdE0 dΩ0

ÐÐ ¼ ϕ, Σ s ðr, EÞ f ðE ! E0 , Ω ! Ω0 Þϕ∗ ðr, E0 , Ω0 ÞdE0 dΩ0 (1.120)

Thus, according to Eqs. (1.118)–(1.120) and the definition of the adjoint operator, the adjoint operator L∗ of L is obtained as L∗ ϕ ¼ Ωrϕ∗ + Σ t ϕ∗ ð∞ ð  Σ s ðr, EÞ ϕ∗ ðr, E0 , Ω0 Þf ðE ! E0 , Ω ! Ω0 ÞdE0 dΩ0

(1.121)

Ω0

0

Similarly, the operator F has the following formula: ð ðϕ∗ , FϕÞ ¼

ð dr dE



ð dΩϕ∗ ðr, E, ΩÞ

χ ð EÞ 4π

ðð

0

0

0

νΣ f ðr, E , Ω ÞdE dΩ

0



ð ðð ð ð χ ð EÞ ∗ ϕ ðr, E, ΩÞdEdΩ ¼ dr dE0 νΣ f ðr, E0 Þϕðr, E0 , Ω0 ÞdΩ0 4π   ðð νΣ f ðr, EÞ χ ðE0 Þϕ∗ ðr, E0 , Ω0 ÞdE0 dΩ0 ¼ ϕ, 4π (1.122)

32

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Then the adjoint operator F∗ of F is F∗ ϕ ¼

νΣ f ðr, EÞ 4π

ð∞ ð Ω0

0

χ ðE0 Þϕ∗ ðr, E0 , Ω0 ÞdE0 dΩ0

(1.123)

In this way, the adjoint neutron transport equation of the reactor is finally obtained. Ωrϕ∗ + Σ t ϕ∗ ¼ νΣ f ðr, EÞ + 4πk

ð∞ ð

ð∞ ð 0

Ω0

0

Ω

0

Σ s ðr, EÞf ðE ! E0 , Ω ! Ω0 Þϕ∗ ðr, E0 , Ω0 ÞdE0 dΩ0

χ ðE0 Þϕ∗ ðr, E0 , Ω0 ÞdE0 dΩ0 (1.124)

Satisfying boundary conditions on the outer surface Γ ϕ∗ ðr s , E, ΩÞ ¼ 0, ðΩ  nÞ > 0, rs Γ

(1.125)

Obviously, M∗ and F∗ are the adjoint operators of M and F, respectively.

1.7.3 Neutron importance function It can be proved that the previously derived neutron adjoint flux density ϕ∗(r, E, Ω) and adjoint transport equation (1.124) have clear physical meanings. In a critical reactor, the reactor has a stable power level due to fission of various neutrons in different states (r, E, Ω). Obviously, the contribution of neutrons at different positions r, with different energies E, and direction of motion Ω to the reactor power is different. The phase space (r, E, Ω) is located at a certain point r, and the contribution of the energy E and the neutron in the Ω direction to the stable power of the critical reactor is called the neutron importance function ϕ∗(r, E, Ω). Obviously, at the outer boundary Γ of the reactor, the emission of neutrons into the vacuum [(Ω  n) > 0] is unlikely to generate any power in the reactor, that is, ϕ∗ ðr, E, ΩÞ ¼ 0 r Γ and ðΩ  ns Þ > 0

(1.126)

According to the definition of neutron importance function, it can be seen that the neutron importance has a superposition nature. Assuming that the importance function of a neutron is ϕ∗(r, E, Ω), then n such state neutrons have the importance equal to nϕ∗(r, E, Ω). On the other hand, for critical reactors, the power remains stable. Therefore, the contribution of neutrons to stable power at a certain moment equals the contribution of all neutrons generated by these neutrons during later time, so the neutron importance is conserved. According to the principle of conservation of neutron importance, the neutron transports a small distance ds along the direction of its motion, and its importance should be balanced with the neutron importance produced and disappeared during this transport process. In the following, the equation of conservation of neutron

Neutron transport equation

33

importance, the neutron transport adjoint equation, is established based on the physical concept of conservation of neutron importance. Observe the neutron shown in Fig. 1.6 where at point r, energy is E and the direction of motion is Ω. Its neutron importance function is ϕ∗(r, E, Ω). These neutrons 0 move ds in the direction of Ω, and the  number  of neutrons arriving at r ¼ r + dsdΩ

point without collision is equal to n 1  ds λt , where λt ¼ 1=Σ t is the total mean free

path. Thus, when the r0 point is reached, the changed neutron importance function is equal to n½ð1  Σ t dsÞϕ∗ ðr + dsΩ, E, ΩÞ  ϕ∗ ðr, E, ΩÞ

(1.127)

L ¼ n½ϕ∗ ðr + dsΩ, E, ΩÞ  ϕ∗ ðr, E, ΩÞ

(1.128)

Let

It is the increment of the importance of the neutron when transporting from r point to r0 point. R ¼ nΣ t dsϕ∗ ðr + dsΩ, E, ΩÞ

(1.129)

is the loss of neutron importance due to collisions on the ds path. It is associated with neutron capture and neutron transport to other states. Among these collision neutrons, nΣ t ds neutrons are transported to other states by scattering collisions (changing direction and energy). Let the scattering function be f(E ! E0 , Ω ! Ω0 ), then the neutron importance function of this partially scattered neutron is I ¼ ndsΣ s ðr, EÞ

ð∞ ð

f ðE ! E0 , Ω ! Ω0 Þϕ∗ ðr, E, ΩÞdE0 dΩ0

(1.130)

0

Assuming that the angular distribution of fission neutrons is isotropic, then the importance function of nνΣ f ds neutrons due to fission is nνΣ f ds Q¼ 4π

ð∞ ð 0

Ω0

χ ðE0 Þϕ∗ ðr, E0 , Ω0 ÞdE0 dΩ0

(1.131)

If the reactor is critical, there should be a neutron balance equation LR+I +Q¼0

(1.132)

If the reactor is not critical, then Q can be divided by k, that is, LR+I +

Q ¼0 k

Multiply the formula by nds and notice that

(1.133)

34

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

lim

ds!0

ϕ∗ ðr + dsΩ, E, ΩÞ  ϕ∗ ðr, E, ΩÞ ¼ Ω  rϕ∗ ðr, E, ΩÞ ds

(1.134)

Substitute the relevant items Ω  rϕ∗ ðr, E, ΩÞ + Σ t ϕ∗ ðr, E, ΩÞ ¼

ð∞ ð 0

+

Ω

0

Σ s ðr, EÞf ðE ! E0 ; Ω ! Ω0 Þϕ∗ ðr, E0 , Ω0 ÞdE0 dΩ0

νΣ f ðr, EÞ 4πk

ð∞ ð 0

Ω0

(1.135)

χ ðE0 Þϕ∗ ðr, E0 , Ω0 ÞdE0 dΩ0

ϕ∗ ðr s , E, ΩÞ ¼ 0, ðΩ  nÞ > 0, rs Γ

(1.136)

It can be seen Eqs. (1.124) and (1.125) are neutron transport adjoint equations. It can be proved that under P1 approximation, the adjoint equation of multigroup neutron diffusion also has the meaning of conservation of the neutron importance function of the G-group. Thus, from a physical point of view, the reactor adjoint equation is an equation that characterizes the conservation of neutron importance, and the adjoint flux distribution is also a function distribution of neutron importance.

References [1] E.E. Lewis, W.F. Miller, Computational Methods of Neutron Transport, John Wiley & Sons, 1984. [2] A. Hebert, Applied Reactor Physics, Presses International Polytechnique, 2009. [3] Y.Y. Azmy, E. Sartori, Nuclear Computational Science—A Century in Review, Springer, 2010.

Collision probability method Yunzhao Li Xi’an Jiaotong University, Xi’an, People’s Republic of China

2.1

2

Integral-form neutron transport equation

Integral-form neutron transport equation (NTE) has been introduced in Section 1.5. Together with the fission source treatment in Section 1.6.1 and the multigroup approximation for the neutron energy variable in Section 1.6.2, it can be written as follows: Z

l0

ϕg ðr, ΩÞ ¼

Qg ðr  l0 Ω, ΩÞeg ðr  l0 Ω ! r Þdl0

0

+ ϕg ðr  l0 Ω, ΩÞeg ðr  l0 Ω ! r Þ Qg ðr, ΩÞ ¼

XZ g0

Σ s, gg0 ðr, Ω0 ! ΩÞϕg0 ðr, Ω0 ÞdΩ0 + Sg ðr, ΩÞ

(2.1) (2.2)

4π

where Sg(r, Ω) is the neutron source in energy group g caused by fission and/or external source; weakening factor or contribution coefficient is defined as   eg ðr  l0 Ω ! r Þ ¼ exp τg ðr  l0 Ω ! r Þ  Z l0  ¼ exp  Σ t, g ðr  l0 ΩÞdl0

(2.3)

0

Accordingly, the neutron angular flux ϕg(r, Ω) can be obtained from two aspects. Firstly, it is the neutron source of Qg(r  l0 Ω, Ω), where l0  [0, l0) refers to the distance from the source location to the targeted location and l0 stands for the distance from the targeted location to the domain boundary along the direction of Ω. Secondly, it is the incoming angular flux at the boundary ϕg(r  l0Ω, Ω). Both these two contributions have weakened exponentially due to the increase in its distance to the targeted location. With isotropic source and isotropic scattering approximations, Qg ðr, ΩÞ ¼

1 1X 1 Q g ðr Þ ¼ Σ s, gg0 ðr ÞΦg0 ðr Þ + Sg ðr Þ 4π 4π g0 4π

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation. https://doi.org/10.1016/B978-0-12-818221-5.00001-5 Copyright © 2021 Elsevier Ltd. All rights reserved.

(2.4)

36

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

the neutron scalar flux is also contributed by two components Z Φ g ðr Þ ¼

e g ðr 0 ! r Þ 0 Q g ðr 0 Þ dv 4πjr0  r j2  V   Z r  rs r  r s e g ðr s ! r Þ  ns ϕg r s , + ds jr  r s j jr  rs j2 S jr  r s j

(2.5)

 s where Φg(r) represents the scalar flux of energy group g, ϕg r s , jrr rrs j is the incoming angular flux of energy group g defined by the corresponding boundary condition. As a close integral-differential equation system, boundary conditions would be required. Here, the albedo boundary condition is taken as an illustrative example.



 

 φg ðr s , ΩÞ ¼ βs φg r s , 1  2 nTs Ω Ω , nTs Ω < 0

(2.6)

where βs is the albedo on surface s and ns is the unit outward normal vector of surface s.

2.2

Collision probability for solving NTE

To discretize the continuous neutron transport equation, the entire problem domain can be divided into I nonoverlapped regions as i ¼ 1 I with its region volume as Vi. Within each region, materials or cross sections are all constants, which can be satisfied as long as the region mesh is smaller or equal to the size of the material mesh of the problem. Correspondingly, the entire domain surface is also divided into M nonoverlapped region surfaces as m ¼ 1  M with its surface area as Sm. Multiplying Eq. (2.5) by the total cross section of the i0 th region, integrating the corresponding equation over the i0 th region, separating the volume integration over the entire domain in the first right-hand side term into the sum of integrations over each region, separating the surficial integration over the entire domain surface into the sum of integrations over each region surface, introducing the flat source approximation in each region and the flat incoming angular flux on each outer boundary surface yield the following expression [1]: Σ t,g, i Φg, i Vi ¼

I X

Qg, i0 Pg, ii0 Vi0 +

i0 ¼1

M X

Jg,m Pg, im Sm

(2.7)

m¼1

where region-wise averaged neutron flux density is Φg, i ¼

1 Vi

Z Φg ðr Þdv Vi

(2.8)

Collision probability method

37

Region-wise averaged neutron source density is Qg, i ¼

1 Vi

Z Qg ðr Þdv

(2.9)

Vi

It can be obtained by using the relationship between neutron source and neutron flux as in Eq. (2.4). Region boundary surface averaged neutron incoming partial current is Jg,m ¼

1 Sm

Z Z Sm nΩ 0 ηm > 0

(7.16)

The difference equation is solved in discrete angular directions, so the mesh boundary j j fluxes ψ i1/2, and ψ i1/2, are equivalent to the incident flux or the outgoing flux for a m m given mesh and direction Ωm. Since the incident flux is usually known, the scanning calculation for all grids can be done either according to boundary conditions or according to angular flux continuous conditions of adjacent grids. In practice, firstly divide {Ωm} into four groups according to the quadrant distribution of (μ, η) (as shown in Fig. 7.4), then look at a typical grid (Fig. 7.5). Obviously, in a certain direction Ωm of quadrant I, the grid boundaries C and D are incident boundaries. Then j ψ i1/2, and ψ i,m j1/2 are incident fluxes, so one can calculate ψ ijm by using Eq. (7.16), m j and then calculate the outgoing fluxes ψ i+1/2, and ψ i,m j+1/2 according to the diamond m difference formula. These values can be used as the incident flux for the next grid, and then complete the calculations for all grids in the Ωm direction. Fig. 7.6 describes the scanning process for a typical structured differential mesh, assuming that the initial flux in the current direction is already known. h

Fig. 7.4 Quadrant of two-dimensional angular geometry.

II

I ^ Wm

m

III

IV

The discrete ordinate method

A

yj + 1/2

D

yj – 1/2

173

Fig. 7.5 Spatial grid.

B

(i, j)

C xi – 1/2

xi + 1/2

xi

^ W

y

y

Sweep 1 (III) B

5 2

1

B

B

2

B

4

3

3

3 3 2

1

y

4

x

5 2

2

4

B

B 3

8

8 ^ B W

B 7

7

8

8 6

5 B 5 B

x

Sweep 4 (I)

7

5

4

y

Sweep 4 (II) 7

2

2

1

1

B

2

1

1 1 2 1 2

Sweep 2 (IV)

B

2 3 2

^ W

5 B

Fig. 7.6 Spatial angle scanning process.

^ W x

6

B 6

6 B

B

x

174

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

7.2

SN finite element method for unstructured geometry

The finite element method applied to transport equation can be divided into two categories: one is the finite element method for the first-order Boltzmann equation, and the other is the finite element method for the second-order neutron transport equation. The finite element for second-order equations can be based on the traditional Galerkin variation or Ritz variation, the typical methods include the finite element discretization for the second-order even-parity equation [5] and the finite element discretization based on the second-order SAAF equation [6]. However, because the second-order neutron transport equation has a cross section in the denominator, it cannot accurately describe the vacuum medium (the cross section is zero), which is the disadvantage of solving transport equation with vacuum region. The finite element methods for first-order transport equation mainly have discontinuous finite element method (DFEM) [7, 8] and least-squares finite element method (LSFEM) [9]. The main disadvantage of DFEM is that due to the precise processing of transport equation for the heterogeneous medium, many unknowns would be added in the multidimensional case, thereby increasing the computational complexity. LSFEM can avoid this issue, and compared with the traditional Galerkin variation method, the symmetric stiffness matrix produced by LSFEM is convenient for the storage of one-dimensional array and fast iterative method. This section discusses LSFEM based on the first-order Boltzmann equation.

7.2.1 Least-squares variation In three-dimensional Cartesian coordinates, Ω  rψ g ðr, ΩÞ ¼ Ωx

∂ψ g ðr, ΩÞ ∂ψ g ðr, ΩÞ ∂ψ g ðr, ΩÞ + Ωy + Ωz ∂x ∂y ∂z

(7.17)

∂ψ m,g ðr Þ ∂ψ m,g ðr Þ ∂ψ m,g ðr Þ + ηm + ξm ∂x ∂y ∂z

(7.18)

For SN method, h

Ω  rψ g ðr, ΩÞ

i m

¼ μm

Only the case in which the scattering source term is isotropic is discussed here: μm

∂ψ m,g ðr Þ ∂ψ m,g ðr Þ ∂ψ m,g ðr Þ + ηm + ξm + Σt,g ðr Þψ m,g ðr Þ ∂x ∂y ∂z G X M X 1 ωm0 ψ m0 ,g0 ðr Þ ¼ Σs, g0 g ðr Þ  4π g0 ¼1 m0 ¼1 +

G M X χg X υg0 Σf ,g0 ðr Þ  ωm0 ψ m0 ,g0 ðr Þ + Sm,g ðr Þ 4π g0 ¼1 m0 ¼1

(7.19)

The discrete ordinate method

175

If Eq. (7.19) is written in operator form: Lψ m,g ¼ F

(7.20)

where 8   ∂ ∂ 1 > > + η + Σ Σ L ¼ μ ð r Þ  ð r Þω > t,g s, gg m m > > ∂x m ∂y 4π > < G M X 1 X Σs, g0 g ðr Þ F¼  > > 4π g0 ¼1 > > m0 ¼ 1 > > : 0 m 6¼ mðg0 ¼ gÞ ωm0 ψ m0 , g0 ðr Þ +

G M X χg X υg0 Σf ,g0 ðr Þ  ωm0 ψ m0 ,g0 ðr Þ + Sm,g ðr Þ: 4π g0 ¼1 m0 ¼1

(7.21)

Then, the neutron transport equation can be expressed as follows: 

Lψ m,g ðx, yÞ ¼ F ðx, yÞR2 ψ m,g ðx, yÞ ¼ g ðx, yÞ∂R2 ^ nðx, yÞ  Ωm < 0

(7.22)

The least-squares minimization function of this problem is i2     ð h e e Lψ m,g  F dxdy min F ψ m, g , F ψ m,g ≔

ψ m, g V

(7.23)

R2

where  

V≔ vL2 R2 ; Bv ¼ g ðx, yÞ∂R2 ^ nðx, yÞ  Ωm < 0

(7.24)

Then, the least-squares variation problem can be written as follows: (

Find ψ m , g V s:t   ae ψ m,g , ψ 0 ¼ beðψ 0 Þ

8ψ 0 V

(7.25)

where 8     ð > > < ae ψ m,g , ψ 0 ¼ Lψ m,g , Lψ 0 ¼ Lψ m,g  Lψ 0 dxdy R2 ð > 0 0 > ¼ F  Lψ dxdy : beðψ Þ R2

(7.26)

176

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Therefore, the least-squares finite element variational form of neutron transport equation is written as    1 + Σt, g ðr Þ  Σs, gg ðr Þωm ψ m, g ðrÞ  ∂x ∂y ∂z 4π     0 0 0 ∂ψ ðrÞ ∂ψ ðr Þ ∂ψ ðr Þ 1 μm + ηm + ξm + Σt, g ðr Þ  Σs, gg ðr Þωm ψ 0 ðr Þ dxdydz ∂x ∂y ∂z 4π 8 > > > > > > > > G M G X X χg X ÐÐÐ< 1 Σs, g0 g ðr Þ υg0 Σf , g0 ðr Þ  ¼ ωm0 ψ m0 , g0 ðr Þ + > 4π 0 4π > > 0 ¼1 g0 ¼1 g ¼1 > m > > > > :   m0 6¼ m g0 ¼ g ÐÐÐ



μm

∂ψ m, g ðrÞ

M X



+ ηm

∂ψ m, g ðrÞ

+ ξm

∂ψ m, g ðrÞ

ωm0 ψ m0 , g0 ðr Þ + Sm, g ðrÞg

m0 ¼1

 μm

   ∂ψ 0 ðrÞ ∂ψ 0 ðr Þ ∂ψ 0 ðr Þ 1 + ηm + ξm + Σt, g ðr Þ  Σs, gg ðr Þωm ψ 0 ðr Þ dxdydz ∂x ∂y ∂z 4π (7.27)

7.2.2 Spatial discretization The spatial discretization is introduced by taking triangular unit as an example. In Fig. 7.7, all discrete points in the region are called nodes. All nodes in the region are numbered according to a certain rule, and these nodes are called global nodes. In each subdivision unit, the nodes also have corresponding sequential coding, which is called local coding. The polynomial function with a value of 1 at current node and 0 at the other nodes is called the basis function or coordinate function at this node, denoted as Ui(x, y). Therefore, there are n basis functions corresponding to n nodes, thus forming a basis function system {Ui}ni¼1, Ui ðxk , yk Þ ¼ δi,k i ,k ¼ 1,2, ⋯, n

(7.28)

where (xk, yk) is the coordinate of the kth node. The function space is linearly independent and can form a subspace called finite element subspace Wh. It can be proved that the neutron flux ψ m, g(x, y) and any function ψ 0(x, y) can be approximated by the basis function in the finite element subspace Wh when the size of the partition unit is sufficiently small, namely, ψ m,g ðx, yÞ 

n X i¼1

ψ i Ui ðx, yÞ

(7.29)

The discrete ordinate method

177 13 10

12

9

11

4

8 7

6

5

3 2

1

Fig. 7.7 Triangulation diagram.

ψ 0 ðx, yÞ 

n X

ψ 0j Uj ðx, yÞ

(7.30)

j¼1

where ψ i and ψ 0j are the real numbers, which are, respectively, called the approximation coordinate values of ψ m,g(x, y), and ψ 0(x, y). Substituting Eqs. (7.29), (7.30) into Eq. (7.25) yields ae

n X

ψ i Ui ðx, yÞ,

i¼1

n X

! ψ 0j Uj ðx, yÞ

¼ be

n X

! ψ 0j Uj ðx, yÞ

(7.31)

n X   be Uj ðx, yÞ ψ 0j

(7.32)

j¼1

j¼1

Namely, n X

ae

n X

j¼1

! ψ i Ui ðx, yÞ, Uj ðx, yÞ ψ 0j ¼

i¼1

j¼1

Because of the arbitrariness of ψ 0(x, y), ψ 0j is also arbitrary. Therefore, the coefficients corresponding to ψ 0j should be identical, that is, n X

    ψ i ae Ui ðx, yÞ, Uj ðx, yÞ ¼ be Uj ðx, yÞ j ¼ 1, 2, ⋯, n

(7.33)

i¼1

Let coefficient matrix K ¼ (Kij)nn, where   Kij ¼ ae Ui ðx, yÞ, Uj ðx, yÞ

(7.34)

Thus Eq. (7.31) can be written as a matrix Kψ ¼ F

(7.35)

178

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

where the right term is   F ¼ col beðU1 Þ, beðU2 Þ, ⋯, beðUn Þ

(7.36)

The unknown vector is   ψ ¼ col ψ 1, ψ 2, ⋯, ψ n

(7.37)

If K and F are determined, the vector ψ can be solved using Eq. (7.35), and the approximate solution of neutron flux can be obtained Let the coordinate of the kth node be (xk, yk), and the value of the neutron flux at the kth node can be obtained from the basis function property––Eq. (7.38), ψ m,g ðxk , yk Þ 

n X

ψ i Ui ðxk , yk Þ ¼ ψ k

(7.38)

i¼1

Therefore, the required vector ψ (or approximate coordinate) is the vector formed by the value of neutron flux ψ at each node, so the next step is to solve the geometric partition of the region and determine the basis function system {Ui}ni¼1. In order to construct the basis function on each node, the shape function on each unit must be constructed first. The polynomial whose value is equal to 1 at the node k of an element and 0 at other nodes is called the shape function at the node k of the element, denoted as Lk (x, y, z). The basis function here selects a first-order linear polynomial, three nodes must be specified on the triangular unit, and three vertices of the triangle, A1, A2, and A3, are selected as nodes, as shown in Fig. 7.8. The number of vertices of the triangular unit is called local coding. In order to standardize the shape functions on triangular unit with different positions and shapes, area coordinates are introduced. Set one arbitrary point P on triangular unit Tr, P and the triangular vertices form three small triangles, area is, respectively, Δ1,Δ2,Δ3, as shown in Fig. 7.8, set as λ1 ¼ Δ1 =Δ, λ2 ¼ Δ2 =Δ, λ3 ¼ Δ3 =Δ

(7.39)

Here Δ is the area of the triangular unit Tr, obviously, λ1 + λ2 + λ3 ¼ 1, 0  λi  1

(7.40)

And the rectangular coordinates (x, y) of point P correspond to (λ1, λ2, λ3) one by one, so (λ1, λ2, λ3) is called as the area coordinate or barycenter coordinate of point P. Then, the area coordinates of the three vertices of the triangular unit are, respectively, A1 ¼ ð1, 0, 0Þ, A2 ¼ ð0, 1, 0Þ, A3 ¼ ð0, 0, 1Þ

The discrete ordinate method

179

Fig. 7.8 Area coordinate diagram.

Let the coordinates of each vertex A1, A2, A3 of triangular unit Tr in the Cartesian coordinate system be, respectively, (x1, y1), (x2, y2), (x3, y3), then the relationship between the coordinates of point P and the area coordinates is 2 3 2 32 3 1 1 1 1 λ1 4 x 5 ¼ 4 x1 x2 x3 54 λ2 5 y y1 y2 y3 λ3

(7.41)

The coefficient matrix on the right-hand side of Eq. (7.41) is called the Jacobian transformation matrix of triangular unit. The relationship between (λ1, λ2, λ3) and (x, y) can be solved, 2

c1 2 3 6D 6 λ1 6 4 λ2 5 ¼ 6 c2 6D 6 λ3 4 c3 D

a1 D a2 D a3 D

3 b1 2 3 D7 7 1 7 b2 74 5 x D7 7 y 5 b3 D

(7.42)

where 8 a1 ¼ y 2  y 3 a2 ¼ y3  y1 a3 ¼ y 1  y 2 > > > < b ¼ ðx  x Þ b ¼ ðx  x Þ b ¼ ðx  x Þ 1 2 3 2 3 1 3 1 2 > c ¼ x y  x y c ¼ x y  x y c ¼ x y  x y 1 2 3 3 2 2 3 1 1 3 3 1 2 2 1 > > : D ¼ c1 + c2 + c3

(7.43)

Thus, there are three corresponding shape functions L1, L2, L3 for the three vertices corresponding to triangular unit Tr. According to the need of calculating precision,

180

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

shape functions of different orders can be selected. Here, only the first-order Lagrange basis function is introduced, as shown in the following formula: Lk ðλ1 , λ2 , λ3 Þ ¼ λk , k ¼ 1, 2,3

(7.44)

For three-dimensional problems, tetrahedral grids are generally selected. The shape function of first-order Lagrange is Lk ðλ1 , λ2 , λ3 , λ4 Þ ¼ λk , k ¼ 1,2, 3,4

(7.45)

Since the basis function system {Ui}ni¼1 needs to satisfy the condition that Ui is 1 at the ith node and 0 at other nodes, the basis function of all the ith node can be superimposed by the shape function of all elements in the influencing element set Ri (the element set with this node as the vertex) at this point, i.e., Ui ðx, y, zÞ ¼

X

LrkðiÞ ðx, y, zÞ=Mðx, y, zÞ i ¼ 1, 2, ⋯, n

(7.46)

rRi

where M (x, y, z) refers to the number of all elements containing the point (x, y, z), and r Lk(i ) refers to the shape function at the vertex in the set Ri where the local coding of the rth triangular unit is k (the total coding is i). Substitute the expression of the basis function (7.46) into the coefficient matrix (7.34). Since M (x, y, z) is only greater than or equal to 1 at the vertex or edge of the triangular unit, it has no contribution to the integral value of the entire integral region, so 8 < 0X   i , jnodes do not belong to the same triangle unit r r Ki, j ¼ e , L a L kðiÞ kð jÞ i, jnodes belong to the same triangle unit : rRi, j

(7.47) where Ri,j represents a set of triangular units with the nodes of the global coding i and j as the vertices. Similarly, the expression for the right term of Eq. (7.35) can be obtained using Eq. (7.21) and the characteristics of M (x, y, z)        F ¼ col be Lrkð1Þ , ⋯, be LrkðiÞ , ⋯, be LrkðnÞ

(7.48)

In the calculation of stiffness matrix and right vector, it is only necessary to know the shape function of triangular unit, not the specific expression of basis function, and these shape functions are the standard form of area coordinate expression, which is very convenient to calculate.

The discrete ordinate method

181

7.2.3 Treatment of boundary condition (1) Reflective boundary condition On reflective boundary, the neutron angular flux satisfies   ψ g ðx, y, z, ΩÞ ¼ ψ g x, y, z, Ω0

(7.49)

where Ω0 is the mirror reflective direction of Ω. Given the invariance of coordinate rotation in Cartesian coordinates, this section discusses only the case where the boundary is perpendicular to the x-axis. In the two-dimensional Cartisian coordinate, the reflective boundary is perpendicular to the x-axis, as shown in Fig. 7.9. Since the discrete ordinate method discretized the direction variable Ω into several directions Ωm, the reflective boundary conditions are easy to deal with. When the reflective boundary condition is perpendicular to the x-axis, the incident neutron angular flux density along the direction m satisfies the following equation: ψ g ðx, y, μm , ηm Þ ¼ ψ g ðx, y,  μm , ηm Þ

(7.50)

(2) Vacuum boundary condition When the boundary condition is vacuum boundary, even if the boundary is perpendicular to any coordinate axis, the incident neutron angular flux density along the direction m satisfies the equation: ψ g ðx, y, z, μm , ηm , ξm Þ ¼ 0

(7.51)

7.2.4 Solution of the stiffness matrix For the neutron transport problem of the reactor, the finite element stiffness matrix is often a large symmetric positive definite sparse (banded) matrix of n  n, where n is the number of nodes, generally up to 104–107. If conventional methods such as

Fig. 7.9 Diagram of the direction of the reflective boundary perpendicular to the x-axis.

182

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

chasing, iterative, etc., are used to solve such linear equations, it will take a lot of computational time, which is difficult to accept, and the conjugate gradient (CG) method is one of the most effective methods to solve the symmetrical positive definite sparse matrix equations. This section will use the CG method to solve the finite element equations (7.27). The theoretical basis of the CG method is to solve the variational principle of the linear system: the solution of the Eq. (7.27) is equivalent to the solution of ψ*, so that it satisfies φðψ ∗ Þ ¼ minn φðψ Þ ψR

(7.52)

where the functional 1 φðψ Þ ¼ ðψ, Kψ Þ  ðψ, FÞ 2

(7.53)

Let p1, p2  Rn be two directions. When pT1 Kp2 ¼ 0, p1 and p2 are called the conjugate direction of K. The conjugate gradient method uses a gradient of the functional to construct a set of conjugate directions, and then minimizes the functional in the conjugate direction. Let p0 ¼ r0 ¼ F  Kψ ð0Þ

(7.54)

where ψ (0) is any given initial value. If ψ (i ) and pi1 are known, then ψ (i+1) and pi can be determined using following equations: ψ ði + 1Þ ¼ ψ ðiÞ + αi pi pi ¼ γ i + βi pi1

(7.55)

where γ i ¼ F  Kψ ðiÞ αi ¼

ð γ i , pi Þ ðγ i , γ i Þ , βi ¼ ðpi , Kpi Þ ðγ i1 , γ i1 Þ

(7.56) (7.57)

Then, the whole solution process is as follows: Set, ψ 0 ¼ 0, γ 0 ¼ F  Kψ 0 , p0 ¼ γ 0

(7.58)

For i ¼ 0, 1, 2, ⋯ αi ¼ ðγ i , γ i Þ=ðpi , Kpi Þ

(7.59)

The discrete ordinate method

183

ψ i + 1 ¼ ψ i + αi pi

(7.60)

γ i + 1 ¼ γ i  αi Kpi

(7.61)

βi + 1 ¼ ðγ i , γ i Þ=ðγ i + 1 , γ i + 1 Þ

(7.62)

pi + 1 ¼ γ i + β i + 1 pi

(7.63)

until the convergence is complete. Formulas (7.58)–(7.63) constitutes the calculation formula of the CG algorithm. Since the CG method is a widely used method in the linear system solution, detailed proof of convergence is not given here.

7.2.5 Numerical results Based on the theoretical model described above, this chapter develops a threedimensional least-squares finite element transport calculation code LESFES [12]. This section gives the results of typical cases calculated by the code. (1) TAKEDA benchmark This example is selected from the second example in the 3D TAKEDA benchmark published by OECD/NEA [13]. This problem simulates the neutron transport problem of a small fast breeder reactor in a three-dimensional Cartesian coordinate system. Its geometry is shown in Fig. 7.10. See Appendix 1 for specific cross-section parameters. The TAKEDA benchmark provides reference values calculated by several different methods. For the sake of simplicity, this chapter only lists the values obtained by the Monte Carlo (MC) method and the spherical harmonic method [14] (MARK-PN) when comparing the results. The average neutron flux results for each region in the case of full control rods out are listed in Table 7.2.The calculation results of the keff values are listed in Table 7.3. Although the LESFES calculation uses a fully symmetric S2 quadrature set, it can be seen from Table 7.3 that the error of keff result is 0.22% compared to the Monte Carlo result, which is less than the error of MARK-PN result (0.64%). As can be seen from Table 7.2, most of the error between the LESFES result and the Monte Carlo reference value is less than 3.0%, and the maximum value appears in the fourth energy group in sodium-filled region, which is 13.8%, quite similar to the MARK-PN code. (2) MOSTELLER benchmark The schematic diagram of the geometry of the MOSTELLER benchmark is shown in Fig. 7.11 [15]. The cross section of the problem and the detailed description are provided in Appendix 2. The results are presented in Table 7.4. As can be seen from Table 7.4, for the MOSTELLER benchmark, the results of LESFES code are very close to the results of the DRAGON code [16], which are also very close to the reference values. (3) MTR-type reactor plate element benchmark Fig. 7.12 shows the geometrical diagram of the MTR-type reactor plate element benchmark [17]. Parameters including cross section are provided in Appendix 3. There are a total of 23 plates in the problem, including 17 fuel plates, 4 aluminum plates, and 2 water gap plates.

184

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

0

Vacuum 35 45

0

55

70 x (cm)

Axial blanket 20 y (cm)

Vacuum

70

Fuel

CR Radial blanket

Reflective

75

Middle line

Vacuum

Vacuum

Reflective

Radial blanket

55

Fuel CR/CRP

Fuel

0 0

35

45

55

70

CRP

x (cm)

Reflective 130 CR: Control rod CRP: Control rod position with Na

Axial blanket 150 Vacuum z (cm)

Fig. 7.10 Small fast neutron breeder reactor geometry. Table 7.2 Comparison of regional average neutron flux in FBR benchmark.

Code Monte-Carlo

MARK-PN

LESFES

Energy group 1 2 3 4 1 2 3 4 1 2 3 4

Fuel region

Axial blanket region

Radial blanket region

Sodium filled region

4.2814E  5 2.4081E  4 1.6411E  4 6.2247E  6 4.2370E  5 2.3926E  4 1.6679E  4 6.2855E  6 4.1575E  5 2.3395E  4 1.5862E  4 5.9180E  6

5.1850E  6 4.6912E  5 4.6978E  5 3.7736E  6 5.4492E  6 5.3696E  5 5.7551E  5 5.0063E  6 5.3595E  6 4.5999E  5 4.5237E  5 3.5048E  6

3.3252E 6 3.0893E 5 3.2834E 5 2.0473E 6 3.5595E 6 3.5354E 5 4.0261E 5 2.6785E 6 3.4131E 6 3.0375E 5 3.1723E 5 1.9999E 6

2.5344E  5 1.6658E  4 1.2648E  4 6.9840E  6 2.5130E  5 1.6795E  4 1.3187E  4 7.9182E  6 2.8020E  5 1.6625E  4 1.2028E  4 6.0195E  6

Table 7.3 Comparison of keff calculation results in FBR benchmark.

keff

Monte-Carlo

MARK-PN

LESFES

0.9732

0.9794

0.9711

1 1 2 3

Fig. 7.11 Geometric diagram of the Mosteller benchmark. Table 7.4 Comparison of keff results in the MOSTELLER benchmark. Fuel enrichment (wt%)

Fuel temperature (K)

keff (ref )

keff (DRAGON)

S4

S8

0.711

600 900 600 900 600 900 600 900 600 900

0.6638 0.6567 0.9581 0.9484 1.0961 1.0864 1.1747 1.1641 1.2379 1.2271

0.661623 0.653266 0.955832 0.944818 1.093464 1.081589 1.171287 1.159059 1.234427 1.221573

0.661271 0.652784 0.955234 0.944178 1.092531 1.080747 1.170421 1.158248 1.233512 1.220678

0.661387 0.652947 0.955542 0.944347 1.092945 1.080975 1.170867 1.158467 1.233978 1.220986

1.6 2.4 3.1 3.9

8.05

Unit: cm

LESFES

Fig. 7.12 Geometry of the MTR plate-type component.

6.30

6.65 7.60

186

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

In this calculation, the quarter core is modeled. Fig. 7.13 is a schematic diagram of the division of the MTR-type reactor plate element using triangular elements. The LESFES code was used to calculate the transport problems in the following cases: 1. When the fuel assembly is homogenized into one zone and one group, the calculation results are as presented in Table 7.5.

Fig. 7.13 Schematic diagram of the MTR-type reactor plate component. Table 7.5 Fuel assembly k∞ of the HEU. Different institutions (codes) LESFES

235

U burnup

ANL

EIR

OESGAE

CNEA

DRAGON

S2

S4

S6

0 5 10 30 50

1.7370 1.6370 1.6165 1.5223 1.3876

1.7497 1.6530 1.6316 1.5367 1.40562

1.743 1.641 1.621 1.524 1.390

1.7422 1.6438 1.6223 1.5254 1.3909

1.7454 1.6474 1.6259 1.5280 1.3923

1.7400 1.6412 1.6202 1.5211 1.3882

1.7415 1.6433 1.6219 1.5231 1.3897

1.7425 1.6444 1.6231 1.5252 1.3901

OSGAE, ANL, EIR, CNEA stand for Osterreichische Studiengesellschaft fur Atomenergie (Austria), Argonne National Laboratory (USA), Eidg. Institut fur Reaktorforschung (Switzerland), Commision Nacional de Energia Ato´mica (Argentina).

The discrete ordinate method

187

Table 7.6 Fuel assembly k∞ of the HEU. k∞ LESFES

235

U burnup

ANL

EIR

OESGAE

CNEA

DRAGON

S2

S4

S6

0 5 10 30 50

1.7370 1.6370 1.6165 1.5223 1.3876

1.7497 1.6530 1.6316 1.5367 1.4056

1.743 1.641 1.621 1.524 1.390

1.7422 1.6438 1.6223 1.5254 1.3909

1.7454 1.6474 1.6259 1.5280 1.3923

1.7405 1.6417 1.6209 1.5218 1.3891

1.7418 1.6440 1.6225 1.5237 1.3901

1.7429 1.6451 1.6236 1.5259 1.3908

2. When the fuel assembly is homogenized into one zone and three groups, the calculation results are as presented in Table 7.6. 3. When the fuel assembly is homogenized into two zones (the fuel and the peripheral aluminum are combined into zone 1, and the moderator water is zone 2), the results are as presented in Table 7.7. 4. When the fuel assembly is homogenized into three zones (fuel for zone 1, aluminum for zone 2, moderator water for zone 3), three groups, the calculation results are as presented in Table 7.7.

As can be seen in Tables 7.5–7.7, the results of the LESFES code are in good agreement with the reference values.

7.3

SN nodal method for triangular prism meshes

Due to the flexible geometric modeling of triangular meshes, the finite element method has obvious advantages in the calculation of neutron transport for unstructured geometry. However, the fine mesh method also has obvious shortcomings,

Table 7.7 Fuel assembly k∞ of the HEU. LESFES (2) 235

0 5 10 30 50

U burnup

LESFES (3)

LESFES (4)

DRAGON

S4

S6

S4

S6

S4

S6

1.7454 1.6474 1.6259 1.5280 1.3923

1.7418 1.6440 1.6225 1.5237 1.3901

1.7429 1.6451 1.6236 1.5259 1.3908

1.7420 1.6443 1.6230 1.5242 1.3902

1.7431 1.6455 1.6239 1.5261 1.3911

1.7423 1.6451 1.6236 1.5245 1.3905

1.7433 1.6458 1.6242 1.5269 1.3913

188

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

such as the disadvantages of large computational memory consumption and low computational efficiency. These shortcomings make the calculation cost too much to be accepted by the engineering design when calculating the reactor core for the three-dimensional transport calculation of the reactor core, a two-step calculation strategy based on the homogenization theory is usually adopted, and the core adopts the coarse mesh methods/nodal method for transport calculation to improve the calculation efficiency. However, the traditional nodal methods use cubic or hexagonal prism meshes and can only calculate the traditional structured geometry core. For some special reactors with unstructured geometric design, the abovementioned nodal method is difficult to accurately model. To solve this problem, this section introduces an SN nodal method based on a triangular prism grid, which not only inherits the high efficiency of the traditional nodal method, but also meets the needs of unstructured geometry modeling.

7.3.1 Spatial discretization Similar to the traditional nodal methods based on transverse integration technique, in the triangular prism geometry, the three-dimensional equation can also be transformed into four one-dimensional equations by the transverse integration technique, namely three radial and one axial one-dimensional equations. First, the derivation of the radial equation is introduced. Similar to the finite element method, an arbitrary triangle is firstly transformed to a standard triangle by the area coordinate transformation, which is shown in Figs. 7.14 and 7.15. The transformation formula is

y

x Fig. 7.14 Arbitrary triangle under the rectangular coordinate system (x, y).

The discrete ordinate method

189

y′

A3′

1 3 3′ 3

1

x′

A2′

1 3′

u 2

A1′

2 0 3′

3

ys (x)

v 3 3

Fig. 7.15 Equilateral triangle under the rectangular coordinate system (x’, y’).

3 2 1 0 0 2 3 2 3 p ffiffi ffi 61 1   3 7 1 7 1 6 xn + xp 76 0 7 6 7 6 xk + xn + xp xk + xn + xp 4x 5 ¼ 63 74 x 5 2 2 7 6 pffiffiffi 5 y0 4       3 1 1 y yk + yn + yp yk + yn + yp yn + yp 3 2 2

(7.64)

At the same time, the z-direction is also normalized and transformed, and the local coordinate system with the center of the regular triangular prism as the coordinate origin is obtained. The discrete transport equation in the local coordinate system can be written as ∂ψ ðx0 , y0 , zÞ ∂ψ ðx0 , y0 , zÞ ξ ∂ψ ðx0 , y0 , zÞ + Σt ψ ðx0 , y0 , zÞ + η + x ∂x0 ∂y0 hz ∂z ¼ Qðx0 , y0 , zÞ

μx

(7.65)

In the formula: 

   yn + yp μ + xn  xp η μx ¼ 2Δ

  1 1 1 1 xk + xn + xp η + yk  yn  yp μ 2 2 2 2 pffiffiffi ηx ¼ 3Δ

(7.66)

(7.67)

pffiffiffi 2/3  x0  1/3,  ys(x0 )  y0  ys(x0 ), ys ðx0 Þ ¼ ðx0 + 2=3Þ= 3, 1/2 < z < 1/2, where Δ represents the actual area of the triangle, and hz is the actual height of the triangular prism.

190

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

For simplicity, the superscripts of x0 and y0 are omitted in the following derivations. For formula (7.65), performing integration in the interval  ys(x) < y < ys(x) and 1/2 < z < 1/2, the transverse integral equation in the x direction can be obtained: μx

d m ½ys ðxÞψ x ðxÞ + Σ t ys ðxÞψ m x ðxÞ ¼ ys ðxÞQx ðxÞ  Lx ðxÞ dx

(7.68)

In the formula: ð ys ðxÞ

ψm x ðxÞ ¼

ys ðxÞ

ð Q x ðxÞ ¼

ys ðxÞ

ys ðxÞ

ψ ðx, y, zÞdy=

ð ys ðxÞ

ð Qðx, y, zÞdy=

dy

(7.69)

dy

(7.70)

ys ðxÞ y s ð xÞ ys ðxÞ

Lx ðxÞ ¼ ys ðxÞLzx ðxÞ + Lr x ðxÞ

(7.71)

ξ ½ψ ðxÞ  ψ z ðxÞ hz z + ð ys ψ z ðxÞ ¼ ψ ðx, y,  1=2Þdy=ð2ys Þ

(7.73)

pffiffiffi Lr x ðxÞ ¼ ðμu ψ u ðxÞ + μv ψ v ðxÞÞ= 3

(7.74)

Lzx ðxÞ ¼

(7.72)

ys

ð ψ u ðxÞ ¼

1=2

ð ψ v ðxÞ ¼

1=2

1=2 1=2

ψ ðx, y, zÞjy¼ys ðxÞ dz

(7.75)

ψ ðx, y, zÞjy¼ys ðxÞ dz

(7.76)

 pffiffiffi  μu ¼  μx  3ηx =2

(7.77)

 pffiffiffi  μv ¼  μx + 3ηx =2

(7.78)

μx ¼ μ

(7.79)

The radial integral equation of the node can be obtained from Eq. (7.68) for μ > 0, ys ðxÞψ x ðxÞ ¼

1 μx

ðx 2=3

Σ  μ t ðxx0 Þ

½ys ðx0 ÞQx ðx0 Þ  Lx ðx0 Þ e

x

dx0

(7.80)

The discrete ordinate method

191

Let x ¼ 1/3, then the formula can be obtained as pffiffiffi ð 1=3 Σ 3  t ð1=3xÞ ψx ¼ ½ys ðxÞQx ðxÞ  Lx ðxÞe μx dx μx 2=3

(7.81)

Similar equations are obtained for the u and v directions. For Eq. (7.65), integrate in the interval 1/2 < x < 1/2 and  ys(x) < y < ys(x) to obtain the transverse integral of the z direction. ξ d ψ ðzÞ + Σ t ψ z ðzÞ ¼ Qz ðzÞ  Lz ðzÞ hz dz z

(7.82)

In the formula: ψ z ðzÞ ¼

Q z ðzÞ ¼

ð 1=3 ð ys ðxÞ 2=3 ys ðxÞ

ð 1=3 ð ys ðxÞ 2=3 ys ðxÞ

ψ ðx, y, zÞdydx=

Qðx, y, zÞdydx=

ð 1=3 ð ys ðxÞ dydx

(7.83)

dydx

(7.84)

2=3 ys ðxÞ

ð 1=3 ð ys ðxÞ 2=3 ys ðxÞ

Lz ðzÞ ¼ 2½μx ψ x ðzÞ + μu ψ u ðzÞ + μv ψ v ðzÞ  ð ys ðxÞ   ψ x ðzÞ ¼ ψ ðx, y, zÞdy  ys ðxÞ ð ψ v ðzÞ ¼

1=3

2=3

x¼1=3

 . ð y s ð xÞ   dy  ys ðxÞ

ψ ðx,  ys ðxÞ, zÞdx

(7.85) (7.86) x¼1=3

(7.87)

The axial integral equation of the node can be obtained from Eq. (7.82). The radial flux and source terms of the node are expanded using the following weighted orthogonal polynomials:  hi ðxÞ ¼

2 1 1, x, x + x  15 18



2

(7.88)

and ð 1=3 2=3

ys ðxÞhi ðxÞhj ðxÞdx ¼ 0, i 6¼ j

(7.89)

A flat approximation is made for the radial leakage term Lzx(x) of the radial equation and the leakage term Lz(x) of axial equation.

192

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

For the radial equation, the radial leakage term Lrx(x) is different from the flat approximation, where a special modulation method is used, which assumes that the radial flux distribution in an equilateral triangle satisfies   ψ ðx, yÞ ¼ c1 x2 + y2 + c2 x + c3 y + c4

(7.90)

In order to determine the four coefficients c1, c2, c3, and c4, the average neutron angular flux density ψ x , ψ u , and ψ v of the three surfaces and a mean neutron angular flux density ψ are used as a known condition. By the obtained flux distribution, the fluxes at the three corner points of a triangle can be determined as ψ px ¼ 5ðψ u + ψ v Þ=3  ψ x =3  2ψ ψ pu ¼ 5ðψ v + ψ x Þ=3  ψ u =3  2ψ

(7.91)

ψ pv ¼ 5ðψ x + ψ u Þ=3  ψ v =3  2ψ The second-order distribution of transverse flux can be obtained from the corner point flux and the average surface flux:       ψ u ðxÞ ¼ 4ψ u  ψ px =3 + 2 ψ pv  ψ u x + 3 ψ px + ψ pv  2ψ u x2

(7.92)

      ψ v ðxÞ ¼ 4ψ v  ψ px =3 + 2 ψ pu  ψ v x + 3 ψ px + ψ pu  2ψ v x2

(7.93)

Thus a second-order approximation of Lrx(x) is obtained:   Lr x ðxÞ ¼ Lr x0 + Lr x1 x + Lr x2 x2

(7.94)

 1   1  Lr x0 ¼ μu 4ψ u  ψ px + μv 4ψ v  ψ px 3 3     Lr x1 ¼ 2μu ψ pv  ψ u + 2μv ψ pu  ψ v     Lr x2 ¼ 3μu ψ px + ψ pv  2ψ u + 3μv ψ px + ψ pu  2ψ v

(7.95)

Using the expressions for flux, source, and leakage terms and the nodal integral equation, the following discrete form of the radial equation can be obtained: ψ x ¼ Px0 ðQx0  Lzx Þ +

2 X

Pxi Qxi +

i¼1

ψ xi ¼ Fxi0 ðQx0  Lzx Þ +

2 X j¼1

2 X

Rxi Lr xi

(7.96)

i¼0

Fxij Qxj +

2 X j¼0

Gxij Lr xj , μx > 0 , i ¼ 0,1, 2

(7.97)

The discrete ordinate method

193

ψ xi ¼ Fxi0 ðQx0  Lzx Þ +

2 X

Fxij Qxj +

j¼1

2 X

Gxij Lr xj + Hxi ψ x , μx < 0,

j¼0

(7.98)

i ¼ 0,1, 2 In the formula: pffiffiffi ð 1=3 Σ 3  t ð1=3xÞ Pxi ¼ ys ðxÞhi ðxÞe μx dx, i ¼ 0, 1,2 μx 2=3 1 Rxi ¼  μx

ð 1=3 xi e

Σ  μ t ð1=3xÞ x

2=3

dx, i ¼ 0,1, 2

8 ð 1=3 ð x Σ >  t ðxx0 Þ 0 > > ys ðx0 Þhj ðx0 Þ e μx dx hi ðxÞdx > > 1 2=3 2=3 > > > , μx > 0 ð 1=3 > > μx > 2 > > ys ðxÞðhi ðxÞÞ dx > < 2=3 Fxij ¼ ð 1=3 ð 1=3 Σ > >  jμ t jðx0 xÞ 0 > 0 0 > x y ð x Þh ð x Þ e dx hi ðxÞdx > s j > > 1 2=3 x > > , μx < 0 ð 1=3 > > jμx j > > 2 > ys ðxÞðhi ðxÞÞ dx :

(7.99)

(7.100)

(7.101)

2=3

8 ð 1=3 ð x Σt 0 > > 0 j  μx ðxx Þ 0 > pffiffiffi ð x Þ e dx hi ðxÞdx > > > 3 2=3 2=3 > > , μx > 0  ð 1=3 > > 3μx > 2 > > y ð x Þ ð h ð x Þ Þ dx > s i < 2=3 Gxij ¼ ð 1=3 ð 1=3 Σt 0 > > ðx xÞ 0 j  > > p ffiffi ffi ðx0 Þ e jμx j dx hi ðxÞdx > > 3 > 2=3 x > > , μx < 0 ð 1=3 > > 3j μ x j > 2 > > ys ðxÞðhi ðxÞÞ dx :

(7.102)

2=3

Hxi ¼

pffiffiffi ð 1=3 ð 1=3 Σ 3  t ð1=3xÞ ψx hi ðxÞe jμx j dx= ys ðxÞðhi ðxÞÞ2 dx 3 2=3 2=3

(7.103)

By changing the subscript x to u and v, three basic equations in the u direction and three basic equations in the v direction are obtained, respectively. For the axial equation, after discretization, we get ψ z + ¼ Pz0 ðQz0  Lz Þ +

2 X i¼1

Pzi Qzi + Tψ z , ξ > 0

(7.104)

194

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

ψ z ¼ Pz0 ðQz0  Lz Þ +

2 X

Pzi Qzi + Tψ z + , ξ < 0

(7.105)

Fzij Qzj + Hzi ψ z , ξ > 0

(7.106)

Fzij Qzj + Hzi ψ z + , ξ < 0

(7.107)

i¼1

ψ zi ¼ Fzi0 ðQz0  Lz Þ +

2 X j¼1

ψ zi ¼ Fzi0 ðQz0  Lz Þ +

2 X j¼1

In the formula: 8 ð Σh > h 1=2  tξ z ð1=2zÞ > > z dz, ξ > 0 > < ξ 1=2 fi ðzÞe Pzi ¼ ð Σh > hz 1=2 >  jtξj z ð1=2 + zÞ > > f ð z Þe dz, ξ < 0 i : jξj 1=2

T

m

8
0

(7.108)

(7.109)

ξ  tξ z ðzz0 Þ 0 0 > > fj ðz Þe dz fi ðzÞdz ðfi ðzÞÞ2 dz, ξ > 0 < ξ 1=2 1=2 1=2 Fzij ¼  ð 1=2 ð ð > hz 1=2 1=2 0  Σjtξhj z ðz0 zÞ 0 > > fj ðz Þe dz fi ðzÞdz ðfi ðzÞÞ2 dz, ξ < 0 : jξj 1=2 z 1=2 (7.110) 8 ð 1=2  ð 1=2 Σh hz >  tξ z ðz + 1=2Þ > > e fi ðzÞdz ðfi ðzÞÞ2 dz, ξ > 0 < ξ 1=2 1=2 Hzi ¼  ð 1=2 ð > hz 1=2  Σjtξhj z ð1=2zÞ > > e fi ðzÞdz ðfi ðzÞÞ2 dz, ξ < 0 : jξj 1=2 1=2

(7.111)

7.3.2 Numerical solution process In order to improve the calculation efficiency, the nodal coupling equation can be transformed into an equivalent finite difference form for numerical solution. For a triangular prism geometry with arbitrary triangular shape in the radial direction, for each discrete direction, the relationship between the discrete direction and the mesh surface can be used to determine a set of grid scan orders. It can be found that for radial triangles, there may be one or two incident surface.

The discrete ordinate method

195

Here we only discuss the case of ξ > 0:   2 2 X X Px0 Px0 Pxi  Fx0i Qxi + 2μx Rxi  Gx0i Lr xi  Fx00 Fx00 i¼1 i¼0

 2 ξX Pz0 2μu ψ u  2μv ψ v  Pzi  Fz0i Qzi + hz i¼1 F

  z00  ξ Pz0 Px0 ξ Pz0 T Hz0  1 ψ z + + Σt 2μx hz Fz00 Fx00 hz Fz00

ψ ¼ Q  2μx

(7.112) For an incident surface:   2 2 X X Px0 Px0 ψ ¼ Q  2μx Pxi  m Fx0i Qxi + 2μx Rxi  m Gx0i Lr xi  Fx00 Fx00 i¼1 i¼0   2 2 X X Pu0 Pu0 2μu Pui  m Fu0i Qui + 2μu Rui  Gu0i Lr ui  Fu00 Fu00 i¼1 i¼0

 2 ξX Pz0 2μv ψ v  Pzi  m Fz0i Qzi + hz i¼1 F

 z00  ξ Pz0 Px0 Pu0 ξ Pz0 T Hz0  1 ψ z + + Σt 2μx m + 2μu hz Fz00 Fx00 Fu00 hz Fz00 (7.113) The radial outgoing surface flux can be written as   2 2 X X Px0 Px0 Px0 ψx ¼ ψ+ Pxi  m Fx0i Qxi + Rxi  Gx0i Lr xi Fx00 Fx00 Fx00 i¼1 i¼0

(7.114)

The relationship for updating the radial high-order moments is  2 X Fm Fm m xi0 m xi0 m ψ + F  F xij x0j Qxj Fm Fm x00 x00 j¼1  2 X Fm m m xi0 m + Gxij  m Gx0j Lr xj , μx > 0 F x00 j¼0

ψm xi ¼

 2 X Fxi0 Fxi0 ψ xi ¼ m ψ + Fxij  m Fx0j Qxj + Fx00 Fx00 j¼1



 2 X Fxi0 Fxi0 Gxij  Gx0j Lr xj + Hxi  Hx0 ψ x , μx < 0 Fx00 Fx00 j¼0

(7.115)

(7.116)

196

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

The relationship for updating the axial outgoing surface is 

 2 X Pz0 Pz0 Pz0 ψ+ Pzi  Fz0i Qzi + T  Hz0 ψ z , ξ > 0 Fz00 Fz00 Fz00 i¼1

(7.117)



 2 X Pz0 Pz0 Pz0 ψ z ¼ ψ+ Pzi  Fz0i Qzi + T  Hz0 ψ z + , ξ < 0 Fz00 Fz00 Fz00 i¼1

(7.118)

ψz+ ¼

The relationship for updating the axial high-order moments is 

 2 X Fzi0 Fzi0 Fzi0 ψ zi ¼ ψ+ Fzij  Fz0j Qzj + Hzi  Hz0 ψ z , ξ > 0 Fz00 Fz00 Fz00 j¼1

(7.119)



 2 X Fzi0 Fzi0 Fzi0 ψ zi ¼ ψ+ Fzij  Fz0j Qzj + Hzi  Hz0 ψ z + , ξ < 0 Fz00 Fz00 Fz00 j¼1

(7.120)

7.3.3 Selection of the quadrature set For the neutron transport problem at rectangular boundaries, the full symmetric quadrature set can well meet the calculation requirements. However, for unstructured geometry, it is often the case that the boundary is a hypotenuse. In this section, the applicable 60-degree symmetric quadrature set is introduced to solve the geometrical relationship with 60 or 120 degrees symmetrical neutron transport problem. Fig. 7.16 shows the 1/4 sphere in the range 1  ξ  1 and 0  φ  π2. If finding the direction of symmetry with respect to φ ¼ π3 at each layer of ξi, we can get the quadrature of symmetry with respect to 60 degree. Considering 0  φ  π, the independent variable Ω is integrated on the 1/2 sphere: ð ð ð 1 1 1 π dΩ ¼ dφdξ 2π 2π 1 0

(7.121)

pffiffiffiffiffiffiffiffiffiffiffiffi 1  ξ2 cos φ pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi η ¼ 1  ξ2 sin φ ¼ 1  μ2  ξ2

(7.122)

A¼ due to μ¼

Substituting formula into formula for variable substitution, we get 1 A¼ 2π

ffiffiffiffiffiffiffi2ffi ð 1 ð p1ξ 1 

pffiffiffiffiffiffiffi2ffi 1ξ

1 dμdξ 1  μ2  ξ2

(7.123)

The discrete ordinate method

197

Fig. 7.16 60 degree symmetric quadrature set.

Let y ¼ μ= A¼

pffiffiffiffiffiffiffiffiffiffiffiffi 1  ξ2 , then formula becomes

1 2π

ð

1 1

ð

1

1 pffiffiffiffiffiffiffiffiffiffiffiffi dydξ 1  y2 1

(7.124)

For Eq. (7.124), the integral about ξ can be done using Gauss-Legendre quadrature pffiffiffiffiffiffiffiffiffiffiffiffi formula. The integral about y is the integral with the right function 1= 1  y2 in the interval (1, 1), which, according to the numerical integral theory, can be completed by Gauss-Chebyshev product-saving formula. Considering 0  ξ  1, a set of Gauss-Legendre quadrature set {ξ1, ⋯, ξN/2} is determined by the Legendre polynomial PN/2(ξ) ¼ 0. For 1  ξ  0, the Gauss-Legendre quadrature set is { ξN/2, ⋯,  ξ1}. As shown in Fig. 7.16, there are N latitude layers of ξi in the interval 1  ξ  1, and the corresponding weight coefficient of each layer is {ωN/2, ⋯, ω1, ω1, ⋯, ωN/2}. When integrating y at the given ith latitude layer, n quadrature points {μi1, ⋯, μin} are determined by Chebyshev polynomial Tn(y) ¼ 0(y ¼ cos φ), which is called Gauss-Chebyshev quadrature set; n is the number of accumulated points on the ith latitude layer, which can be arbitrarily selected. In order to obtain the symmetric quadrature set on the

surface of 60 degrees, n ¼ 6 is selected, π π 5π 7π 3π 11π namely, six directions 12 , 4 , 12 , 12 , 4 , 12 are selected from the interval of (0, π). For each latitude layer, the weight coefficients of each Chebyshev points are considered to be equal: ωij ¼ ωi =n, j ¼ 1,⋯,6

(7.125)

198

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Since the solutions of Tn(y) ¼ 0 (quadrature points) are

 2j  1 π , j ¼ 1,⋯,6 yj ¼ cos 12 According to μ ¼ y μij ¼

(7.126)

pffiffiffiffiffiffiffiffiffiffiffiffi 1  ξ2 , one can get

 qffiffiffiffiffiffiffiffiffiffiffiffi 2j  1 π , i ¼ 1,2, ⋯N=2, j ¼ 1, ⋯, 6 1  ξ21 yj ¼ cos 12

(7.127)

Also available: ηij ¼

 qffiffiffiffiffiffiffiffiffiffiffiffi 2j  1 π , i ¼ 1, 2,⋯N=2, j ¼ 1,⋯,6 1  ξ21 yj ¼ sin 12

(7.128)

To generalize to the whole sphere, simply change the value range of j in Eqs. (7.123), (7.127), (7.128) to j ¼ 1, ⋯, 12. As can be seen from Eq. (7.126), the resulting quadrature is also symmetric about 0, 90, 120, and 180 degrees. According to the above model, Table 7.8 presents the quadrature set of symmetric 60 degree within an octagonal limit.

7.3.4 Numerical results Based on the theoretical model described earlier, a three-dimensional triangular-z nodal SN transport calculation code called DNTR is developed [18, 19]. This section gives the results of typical examples calculated by the code. (1) TAKEDA benchmark

This example uses the same example as in Section 7.2.5 and calculates two cases: Case 1: full withdrawal of all control rods; Case 2: half the control rod insertion. For case 1, only 1/8 core is calculated. For the DNTR code, the radial section is divided into 420 triangular meshes, and the axial section is divided into 14 layers. The TDOT [20] code uses meshes of 1 cm  1 cm  1 cm.For case 2, only 1/4 core is calculated. For the DNTR code, the radial section is also divided into 420 triangular meshes, and the axial section is divided into 32 layers. The TDOT code uses meshes of 5 cm  5 cm  5 cm, which is limited by the computer memory. Table 7.9 lists the effective multiplication factor results calculated by the codes in both cases. The calculation results of the DNTR code agree well with the reference values provided by the Monte Carlo code, and the relative error of the control rod value is smaller than the TDOT code. In terms of computational time, the one of the DNTR code is less than 1/7 of that of the TDOT code when all control rods are withdrawn. When the control rods are inserted half, the calculation time of the DNTR code is about 1/6 that of the TDOT code. The average neutron flux of each zone calculated by the Monte Carlo code when the control rod is withdrawn and the relative errors of the TDOT code and DNTR code are listed

The discrete ordinate method

199

Table 7.8 60 degree symmetric quadrature group (with one octagonal limit). SN order

Discrete direction m

Latitude layer i

μ

η

ξ

ω

2

1 2 3 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 1 1 2 2 2 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 4 4 4

0.78867 0.57735 0.21132 0.49105 0.35947 0.13157 0.90838 0.66498 0.24340 0.34893 0.25544 0.09349 0.72463 0.53047 0.19416 0.93802 0.68668 0.25134 0.26949 0.19728 0.07221 0.58382 0.42738 0.15643 0.82178 0.60158 0.22019 0.94953 0.69510 0.25442

0.21132 0.57735 0.78867 0.13157 0.35947 0.49105 0.24340 0.66498 0.90838 0.09349 0.25544 0.34893 0.19416 0.53047 0.72463 0.25134 0.68668 0.93802 0.07221 0.19728 0.26949 0.15643 0.42738 0.58382 0.22019 0.60158 0.82178 0.25442 0.69510 0.94953

0.57735

0.33333

0.86113

0.11595

0.33998

0.21738

0.93246

0.05710

0.66120

0.12025

0.23861

0.15597

0.96028

0.03374

0.79666

0.07412

0.52553

0.10456

0.18343

0.12089

4

6

8

in Table 7.10. It can be seen that the calculation result of the DNTR code agrees well with the reference value, and also agrees well with the calculation result of the TDOT code. Table 7.11 lists the average neutron flux calculated by the Monte Carlo code in case 2 and the relative errors of the TDOT code and DNTR code. The result of the DNTR code is slightly better than that of the TDOT code. For the axial blanket zone, the maximum relative error of the neutron flux density of each group of the DNTR code is 0.45%, while the maximum relative error of the neutron flux density of each group of the TDOT procedure is 1.36%. For control rods, the results of the DNTR code are better than the TDOT code in each energy group. In addition, the maximum relative error occurred in the fourth group of the sodium-filled zone, and the relative errors of the DNTR code and the TDOT code were 5.12% and 5.81%, respectively.

200

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Table 7.9 Comparison of effective multiplication factors with reference values in DNTR calculation benchmark 2. Case 1 CPU time (s)

Code

keff

Monte Carlo TDOT

0.9732 0.0002a 0.9732 0.00b 0.9736 0.04b

DNTR

a b

Case 2

–

keff 0.9594  0.0002 0.9588  0.06 0.9597 0.03

763 97

CPU time (s)

Control rod worth

–

1.47E 02  0.03E  02 1.54E 02 4.76 1.49E 02 1.36

1404 220

Relative standard deviation. Relative error between calculated and reference values/%.

Table 7.10 Comparison of average neutron flux density and reference value in case 1 of DNTR calculation benchmark 2. Monte Carlo

TDOT

DNTR

Fuel zone 1G 2G 3G 4G

4.2814E 05 2.4081E 04 1.6411E 04 6.2247E 06

(0.06)a (0.05) (0.06) (0.20)

0.04b 0.02 0.02 0.24

0.03b 0.03 0.01 0.25

5.1850E 06 4.6912E 05 4.6978E 05 3.7736E 06

(0.27) (0.14) (0.16) (0.46)

0.03 0.04 0.12 0.97

0.06 0.26 0.31 1.12

3.3252E 06 3.0893E 05 3.2834E 05 2.0473E 06

(0.18) (0.10) (0.10) (0.34)

0.08 0.20 0.22 0.23

0.31 0.21 0.37 0.50

2.5344E 05 1.6658E 04 1.2648E 04 6.9840E 06

(0.33) (0.18) (0.23) (0.17)

0.66 0.07 0.13 0.05

0.89 0.01 0.10 0.38

Axial blanket zone 1G 2G 3G 4G

Radial blanket zone 1G 2G 3G 4G

Sodium-filled zone 1G 2G 3G 4G a b

Relative standard deviation. Relative error between calculated and reference values/%.

The discrete ordinate method

201

Table 7.11 Comparison of average neutron flux density and reference value in case 2 of DNTR calculation benchmark 2. Monte Carlo

TDOT

DNTR

4.3482E  05 (0.06)a 2.4171E  04 (0.05) 1.6200E  04 (0.06) 6.0438E  06 (0.21)

0.03b 0.03 0.01 0.35

0.10b 0.05 0.06 0.24

5.2209E  06 (0.27) 4.6772E  05 (0.14) 4.6190E  05 (0.16) 3.6287E  06 (0.45)

1.36 1.03 0.55 0.06

0.45 0.21 0.31 0.20

3.3176E  06 (0.18) 3.0438E  05 (0.10) 3.2126E  05 (0.11) 2.0016E  06 (0.34)

0.47 0.84 0.40 0.30

0.08 0.13 0.43 0.91

2.5902E  05 (0.46) 1.6779E  04 (0.25) 1.2551E  04 (0.31) 7.0648E  06 (1.79)

0.93 0.07 0.50 5.81

0.24 0.39 0.73 5.12

1.6556E  05 (0.54) 9.1050E  05 (0.26) 5.1815E  05 (0.30) 1.1073E  06 (1.00)

1.38 1.51 1.14 3.11

0.71 0.47 0.31 1.03

Fuel zone 1G 2G 3G 4G

Axial blanket zone 1G 2G 3G 4G

Radial blanket zone 1G 2G 3G 4G

Sodium-filled zone 1G 2G 3G 4G

Control rod 1G 2G 3G 4G a b

Relative standard deviation. Relative error between calculated and reference values/%.

(2) KNK-II experimental fast reactor benchmark The benchmark simulates the KNK-II experimental fast neutron reactor [13]. The shape of the assembly is hexagonal, the geometry is shown in Fig. 7.17, and the four groups of cross sections are provided in Appendix 4. The whole core consists of 12 material zones, which are test zone, control rod, control rod tracker, steel, axial blanket zone, axial reflection zone, drive zone without moderator, drive zone with moderator, reflection zone without moderator, reflection zone with moderator, KNK-1 reflection zone, and sodium/steel zone. This example calculates two cases: Case 1: full withdrawal of all control rods; Case 2: half the control rod insertion. When the DNTR code is calculated, the angular variable is discretized by S4 full symmetric quadrature. For case 1 and case 2, the DNTR code only calculates 1/4 core, the radial

202

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

7.5 cm Test zone Control rod/follower Driver without moderator Driver with moderator Reflector without moderator Reflector with moderator KNK-I reflector Sodium/Steel

(A) (cm)

Control rod position

190

1 155 145

1

1

3

3

6

5

1

1

1

1

1

2

2

3

3

4

4

5

6

11

125

95

10 9

8

7

4

7

8

9 10

12 12

65 3

3

2

2

2

3

3

1

1

1

1

1

1

1

45 1

1

1 Steel 1 2 Axial blanket 3 Axial reflector 4 Test zone 5 Driver without moderator 11 6 Driver with moderator 7 Reflector without moderator 8 Reflector with moderator 9 KNK-1 reflector 10 Sodium/steel 11 Control rod 1 12 Control rod follower

0

(B)

CR-out CR-half-in CR-in

Fig. 7.17 Geometry of the KNK-II experimental fast reactor benchmark. (A) Radial direction. (B) Axial direction. section is divided into 404 triangular meshes, and the axial section is divided into 32 layers. The reference value is provided by the Monte Carlo code called GMVP [21]. In addition, the calculation results of the hexagonal nodal SN code called NSHEX [21] are also given as a reference. Table 7.12 presents the results of the effective multiplication factors, calculational time, control rod values and their relative errors with reference values which are calculated by the DNTR code in both cases. It can be seen that the relative errors between the calculation result of the DNTR code and the reference value are not more than 0.05%. The DNTR code has the accuracy comparable to the NSHEX code. In addition, the relative errors between the control rod value and the reference value obtained by the DNTR code are small. The results of DNTR have a good consistency with that of the GMVP code. Table 7.13 presents the average neutron flux calculated by the GMVP code and the relative errors of the NSHEX code and DNTR code in case 1. From the comparison of

The discrete ordinate method

203

Table 7.12 Comparison of effective multiplication factors with reference values in DNTR calculation benchmark 4. Case 1 CPU time (s)

Code

keff

GMVP

1.0955  0.0005a 1.0966  0.10b 1.0960 0.05b

NSHEX DNTR

a b

– – 198

Case 2 keff 0.9839  0.0004 0.9842 0.03 0.9844 0.05

CPU time (s)

Control rod worth

–

22.3E  02

–

22.4E  02 0.45 22.3E  02 0.00

247

Relative standard deviation. Relative error between calculated and reference values/%.

Table 7.13 Comparison of average neutron flux density and reference value in case 1 of DNTR calculation benchmark 4. GMVP

NSHEX

DNTR

3.51E 5 (0.31)a 4.96E 5 (0.27) 2.61E 5 (0.33) 1.20E 5 (0.44)

0.03b 0.62 0.65 0.65

0.33b 0.23 0.39 0.52

1.34E 4 (0.15) 1.09E 4 (0.17) 3.09E 5 (0.23) 3.87E 6 (0.37)

0.17 0.17 0.22 0.76

0.40 0.33 0.05 0.42

9.97E 5 (0.10) 8.00E 5 (0.16) 2.98E 5 (0.20) 6.82E 6 (0.23)

0.47 0.00 0.36 0.19

0.31 0.06 0.45 0.07

5.25E 5 (0.14) 5.06E 5 (0.15) 2.26E 5 (0.21) 1.04E 5 (0.29)

0.50 0.48 0.16 0.85

0.04 0.12 0.29 0.25

Axial blanket zone 1G 2G 3G 4G

Test zone 1G 2G 3G 4G

Drive zone without moderator 1G 2G 3G 4G

Control rod tracker 1G 2G 3G 4G a b

Relative standard deviation. Relative error between calculated and reference values/%.

204

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

the results, it can be observed that the relative errors of the average neutron flux density of each region calculated by the DNTR code and the NSHEX code are less than 1.0%. The maximum relative errors of the DNTR code and the NSHEX code are 0.52% and 0.85%, respectively. Table 7.14 presents the average neutron flux calculated by the GMVP code and the relative errors of the NSHEX code and DNTR code in case 2. From the comparison of the results, it can be observed that the relative errors of the average neutron flux density of each region calculated by the DNTR code and the NSHEX code are less than 1.0%. The maximum relative errors of the DNTR code and the NSHEX code are 0.84% and 0.94%, respectively.

Table 7.14 Comparison of average neutron flux density and reference value in case 2 of DNTR calculation benchmark 4. GMVP

NSHEX

DNTR

Axial blanket zone 1G 2G 3G 4G

3.70E 5 4.84E 5 2.22E 5 8.63E 6

(0.28)a (0.31) (0.44) (0.57)

0.09b 0.01 0.58 0.18

0.09b 0.06 0.58 0.78

1.49E 4 1.12E 4 2.64E 5 2.45E 6

(0.15) (0.17) (0.27) (0.45)

0.24 0.15 0.02 0.61

0.24 0.45 0.19 0.52

Test zone 1G 2G 3G 4G

Drive zone without moderator 1G 2G 3G 4G

1.05E 4 7.94E 5 2.56E 5 5.35E 6

(0.11) (0.14) (0.21) (0.27)

0.41 0.07 0.06 0.29

0.48 0.25 0.35 0.06

4.43E 5 3.56E 5 7.66E 6 5.99E 7

(0.21) (0.18) (0.29) (0.42)

0.00 0.37 0.21 0.54

0.33 0.25 0.13 0.18

1.19E 4 1.00E 4 3.69E 5 1.10E 5

(0.17) (0.18) (0.28) (0.39)

0.65 0.29 0.64 0.94

0.29 0.48 0.84 0.21

Control rod 1G 2G 3G 4G

Control rod tracker 1G 2G 3G 4G a b

Relative standard deviation. Relative error between calculated and reference values/%.

The discrete ordinate method

205 y (cm)

x (cm)

Fig. 7.18 Geometry of a hexagonal boundary. Table 7.15 Cross section of pressurized water reactor assembly. νΣgf Energy group g Materials (cm21)

Σg21 s (cm21)

Σg22 s (cm21)

Σgt (cm21)

χg

1

1.780E  1 1.995E  1 1.089E  3 1.558E  3

1.002E  2 2.188E  2 5.255E  1 8.783E  1

1.9665E 1 2.2206E 1 5.9616E 1 8.8787E 1

1.0 1.0 0.0 0.0

2

1 2 1 2

6.203E 3 0.0 1.101E 1 0.0

(3) Complex geometric problems The problem consists of a hexagonal cell, a central natural uranium rod, and 12 fuel rods distributed symmetrically around it, with the outer boundary being the reflective boundary condition. The geometry is shown in Fig. 7.18 [22]. Two groups of cross section are presented in Table 7.15. Material 1 in the table corresponds to material 1 and material 3 in Fig. 7.15, and material 2 in the table corresponds to material 2 in Fig. 7.18. Since the problem is symmetric, the DNTR code only calculates 1/6 of the hexagonal cells. The average neutron flux density and effective multiplication factor for each zone calculated by a multigroup Monte Carlo code, the TEPFEM code [23], the TPTRI code [22], and the DNTR code are listed in Table 7.16. In general, it can be observed that the calculation result of the DNTR code lies between the results of TPTRI and TEPFEM codes, which is closer to th e TPTRI codes, and the results of the three codes have certain errors compared with the results of multigroup Monte Carlo code. The maximum relative error of the average neutron flux calculated by the DNTR code and the TPTRI code occurred in the third zone of the thermal group, which is 0.66%, and the relative error of the effective multiplication factors of the two codes was 0.016%. Therefore, it can be concluded that the DNTR code has good accuracy in the neutron transport problem in unstructured geometry.

206

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Table 7.16 Comparison of geometric results of hexagonal boundary. Average neutron flux density in each fast groupa

Average neutron flux density in each zone of the thermal group

Code

Zone 1

Zone 2

Zone 3

Zone 1

Zone 2

Zone 3

keff

MG-MCb TEPFEM TPTRI DNTR(S8)

1.0 1.0 1.0 1.0

0.95835 0.99218 0.98190 0.98261

1.00199 1.01050 1.01939 1.01536

0.74686 0.76520 0.75776 0.75724

0.74725 0.77580 0.77293 0.77301

0.72761 0.75169 0.73991 0.74482

1.09080 1.08660 1.08578 1.08595

a b

Neutron flux normalization: average fast neutron flux density in zone 1 is equal to 1.0. Multigroup Monte-Carlo.

Appendix 1 Table A1.1 Cross-sections of reactor core. Group P 1 a (cm ) P νP f (cm1) 1 Pt (cm ) 1 Ps,g1 (cm1) Ps,g2 (cm1) Ps,g3 (cm1) s,g4 (cm )

1

2

3

4

7.45551E 03 2.06063E 02 1.14568E 01 7.04326E 02 3.47967E 02 1.88282E 03 0.00000E +00

3.52540E  03 6.10571E  03 2.05177E  01 0.00000E + 00 1.95443E  01 6.20863E  03 7.07208E  07

7.80136E  03 6.91403E  03 3.29381E  01 0.00000E + 00 0.00000E + 00 3.20586E  01 9.92975E  04

2.74496E  02 2.60689E  02 3.89810E  01 0.00000E + 00 0.00000E + 00 0.00000E + 00 3.62360E  01

Table A1.2 Cross-sections of radial blanket. Group P 1 a (cm ) P νP f (cm1) 1 Pt (cm ) 1 Ps,g1 (cm1) Ps,g2 (cm1) Ps,g3 (cm1) s,g4 (cm )

1

2

3

4

7.43283E  03 1.89496E  02 1.19648E  01 6.91158E  02 4.04132E  02 2.68621E  03 0.00000E + 00

1.99906E  03 1.75265E  04 2.42195E  01 0.00000E + 00 2.30626E  01 9.57027E  03 1.99571E  07

6.79036E 03 2.06978E 04 3.56476E 01 0.00000E + 00 0.00000E + 00 3.48414E 01 1.27195E 03

1.58015E 02 1.13451E 03 3.79433E 01 0.00000E +00 0.00000E +00 0.00000E +00 3.63631E 01

The discrete ordinate method

207

Table A1.3 Cross-sections of radial reflector. Group P 1 a (cm ) P νP f (cm1) 1 Pt (cm ) 1 Ps,g1 (cm1) Ps,g2 (cm1) Ps,g3 (cm1) s,g4 (cm )

1

2

3

4

1.13305E  03 0.00000E + 00 1.71748E  01 1.23352E  01 4.61307E  02 1.13217E  03 0.00000E + 00

4.90793E 04 0.00000E + 00 2.17826E 01 0.00000E + 00 2.11064E 01 6.27100E 03 1.03831E 06

1.94500E 03 0.00000E +00 4.47761E 01 0.00000E +00 0.00000E +00 4.43045E 01 2.77126E 03

5.70263E 03 0.00000E+ 00 7.95199E 01 0.00000E+ 00 0.00000E+ 00 0.00000E+ 00 7.89497E 01

Table A1.4 Cross-sections of axial blanket. Group P 1 a (cm ) P νP f (cm1) 1 Pt (cm ) 1 Ps,g1 (cm1) Ps,g2 (cm1) Ps,g3 (cm1) s,g4 (cm )

1

2

3

4

5.35418E  03 1.31770E  02 1.16493E  01 7.16044E  02 3.73170E  02 2.21707E  03 0.00000E + 00

1.48604E 03 1.26026E 04 2.20521E 01 0.00000E + 00 2.10436E 01 8.59855E 03 6.68299E 07

5.35300E 03 1.52380E 04 3.44544E 01 0.00000E +00 0.00000E +00 3.37506E 01 1.68530E 03

1.34694E 02 7.87302E 4 3.88356E 01 0.00000E+ 00 0.00000E+ 00 0.00000E+ 00 3.74886E 01

Table A1.5 Cross-sections of axial reflector. Group P 1 a (cm ) P νP f (cm1) 1 Pt (cm ) 1 Ps,g1 (cm1) Ps,g2 (cm1) Ps,g3 (cm1) s,g4 (cm )

1

2

3

4

6.39154E  04 0.00000E + 00 1.65612E  01 1.15653E  01 4.84731E  02 8.46495E  04 0.00000E + 00

4.06876E 04 0.00000E + 00 1.66866E 01 0.00000E + 00 1.60818E 01 5.64096E 03 6.57573E 07

1.20472E 03 0.00000E +00 2.68633E 01 0.00000E +00 0.00000E +00 2.65011E 01 2.41755E 03

4.36382E 03 0.00000E+ 00 8.34911E 01 0.00000E+ 00 0.00000E+ 00 0.00000E+ 00 8.30547E 01

Table A1.6 Cross-sections of void zone. Group P 1 a (cm ) P νP f (cm1) 1 Pt (cm ) 1 Ps,g1 (cm1) Ps,g2 (cm1) Ps,g3 (cm1) s,g4 (cm )

1

2

3

4

7.49800E  05 0.00000E + 00 1.36985E  02 9.57999E  03 3.95552E  03 8.80428E  03 0.00000E + 00

3.82435E 05 0.00000E + 00 1.69037E 02 0.00000E + 00 1.64740E 02 3.91394E 04 7.72254E 08

1.39335E 04 0.00000E +00 3.12271E 02 0.00000E +00 0.00000E +00 3.09104E 02 1.77293E 04

4.95515E 04 0.00000E+ 00 6.29537E 02 0.00000E+ 00 0.00000E+ 00 0.00000E+ 00 6.24581E 02

208

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Table A1.7 Fission spectrum. Group

1

2

3

4

Spectrum

0.583319

0.405450

0.011231

0.0

Appendix 2 Table A2.1 Composition of material 2. Temperature (K)

91

600

3.89087E 2

Zr number density

Table A2.2 Composition of material 3. Temperature (K)

1

600

4.42326E  2

H number density

16

O number density

2.21163E  2

11

B number density

5.13231E 5

Table A2.3 Composition of material 1. Fuel enrichment (wt%) 0.711 1.6 2.4 3.1 3.9

Number density Fuel temperature (K)

235

600 900 600 900 600 900 600 900 600 900

1.66078E  4 1.64729E  4 3.73729E  4 3.70693E  4 5.60588E  4 5.56033E  4 7.24086E  4 7.18202E  4 9.10933E  4 9.03532E  4

U

238

U

2.28994E  2 2.27133E  2 2.26940E  2 2.25096E  2 2.25093E  2 2.23264E  2 2.23476E  2 2.21660E  2 2.21163E  2 2.19827E  2

16

O

4.61309E 2 4.57561E 2 4.61355E 2 4.57607E 2 4.61397E 2 4.57648E 2 4.61433E 2 4.57684E 2 4.61475E 2 4.57725E 2

The discrete ordinate method

209

Appendix 3 Table A3.1 Composition of MTR cell benchmark.

Fuel density (g/cm3) Fuel temperature (K) Fuel composition (wt%) 235 U 238 U 27 AL AL density (g/cm3) AL temperature (K) AL composition (wt%) 27 AL Moderator density (g/cm3) Moderator temperature (K) Moderator composition (wt%) 1 H in H2O 16 O Power density (kW/kg)

HEU (93%)

MEU (45%)

LEU (20%)

3.233 293.16

4.009 293.16

6.108 293.16

19.53 14.70 79.00 2.7 293.16

18.00 22.00 60.00 2.7 293.16

14.40 57.60 28.00 2.7 293.16

100 0.9982 293.16

100 0.9982 293.16

100 0.9982 293.16

11.19 88.81 37323.6

11.19 88.81 18029.03

11.19 88.81 7998.3

Appendix 4 Table A4.1 Cross-sections of test zone. Group

1

2

3

4

Σga (cm1) νΣgf (cm1) Σgt (cm1) Σg1 (cm1) s g2 Σs (cm1) Σg3 (cm1) s g4 Σs (cm1)

7.14117E  03 1.79043E  02 1.24526E  01 1.05964E  01 1.12738E  02 1.46122E  04 9.62178E  07

8.00576E  03 1.59961E  02 2.01025E  01 0.00000E + 00 1.89370E  01 3.64847E  03 1.06888E  06

1.45876E 02 2.40856E 02 2.86599E 01 0.00000E +00 0.00000E +00 2.70207E 01 1.80479E 03

4.98120E 02 7.33104E 02 3.68772E 01 0.00000E+ 00 0.00000E+ 00 0.00000E+ 00 3.18960E 01

Table A4.2 Cross-sections of control rod. Group

1

2

3

4

Σga (cm1) νΣgf (cm1) Σgt (cm1) Σg1 (cm1) s g2 Σs (cm1) Σg3 (cm1) s g4 Σs (cm1)

8.62218E  03 0.00000E + 00 1.39085E  01 1.17722E  01 1.26066E  02 1.33314E  04 1.08839E  06

2.91302E  02 0.00000E + 00 2.28152E  01 0.00000E + 00 1.94699E  01 4.32219E  03 1.85491E  07

7.40851E 02 0.00000E +00 3.18806E 01 0.00000E +00 0.00000E +00 2.44352E 01 3.68781E 04

3.12550E 01 0.00000E+ 00 6.27366E 01 0.00000E+ 00 0.00000E+ 00 0.00000E+ 00 3.14816E 01

210

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Table A4.3 Cross-sections of driver without moderator. Group

1

2

3

4

Σga (cm1) νΣgf (cm1) Σgt (cm1) Σg1 (cm1) s g2 Σs (cm1) Σg3 (cm1) s g4 Σs (cm1)

7.09892E 03 1.59878E 02 1.40226E 01 1.19887E 01 1.30790E 02 1.59938E 04 1.07166E 06

9.02877E  03 1.64446E  02 2.28245E  01 0.00000E + 00 2.15213E  01 4.00117E  03 1.82716E  06

1.72478E  02 2.71451E  02 3.25806E  01 0.00000E + 00 0.00000E + 00 3.06885E  01 1.67341E  03

5.74211E 02 8.45807E 02 4.18327E 01 0.00000E +00 0.00000E +00 0.00000E +00 3.60906E 01

Table A4.4 Cross-sections of driver with moderator. Group

1

2

3

4

Σga (cm1) νΣgf (cm1) Σgt (cm1) Σg1 (cm1) s g2 Σs (cm1) Σg3 (cm1) s g4 Σs (cm1)

4.67223E 03 1.01663E 02 1.41428E 01 1.14337E 01 2.09664E 02 1.39132E 03 6.10281E 05

5.57965E  03 9.46359E  03 2.45394E  01 0.00000E + 00 2.12006E  01 2.67269E  02 1.08186E  03

1.32590E  02 1.87325E  02 3.98255E  01 0.00000E + 00 0.00000E + 00 3.52093E  01 3.29030E  02

6.51184E 02 8.25335E 02 4.35990E 01 0.00000E +00 0.00000E +00 0.00000E +00 3.70872E 01

Table A4.5 Cross-sections of reflector without moderator. Group

1

2

3

4

Σga (cm1) νΣgf (cm1) Σgt (cm1) Σg1 (cm1) s Σg2 (cm1) s Σg3 (cm1) s g4 Σs (cm1)

4.64814E 04 0.00000E+ 00 1.59346E 01 1.47969E 01 1.06607E 02 2.49956E 04 1.82565E 06

4.76496E  04 0.00000E + 00 2.16355E  01 0.00000E + 00 2.10410E  01 5.46711E  03 1.00157E  06

1.23810E  03 0.00000E + 00 3.48692E  01 0.00000E + 00 0.00000E + 00 3.42085E  01 5.36879E  03

4.94333E 03 0.00000E +00 6.24249E 01 0.00000E +00 0.00000E +00 0.00000E +00 6.19306E 01

Table A4.6 Cross-sections of reflector with moderator. Group

1

2

3

4

Σga (cm1) νΣgf (cm1) Σgt (cm1) Σg1 (cm1) s Σg2 (cm1) s g3 Σs (cm1) Σg4 (cm1) s

3.97516E 04 0.00000E+ 00 1.39164E 01 1.05911E 01 2.96485E 02 3.06502E 03 1.41697E 04

3.02674E  04 0.00000E + 00 2.46993E  01 0.00000E + 00 1.84820E  01 5.91780E  02 2.69229E  03

1.22034E  03 0.00000E + 00 4.52425E  01 0.00000E + 00 0.00000E + 00 3.73072E  01 7.81326E  02

2.41527E 02 0.00000E +00 5.36256E 01 0.00000E +00 0.00000E +00 0.00000E +00 5.12103E 01

The discrete ordinate method

211

Table A4.7 Cross-sections of KNK-1 reflector. Group

1

2

3

4

Σga (cm1) νΣgf (cm1) Σgt (cm1) Σg1 (cm1) s g2 Σs (cm1) Σg3 (cm1) s g4 Σs (cm1)

4.58692E  04 0.00000E + 00 1.51644E  01 1.38427E  01 1.23901E  02 3.66930E  04 1.69036E  06

4.59443E  04 0.00000E + 00 1.42382E  01 0.00000E + 00 1.37502E  01 4.41927E  03 1.63280E  06

1.07883E 03 0.00000E +00 1.65132E 01 0.00000E +00 0.00000E +00 1.60722E 01 3.33075E 03

5.91325E 03 0.00000E+ 00 8.04845E 01 0.00000E+ 00 0.00000E+ 00 0.00000E+ 00 7.98932E 01

Table A4.8 Cross-sections of sodium/steel. Group

1

2

3

4

Σga (cm1) νΣgf (cm1) Σgt (cm1) Σg1 (cm1) s g2 Σs (cm1) Σg3 (cm1) s g4 Σs (cm1)

2.25039E  04 0.00000E + 00 9.65097E  02 8.83550E  02 7.73409E  03 1.94719E  04 8.89615E  07

2.33696E  04 0.00000E + 00 9.87095E  02 0.00000E + 00 9.52493E  02 3.22568E  03 7.98494E  07

5.39303E 04 0.00000E +00 1.34200E 01 0.00000E +00 0.00000E +00 1.30756E 01 2.90481E 03

3.03759E 03 0.00000E+ 00 4.12670E 01 0.00000E+ 00 0.00000E+ 00 0.00000E+ 00 4.09632E 01

Table A4.9 Cross-sections of steel. Group

1

2

3

4

Σga (cm1) νΣgf (cm1) Σgt (cm1) Σg1 (cm1) s Σg2 (cm1) s g3 Σs (cm1) Σg4 (cm1) s

2.16042E  04 0.00000E + 00 9.83638E  02 9.06050E  02 7.42377E  03 1.18163E  04 8.25890E  07

2.06601E  04 0.00000E + 00 1.35140E  01 0.00000E + 00 1.30581E  01 4.35250E  03 3.41675E  07

5.56175E 04 0.00000E +00 2.24749E 01 0.00000E +00 0.00000E +00 2.19547E 01 4.64594E 03

2.40965E 03 0.00000E+ 00 2.83117E 01 0.00000E+ 00 0.00000E+ 00 0.00000E+ 00 2.80707E 01

Table A4.10 Cross-sections of axial blanket. Group

1

2

3

4

Σga (cm1) νΣgf (cm1) Σgt (cm1) Σg1 (cm1) s Σg2 (cm1) s g3 Σs (cm1) Σg4 (cm1) s

1.93752E  03 2.96101E  03 1.40462E  01 1.23805E  01 1.45483E  02 1.70276E  04 9.37083E  07

1.47927E  03 6.56171E  05 2.25534E  01 0.00000E + 00 2.17260E  01 6.78885E  03 6.04793E  06

4.72919E 03 1.14630E 04 3.27065E 01 0.00000E +00 0.00000E +00 3.17948E 01 4.38782E 03

9.94260E 03 4.93483E 04 3.41224E 01 0.00000E+ 00 0.00000E+ 00 0.00000E+ 00 3.31281E 01

212

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Table A4.11 Cross-sections of axial reflector. Group

1

2

3

4

Σga (cm1) νΣgf (cm1) Σgt (cm1) Σg1 (cm1) s Σg2 (cm1) s g3 Σs (cm1) Σg4 (cm1) s

3.30045E 04 0.00000E+ 00 1.32933E 01 1.22995E 01 9.41231E 03 1.93791E 04 1.39307E 06

3.36186E  04 0.00000E + 00 1.78531E  01 0.00000E + 00 1.73095E  01 5.09881E  03 7.05075E  07

8.61269E  04 0.00000E + 00 2.83151E  01 0.00000E + 00 0.00000E + 00 2.77194E  01 5.09601E  03

3.56939E 03 0.00000E +00 4.62167E 01 0.00000E +00 0.00000E +00 0.00000E +00 4.58598E 01

Table A4.12 Cross-sections of control rod follower. Group

1

2

3

4

Σga (cm1) νΣgf (cm1) Σgt (cm1) Σg1 (cm1) s g2 Σs (cm1) Σg3 (cm1) s g4 Σs (cm1)

9.06964E 05 0.00000E+ 00 7.27587E 02 6.63634E 02 6.23393E 03 7.02121E 05 4.16388E 07

8.04746E  05 0.00000E + 00 1.00218E  01 0.00000E + 00 9.61236E  02 4.01375E  03 1.26939E  07

1.96101E  04 0.00000E + 00 1.60703E  01 0.00000E + 00 0.00000E + 00 1.56016E  01 4.49111E  03

1.20796E 03 0.00000E +00 1.51576E 01 0.00000E +00 0.00000E +00 0.00000E +00 1.50368E 01

Table A4.13 Fission spectrum. Group

1

2

3

4

Spectrum

0.908564

0.087307

0.004129

0.000000

References [1] B.G. Carlson, Transport Theory: Discrete Ordinate Quadrature over the Unit Sphere, Los Alamos Scientific Lab, N. Mex. No. LA-4554, 1970. [2] W.W.J. Engle, The User’s Manual for ANISN, Oak Ridge Gaseous Diffusion Plant, USA, 1967. K-1694. [3] W.A. Rhoades, F.R. Mynatt, The DOT-IV: Two-Dimensional Discrete Ordinates Transport Code, Oak Ridge National Laboratory, USA, No. ORNL-TM–6529, 1979. [4] J. Wood, M.M.R. Williams, Recent progress in the application of the finite element method to the neutron transport equation, Prog. Nucl. Energy 11 (1) (1984) 21–40. [5] E.E. Lewis, Finite element approximation to the even-parity transport equation, Adv. Nucl. Sci. Technol. 13 (1981) 155–325. [6] G.J. Gesh, Finite Element Method for Second Order Forms of the Transport Equation, Texas A&M University, Texas, 1999. [7] R.T. Ackroyd, O.A. Abuzid, A.M. Mirza, Discontinuous finite element solutions for neutron transport in X-Y geometry, Ann. Nucl. Energy 22 (3–4) (1995) 181–201.

The discrete ordinate method

213

[8] T.A. Wareing, J.M. McGhee, J.E. Morel, et al., Discontinuous finite element Sn methods on three-dimensional unstructured grids, Nucl. Sci. Eng. 138 (3) (2001) 256–268. [9] T.A. Manteuffel, K.J. Ressel, G. Starke, Least-squares finite-element solution of the neutron transport equation in diffusive regimes, SIAM J. Numer. Anal. 35 (2) (1998) 806–835. [10] H.L. Lu, H.C. Wu, L.Z. Cao, Two-dimensional nodal transport method for triangular geometry, Ann. Nucl. Energy 34 (5) (2007) 424–432. [11] H.L. Lu, H.C. Wu, A nodal SN transport method for the three-dimensional triangular-z geometry, Nucl. Eng. Des. 237 (8) (2007) 830–839. [12] H.T. Ju, H.C. Wu, Y.Q. Zhou, et al., A least-squares finite-element SN method for solving first-order neutron transport equation, Nucl. Eng. Des. 237 (8) (2007) 823–829. [13] T. Takeda, H. Ikeda, 3-D Neutron Transport Benchmarks, Department of Nuclear Engineering, Osaka University, Japan, March, NEACRP-L-330, 1991. [14] A.K. Ziver, M.S. Shahdatullah, M.D. Eaton, et al., Finite element spherical harmonics(PN) solutions of the three-dimensional takeda benchmark problems, Ann. Nucl. Energy 32 (9) (2005) 925–948. [15] R.D. Mosteller, L.D. Eisenhart, Benchmark calculations for the doppler coefficient of reactivity, Nucl. Sci. Eng. 107 (3) (1991) 265–271. [16] G. Marleau, A. Hebert, R. Roy, A User’s Guide for DRAGON, Institut de genie nucleaire,  Departement de genie physique, Ecole Polytechnique de Montreal, France, February. IGE–174, 2016. [17] IAEA-TECDOC, Research Reactor Core Conversion from the Use of Highly Enriched Uranium to the Use of Low Enriched Uranium Fuels Guidebook, 1980, International Atomic Energy Agency, Vienna, IAEA-TECDOC-233. [18] A. Badruzzaman, An efficient algorithm for nodal-transport solutions in multidimensional geometry, Nucl. Sci. Eng. 89 (3) (1985) 281–290. [19] H. Ikeda, T. Takeda, A new nodal SN transport method for three-dimensional hexagonal geometry, J. Nucl. Sci. Technol. 31 (6) (1994) 497–509. [20] Q.W. Liu, Discrete Ordinates Solutions of the Time-Independent and Time-Dependent Neutron Transport Equations in Three-Dimensional Geometry, (Master thesis), Xi’an Jiaotong University, 2006. [21] T. Takeda, H. Ikeda, 3-D neutron transport benchmark, J. Nucl. Sci. Technol. 28 (7) (1991) 656–669. [22] H.C. Wu, P.P. Liu, Y.Q. Zhou, et al., Transmission probability method based on triangle meshes for solving unstructured geometry neutron transport problem, Nucl. Eng. Des. 237 (1) (2007) 28–37. [23] L.Z. Cao, H.C. Wu, Spherical harmonics method for neutron transport equation based on unstructured-meshes, Nucl. Sci. Technol. 15 (6) (2004) 335–339.

Mesh-free method Haochun Zhang and Yining Zhang Harbin Institute of Technology, Harbin, People’s Republic of China

8.1

8

Introduction of mesh-free method

The calculation procedures of most traditional methods, such as discrete ordinate method or finite volume method, rely on grids which need to be meshed beforehand and the mesh work always requires considerable energy and time. Thus, the mesh-free or meshless method has attracted more interest. One of the most useful features of mesh-free method is its geometric adaptability and ease of application for three dimensions. Besides, this method also eliminates the labor-intensive step of mesh generation. According to the formulation procedure, meshless methods can be divided into three types: weak form (e.g., element-free Galerkin method (EFGM), and meshless local Petrov-Galerkin method (MLPG)), strong form (e.g., collocation method), and the combination of weak and strong form (e.g., meshless weak-strong (MWS) form method). The collocation meshless method is a classical type of strong form, which means that the direct approximate solution for the partial differential equation is adopted. All the governing equations and boundary equations are discretized at some field nodes. It has the advantages of simple algorithm, computational efficiency, and truly meshless. In 1990, Kansa [1] proposed meshless radial basis function collocation method (RBFCM) to solve partial differential equation. Then this method was applied to many areas during the next decades, such as solving of Navier-Stokes equations, radiative transport equation, and analyzing structural mechanics. This method was also applied to laminar flame propagation, to calculate the band structures of periodic composite structures, and so on. In Kansa’s method, the radial basis function (RBF) is interpolated at all discrete nodes. Thus, this method is known as global RBF collocation method (GRBFCM), which is a mesh-less scheme of full matrices and may cause ill-conditioning under some special conditions. In order to overcome the above problems, the local radial basis function collocation method (LRBFCM) is developed, in which the RBF is interpolated on discrete nodes in the support domain, instead of the whole solution domain, and the interpolation matrix is sparse. But the accuracy of LRBFCM is not as high as the GRBFCM method. In 2014, Tanbay and Ozgener [2] used the GRBFCM method to solve the neutron diffusion equation (NDE) for the first time. They verified the applicability of this method in a square homogenous geometry. Then, they compared the accuracy and efficiency of this method with finite element method (FEM) and boundary element method (BEM) using the same

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation. https://doi.org/10.1016/B978-0-12-818221-5.00008-8 Copyright © 2021 Elsevier Ltd. All rights reserved.

216

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

homogenous geometry with different neutron source and nuclear cross section data. They focused on the influence of shape parameter on the stability and accuracy of numerical procedure. In 2018, Zhang [3] et al. proposed a block meshless method to solve the NDE to improve the speed and accuracy of traditional global and local meshless method. In their study, some satisfied numerical results have been obtained.

8.2

Theory of mesh-free method

8.2.1 Support domain Kansa’s global method is the original collocation mesh-free method and in this method, the support domain is the whole structural domain for every boundary node and interior node, as shown in Fig. 8.1A. For each discrete node, all of the discrete nodes in its support domain provide supporting data, which causes the interpolation matrix full and the calculation time consumption huge. To overcome the calculation cost and probable ill-condition problem, the local method is proposed. In the local method, the support domain of each discrete node is a local domain surrounding the node itself, rather than the whole solution domain, as shown in Fig. 8.1B. In this example, the support domain of representative node A or B is a roundness domain with radius rm and the circular center is the discrete node itself. Thus, the interpolation matrix is band sparse and the application of large-scale problem is feasible. But the disadvantage of this method is the difficulty to solve the multimaterial problem. The radial basis function interpolation of a discrete node may cause obvious error, when its support domain crosses two types of materials. By Zhang’s block method, the whole domain is divided into several blocks and each block shares the same homogeneous material. The support domain of every interior and exterior boundary node is chosen as the geometric block in which the discrete node located. Interface boundary nodes are additionally introduced to impose interface boundary conditions (zero- and one-order continuous conditions) between two types of materials. The node collocations and support domains of representative nodes are shown in Fig. 8.1C. The detailed mathematical method is described in the following. Normally, a reactor core consists of many fuel assembles (FA) and several kinds of materials. Meanwhile, the support domain of a discrete node is just chosen as the FA or sub-FA area (both can be named as block here) in the block method. Thus, the core neutronics problems can be easily solved by block method with the advantages of ease to build core structure and set complicated properties, avoiding time-consuming mesh work and applying to large-scale problem as its interpolation matrix is sparse. The following introduction is mainly based on the NDE, but it is very easy to extend the method to neutron transport equation (NTE).

Mesh-free method

(A)

217

(B)

(C) Fig. 8.1 The differences of node collocations and support domains of representative nodes between GRBFCM, LRBFCM, and BRBFCM methods (the green areas represent support domains, the black and blue nodes represent the interior and boundary discrete nodes, respectively). (A) Global method, (B) Local method, and (C) Block method.

8.2.2 Formulation Based on the multigroup neutron diffusion theory, the general multidimensional time-dependent NDEs with delayed neutron precursors can be established. The NDE for group g can be written as

218

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation


 ! h!
i

 1 ∂φg r , t 1 ! ! ! + r  U φg r , t ¼ r  Dg r rφg r , t ∂t vg vg





 X   G G X X ! ! ! ! ! ! +χ p, g ð1  βÞ ðυΣÞf , g0 r φg0 r , t + r φg0 r , t  Σr, g r φg r , t g0 !g g0 ¼1

+

l X




!

g0 ¼1



χ d , i, g λi Ci r , t + S g

i¼1

(8.1)

The delayed neutron precursor balance equation of family i can be written as
 ! ∂Ci r , t ∂t

G h!
i 


X ! ! ! ! + r  U Ci r , t ¼ β i ðυΣÞf , g0 r φg0 r , t  λi Ci r , t

(8.2)

g0 ¼1

where φ represents the neutron scalar flux density, neutronscm2 s1; Ci represents the delayed neutron precursor concentration of family i, precursorscm3; Sg is the !

extra neutron source, neutronscm3 s1; v is the neutron velocity, cms1; U is the liquid fuel velocity, cms1; D is the neutron diffusion coefficient, cm; Σ f is the macroscopic fission cross section, cm1; Σ g0 ! g is the macroscopic scattering cross section from g0 energy group to g energy group, cm1; Σ r is the macroscopic removal cross section, cm1; υ is the average number of neutrons that are emitted from each fission process; χ p is the fission spectrum of the prompt neutrons; χ d is the fission spectrum of the delayed neutrons; λi is the decay constant of the delayed neutron precursor of fraction of the delayed neutron precursor of family i; and β is the family i; βi is the P sum of all βi, β ¼ liβi. !

The terms containing U in Eqs. (8.1), (8.2) represent the convection terms such as the flow effect of the liquid fuel on neutron flux and delayed neutron precursor concentration. These terms has obvious influence only on liquid fuel reactors, such as molten salt reactor (MSR). While calculating, the power iteration method is adopted and an eigenvalue named the effective multiplication factor keff is introduced, which is defined as ðX G g0 ¼1

ðnÞ

keff ¼

1 ðn1Þ

keff

ðnÞ

ðυΣÞf , g0 φg0 dV

ðX G g0 ¼1

(8.3) ðn1Þ ðυΣÞf , g0 φg0 dV

where (n) and (n 1) represent the current and last iteration step. The NDE with delayed neutrons for solid fuel reactors can be written as

Mesh-free method

219


 ! G
 1 


X 1 ∂φg r , t ! ! ! ! χ p, g ð1  βÞ ðυΣÞf , g0 r φg0 r , t ¼ r  Dg r rφg r , t + ∂t vg keff g0 ¼1 G X l   X




X ! ! ! ! ! r r , t  Σ r r , t + χ λ C r , t + Sg φ φ + 0 r , g i i g g d , i , g 0 g !g g0 ¼1

i¼1

(8.4)


 ! ∂Ci r , t ∂t

¼

G 


1 X ! ! ! βi ðυΣÞf , g0 r φg0 r , t  λi Ci r , t keff g0 ¼1

(8.5)

For steady neutronics problem without delayed neutrons and upscattering, Eq. (8.4) can be simplified as



 ! ! ! ! r  Dg r rφg r + Σr, g r φg r g1 X G

 X

 X 1 ! ! ! ! χ p, g ðυΣÞf , g0 r φg0 r + r φg 0 r + S g ¼ g0 !g keff 0 0 g ¼1 g ¼1

(8.6)

The iterative algorithm is continued using Eqs. (8.3)–(8.5), until a predetermined convergence criterion is satisfied   ðnÞ k  kðn1Þ    eff eff  > >
> ! > : ∂ψ j r ki =∂n

,

j¼1,2,…M + N

for vacuum boundary (8.17b) , for reflective boundray

For interface boundary, zero- and one-order continuous conditions need to be met in functional space:

222

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation M +N X

M +N

 X ! ! p, ðnÞ q, ðnÞ ψ j r pi aj,g  ψ j r qi aj, g ¼ 0

j¼1

(8.18a)

j¼1

M +N X

Dpg


 ! ∂ψ j r pi ∂n

j¼1

apj, g 

M +N X

Dqg

j¼1


 ! ∂ψ j r qi ∂n

aqj, g ¼ 0

(8.18b)

where p and q are sequence number of two neighbor blocks, which share a same interface boundary. The matrix form of Eq. (8.18) can be written as " ½ApT where

AqT 

# p, ðnÞ ag q, ðnÞ ¼ 0 ag

(8.19)


3 ! ApT0, j r pi 7 6 ApT ¼ 4
5 !p p AT1, j r i


3 ! AqT0, j r qi 7 6 AqT ¼ 4
5 !q q AT1, j r i

2

h
i ! ApT0, j ¼ ψ j r pi h

2

, i¼1, 2, …, NTk

j¼1, 2, …M + N




!



ApT1, j ¼ Dpg  ∂ψ j r pi =∂n

h




!

, AqT0, j ¼ ψ j r qi

i

i

h

j¼1, 2, …M + N

i¼1, 2, …, NTk

j¼1, 2, …M + N


 iT ! , AqT1, j ¼ Dqg  ∂ψ j r qi =∂n

j¼1, 2, …M + N

(8.20)

Combining Eqs. (8.12), (8.16), (8.19), the final discretized system of liner equations can be written in a matrix form as A  aðgnÞ where



Sð n Þ ¼ g 0

(8.21)

2

A1T 0 AsT

⋯

0

3

7 6 7 6

T p q 7 ⋮ ⋱ A 0 A ⋮ A ¼ AI AE AT mðM + N ÞmðM + N Þ , AT ¼ 6 7 6 T T 5 4 t m 0 ⋯ ⋱ AT 0 AT N mðM + N Þ T 3 3 2 1 2 1 ⋯ 0 ⋯ 0 AI AE 7 7 6 6 7 7 6 6 ⋱ ⋱ 7 7 6 6 7 7 6 6 k k AI AE AI ¼ 6 ⋮ ⋮ 7 AE ¼ 6 ⋮ ⋮ 7 7 7 6 6 7 7 6 6 ⋱ ⋱ 5 5 4 4 0 ⋯ Am 0 ⋯ Am I mMmðM + N Þ E NE mðM + N Þ h iT h iT ðnÞ ð n Þ ag ¼ a1g, ðnÞ ⋯ agk, ðnÞ ⋯ agm, ðnÞ , Sg ¼ S1g, ðnÞ ⋯ Sgk, ðnÞ ⋯ Sgm, ðnÞ (8.22)

Mesh-free method

223

In matrix AT, superscripts s and t represent two blocks which are the neighbor of blocks 1 and m, respectively. The line amounts of matrix AT and AE are NT ¼

m P

k¼1

NTk , NE ¼

m P

k¼1

NEk , and NT + NE ¼ mN. Thus, the matrix A is nonsingular and

shown to be constant for each energy group. The corresponding coefficients matrix of energy group g can be obtained: " aðgnÞ

¼A

1



SðgnÞ

# (8.23)

0

Finally, substituting ak,(n) (the corresponding coefficients in block k) into Eq. (8.8), the g neutron flux density at each point can be solved. If the neutron delay effect needs to be considered, the delayed neutron precursor balance equation, e.g., Eq. (8.5), can also be derived by semi-implicit time discretization scheme:
 ðnÞ ! Ci r ¼


 ðn1Þ !

 C r i βi ! ðn1Þ ! ðυΣÞf ,g0 r φg0 r + λi Δt + 1 keff ðλi + 1=ΔtÞ g0 ¼1 G X

(8.24)

For NTE, the governing equation is h

G X  X
i


 ! ! ^ ! ! ^  r + Σt, g ! Ω r ϕg r , Ω ¼ r φg0 r 0 s, g !g g0 ¼1

G

 1 X ! ! υg0 Σf , g0 r φg0 r + χg λ g0 ¼1
 ! ^ + Se, g r , Ω

(8.25)

The angular space is discretized into M  N directions, ΔΨ ¼ π/M, Δθ ¼ 2π/N. The weight of every direction is wmm0 ¼

ð θm0 Δθ=2 ð ψ m0 Δψ =2 θm0 Δθ=2 ψ m0 Δψ=2

f ðθm , ψ m , θ0 , ψ 0 Þdψ 0 dθ0

(8.26)

8.2.4 Numerical procedure and core development For the convenience of model establishing, blocks dividing, and properties setting, a user-friendly code named block radial basis function collocation meshless for two dimension (BRBFCM-2D) is developed by C# programming language. The main code used for numerical calculation is programmed by C language and the numerical procedure of BRBFCM is carried out in the following ways:

224

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Establish the blocks structure and set the properties of all the materials. Generate interior and boundary nodes. Calculate all the elements in the interpolation matrix A, according to Eq. (8.20). Set or update the iteration initial value for neutron flux φ(n1) and the eigenvalue k(n1) . eff Calculate the elements in the source matrix S, according to Eq. (8.13d). Solve the linear equation (8.23) to obtain the corresponding coefficients matrix a for every group g, then calculate the neutron flux φ(n) by RBF interpolation function Eq. (8.8). Step 7. Calculate the eigenvalue k(n) eff and the relative error εr. If the convergence criterion is satisfied, terminate the iteration process and go to step 8; otherwise, go back to step 4. Step 8. If the time t reaches the preset scope, terminate the iteration; otherwise, accumulate new time as t(n) ¼ t(n1) + Δt and go back to step 4. Step 9. If a transient problem needs to be solved, steps 1–8 should be carried out. Otherwise, step 8 should be ignored for steady problems. The procedure flowchart and relationship between databases and executable programs are shown in Fig. 8.3. Step 1. Step 2. Step 3. Step 4. Step 5. Step 6.

8.3

Numerical results

Seven examples are simulated to verify the numerical performances of the block mesh-free method for solving the neutronics problem, among which the first five problems are diffusion problems and the latter two problems are transport problems. All the node collocations and numerical calculations are implemented by the BRBFCM-2D codes. The eigenvalue keff and some physical fields, such as neutron flux and normalized power distribution, are compared between the numerical results of the block mesh-free method and the benchmark reference.

Fig. 8.3 Software procedure flowchart of BRBFCM-2D.

Mesh-free method

225

8.3.1 2D2G problem Firstly, to assess the applicability and accuracy of the block mesh-free method, a twodimensional-two-group (2D2G) problem is studied, before applying this method for core scale problem. Meanwhile, some key parameters of this method, such as the shape factor c and block quantities m, are also investigated. The geometric and physical structure of 2D2G problem is illustrated in Fig. 8.4. It is a square geometry with two vacuum boundaries and two reflective boundaries. Neutron energy spectrum is divided into two energy groups with fission source and without extra source. The length l of each side of the square is 25 cm and nuclear physical data are presented in Table 8.1. This problem has an analytical solution of keff, anl ¼ 1.96293774. To solve this problem, four groups of blocks and node collocations are applied. For comparison, the total number of interior nodes are kept invariant. The number of blocks, interior nodes, and boundary nodes of each group are presented in Table 8.2. In each collocation group, the distance between every two neighbor interior nodes are set as a constant h. For boundary nodes, the distance from a boundary node to its nearest interior node is h/2 and two neighbor boundary nodes also share the same interval h. The blocks and node collocation diagram of No. 2 group for 2D2G problem are illustrated in Fig. 8.5. Reflective boundary Vacuum boundary

Reflective boundary

Fig. 8.4 Geometric and physical structure of 2D2G problem. Table 8.1 Nuclear data of 2D2G problem. P 21 g D (cm) υ χ ) f (cm

P

1 2

0.13552 0.08228

1.2245 1.2245

2.65 2.55

0.575 0.425

0.06300 0.06776

r

(cm21)

P

s, g!g+1

(cm21)

0.0676 –

Table 8.2 Blocks and collocated nodes for 2D2G problem. No.

blocks

Interior nodes

Boundary nodes

Total nodes

1 2 3 4

22¼4 4  4 ¼ 16 6  6 ¼ 36 8  8 ¼ 64

12  12  4 ¼ 576 6  6  16 ¼ 576 4  4  36 ¼ 576 3  3  64 ¼ 576

144 240 336 432

720 816 912 1008

226

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Fig. 8.5 Node collocation of m ¼ 4  4 blocks for 2D2G problem.

The quantity named relative error εr is used to measure the difference between the numerical solution keff, num and the analytical solution keff, anl:   keff ,num  keff , anl   εr ¼   keff ,anl

(8.27)

Relative error of keff

Fig. 8.6 shows the relative error εr as a function of the shape factor square c2 using four different blocks and node collocations. The exponential convergence can be observed

2 ¥ 2 blocks 4 ¥ 4 blocks 6 ¥ 6 blocks 8 ¥ 8 blocks

2

Shape factor square, c

Fig. 8.6 Relative error of keff as a function of the shape factor square for spatial resolutions m ¼ 2  2 (square symbols), m ¼ 4  4 (round symbols), m ¼ 6  6 (inverted triangle symbols), m ¼ 8  8 (positive triangle symbols).

Mesh-free method

227

clearly and the final accuracy is high, resulting in relative error of the global eigenvalue keff of the order of 105 for m ¼ 2  2 and m ¼ 4  4 blocks. It can also be observed that, for higher c2 values, the liner system becomes ill-conditioned and the calculation cannot be converged. This example shows convergence regular and the relationship between accuracy and shape factor. In the following problems, the principle of setting shape factor is that this number should as large as possible until the iterative equations are ill-conditioned and the calculation is diverging. In order to compare the numerical performance between GRBFCM, LRBFCM, and our BRBFCM, four groups of calculations are carried out in this section. For LRBFCM, the radius of support domain rm is a key parameter, so two different sizes of it are adopted, i.e., rm ¼ 25 cm and rm ¼ 15 cm. In order to eliminate the effect of discrete node quantities, the total nodes of the four calculations are generated approximately equally. The calculation results, relative errors, and time consumptions of different numerical methods are listed in Table 8.3, and the numerical results of fast and thermal neutron flux on the diagonal line are plotted in Fig. 8.7. It can be seen that the accuracies of GRBFCM and BRBFCM are of the same level since the relative error εr of these two methods are of the order of 105. But the time consumptions differ sharply. The BRBFCM only takes 1.029 s to achieve convergence, compared to 35.955 s of GRBFCM, which means that 97.1% calculation time is saved. The reason Table 8.3 Calculation results, relative errors and time consumptions of 2D2G problem by GRBFCM, LRBFCM and BRBFCM. No.

Numerical method

Total nodes

keff,num

εr (%)

Time consumption (s)

1 2 3 4

GRBFCM LRBFCM (rm ¼ 25 cm) LRBFCM (rm ¼ 15 cm) BRBFCM (m ¼ 6  6)

900 900 900 912

1.962904 1.968107 1.994903 1.962915

0.0017 0.26 1.6 0.0012

35.955 21.791 16.638 1.029

GRBFCM LRBFCM (rm=25 cm)

Fast neutron flux

5

LRBFCM (rm=15 cm) 4

BRBFCM

3 2

12 Thermal neutron flux

6

(A)

LRBFCM (rm=15 cm) 8

BRBFCM

6 4 2

1 0

GRBFCM LRBFCM (rm=25 cm)

10

0

0

5

10 15 Distance (cm)

20

25

(B)

0

5

10 15 Distance (cm)

20

Fig. 8.7 Calculation results of neutron flux on the diagonal line for 2D2G problem by GRBFCM, LRBFCM, and BRBFCM, (A) Fast neutron flux, (B) Thermal neutron flux.

25

228

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

is that the interpolation matrix is full by GRBFCM, while it is very sparse by BRBFCM. For LRBFCM, although the time consumption reduces as the radius of support domain decreases, the accuracy becomes worse. In total, the BRBFCM has the advantages in both the accuracy and the efficiency. This problem simulates the neutronics behaviors in a one-quarter square nuclear fuel model, and the neutrons accumulative effect can be observed clearly at the symmetric center, causing a neutron flux peak at distance¼ 0 in Fig. 8.7A and B, for both fast and thermal neutrons. Additionally, the fission spectrum distributes in both fast and thermal group (χ ¼ 0.575 for fast group and χ ¼ 0.425 for thermal group) for this problem, and the scatter reaction only occurs from fast group to thermal group, indicating that the thermal neutron flux is obviously higher than the fast neutron flux, as shown in Fig. 8.7A and B.

8.3.2 2D4G problem In order to show the applicability and accuracy of BRBFCM in more energy groups, another 2D square problem in introduced in this section. It is a two-dimensional fourgroup (2D4G) problem and has the same geometry and boundary conditions as the 2D2G problem. The difference is that the neutron energy spectrum is divided into four groups. The group constants for the 2D4G problem are listed in Table 8.4. Four calculations are carried out using the node collocations presented in Table 8.2. Meanwhile, the reference value of effective multiplication factor is keff,ref ¼ 0.87227. The calculation results, relative errors, and time consumptions are listed in Table 8.5. It can also be observed that as more blocks are divided, more calculation time can be saved. The reason for this is the small blocks meaning narrow bandwidths of interpolation matrix, thus reduced calculation cost. Altogether, a good agreement between numerical and reference values has been obtained, which verifies the applicability of BRBFCM in four energy group situation.

8.3.3 2D-IAEA problem This problem is a two-dimensional pressurized water reactor (PWR) benchmark problem published by the International Atomic Energy Agency (IAEA) and named the 2D-IAEA problem. It is a typical simplified PWR which is widely used in the verification of neutron diffusion calculation codes. The reactor has a two-zone core and is reflected by water radially. There are 177 groups of FAs arranged in the core and nine groups of these FAs are inserted control rods. Four kinds of materials are applied in this core. A quarter of 2D-IAEA problem is illustrated in Fig. 8.8. The neutron energy spectrum is divided into two groups: fast and thermal neutrons. The nuclear data are presented in Table 8.6. Two groups of node collocations are generated and applied for this problem by BRBFCM. In both schemes, blocks are set of same scale as 10  10 cm. The difference is that the interior node quantity for one scheme is M ¼ 2  2, and for the other is M ¼ 4  4. The node collocation diagram of M ¼ 2  2 for 2D-IAEA problem is illustrated in Fig. 8.9. In GRBFCM, the scale of this problem is so large that the time and cost are unbearable. This method is not suitable for the multimaterial reactor core scale problem. In LRBFCM, one group of node collocation is generated and the node

Table 8.4 Nuclear data of 2D4G problem. g

D (cm)

υ

Σf,g (cm21)

Σr,g (cm21)

Σs,g!g+1 (cm21)

Σs,g! g+2 (cm21)

Σs,g! g+3 (cm21)

χ

1 2 3 4

2.876787 1.570845 0.722486 0.964199

2.4 2.4 2.4 2.4

0.0049492 0.0022188 0.0043629 0.0110879

0.028204 0.005275 0.017612 0.026546

0.023597 0.001615 0.004684 0

4.079E 6 4.231E  8 0 0

4.449E  8 0 0 0

0.768 0.232 0 0

230

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Table 8.5 Calculation results, relative errors and time consumptions of 2D4G problem. No.

Numerical method

Total nodes

keff,num

εr (%)

Time consumption (s)

1 2 3 4

BRBFCM (m ¼ 2  2) BRBFCM (m ¼ 4  4) BRBFCM (m ¼ 6  6) BRBFCM (m ¼ 8  8)

720 816 912 1008

0.872188 0.872209 0.872212 0.872220

0.0094 0.0070 0.0067 0.0058

3.286 1.810 1.100 1.003

Fig. 8.8 One quarter of the 2D-IAEA reactor.

Table 8.6 Two-group nuclear data for 2D-IAEA problem. Dg Region Group g (cm)

Σa,g (cm21)

υΣf,g (cm21)

χ

Σs,g! g+1 (cm21)

A

0.01 0.08 0.01 0.085 0.01 0.13 0 0.01

0 0.135 0 0.135 0 0.135 0 0

1.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0

0.02 0 0.02 0 0.02 0 0.04 0

B C D

1 2 1 2 1 2 1 2

1.5 0.4 1.5 0.4 1.5 0.4 2.0 0.3

Material Fuel 1 Fuel 2 Fuel 2 + rod Reflector

Mesh-free method

231

Fig. 8.9 Node collocation of M ¼ 2  2 for 2D-IAEA problem.

quantity is approximated to the M ¼ 2  2 scheme of BRBFCM. Two sizes of support domain are adopted by LRBFCM: rm ¼ 50 cm and rm ¼ 100 cm. The calculation results, relative errors, and time consumptions are listed in Table 8.7. The normalized fast and thermal neutron flux distributions on the diagonal line of the calculation domain are also plotted in Fig. 8.10. Firstly, the BRBFCM has a great advantage over LRBFCM in the aspect of time consumption. When compared the same node quantity level calculation No. 1, No. 2, and No. 3 in Table 8.7, the time consumptions can be reduced from 724 s (calculation No. 1) and 3456s (calculation No. 2) to 6.602 s (calculation No. 3), which means that 99% and 99.8% calculational time is saved, respectively. Secondly, the accuracy of BRBFCM is also better than LRBFCM, and as more nodes used, a higher degree of accuracy could also be gained, as we expected. The

Table 8.7 Calculation results, relative errors and time consumptions of 2D-IAEA problem by LRBFCM and BRBFCM. No.

Numerical method

Total nodes

keff,num

εr (%)

Time consumption (s)

1 2 3 4

LRBFCM (rm ¼ 50 cm) LRBFCM (rm ¼ 100 cm) BRBFCM (M ¼ 2  2) BRBFCM (M ¼ 4  4)

2272 2272 1996 5920

1.042859 1.037480 1.029247 1.029563

1.29 0.77 0.033 0.0021

724 3456 6.602 17.267

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

50

20

40

16

Thermal neu ron flux

Fast neu ron flux

232

30 20

Reference LRBFCM (rm=50 cm)

10

LRBFCM (rm=100 cm) BRBFCM (M=2 ¥ 2) BRBFCM (M=4 ¥ 4)

0 0

(A)

20

40

60

80

Distance(cm)

100

Reference LRBFCM (rm=50 cm) LRBFCM (rm=100 cm) BRBFCM (M=2 ¥ 2) BRBFCM (M=4 ¥ 4)

12 8 4 0

120

0

(B)

20

40

60

80

100

120

Distance(cm)

Fig. 8.10 Neutron flux on the diagonal line for 2D-IAEA problem. (A) Fast neutron flux, (B) Thermal neutron flux.

reason for accuracy improvement from LRBFCM to BRBFCM is that LRBFCM cannot treat the interface between two materials well and BRBFCM is particularly improved here, since zero-order flux continuous condition and one-order beam continuous condition are taken into consideration. As shown in Fig. 8.10 the maximum errors of LRBFCM appear at the interface between different materials, such as the peaks at distance 110 cm in Fig. 8.10B, which is the interface between water reflector area and fuel area. Besides, several neutron flux sinks can also be observed clearly in Fig. 8.10A and B at distance 0–10 and 70–90 cm, where the control rods are located, which proves that the power flattening function of these control rods are effective. A comparison of the normalized FA power densities between the reference values and two BRBFCM solutions are presented in Fig. 8.11. It can be seen that there is a

¥ ¥

Fig. 8.11 Normalized FA power density for the 2D-IAEA problem.

Mesh-free method

233

good match between the numerical results and reference. As more nodes set, the numerical results become closer to the reference, for all the FAs in this problem.

8.3.4 2D-TWIGL problem The 2D-TWIGL benchmark problem is a simplified two-dimensional two-group seedblanket reactor with one delayed neutron precursor family. One-quarter of the core is illustrated in Fig. 8.12, including three regions: region 1 is perturbed seed region, region 2 is unperturbed seed region, and region 3 is blanked region. Both region 1 and region 2 contain primary fissile materials, and the difference is that the nuclear properties (the thermal absorption cross section in this case) of region 1 are timedependent in transient calculation, while those of region 2 are not. Due to the symmetry, left and bottom boundaries are reflective boundaries, while the other two external boundaries are vacuum. The two-group nuclear data and delayed neutron data for 2D-TWIGL problem are presented in Tables 8.8 and 8.9.

Fig. 8.12 One-quarter of the 2D-TWIGL reactor.

Table 8.8 Two-group nuclear data for 2D-TWIGL problem. Region

Group g

Dg (cm)

Σa,g (cm21)

υΣf,g (cm21)

χ

Σs,g! g+1 (cm21)

A

1 2 1 2 1 2

1.4 0.4 1.4 0.4 1.4 0.4

0.01 0.15 0.01 0.15 0.008 0.05

0.007 0.2 0.007 0.2 0.003 0.06

1.0 0.0 1.0 0.0 1.0 0.0

0.01 0 0.01 0 0.01 0

B C

234

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Table 8.9 Delayed neutron data for 2D-TWIGL problem. Family, i

Delay fraction, βi

Decay constant, λi (s21)

1

0.0075

0.08

To solve the 2D-TWIGL problem by BRBFCM, two groups of node collocations are generated and applied. In both schemes, the whole calculation domain is divided into 10  10 blocks, according to the core geometrical structure. The difference is that each block is represented by M ¼ 3  3 interior nodes in one scheme, while in the other scheme the interior node quantity is M ¼ 5  5. The node collocation of M ¼ 3  3 is shown in Fig. 8.13. For LRBFCM, two schemes with support domain rm ¼ 30 or 50 cm are also carried out. These two schemes share same node collocation with total nodes equal to 2741. All the reference values of this problem can be found in Reference. Firstly, the steady-state calculation is carried out. The calculation results are listed in Table 8.10 and the neutron fluxes on the diagonal line are plotted in Fig. 8.14. As can be seen, the BRBFCM has advantages over LRBFCM both in the aspect of time consumption and in the aspect of accuracy for solving this problem. Taking calculation No. 3 as an example, BRBFCM uses a smaller node quantity (2100) and a shorter time consumption (1.879 s) and gets a better calculation results (0.011%), compared with calculation No. 1 and No. 2 by LRBFCM. Since the steady-state 2D-TWIGL problem is not so complex as the 2D-IAEA problem, the LRBFCM performs better in this problem than in the 2D-IAEA problem. The maximum errors of LRBFCM appears at the distance from 0 to 30 cm in

Fig. 8.13 Node collocation of M ¼ 3  3 for 2D-TWIGL problem.

Mesh-free method

235

Table 8.10 Calculation results, relative errors and time consumptions of 2D-TWIGL problem by LRBFCM and BRBFCM. No.

Numerical method

Total nodes

keff,num

εr (%)

Time consumption (s)

1 2 3 4

LRBFCM (rm ¼ 30 cm) LRBFCM (rm ¼ 50 cm) BRBFCM (M ¼ 3  3) BRBFCM (M ¼ 5  5)

2741 2741 2100 3600

0.918118 0.911356 0.913116 0.913200

0.54 0.20 0.011 0.0016

176.7 1415 1.879 6.955

180

30

Thermal neutron flux

Fast neutron flux

150 120 90

LRBFCM (rm=50 cm)

30 0

(A)

Reference LRBFCM (rm=30 cm)

60

BRBFCM (M=3 ¥ 3) BRBFCM (M=5 ¥ 5)

0

20

40

60

80

100

120

24

LRBFCM (rm=50 cm) BRBFCM (M= 3 ¥ 3) BRBFCM (M= 5 ¥ 5)

18 12 6 0

(B)

Distance(cm)

Reference LRBFCM (rm=30 cm)

20

40

60

80

100

120

Distance(cm)

Fig. 8.14 Neutron flux on the diagonal line for 2D-TWIGL problem. (A) Fast neutron flux, (B) Thermal neutron flux.

Fig. 8.14A and B, mainly due to the error accumulation of neutron flux solving near the symmetric boundary. A significant error peak by LRBFCM can be also observed near distance 80 cm at Fig. 8.14B, which is the interface corner between two different materials. In BRBFCM, both schemes perform well and the relative error of keff can reach 104 and 105 order for M ¼ 3  3 and M ¼ 5  5, respectively. A comparison of the normalized region power density by BRBFCM with the reference values are also presented in Fig. 8.15, and a good agreement can be seen in both M ¼ 3  3 and M ¼ 5  5 scheme.

¥ ¥

Fig. 8.15 Normalized FA power density for the 2D-TWIGL problem.

236

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Relative power

The next step consists of two transient state problems, which initiated from the steady state obtained earlier, namely the step perturbation and the linear ramp perturbation transient problem. In the step perturbation transient problem, the thermal neutron absorption cross section in region 1 decreases by 0.0035 cm1 at 0 s and stays constant. While in the linear ramp perturbation transient problem, the thermal neutron absorption cross section in region 1 is linearly decreased by 0.0035 cm1 over 0.2 s. Considering the time-consumption endurance, only BRBFCM method with the node collocation of M ¼ 3  3 is adopted to transient calculation. The total calculated time scope is 0.5 s. The results of relative power of the two transient problems as a function of time are shown in Figs. 8.16 and 8.17. Besides, the numerical data of relative power between BRBFCM and reference are presented in Tables 8.11 and 8.12, respectively. A good agreement can be seen from Figs. 8.16 and 8.17. The numerical data and relative errors in Tables 8.11 and 8.12 also

2.0 Reference BRBFCM

1.6 1.2 0.0

0.1

0.2

0.3

0.4

0.5

Time (s)

Relative power

Fig. 8.16 Relative power for the 2D-TWIGL step perturbation transient.

2.0 Reference BRBFCM

1.6 1.2 0.0

0.1

0.2

0.3

0.4

0.5

Time (s)

Fig. 8.17 Relative power for the 2D-TWIGL linearly ramp perturbation transient. Table 8.11 Comparison of the relative power for the 2D-TWIGL step perturbation transient. Times (s)

BRBFCM

Reference

Error (%)

0.1 0.2 0.3 0.4 0.5

2.062 2.080 2.097 2.115 2.133

2.061 2.078 2.095 2.113 2.131

0.08 0.12 0.14 0.14 0.10

Mesh-free method

237

Table 8.12 Comparison of the relative power for the 2D-TWIGL linearly ramp perturbation transient. Times (s)

BRBFCM

Reference

Error (%)

0.05 0.1 0.2 0.3 0.4 0.5

1.211 1.303 1.949 2.077 2.094 2.111

1.125 1.307 1.957 2.074 2.096 2.109

0.34 0.34 0.41 0.13 0.09 0.12

demonstrate that BRBFCM can solve transient neutronics problems accurately, just with a small node quantity.

8.3.5 4G-LMFBR problem The last case is named four-group liquid metal fast breeder reactor (LMFBR) problem, which is a right circular cylinder in R-Z geometry. This core consists of two core areas, two blanket areas, and a reflector. One quarter of the rotation section is shown in Fig. 8.18, with reflective boundary condition on the symmetry planes and bare boundary condition on the outer boundaries. The neutron energy spectrum is divided into four groups and the nuclear reaction section data is listed in Table 8.13. Besides, the neutron diffusion coefficient can be obtained as Dg ¼ 1/(3Σ t,g). This problem is used for checking the performance of LRBFCM in more energy groups, because only two energy groups are not quite enough for real core calculations. Four calculations are carried out by BRBFCM with different node quantity. While generating the support nodes, the whole calculation domain is divided into z (cm)

r (cm)

Fig. 8.18 One-quarter of the 4G-LMFBR reactor.

238

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Table 8.13 Four-group nuclear data for LMFBR problem. Energy group g Region Core 1

Core 2

Blanket 1

Blanket 2

Outer reflector

Fission spectrum

Cross section (cm)

1

2

3

4

Σ t,g υΣ f,g Σ s,g0 !g g0 ¼ 1 g0 ¼ 2 g0 ¼ 3 g0 ¼ 4 Σ t,g υΣ f,g Σ s,g0 !g g0 ¼ 1 g0 ¼ 2 g0 ¼ 3 g0 ¼ 4 Σ t,g υΣ f,g Σ s,g0 !g g0 ¼ 1 g0 ¼ 2 g0 ¼ 3 g0 ¼ 4 Σ t,g υΣ f,g Σ s,g0 !g g0 ¼ 1 g0 ¼ 2 g0 ¼ 3 g0 ¼ 4 Σ t,g υΣ f,g Σ s,g0 !g g0 ¼ 1 g0 ¼ 2 g0 ¼ 3 g0 ¼ 4 χ

0.116757 0.017811 0.07235 0 0 0 0.116695 0.019505 0.07238 0 0 0 0.12268 0.014126 0.07493 0 0 0 0.132588 0.017301 0.07909 0 0 0 0.11317 0 0.08232 0 0 0 0.588153

0.221928 0.004777 0.03767 0.21435 0 0 0.221781 0.006108 0.03709 0.21375 0 0 0.234088 0.000838 0.04196 0.22763 0 0 0.256029 0.001358 0.04652 0.24868 0 0 0.177615 0 0.03028 0.17456 0 0 0.40819

0.348579 0.00632 0.00019 0.00416 0.33785 0 0.348871 0.008089 0.00018 0.00415 0.33741 0 0.363161 0.001073 0.00022 0.00431 0.35443 0 0.38695 0.001767 0.00025 0.00469 0.37668 0 0.36705 0 0.00007 0.00282 0.36419 0 0.003638

0.350966 0.024478 0 3.00E 07 0.001801 0.32258 0.35633 0.031306 1.36E 08 3.08E 08 0.001803 0.324236 0.345218 0.004205 0 1.76E 07 0.001793 0.329045 0.369594 0.00692 0 6.09E 07 0.001905 0.34972 0.411535 0 0 0 0.00163 0.407033 1.95E 05

12  17 blocks, and each block is represented by M ¼ 1  1, 3  3, 5  5, or 7  7 interior nodes in different node collocation scheme, respectively. Taking M ¼ 3  3 as an example, the node collocation is drawn in Fig. 8.19. The reference value of the effective multiplication factor is keff,ref ¼ 0.94032. The calculation results and relative errors are listed in Table 8.14. It can be seen that the numerical results in BRBFCM method have very small differences with the reference value and the time consumptions are acceptable.

Mesh-free method

239

z (cm)

r (cm)

Fig. 8.19 Node collocation of M ¼ 3  3 for 4G-LMFBR problem.

Table 8.14 Calculation results, relative errors and time consumptions of 4G-LMFBR problem. No.

Numerical method

Total nodes

keff,num

εr (%)

Time consumption (s)

1 2 3 4

BRBFCM (M ¼ 1  1) BRBFCM (M ¼ 3  3) BRBFCM (M ¼ 5  5) BRBFCM (M ¼ 7  7)

1018 4282 9178 15706

0.934224 0.939503 0.939558 0.939561

0.65 0.087 0.081 0.081

4.239 11.529 21.352 39.743

8.3.6 Natelson problem Natelson problem [4] is a neutron transport problem with fixed source. The whole domain contains two kinds of materials and all of the four boundaries are reflective boundaries. The geometry structure is a square with side length L ¼ 3.0 cm and in the bottom-left 1.0 cm  1.0 cm corner area, the fixed neutron source is 1.0 neutroncm3 s1. In the source area, the total section is Σ t ¼ 1.0 cm1 and the scatter section is Σ s ¼ 0.5 cm1. In the other area, the total section is Σ t ¼ 1.0 cm1 and the scatter section is Σ s ¼ 0.25 cm1. The geometry structure and node collocation map are shown in Fig. 8.20. As shown in Fig. 8.20, 25  25 nodes are collocated for the Natelson problem. Besides, the angular space is divided into 36  72 directions. The neutron flux along the sideline of y ¼ 3.0 cm between the calculation result and the reference are shown in Fig. 8.21. It can be seen that the BRBFCM result and the reference match well and the average difference is about 0.8%.

240

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Fig. 8.20 Node collocation for the Natelson problem. 0.045 0.040

Ref BRBFCM

Neutron flux

0.035 0.030 0.025 0.020 0.015 0.010 0.0

0.5

1.0

1.5

2.0

2.5

3.0

x (cm)

Fig. 8.21 Comparison of total flux along line y ¼ 3.0 cm for Natelson problem.

8.3.7 Transient localized source problem This problem is a transient one-dimensional localized source neutron transport problem [5]. The total length is 10.0 cm and the source is located in the center with the range of 5.0 cm. The geometry for the localized source problem is shown in Fig. 8.22. The medium is assumed to have the total section of Σ t ¼ 1.0 cm1 and the scatter section of Σ t ¼ 0.9 cm1. There are no particles within the slab at t ¼ 0 and the source is turned on at t ¼ 0+. The source remains at a constant strength level of 1.0 neutroncm3 s1 throughout the entire simulations. Particles released from the source have a velocity of 1.0 cm s1. While calculating, 40 nodes are collocated along the x line and the angular space is averagely divided by 64 directions. The time step is set as 0.5 s. The calculation results

Mesh-free method

241

Fig. 8.22 Geometry for the localized source problem.

of t ¼ 1.0, 2.5, 5.0, 7.5, 10.0 s are shown in line in Fig. 8.23 and a good matches between the reference point values can be seen.

8.4

Discussions

The calculation of mesh-free method does not rely on the beforehand meshes, making the geometry building work much easier than the mesh methods. The original strong form of mesh-free method is Kansa’s global method, but its time consumption is large. To overcome this problem, the local method is proposed and this method can support about 103–104 field node calculation in normal PC. The block method is proposed by Zhang to improve the speed and accuracy of collocation mesh-free method, and this method can support about 104–105 filed nodes, making this collocation mesh-free method more applicable for some science and engineering calculation situations.

Flux (neutron cm−3 s−1)

6

t = 10.0 s

5

t = 7.5 s

4

t = 5.0 s

3 t = 2.5 s

2 t = 1.0 s

1

0 0

1

2

3

4

5

6

Distance (cm)

Fig. 8.23 Results for the localized source problem.

7

8

9

10

242

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Block method is a more convenient approach for node collocation spatial discretization compared to time-consuming meshwork used in finite volume method (FVM) or finite element method (FEM). Seven problems are tested, such as the 2D2G square problem is used to investigate the influence of the shape factor and node quantity on eigenvalue accuracy, the 2D-IAEA benchmark problem is used to demonstrate the application of BRBFCM in multimaterial steady-state problem, the 2D-TWIGL benchmark problem is used for transient state neutronics problem, and 2D4G square and 4G-LMFBR problems are used for four energy group problem. The results show that the BRBFCM has advantages over traditional GRBFCM and LRBFCM in the aspect of both accuracy and efficiency. The relative error of the effective multiplication factor can be converged to the order of 105, when the node quantity is set at the order of 103, which is acceptable.

References [1] E.J. Kansa, Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics-I surface approximations and partial derivative estimates, Comput. Math. Appl. 19 (8) (1990) 127–145. [2] T. Tanbay, B. Ozgener, A comparison of the meshless RBF collocation method with finite element and boundary element methods in neutron diffusion calculations, Eng. Anal. Bound. Elem. 46 (2014) 30–40. [3] Y.N. Zhang, H.C. Zhang, X. Zhang, et al., Block radial basis function collocation meshless method applied to steady and transient neutronics problem solutions in multi-material reactor cores, Prog. Nucl. Energy 109 (2018) 83–96. [4] F. Inanc, A.F. Rohach, A nodal solution of the multigroup neutron transport equation using spherical harmonics, Ann. Nucl. Energy 15 (10) (1988) 501–509. [5] K.R. Olson, D.L. Henderson, Numerical benchmark solutions for time-dependent neutral particle transport in one-dimensional homogeneous media using integral transport, Ann. Nucl. Energy 31 (13) (2004) 1495–1537.

Other methods Youqi Zheng, Yunzhao Li, and Liangzhi Cao Xi’an Jiaotong University, Xi’an, People’s Republic of China

9

In the past decades, many innovative numerical methods have been proposed to solve the neutron transport equation for complex geometry. In the previous chapters, some well-developed methods are extended from structured meshes to unstructured meshes. Apart from those methods, there are some other innovative methods which were proposed in recent years and have not been well developed. This chapter provides a very brief description of three of these methods. To extend the nodal method to complex geometry, new methods that allow heterogeneity inside nodes are proposed which involve the functional expansion method inside nodes and use of the finite elements embedded in the nodal calculation. In this chapter, we introduce two methods based on such ideas. One is the wavelet expansion method which presents a new trial using wavelets as the basis function [1, 2]. The other is the heterogeneous variational nodal method for neutron transport solution [3, 4]. The Monte Carlo method has been recognized as an ideal solution method for neutron transport in complex geometry. To solve the problem of efficiency of Monte Carlo method, the idea of hybrid Monte Carlo and deterministic neutron transport was tried [5]. In this chapter, we also introduce a new trial of hybrid method, which benefits from the continuous-energy advantage of Monte Carlo method and the highefficiency of deterministic method.

9.1

Wavelet expansion method in neutron transport

9.1.1 Wavelet basis function The wavelet functions are defined by a dilation and translation operation such as ψ n, k ¼ 2n=2 ψ ð2n x  kÞ

(9.1)

for some ψ  L2(R) and (n,k) Z2. Z and R denote the set of integers and real numbers. L2(R) denotes the space of measurable, square-integrable functions. The wavelet functions are generated from the scaling functions ϕn, k(x), which have the same form ϕn, k ¼ 2n=2 ϕð2n x  kÞ

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation. https://doi.org/10.1016/B978-0-12-818221-5.00007-6 Copyright © 2021 Elsevier Ltd. All rights reserved.

(9.2)

244

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

for some ϕ  L2(R). Suppose we define Vn ¼ closure < ϕn, k : kZ >

(9.3)

Wn ¼ closure < ψ n, k : kZ >

(9.4)

and

then the scaling functions and the wavelet functions have the following subspace relations: ϕn, k Vn

(9.5)

ψ n, k Wn

(9.6)

…V1  V0  V1  V2 …

(9.7)

Vn ¼ Vn1 Wn1

(9.8)

[ Vn ¼ L2 ðRÞ

(9.9)

 Wn ¼ L2 ðRÞ

(9.10)

n

and n

where  stands for orthogonal sum. Thus, a wavelet decomposition at scale n becomes n1 X

fn ðxÞ ¼ fnm ðxÞ +

gj ðxÞ, fj Vj , gj Wj

(9.11)

j¼nm

or f n ðxÞ ¼

X

an, k ϕn, k ðxÞ ¼

k

X

anm, k ϕnm, k ðxÞ +

k

n1 X X

bnm, k ψ j, k ðxÞ

(9.12)

j¼nm k

where fn(x) represents the function f at the single scale n. Daubechies discovered that it was possible to develop a wavelet with desired properties suitable for a specific problem [6]. Based on the discrete wavelet transform, a new set of compactly supported orthonormal wavelets was constructed. It is named “Daubechies wavelets,” for which the scaling functions ϕn,k(x) and wavelet functions ψ n,k(x) are represented as ϕn, k ðxÞ ¼

2NX + 2k1 j¼2k

cj2k ϕn + 1, j ðxÞ

(9.13)

Other methods

245

and ψ n, k ðxÞ ¼

2X k1

ð1Þj c1j + 2k ϕn + !, j ðxÞ

(9.14)

j¼2k2N + 2

where n is called “dilation order” and N is called “Daubechies order.” To apply the Daubechies scaling function to expand the angular variable, the basis functions are defined on an interval. Generally, a unit interval is considered without loss of generality. However, the supporting region is related to the dilation order and Daubechies order, which makes them overlap the intervals. This requires a special treatment. Fig. 9.1 illustrates the distribution of scaling functions on a unit interval for Daubechies order N ¼ 2 and dilation order n ¼ 3. A technique to restrict the wavelets on an interval is to consider the “wrapped” wavelets [7], which requires only simple transformation of known scaling functions and wavelet functions. Although it is simple to use, it requires the expanded functions to be periodic, i.e., it forces the values of the unknown functions to be equal at two edges. This phenomenon is called “edge effect.” It may significantly change the values of unknown functions at the interface between two regions. To avoid the edge effect, the Daubechies wavelets on the interval constructed by Cohen et al. [8] is adopted. Define the boundary scaling function for dilation order n ¼ 0 as

1.6 1.4

Scaling function (N=2, n=3)

1.2 1.0 0.8 0.6 0.4 0.2 0.0 –0.2 –0.4 –0.6 0.0

0.2

0.4

0.6

0.8

1.0

Distance

Fig. 9.1 Distribution of original scaling function for N ¼ 2 and n ¼ 3 on a unit interval.

246

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation N 2 X

ϕleft 0 ðxÞ ¼

N 2 X

ϕð x  k Þ ¼

k¼∞

ϕð x  k Þ

(9.15)

k¼N + 1

then the boundary scaling function has compact support and is orthogonal to all the interior scaling functions ϕ0,k. In the same way, we can also get the boundary scaling on the other half line (∞, 0]. The boundary scaling functions ϕleft function ϕright 0 0 , right ϕ0 , and interior scaling functions ϕ0,k (that are local inside the interval) are used to construct a new set of scaling functions, i.e., the Daubechies scaling functions on the interval: ϕleft n, k ¼

N 1 X

Hk,leftl ϕleft n + 1,l +

N 1 X

left Gleft k, l ϕn + 1,l +

1 X

NX + 2k

right Hk,right l ϕn + 1,l +

right

gleft k,m ϕn + 1,m

(9.17)

1 X

N1 + 2k + 2 X

hright k,m ϕn + 1,m

(9.18)

m¼N1

l¼N

ψ n, k ¼

(9.16)

m¼N

l¼0

ϕright n, k ¼

hleft k, m ϕn + 1,m

m¼N

l¼0

ψ left n, k ¼

NX + 2k

right right

Gk, l ϕn + 1, l +

N1 + 2k + 2 X

right

gk, m ϕn + 1,m

(9.19)

m¼N1

l¼N

where ϕn,k and ψ n,k stand for the scaling function and wavelet function, respectively. Hk,l, hk,l, Gk,l, and gk,l denote the filter coefficients constructed by Daubechies.

9.1.2 Wavelet expansion method This section starts from an angularly discretized equation also using the Daubechies wavelet expansion [9]. The two-dimensional multigroup neutron transport equation in X-Y geometry is discussed as follows: Ωx

∂ϕg ðr, ΩÞ ∂ϕg ðr, ΩÞ + Ωy + Σ t ϕg ðr, ΩÞ ¼ qg ðr, ΩÞ ∂x ∂y

(9.20)

where ϕ is the flux and q is the source distribution. The angular flux at discrete polar angle is expanded by the Daubechies scaling function in every angular subdomain, as in Eq. (9.21). The definition of angular subdomain is shown in Fig. 9.2. ϕg, m ðr, φÞ ¼

np X

2 ϕg, mp ðr Þψ p ðξÞ, ξ ¼ φ, and ξ½0, 1Þ π p¼1

(9.21)

Other methods

247

Fig. 9.2 Definition of angular subdomains in the X-Y geometry.

Fourth angular subdomain First angular subdomain

Third angular subdomain

Second angular subdomain

Finally, the decoupled angular discretization scheme is obtained. The spatial variable is omitted in the equation, and the angular flux is finally discretized as np X

k

a Dx, mpp0

∂ϕkg, mp ∂x

p¼1

i

+ b Dy, mpp0

∂ϕkg, mp

!

∂y

+ Σ t ϕkg, mp0 ¼ qkg, mp0 , p0 ¼ 1, np

(9.22)

where k is the index of angular subdomain, ak and bk are defined as a1 ¼ 1, a2 ¼ 1, a3 ¼ 1, a4 ¼ 1

(9.23)

b1 ¼ 1, b2 ¼ 1, b3 ¼ 1, b4 ¼ 1

(9.24)

and the coefficients are defined as qffiffiffiffiffiffiffiffiffiffiffiffiffi ð π 1  μ2m cos ξψ p ðξÞψ p0 ðξÞdξ 2 ð qffiffiffiffiffiffiffiffiffiffiffiffiffi π Dy, mpp0 ¼ 1  μ2m sin ξψ p ðξÞψ p0 ðξÞdξ 2

Dx, mpp0 ¼

qg, mp0 ¼

ð1

ψ p0 ðξÞSg, m ðr, ξÞdξ

(9.25)

(9.26)

(9.27)

0

Since the angularly discretized equations are coupled by a series of expansion coefficients Φkg,mp, i.e., the iterative method is applied to decouple the coefficients as ak Dx, mp0 p0

l ∂ϕk, g, mp0

∂x

k, l1 ¼ qg,mp 0  k, l1 ¼ q0 g, mp0

np X

p¼1,

p6¼p0

∂ϕk,l g, mp0

+ Σ t ϕk,l g, mp0 ∂y ! k,l1 k, l1 ∂ϕg,mp ∂ϕg, mp k k + b Dy, mpp0 a Dx, mpp0 ∂x ∂y

+ bi Dy, mp0 p0

p0 ¼ 1, np

(9.28)

248

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Fig. 9.3 Configuration of the slab geometry.

where l denotes the iteration index, and the spatial variables are omitted here. In Eq. (9.28), the unknowns are only Φig,mp0 , not related to other angular expansion coefficients. The variation method is used on the nodes divided as shown in Fig. 9.3 in the X- or Y-dimension. In the variation, the weighting function is multiplied on both sides of Eq. (9.28), then, the integration is done over the given node as ð by ð bx

k

a Dx, mp0 p0 ay

ax

∂ϕkg, mp0 ∂x

k

+ b Dy, mp0 p0

∂ϕkg, mp0 ∂y

! + Σ t ϕkg, mp0

k  q0 g, mp0

ϕ0 dxdy p0 ¼ 1,np (9.29)

¼ 0,

where Φ0 is the weighting function, and iteration index is omitted in Eq. (9.29). The trialing function and the weighting function are both represented by the Daubechies’ scaling function in terms of tensor product as ϕkg, mp0 ðx, yÞ ¼

npj npi X X

ϕkg, mp0 , ij ψ i ðxÞψ j ðyÞ

(9.30)

i¼1 j¼1

ϕ0 ðx, yÞ ¼ ψ i0 ðxÞψ j0 ðyÞ

(9.31)

where npi and npj are the total number of expansion coefficients, determined by the expansion order in the X- and Y-dimension, respectively. Generally, a given node was transferred into the normalized nodes firstly, the wavelet expansion is done in the normalized nodes. In each one the variation can be described as     dψ i ðξx Þ i dψ j ξy k a ξy + b Dy, mp0 p0 ϕg, mp0 , ij ψ i ðξx Þ dξy dξx 0 i¼1 j¼1 !       + Σ t ϕkg, mp0 , ij ψ i ðξx Þψ j ξy  q0 kg, mp0 ψ i0 ðξx Þψ j0 ξy J ξx , ξy dξx dξy

ð1 ð1 X npj npi X 0

¼ 0,

i

Dx, mp0 p0 ϕkg, mp0 , ij ψ j

i0 ¼ 1,npi; j0 ¼ 1, npj; p0 ¼ 1, np (9.32)

where jJ(ξx, ξy)j is the Jacobian matrix, defined as

Other methods

249

  ∂x      ∂ξx J ξx , ξy  ¼   ∂x   ∂ξy

 ∂y   ∂ξx   ∂y   ∂ξy 

(9.33)

Considering the orthonormality property of Daubechies scaling functions, the variation can be simplified as ð1 npi X dψ i ðξx Þ ϕkg, mp0 , ij0 ψ i0 ðξx Þdξx ak ΔyDx, mp0 p0 0 dξx i¼1   ð1 npj X   dψ j ξy k k + b Δx Dy, mp0 p0 ϕg, mp0 , i0 j ψ j0 ξy dξy + Σ t ϕg, mp0 ,i0 j0 ΔxΔy dξy 0 j¼1 ¼ q0 kg, mp0 ΔxΔy

ð1 0

ψ i0 ðξx Þdξx

ð1

  ψ j0 ξy dξy , i0 ¼ 1,npi; j0 ¼ 1, npj; p0 ¼ 1, np

0

(9.34) where Δx, Δy are the size of original nodes, and the derivation of Daubechies scaling function is calculated using difference instead as      ψ i x0 + d=2n + 1  ψ i x0  d=2n + 1 dψ i ðξx Þ ¼ dξx ξx ¼x0 d=2n

(9.35a)

and   dψ j ξy   dξy 

ξy ¼y0

    ψ j y0 + d=2n + 1  ψ j y0  d=2n + 1 ¼ d=2n

(9.35b)

where d/2n is the construction distance divided by the dilation. With all the coefficients in Eq. (9.34) known, the wavelet expansion coefficients can be obtained by solving the matrix: Kψ ¼ b

(9.36)

where Ki, j ¼ ak ΔyDx,mp0 p0 Cx, ii0 ϕkg, mp0 , ij0 + bk ΔyDx, mp0 p0 Cy, jj0 ϕkg, mp0 ,i0 j ( Σ t, g ϕkg, mp0 , i0 j0 ΔxΔy, i ¼ i0 , j ¼ j0 + 0, else h iT ψ ¼ ϕkg, mp0 ,11 , ϕkg, mp0 ,21 , ⋯, ϕkg, mp0 ’, npinpj

(9.37)

(9.38)

250

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

and h iT k k k b ¼ E11 q0 g, mp0 , E21 q0 g, mp0 , ⋯, Enpinpj q0 g, mp0

(9.39)

with ð1

dψ i ðξx Þ Cx, ii0 ¼ Δy ψ i0 ðξx Þdξx and Cy , jj0 ¼ Δx 0 dξx Eij ¼ ΔxΔy

ð1

ψ i ðξx Þdξx

0

ð1

    dψ j ξy ψ j0 ξy dξy dξy 0

ð1

  ψ j ξy dξy

(9.40)

(9.41)

0

Take the reflective boundary condition for example. The boundary condition can be expressed as ϕg ðrb , ΩÞ ¼ ϕg ðrb , Ω0 Þ,

Ωn < m i X X  > χ jn hjγ ðr Þknγ ðΩÞ : φγ ðr, ΩÞ ¼ j

(9.47)

n

where spatial trail functions fi(r) and hjγ (r) are orthogonal polynomials within the node and along the node interface γ, similarly angular trail functions gm(Ω) and knγ (Ω) are spherical harmonics Ypq(Ω) 

gT ðΩÞ ¼ ½Y00 Y22 Y21 Y20 Y21 Y22 Y44 ⋯ kT ðΩÞ ¼ ½ Y10 Y32 Y31 Y30 Y31 Y32 Y54 ⋯ 

(9.48)

defined within the node and along the node interface, where the polar angle is defined with respect to the outward normal. The response matrix equations for VNM can be obtained as three formulas in terms of the partial current moments jg on each interface, and the volume flux and source moments ϕg and sg within each node. The source equation is sg ¼

X g0 6¼g

Σ gg0 ϕg0 +

1X Fgg0 ϕg0 k g0

(9.49)

The within-group current equation is 

 I j  Rg Π jg ¼ Bg sg

(9.50)

and the within-group flux equation is   ϕg ¼ Hg sg  Cg Ij  Π jg

(9.51)

where Bg, Cg, Hg, and Rg are the response matrices determined by the geometry and cross section of the node; Ij and Π stand for the identity and spatial connectivity matrices, to which no energy index is assigned due to the fact that the size of these two matrices automatically fit the size of partial current vector. The entries of the response matrices are interproduct of cross sections and basic functions in both space and angle. It brings two advantages. Firstly, the cross-section distribution within each node can be either constant function or nonconstant function. It is homogeneous VNM if the cross-section distribution is constant, while heterogeneous VNM if the cross-section is not constant. Secondly, the shape of the node can be arbitrary as long as the interproduct can be evaluated.

256

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Substitution of the source equation into the current equations and the flux equations yields 

X  1X Σ gg0 ϕg0 ¼ Bg Fgg0 ϕg0 Ij  Rg Π jg  Bg k g0 g0 6¼g

(9.52)

and ϕg  H g

X

  1X Σ gg0 ϕg0 + Cg Ij  Π jg ¼ Hg Fgg0 ϕg0 k g0 g0 6¼g

(9.53)

Combining all the energy groups then leads to the VNM eigenvalue system 

Ij RΠ  BΣ C Ij  Π Iϕ  HΣ

    j B 1 ¼ Fϕ H k ϕ

(9.54)

where response matrices B, C, H, and R are block diagonal over the energy groups, and Iϕ is an identity matrix associated with the flux vector. The right-hand side contains the neutron fission source, and the left-hand side contains the neutron transport operators. The coefficient matrix is divided into four blocks. Block Ij  RΠ is the block diagonal by energy groups, it represents the spatial transport of neutron streaming from one node to another. The block Iϕ  HΣ is block diagonal by spatial nodes. It represents the energy transport due to neutron scattering from one energy group to another. The other two blocks connect the transports in space and energy. Elimination of currents from the VNM eigenvalue system yields the second form of the VNM eigensystem, with only flux included in the solution vector. 

 Iϕ  TΣ ϕ ¼ Tf

(9.55)

where spatial transport coefficient matrix   1 T ¼ H  C Ij  Π Ij  RΠ B

(9.56)

is also block diagonal over the energy groups and the fission source is f ¼ 1k Fϕ. Traditionally, the power method (PM) is used to solve the eigenvalue problem iteratively. It is usually referred to as the fission source iteration or the outer iteration.

9.2.2 The legacy multigroup Guass-Seidel (MG GS) algorithm To obtain the legacy algorithm, we first separate the scattering matrix into down- and up-scattering Σ ¼ ΣD + ΣU

(9.57)

Other methods

257

The legacy multigroup algorithm is then obtained by iterating the second form of VNM system with known fission source and sweeping the scattering matrix in Gauss-Seidel form

ϕðnÞ ¼ T Σ D ϕðnÞ + Σ U ϕðn1Þ + f

(9.58)

where n indexes the multigroup iteration number. Moving the term with updated flux from the right to the left then yields

 1 ϕðnÞ ¼ Iϕ  TΣ D T Σ U ϕðn1Þ + f

(9.59)

The matrix Iϕ  TΣD is block lower-triangular over the energy groups and can be inverted by one Gauss-Seidel sweep. Consequently, only one iteration is required to solve the VNM multigroup system if no up-scattering is present (ΣU ¼ 0). When up-scattering occurs, several multigroup iterations are required, typically 5–10 for the thermal up-scattering. The multigroup iteration is usually called the up-scattering iteration, or thermal iteration since thermal range is where natural up-scattering usually happens.

9.2.3 The within group red-black Gauss-Seidel (RBGS) algorithm To apply the T matrix in Eq. (9.56), the spatial matrix Ij  RgΠ must be inverted for each energy group in order to solve for the partial currents jg for a given source q g ¼ Bg s g : 

 Ij  Rg Π jg ¼ qg

(9.60)

A red-black response matrix algorithm is used in the VARIANT code for Cartesian geometry. In hexagonal geometry, the two-color schemes must be replaced by a four-color algorithm with exactly the same logic. In the red-black response matrix algorithm, each node is assigned a color, such as red or black, such that the adjacent nodes do not share the same color. Except at the outer boundaries, all the incoming partial currents of the nodes are in one color and the outgoing currents of the nodes with other color(s). By reordering the outgoing currents vector according to the color of the corresponding node, the spatial response in twocolor cases becomes 

I Rb Π br

Rr Π rb I

    jr q ¼ r jb qb

(9.61)

where the energy group g index has been omitted. As a result, the within-group spatial sweep can be done in Gauss-Seidel form by sweeping nodes color by color.

258

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

(

ði + 1Þ

ð iÞ

jr ¼ Rr Π rb jb + qr ði + 1Þ ði + 1Þ jb ¼ Rb Π br jr + qb

(9.62)

where i is the within-group iteration index. In addition, complete sweep within each WG RBGS iteration can be quite time consuming since the dimensions of the response matrices are often large. A partitioned matrix algorithm [6] has been designed to accelerate the WG RBGS iteration. Both vectors and matrices in the WG partial currents equation can be partitioned into lower order and higher order parts "

Ijα  Rαα Π Rαβ Π Rβα Π Ijβ  Rββ Π

#    jα qα β ¼ j qβ

(9.63)

where α and β stand for the lower and higher orders. Before each full sweep, the lower order system

Ijα  Rαα Π jα ¼ qα + Rαβ Πjβ

(9.64)

is solved to update the lower order moments with fixed higher order moments to reduce the total number of full-order sweeps. The lower order system can also be solved by a lower order RBGS algorithm. In transport calculation, this partitioned matrix acceleration is significant due to the large number of higher order space-angle interface terms. In diffusion calculation, the spatial expansion order should be higher than linear partitioned matrix acceleration to have a significant impact.

9.3

Hybrid method of Monte Carlo and method of characteristics

In addition to many kinds of deterministic methods, Monte Carlo method is also a classical method for solving transport equation [12]. It is also called stochastic method which mainly introduces the idea and theory of probability. The stochastic method or Monte Carlo (MC) method [13] transforms the differentialintegral neutron-transport equation into the integral neutron-emitting density equation, in which the solution of the neutron-transport equation can be represented as integrals over the space-angle-energy phase space. Thus, the Monte Carlo method which evaluates integrals by sampling and tallying can be used to solve the neutron-transport equation. Each sampling process is a neutron history, including both neutron-tracking and neutron-collision processes. The neutron history simulation usually requires the neutron cross sections in continuous energy be saved in a nuclear-data library. Once a sufficient

Other methods

259

number of samples are obtained, the target quantity can be estimated by tallying a certain amount of values in the corresponding designated phase space. Because the point-wise continuous energy cross section is applied, there is no resonance selfshielding calculation as in the multigroup deterministic method. Consequently, the Monte Carlo simulation is taken as the reference solution for the deterministic method. According to the law of large numbers and the central limit theorem, the number of histories has to be sufficiently large to obtain accurate results. Hence, the computing time usually becomes the burden for the Monte Carlo method. To improve the computing accuracy of the deterministic method, or to combine the advantages of the deterministic and stochastic methods, a series of hybrid methods have been proposed and investigated. Wagner [14] proposed a method called forward weighted consistent adjoint driven importance sampling (FW-CADIS) method. It uses the deterministic method to provide the source-biasing parameters for the MC simulation to accelerate the convergence of the fission source. Lee [15] used the coarsemesh finite differencing (CMFD) method to accelerate the MC fission source iteration. Yang and Larsen [16] proposed the functional Monte Carlo (FMC) method which firstly uses a low-order equation to calculate the Eddington factor to accelerate the convergence of the high-order equation. Hyunsuk [17] proposed a hybrid method which uses the MC method to simulate the resonance and thermal energy range and uses the method of characteristics (MOC) to sweep fast and thermal groups. It is tested based on an eight-group problem and has been proven to be capable of solving the problem faster than the traditional MC method and providing more accurate results than the pure MOC. Considering the fact that the most problematic issue for the deterministic method is the resonance self-shielding calculation, a deterministic-stochastic energy-hybrid method is proposed. In addition, the space and angle discretizing strategies of the deterministic method include discrete-ordinates method and spherical harmonic function method in angle; fine-mesh difference method and coarse-mesh nodal method in space; and MOC in space and angle [18]. Considering the flexibility in geometry, MOC is chosen as the deterministic method in the hybrid method.

9.3.1 The criticality calculation of transport equation The hybrid method of Monte Carlo and method of characteristics is mainly aimed to solve critical transport equation. The steady-state continuous-energy neutrontransport equation for the criticality calculation is as follows: ð ∞ ð 4π Ωrϕðr, E, ΩÞ + Σ t ðr, EÞϕðr, E, ΩÞ ¼ Σ s ðr, E0 ! E, Ω0 ! ΩÞϕðr, E0 , Ω0 ÞdΩ0 dE0 0 0 ð ð χ ðEÞ ∞ 4π + νðE0 ÞΣ f ðr, E0 Þϕðr, E0 , Ω0 ÞdΩ0 dE0 4πkeff 0 0 (9.65)

260

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

where: Ω0 ¼ direction of incidence neutron Ω ¼ direction of ejection neutron r ¼ position of neutron E0 ¼ energy of incidence neutron (MeV) E ¼ energy of ejection neutron (MeV) Σ s(r, E0 ! E, Ω0 ! Ω) ¼ macroscopic differential scattering cross section (cm1) χ(E) ¼ fission spectrum Σ t(r, E) ¼ macroscopic total cross section (cm1) ν(E0 )Σ f(r, E0 ) ¼ macroscopic neutron-generation cross section (cm2 s1) ϕ(r, E, Ω) ¼ angular flux in position r and energy E (cm2 s1) keff ¼ effective multiplication factor

For such an eigenvalue problem, the power iteration can be used as the fission-source iteration to obtain the fundamental eigenvalue and the eigenvector as the multiplication factor and the neutron-flux distribution. With an initial guess of the eigenvalue and the eigenvector, the fission-source distribution and the total fission-source can be calculated, respectively, as ð ð χ ðEÞ ∞ 4π νðE0 ÞΣ f ðr, E0 Þϕ0 ðr, E0 , Ω0 ÞdΩ0 dE0 4π 0 0 ð∞ ð Qf 0 ðr, E, ΩÞdΩdEdr

Q0f ðr, E, ΩÞ ¼ ð QF ¼ 0

rR 0

(9.66)

(9.67)

4π

With the fission-source distribution as known, the fixed-source equation can be solved by using the deterministic or stochastic method to update the flux distribution Ωrϕn ðr, E, ΩÞ + Σ t ðr, EÞϕn ðr, E, ΩÞ ¼ ð ∞ ð 4π 1 Σ s ðr, E0 ! E, Ω0 ! ΩÞϕn ðr, E0 , Ω0 ÞdΩ0 dE0 + n1 Qn1 ðr, E, ΩÞ keff f 0 0

(9.68)

Once the flux profile is updated, the fission-source distribution and the total fission source can be updated, respectively, as following: Qnf ðr, E, ΩÞ ¼ ð QnF ¼

χ ð EÞ 4π

ð∞ ð

rR 0

4π

ð ∞ ð 4π 0

νðE0 ÞΣ f ðr, E0 Þϕn ðr, E0 , Ω0 ÞdΩ0 dE0

(9.69)

0

Qnf ðr, E, ΩÞdΩdEdr

Consequently, the eigenvalue can be updated:

(9.70)

Other methods

n keff ¼

261

n1 n keff QF

(9.71)

Qn1 F

Then, the iteration can be continued with the updated fission source and keff. The iteration is stopped if it is converged.

9.3.2 Division of energy ranges The neutron energy phase-space can be divided into three energy ranges: the fast energy range denoted as G1, the resonance range G2, and the thermal energy range G3. Other than the upper and lower limit of the energy phase space, two more energy boundaries would be required. They are chosen according to the following considerations. (1) There is no up-scattering from either G2 or G3 to G1. Since the neutron energy is very high, it is very hard for the neutrons in reactor core to obtain energy from the collision with a nucleus. (2) There is no up-scattering from G3 to G2. It means that the energy interface between G2 and G3 has to be high enough to leave the up-scattering in G3. (3) There are no fission neutrons in G3. This is true since most fission neutrons are fast ones. In the actual choice of G3, consideration (2) may not be realized exactly, thus, there may be a little approximation in the energy boundary of G2 and G3. According to the division of the energy ranges, the continuous neutron-transport equation can be written for the fast energy range G1 as Ωrϕ ð ðr, Ω, EÞ + Σ t ϕðr, Ω, EÞ ¼ Σ s ðr, Ω0 ! Ω, E0 ! EÞϕðr, Ω0 , E0 ÞdΩ0 dE0 + Qf ðr, Ω, EÞ,EG1

(9.72)

E0 G1

where Qf(r, Ω, E) is the fission source with ejection neutron energy E. Only the scattering within the energy range appears since there is no up-scattering from either G2 or G3 to G1. There is no need to couple this equation with the ones in the other two energy ranges through scattering. In this energy range, neutron cross-section excitation curve is smooth. The multigroup approximation does not cause large error. Thus, the neutron-transport equation can be written into the multigroup form and solved by using the deterministic method MOC. The multigroup form of the neutron-transport equation in G1 is as follows: Ωrϕg ðr, ΩÞ + Σ t ϕg ðr, ΩÞ ¼

gG1 ð 4π X g0 ¼1

0

Σ s ðr, Ω0 ! Ω, g0 ! gÞϕg0 ðr, Ω0 ÞdΩ0 + Qfg ðr, ΩÞ, gG1 (9.73)

262

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

For the resonance energy range G2, the continuous neutron-transport equation becomes ð Σ s ðr, Ω0 ! Ω, E0 ! EÞϕðr, Ω0 , E0 ÞdΩ0 dE0 Ωrϕðr, Ω, EÞ + Σ t ϕðr, Ω, EÞ ¼ E0 G2

ð

Σ s ðr, Ω0 ! Ω, E0 ! EÞϕðr, Ω0 , E0 ÞdΩ’dE0 + Qf ðr, Ω, EÞ,EG2

+ E0 G1

(9.74) There are two scattering source terms: the scattering within the energy range and the down-scattering from G1. In this energy range, the resonance phenomenon is serious. The Monte Carlo method can be used to solve the equation with a given fission source from the former fission source iteration and a given down-scattering source from the newly updated G1 energy range solution. The sum of these two given source terms can be regarded as a probability density function to sample neutron histories. This probability density function is in multigroup form. After sampling among groups, the particle energy point is sampled uniformly within the energy group. As for the within-energy range scattering, it can be considered implicitly by neutron history simulation. In contrast, for the thermal energy range G3, it can be written as ð Ωrϕðr, Ω, EÞ + Σ t ϕðr, Ω, EÞ ¼ ð

Σ s ðr, Ω0 ! Ω, E0 ! EÞϕðr, Ω0 , E0 ÞdΩ0 dE0 +

E0 G3

Σ s ðr, Ω0 ! Ω, E0 ! EÞϕðr, Ω0 , E0 ÞdΩ0 dE0 +

E0 G1

ð

Σ s ðr, Ω0 ! Ω, E0 ! EÞϕðr, Ω0 , E0 ÞdΩ0 dE0 , EG3

E0 G2

(9.75) where three scattering source terms show up including the scattering within energy range, the down-scattering sources from G1 and G2 respectively. In this energy range, the same multigroup processing method with G1 can be used because the crosssection in this energy range is relatively smooth too. The multigroup neutron-transport equation is Ωrϕg ðr, ΩÞ + Σ t ϕg ðr, ΩÞ ¼

ð 4π gG3 X

Σ s ðr, Ω0 ! Ω, g0 ! gÞϕg0 ðr, Ω0 ÞdΩ0 +

g0 ¼gG2 + 1 0 gG1 ð 4π X g0 ¼1 0

ð

Σ s ðr, Ω0 ! Ω, g0 ! gÞϕg0 ðr, Ω0 ÞdΩ0 +

ð 4π

E0 G2 0

Σ s ðr, Ω0 ! Ω, E0 ! gÞϕðr, Ω0 , E0 ÞdΩ0 dE0 ,gG3 (9.76)

Other methods

263

It can be solved by using MOC, since the down-scattering source from G1 and G2 can be obtained by the fast energy range MOC simulation and the resonance energy range Monte Carlo tallies.

9.3.3 The MOC calculations in the fast and thermal energy ranges For the fast and thermal energy ranges, the neutron-transport equation can be written as an integral form along a characteristic line: ð s 0 0 Σ g, t, i ðr0 + s Ωm Þds + φg, i, k ðr0 + sΩm , Ωm Þ ¼ φg, i, k ðr0 , Ωm Þexp 0 ð s ðs (9.77) Qg, i ðr0 + s0 Ωm , Ωm Þ exp Σ g, t, i ðr0 + s00 Ωm Þds00 ds0 s0

0

where g is the index of energy group, i is the index of source region, k is the index of line segment, m is the index of neutron flying direction, and s is the length of the line segment. The source which includes scattering and fission sources can be written as follows: G 1 X Σ s ðr, g0 ! gÞϕg0 ðr Þ + Qfg ðr, Ωm Þ (9.78) Qg, i ðr, Ωm Þ ¼ 4π g0 ¼1 Assuming constant source and constant material property for each line segment, the equation above can be integrated along the characteristic line and the flux of its ending point can be written as   φg, i, k ðr0 + sΩm , Ωm Þ ¼ φg, i, k ðr0 , Ωm Þexp Σ g, t, i s   Qg, i ðΩm Þ + 1  exp Σ g, t, i s (9.79) Σ g, t, i In this way, the neutron-flux of the ending point can be obtained sequentially. Once all the characteristic lines of the problem are swept, the scalar flux of each flat source region can be obtained by summing all its angular fluxes together with the corresponding weights. As for the within-group scattering and the outer-group scattering, the corresponding within-group and multigroup iterations can be carried out, to obtain the flux of current fission iteration.

9.3.4 The Monte Carlo simulation in the resonance energy range To use the Monte Carlo method, the neutron-transport equation has to be transformed into the form of emission density equation: Qðr, Ω, EÞ ¼ Sðr, Ω, EÞ ð ð Emax ð ∞ + 4π 0

0

K ðr 0 , Ω0 , E0 ! r, Ω, EÞQðr 0 , Ω0 , E0 ÞdrdE0 dΩ0

(9.80)

264

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

where K(r0 , Ω0 , E0 ! r, Ω, E) is the probability density function of sampling. It contains the neutron transport and collision kernels [13]. Once the neutron is generated by using the emission density, its history can be simulated with those two kernels sequentially. In the G2 energy range of each fission source iteration, Monte Carlo simulation is equivalent to solving the above emission density equation.

9.3.5 The hybrid Monte Carlo deterministic method in energy range Once the fixed source problems in G1, G2, and G3, respectively, are solved sequentially, the fission source and eigenvalue should be updated. In the hybrid method, however, the total fission source can be written as follows: QF ¼

R X

0

0

X

@

χg @

r¼1

gG1[G2

ð

ð∞

+ rR 0

ð χ ðEÞ

gG1 ð 4π X g0 ¼1 0





νΣ fg0 ðr Þϕg0 r, Ω0 dΩ0 +

gG3 X

ð 4π

g0 ¼gG2 + 1 0





11

νΣ fg0 ðrÞϕg0 r, Ω0 dΩ0 AAVr

ð 4π       ν E0 Σ f r, E0 ϕ r, E0 , Ω0 dΩ0 dE0 dEdr

E0 G2 0

(9.81)

where g0 and g are the energy group of incident and ejection neutrons, respectively. The fission sources with ejection fission neutrons in G1 and G2, respectively, are in the form of summation due to the multigroup approximation, while the fission source with ejection fission neutrons in G2 is in the form of integral from Monte Carlo tally. In this hybrid method, part of fission source is calculated by Monte Carlo tally together with a variance for each flat source region and each energy group. The other part of fission source is calculated by MOC sweeping without variance. With the error transfer formula [19]: σ ðf ðx ÞÞ ¼ 2

n  X ∂f 2 i¼1

∂xi

σ 2xi

(9.82)

where xi is ith statistic value of function f(x), σ xi is the standard deviation of xi. σ(f(x)) is standard deviation of function f(x). The variance of total fission source can be written as σ 2QF ¼

R X G  X 2 νΣ fg0 ðr Þ σ 2ϕg0 ðr ÞÞVr 2

(9.83)

r¼1 g0 ¼1

For updating of keff, the error transfer formula will be used again and the final variance can be written as

Other methods

σ 2kn + 1 eff

!2  n + 1 2  n 2 n keff keff QnF + 1 QF 2 2 ¼ σ kn + σ Qn + 1 +  σ 2Qn F eff F QnF QnF ðQnF Þ2

265

(9.84)

The final standard deviation for keff is the root of variance above. According to the theoretical derivations described earlier, the simplified flow chart of the hybrid method is shown in Fig. 9.8.

9.3.6 Numerical results In this section, some cases are calculated under hybrid method to verify its efficiency. The geometry of the testing case is a fuel pin with cladding and moderator. The MOC and Monte Carlo methods are coupled in the mesh of flat-source region in space. The dividing of the flat-source region for each pin-cell is shown in Fig. 9.9. In these calculations, both the multigroup and the continuous-energy nuclear-data libraries are originally obtained from ENDF-B/VII.0. A traditional WIMS-69 group

Fig. 9.8 Flow chart of hybrid method.

266

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Fig. 9.9 Flat source region of the hybrid method.

structure is chosen for the deterministic solution mainly due to the consideration of its mature application in the deterministic method. The boundary between energy ranges of G1, G2, and G3 are set as 9118 and 4 eV, respectively. Thus, the first 14 groups are in the fast energy range of G1, while the 15th–27th groups are in the resonance energy range of G2, and 28th–69th groups belong to G3. There are four test cases to compare the pure MOC, the pure MC, and the hybrid methods. For Cases 2–4, the calculation parameters are set as follows: In the pure Monte Carlo calculation, 100,000 particles with 100 inactive cycles and 100 active cycles were set. In the pure MOC calculation, the number of polar angles is 3, while the number of azimuthal angles is 8 and the width of characteristic line is 0.03 cm. In the hybrid method, the settings for the MOC part are the same as the pure MOC, while 1,000,000 particles were sampled for the one Monte Carlo simulation within each power iteration. For Case 1, because the geometry scale is one-ninth of the others, the number of particles for pure MC and hybrid is set to one-ninth of the others in order to keep consistency in computational efficiency and accuracy. Case 1. Single fuel pin problem In the fuel pin with 3.1% U235 enrichment, the structure includes helium gap and the clad. Boundary conditions are all reflective. The results of the three methods are presented in Table 9.2. It can be found that the hybrid method runs faster than the pure Monte Carlo method by a factor of about 4.3. The accuracy of hybrid method is slightly better than the pure MOC mainly because the pure MOC can get a relatively accurate result in this case. The confidence interval of keff in hybrid method is calculated by using the transfer formula. In hybrid method, each Monte Carlo simulation is regarded as a converged fixed source simulation. The number of sampling is large and the standard deviation is small. It can be guaranteed that the statistical standard deviation is under the convergence criterion of fission source iteration in the entire flow.

Other methods

267

Table 9.2 Results of single fuel pin. keff MC MOC Hybrid

Confidence interval

1.23874 0.00264 1.23519 — 1.23741 0.0000005

Time (s)

MC time (s)

MOC time (s)

Discrepancy to MC

122.35 24.35 28.44

— — 8.89

— 11.44 17.02

— 232 87

The energy spectrum of this case is shown in Fig. 9.10. It can be concluded that hybrid method can get an accurate spectrum, therefore, the hybrid method can get accurate flux and fission source in energy range. It proves the efficiency of energy range hybrid. The conclusions are the same as in other three cases. Case 2. 3  3 fuel-pin lattice problem A 3  3 fuel-pin lattice structure was constructed by using the single fuel pin in Case 1 to verify different geometry arrangements. The results of three methods are presented in Table 9.3. The hybrid method runs faster than the pure Monte Carlo method by a factor of about 6.5. The accuracy of hybrid method is slightly better than the pure MOC.

Hybrid MC MOC Hybrid_error MOC_error

0.08

0.4

0.2 0.04 0.0

0.02 0.00

–0.2 –0.02 –0.4

–0.04 0

10

20

30

40

Group

Fig. 9.10 Energy spectrum of single fuel pin.

50

60

70

Error to MC

Normalized flux

0.06

268

Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

Table 9.3 Results of 3  3 fuel-pin lattice. keff MC MOC Hybrid

Confidence interval

1.23875 0.00093 1.23579 — 1.24076 0.0000043

Time (s)

MC time (s)

MOC time (s)

Discrepancy to MC

1334.5 89.27 204.51

— — 105.76

— 75.66 84.77

— 193 131

Fig. 9.11 3  3 fuel-pin lattice with Cd pin inside.

Table 9.4 Results of 3  3 fuel-pin lattice with Gd pin. keff MC MOC Hybrid

Confidence interval

0.88392 0.00069 0.87908 — 0.88390 0.0000062

Time (s)

MC time (s)

MOC time (s)

Discrepancy to MC

1331.5 123.67 238.22

— — 124.7

— 96.67 86.81

— 623 3

Case 3. 3  3 fuel-pin lattice with Gadolinium (Gd) pin in the center The fuel-pin in the center of Case 2 is replaced by a Gd pin. The structure of this case is shown in Fig. 9.11. The absorption of Gd pin is large, which causes strong resonance self-shielding effect. The results in Table 9.4 illustrate the effectiveness of hybrid method for resonance processing. From the results above, the accuracy of the hybrid method is much better than that of MOC, with a 5.6 times acceleration compared with the pure Monte Carlo simulation. Case 4. 3  3 fuel-pin lattice with Ag-In-Cd control rod in the center. The structure is the same as in Fig. 9.11, while an Ag-In-Cd control rod is in the center, instead of the Gd pin. The absorption of Ag-In-Cd control rod is stronger, so does its resonance self-shielding effect. From the results in Table 9.5, it can be observed that the hybrid method is still much better compared with the pure MOC, with a 3.2 times acceleration compared with the pure Monte Carlo method.

Other methods

269

Table 9.5 Results of 3  3 fuel-pin lattice with Ag-In-Cd pin. keff MC MOC Hybrid

Confidence interval

0.74012 0.00059 0.73397 — 0.74228 0.0000057

Time (s)

MC time (s)

MOC time (s)

Discrepancy to MC

1369.4 154.53 427.75

— — 249.01

— 85.52 131.48

— 1132 393

From the numerical results, it can be concluded that the hybrid method can combine the advantages of the deterministic and stochastic methods. In summary, the hybrid method provides a new solution to the neutron-transport problem. In the case of partly retaining the advantage of MC in the hybrid method, the accuracy is higher than pure deterministic method, and the speed is faster than pure MC. Due to the strong geometric adaptability of both MC and MOC, the hybrid method is expected to model and calculate complex geometry cases. With these advantages, it can be applied to scientific research and engineering applications.

References [1] L.Z. Cao, H.C. Wu, Y.Q. Zheng, Solution of neutron transport equation using Daubechies wavelet expansion in the angular discretization, Nucl. Eng. Des. 238 (9) (2008) 2292–2301. [2] Y.Q. Zheng, H.C. Wu, L.Z. Cao, Application of the wavelet expansion method in spatialangular discretization of the neutron transport equation, Ann. Nucl. Energy 43 (2012) 31–38. [3] C.B. Carrico, E.E. Lewis, G. Palmiotti, Three-dimensional variational nodal transport methods for Cartesian, triangular, and hexagonal criticality calculations, Nucl. Sci. Eng. 111 (2) (1992) 168–179. [4] Y.Z. Li, E.E. Lewis, M.A. Smith, H.C. Wu, L.Z. Cao, Preconditioned multi-group GMRES algorithms for the variational nodal method, Nucl. Sci. Eng. 179 (2015) 42–58. [5] J.C. Wagner, D.E. Peplow, S.W. Mosher, et al., Review of hybrid (deterministic/Monte Carlo) radiation transport methods, codes, and applications at Oak Ridge National Laboratory, Prog. Nucl. Sci. Technol. 2, 2011, 808–814. [6] I. Daubechies, Ten Lectures on Wavelets, Siam, Philadelphia, PA, 1992. [7] D.E. Newland, An Introduction to Random Vibrations, Spectral & Wavelet analysis, John Wiley and Sons, New York, 1993. [8] A. Cohen, I. Daubechies, P. Vial, Wavelets on the interval and fast wavelet transforms, Appl. Comput. Harmon. Anal. 1 (1) (1993) 54–81. [9] Y.Q. Zheng, H.C. Wu, L.Z. Cao, An improved three-dimensional wavelet-based method for solving the first-order Boltzmann transport equation, Ann. Nucl. Energy 36 (9) (2009) 1440–1449. [10] A.G. Buchan, C.C. Pain, et al., Linear and quadratic octahedral wavelets on the sphere for angular discretisations of the Boltzmann transport equation, Ann. Nucl. Energy 32 (11) (2005) 1224–1273.

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Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation

[11] G. Palmiotti, E.E. Lewis, C.B. Carrico, VARIANT: Variational Anisotropic Nodal Transport for Multidimensional Cartesian and Hexagonal Geometry Calculation ANL-95/40, Argonne National Laboratory, Lemont, IL, October 1995. LMFBR: Physics (UC-534). [12] Y. Azmy, E. Sartori, Nuclear Computational Science: A Century in Review, Springer Science & Business Media, Germany, 2010. [13] F.B. Brown, Fundamentals of Monte Carlo Particle Transport, Los Alamos National Laboratory, USA, 2005. LA-UR-05-4983. [14] J.C. Wagner, S.W. Mosher, Forward-weighted CADIS method for variance reduction of Monte Carlo reactor analyses, Trans. Am. Nucl. Soc. 103 (2010) 342–345. [15] M.J. Lee, H.G. Joo, D. Lee, K. Smith, Multi-Group Monte Carlo Reactor Calculation With Coarse Mesh Finite Difference Formulation for Real Variance Reduction, Oak Ridge National Lab, Oak Ridge, TN, October 2010, pp. 17–21. [16] E.W. Larsen, J. Yang, A functional Monte Carlo method for k-eigenvalue problems, Nucl. Sci. Eng. 159 (2) (2008) 107–126. [17] H. Lee, S. Choi, D. Lee, A hybrid Monte Carlo/method-of-characteristics method for efficient neutron transport analysis, Nucl. Sci. Eng. 180 (1) (2015) 69–85. [18] Z.Y. Liu, H.C. Wu, L.Z. Cao, Q.C. Chen, Y.Z. Li, A new three-dimensional method of characteristics for the neutron transport calculation, Ann. Nucl. Energy 38 (2–3) (2011) 447–454. [19] P. Liu, A special form and application of error transfer formula, J. Shandong Inst. Technol. 15 (1) (2001) 76–78.

Index Note: Page numbers followed by f indicate figures and t indicate tables. A Adjoint transport equation adjoint neutron transport equation, 30–32 adjoint operator, 29–30, 111 importance of neutron, 32–34 Albedo boundary, 16 Algebraic equations, 125–126 Algebraic subgrid scale (ASGS) method, 126 discrete maximum principles (DMP) of, 129 Angular approximation, 112–116 Angular flux, 95, 97–98f, 113–114 Angular integrals, 133–134 Angular Jacobian matrix, 115, 116f Angular matrix, 132 Angular quadrature set, 79, 83–84 Angular reflective matrix, 133–134 Angular subdomain, definition of, 246–247, 247f Anisotropic leakage approach, 94 Anisotropic scattering, 168–169 Approximation for three-dimensional calculation, 52 for two-dimensional calculation, 46–47, 52 of direction angle variable, 27 Area coordinates, 117–119, 118f, 178, 179f Asymmetric matrix equation, 16–17 AutoCAD, 85, 85f Axial blanket cross-sections, SN method, 207t, 211t Axial reflector cross-sections, SN method, 207t, 212t Azimuthal angle, 81–82 B Block mesh-free method, 216, 217f, 219, 225, 242 Block radial basis function collocation meshless (BRBFCM) 2D-IAEA problem, 228–231 2D-TWIGL problem, 234 2D2G problem, 227–228, 227f, 227t

numerical procedure of, 223–224 Boltzmann transport equation (BTE), 73–75 Boundary conditions, 73, 78, 80f, 120–121, 250 and unstructured-mesh sweeping, 51, 57 Boundary integral, 121–122 Boundary projection matrix, 134 BRBFCM. See Block radial basis function collocation meshless (BRBFCM) BTE. See Boltzmann transport equation (BTE) C C language, 223–224 C5G7 problem, 103–107 Cartesian coordinate system, 10 direction-dependent variable, 73–74 vs. area coordinates, 118 Cartesian mesh, 78, 92–93 Chinese fusion engineering test reactor (CFETR), 158–160 configuration of, 158, 159f meshes of, 158, 159f multigroup scalar flux distribution of, 158, 160f total scalar flux distribution of, 158, 160f typical tritium breeding blanket (TBB) module of, 153–156 Coarse region division, 68–69, 69f Coarsemesh finite differencing (CMFD) method, 259 Coefficient matrix, 256 Collision probability method (CPM), 61 calculation flowchart, 38–39, 38f evaluation, 40–41 for lattices, 40–42f for solving NTE, 36–38 properties, 41–42 Collocation discrete form, 221 Collocation meshless method, 215 Concave surfaces, 15f

272

Conjugate gradient (CG) method, 181–182 Conservation of neutron value, 32–33 Constructive solid geometry (CSG) modeling method, 84–85 Continuous finite element method, 125 Continuous neutron transport equation, 36, 38, 261–262 Continuous-Galerkin (CG) finite element methods, 109 Control rod cross-sections, SN method, 209t Control rod follower, SN method, 212t Convective matrix, 132 Convective-related matrix, integral of, 134 Convex surface, 15, 15f Coordinate system cylindrical coordinate system, 3, 3f rectangular coordinate system, 2, 2f spherical coordinate system, 4, 4f Coupling equations, 61–62 CPM. See Collision probability method (CPM) Current-coupled collision probability (CCCP) method numerical results 2D C5G7 problem, 68–70 regular and irregular cells, problems of, 64–67 theory, 61 coupling equations, 61–62 discretization, 61 response matrices, 63–64 Cylindrical coordinates, 3, 3f, 12 Cylindrical geometry, 12 D Daubechies order, 245–246 Daubechies scaling function, 248–251 Daubechies wavelets, 244–245 expansion method, 246 Definite condition, 13–16 Albedo boundary, 16 free surface, 15 interface boundary, 15 reflecting boundary, 15 white boundary, 16 Delayed neutron precursor balance equation, 218, 223 Deterministic method, 25 Diffusion matrix, 132 Diffusion-related matrix, integral of, 134–135

Index

Dilation order, 245 Dimensionless coefficient, 37 Discontinuous finite element method (DFEM), 174 Discontinuous-Galerkin (DG) finite element method, 109 Discrete maximum principles (DMP), 128–130 Discrete ordinate method (SN method), 27 finite element method, for unstructured geometry, 174–187 boundary condition, treatment of, 181 least-squares variation, 174–176 numerical results, 183–187 spatial discretization, 176–180 stiffness matrix, Solution of, 181–183 multidimensional discrete ordinate equation, Establishment of, 169–170 nodal method, for triangular prism meshes, 187–188 numerical results, 198–205 numerical solution process, 194–196 quadrature set selection, 196–198 spatial discretization, 188–194 one-dimensional discrete ordinate equation, establishment of, 167–169 spatial variable discretization, 171–173 Discrete spatial variables, 28 Discretion of energy variables, 25 Discretization, 61 DMP. See Discrete maximum principles (DMP) Driver with moderator cross-sections, SN method, 210t Driver without moderator cross-sections, SN method, 210t E Edge effect, 245 Eigenvalue decomposition, 121 Eigenvalue problem, 22–24 Energy group, 25, 26f Even neutron angular flux density, 17 F Fine region division, 68–69, 70f Finite difference method (FDM), 28, 251 Finite element implementation, PN method

Index

algebraic equation, evaluation of, 135 assembling process, 137 LHS matrix, 135–136 RHS vector, 136–137 integrals, evaluation of, 131–135 angular integrals, 133–134 angular matrix, 132 convective matrix, 132 diffusion matrix, 132 formulation, 132–133 spatial integrals, 134 spatial matrix, 132 spatial-angular integrals, 135 Finite element method (FEM), 28, 109 Finite element subspace, 176–177 First-order Lagrange, shape function of, 180 First-order transport equation, 6–16, 110 Cartesian coordinates, 10 cylindrical coordinates, 12 definite condition, 13–16 generation rate, 9 loss rate from collision, 8 from streaming, 7 neutron transport equation, 7 spherical coordinates, 13 Fission source term, 73 Fission spectrum, SN method, 208t, 212t Formulation, 132–133, 217–219 Forward weighted consistent adjoint driven importance sampling (FW-CADIS) method, 259 Four-group liquid metal fast breeder reactor (4G-LMFBR) problem, 237–238 Free surface, 15 Functional Monte Carlo (FMC) method, 259 G Galerkin least-squares (GLS) method, 122–124 Galerkin method, 137–143 Gauss-Legendre quadrature set, 197 Geometric entity assembling, 86, 88f Geometric modeling window, 86, 87f Geometry pretreatment method, 84–90 Global algebraic equations, 137 Global method, mesh-free method, 216, 217f Global radial basis function collocation method (GRBFCM), 215–216, 227f, 227t, 228–231

273

GLS method. See Galerkin least-squares (GLS) method GRBFCM. See Global radial basis function collocation method (GRBFCM) Group constants, 26 Group flux density, 26 H Helium cooling ceramic breeder (HCCB), for fusion DEMO, 153 Heterogeneous variational nodal method, 243, 253–258 legacy MG GS algorithm, 256–257 VNM formulations, 254–256 WG RBGS algorithm, 257–258 Hybrid method of Monte Carlo, 258–269 criticality calculation of transport equation, 259–261 deterministic method, in energy range, 264–265 energy ranges, 261–263 numerical results, 265–269 simulation, in resonance energy range, 263–264 I IAEA 5 region fixed source problem, 149–150 Improved leakage splitting method, 95–98, 104, 105–107t Inflow boundary matrix, 133–134 Integral form of transport equation, 20–22 Integral transport theory method, 27–28 Integral-form neutron transport equation, 35–36 Integrals, evaluation of, 131–135 Interface boundary, 15 International Atomic Energy Agency (IAEA), 228 Intrinsic free path (IFP), 128 Irregular 2D geometric problem, 98–100 Irregular cells, CCCP method, 66, 66–67f, 67–68t Isotropic leakage term, 95 J Jacobian matrix, 248–249

274

K KNK-1 reflector cross-sections, SN method, 211t KNK-II experimental fast reactor benchmark, 201 Kobayashi problem 3i, 151–152 L Leakage probability of surface, 47–49, 52–54, 56f Least-squares finite element method (LSFEM), 174 Least-squares variation, 174–176 Legacy MG GS algorithm, 256–257 Legendre polynomials, 114 LHS matrix, algebraic equation evaluation, 135–136 boundary terms, 136 convective terms, 136 diffusion terms, 135 reactive terms, 136 Linear algebraic equation, 38 Linear equations, 63 Linear triangular element, 117, 117f, 119, 119f Local radial basis function collocation method (LRBFCM), 215–216, 227f, 227t 2D-IAEA problem, 228–231 2D-TWIGL problem, 234 Long ray tracing technique, 76–77 LRBFCM. See Local radial basis function collocation method (LRBFCM) M Material composition, SN method, 208t Material definition, 86, 88f Material identity, 86 Maynard Watanabe problem, 143–146 Mesh-free method features of, 215 introduction, 215–216 numerical results, 224 2D-IAEA problem, 228–233 2D-TWIGL problem, 233–237 2D2G problem, 225–228 2D4G problem, 228 4G-LMFBR problem, 237–238

Index

Natelson problem, 239 transient localized source problem, 240–241 theory formulation, 217–219 numerical procedure and core development, 223–224 support domain, 216 types, 215 Method of characteristics (MOC) technique, 258–269 2D/1D fusion method, 90 basic equations, 90–93 improved leakage splitting method, 95–98 potential negative total sources, 93–94 traditional leakage splitting method, 94–95 basic equations, 73–75 calculations, in fast and thermal energy ranges, 263 criticality calculation of transport equation, 259–261 energy ranges, 261–263 numerical results, 265–269 C5G7 problem, 103–107 irregular 2D geometric problem, 98–100 single fuel particle problem, 100–102 three-dimensional, 80–90 3D modular ray tracing technique, 81–83 angular quadrature sets, 83–84 geometry pretreatment method, 84–90 two-dimensional, 75 long ray tracing technique, 76–77 modular ray tracing technique, 78–80 MOC. See Method of characteristics (MOC) technique Modular ray tracing technique, 78–80 angular quadrature set, 79 azimuthal angle restriction, 79, 80f boundary condition, 78, 80f Cartesian mesh, 78 modular ray, 78, 78–79f modularization, 81 Modular segments, 90 Modularization, 86 Monte Carlo (MC) method, 25, 66, 109 for irregular cells, 66, 68t for regular cells, 65–66, 65t

Index

MOSTELLER benchmark, 183, 185f, 185t MTR cell benchmark composition, 209t MTR-type reactor plate element, 185–186f, 186 Multidimensional discrete ordinate equation, 169–170 Multigroup neutron diffusion theory, 217–218 Multigroup neutron-transport equation, 262–263 Multiquadrics (MQ) radial basis function, 220 N Natelson problem, 239 Neutron integral transport equation, 43 Neutron scalar flux density, 17 Neutron transport equation (NTE), 1 approximate solution for, 24–28, 29t collision probability for solving, 36–38 governing equation, 223 numerical methods, 22–28 approximate solution, 24–28, 29t eigenvalue problem and source iteration method, 22–24 Neutron transport problem, 239 Neutron transport theory, 1, 6–7 Neutrons, 1 angular flux, 1, 5, 35 current density, 5, 5f density, 4, 6–7 energy phase-space, 261 flux, 231–232, 232f, 235f flying direction in triangular mesh, 43, 44f in triangular-z mesh, 43, 44f quadrant division, 44–45, 45f flying in mesh, 52–53 generation rate, 9 importance of, 32–34 in spherical symmetrical case, 13, 13f loss rate from collision, 8 loss rate from streaming, 7 motion of, 6–7 source in energy group, 39f trajectories of, 20f value, 32 conservation of, 32–33 Nondeterministic method, 25 Nonnegative-type matrix, 128–129

275

NTE. See Neutron transport equation (NTE) Nuclear fission reaction, 9 Nuclear reactor system, 219 Numerical integration, 115 Numerical methods of NTE, 22–28 approximate solution for neutron transport equation, 24–28 eigenvalue problem and source iteration method, 22–24 Numerical results current-coupled collision probability (CCCP) method 2D C5G7 problem, 68–70 regular and irregular cells, problems of, 64–67 discrete ordinate method (SN method) finite element method, for unstructured geometry, 183–187 nodal method, for triangular prism meshes, 198–205 hybrid method of Monte Carlo, 265–269 mesh-free method, 224 2D-IAEA problem, 228–233 2D-TWIGL problem, 233–237 2D2G problem, 225–228 2D4G problem, 228 4G-LMFBR problem, 237–238 Natelson problem, 239 transient localized source problem, 240–241 method of characteristics (MOC) technique, 265–269 C5G7 problem, 103–107 irregular 2D geometric problem, 98–100 single fuel particle problem, 100–102 spherical-harmonics-finite element method (PN-FEM) method 1D analytical problem, 137–138 1D Reed cell problem, 138–143 Chinese fusion engineering test reactor (CFETR), 158–160 helium cooling ceramic breeder, for fusion DEMO, 153 IAEA 5 region fixed source problem, 149–150 Kobayashi problem 3i, 151–152 Maynard Watanabe problem, 143–146 strong absorption problem, 148 strong scattering problem, 146–147

276

Numerical results (Continued) typical tritium breeding blanket (TBB) module, of CFETR, 153–157 transmission probability method, 57–59 2D problem, 57 3D problem, 57–59 transmission probability method (TPM), 57–59 2D problem, 57 3D problem, 57–59 wavelet expansion method, in neutron transport, 251–253 Numerical scheme, mesh-free method, 219–223 Numerical solution process, SN nodal method, 194–196 axial high-order moments, 196 axial outgoing surface, 196 incident surface, 195 radial high-order moments, 195–196 radial outgoing surface flux, 195 O Odd neutron angular flux density, 17 One dimensional plane coordinate system, 11–12, 11f One-dimensional (1D) analytical problem, 137–138 One-dimensional (1D) Reed cell problem, 138–143 One-dimensional (1D) SN calculations, 93 One-dimensional (1D) SN equation, on Cartesian mesh, 92–93 One-dimensional discrete ordinate equation, 167–169 Outflow boundary matrix, 133–134 P Petrov-Galerkin method, 121–124 Phase space boundary, 110 Pin-homogenized angular flux, 95–96, 96f, 98 Planar method of characteristics calculation, 93 equation, 95 Polar angle, 82 Potential negative total sources, 93–94 Pressurized water reactor (PWR), 228 Probability for three-dimensional calculation, 52–53

Index

for two-dimensional calculation, 47–48 PWR. See Pressurized water reactor (PWR) Q Quadrature set selection, 196–198 R Radial basis function collocation method (RBFCM), 215–216 Radial blanket cross-sections, SN method, 206t Radial reflector cross-sections, SN method, 207t Reactive matrix, 132 Reactive-related matrix, integral of, 134 Reactor core cross-sections, SN method, 206t Real-form spherical harmonic function, 112 Rectangular coordinate system, 2, 2f arbitrary triangle under, 188–189, 188f equilateral triangle under, 188–189, 189f Reed cell benchmark problem description, 251, 251f flux distribution, comparison of, 251, 252f Reflective boundary condition, 15, 120–121, 181 Reflector with moderator cross-sections, SN method, 210t without moderator cross-sections, SN method, 210t Region-averaged scalar flux, 75 Region-wise averaged neutron flux density, 37 source density, 37 Regular cells problems, CCCP method, 64–66, 64f, 65t Removal matrix, 115 Response matrices, 62–64 RHS vector, algebraic equation evaluation, 136–137 source convective terms, 137 source reactive terms, 137 Riemann decomposition, 121 S Scattering function, 9 Scattering probability function, 114 Scattering source term, 73

Index

Second-order even symmetric neutron transport equation, 17–19 Second-order self-conjugate SAAF transport equation, 19–20 Second-order transport equation, 16–20 even symmetric neutron transport equation, 17–19 self-conjugate SAAF transport equation, 19–20 Semi-implicit time discretization scheme, 223 SGS methods. See Subgrid scale (SGS) methods Single fuel particle problem, 100–102 60 degree symmetric quadrature set, 196, 197f, 198, 199t SN method. See Discrete ordinate method (SN method) Sodium/steel cross-sections, SN method, 211t Source iteration method, 22–24 Spatial angle scanning process, 172, 173f Spatial discretization, 116–120 SN finite element method, for unstructured geometry, 176–180 SN nodal method, for triangular prism meshes, 188–194 Spatial discretization method, 93 Spatial grid, 172, 173f Spatial integrals, 134 convective-related matrix, 134 diffusion-related matrix, 134–135 reactive-related matrix, 134 Spatial matrix, 132 Spatial variable discretization, 171–173 Spatial wavelet expansion solution (WEM), 251–252 Spatial-angular integrals, 135 convective matrix, 135 diffusion matrix, 135 reactive matrix, 135 Spherical coordinate system, 4, 4f, 13 Spherical harmonic expansion approximation, 27 Spherical-harmonics-finite element method (PN-FEM) method basic equations first-order transport equation, 110 weak formulation, from weighted residual method, 111–112 equation, discretization of, 112 angular approximation, 112–116

277

boundary conditions, 120–121 spatial discretization, 116–120 finite element implementation (see Finite element implementation) numerical results 1D analytical problem, 137–138 1D Reed cell problem, 138–143 Chinese fusion engineering test reactor (CFETR), 158–160 helium cooling ceramic breeder, for fusion DEMO, 153 IAEA 5 region fixed source problem, 149–150 Kobayashi problem 3i, 151–152 Maynard Watanabe problem, 143–146 strong absorption problem, 148 strong scattering problem, 146–147 typical tritium breeding blanket (TBB) module, of CFETR, 153–157 recurrence relation, PN, 161–163 shape function, of simplex element, 162–163 line segment, 162 tetrahedron, 162–164 triangle, 162 stabilized finite element method (see Stabilized finite element method) Stabilization parameter, 127–130 Stabilized finite element method, 121–122 Galerkin least-squares (GLS) method, 122–124 parameter values, 130, 131t stabilization parameter, value of, 127–130 streamline upwinding Petrov-Galerkin (SUPG) method, 122 subgrid scale (SGS) methods, 124–126 Taylor-Galerkin (TG) method, 124 unified expression of, 126–127, 130 Steady-state Boltzmann equation, 20–21 Steady-state multigroup neutron transport equation, 254 Steady-state neutron transport equation, 10, 12, 19 Steel cross-sections, SN method, 211t Stiffness matrix solution, 181–183 Stochastic method, 258–260 Streamline upwinding Petrov-Galerkin (SUPG) method, 122 Strong absorption problem, 148 Strong scattering problem, 146–147

278

Subgrid scale (SGS) methods, 124–126 Sun region division, 68–69, 70f SUPG method. See Streamline upwinding Petrov-Galerkin (SUPG) method Surface neutron angular flux, 52 T TAKEDA benchmark, 183, 198 Taylor-Galerkin (TG) method, 124, 128 Test zone cross-sections, SN method, 209t TG method. See Taylor-Galerkin (TG) method 3D MOC. See Three-dimensional method of characteristics (3D MOC) Three-dimensional (3D) modular method, 84–85 Three-dimensional (3D) modular ray tracing technique, 81–83 Three-dimensional calculation, transmission probability method, 51–57 approximation, 52 boundary conditions and unstructured-mesh sweeping, 57 leakage probability, 53–54 probability, 52–53 transmission probability, 55–56 Three-dimensional method of characteristics (3D MOC), 80–90 TPM. See Transmission probability method (TPM) Traditional leakage splitting method, 94–95 Transient localized source pro, 240–241 Transmission probability method (TPM), 43 basic equations, 43–46 numerical results, 57–59 2D problem, 57 3D problem, 57–59 three-dimensional calculation, 51–57 approximation, 52 boundary conditions and unstructured-mesh sweeping, 57 leakage probability, 53–54 probability, 52–53 transmission probability, 55–56 two-dimensional calculation, 46–51 approximation, 46–47 boundary conditions and unstructured-mesh sweeping, 51

Index

leakage probability, 48–49 probability, 47–48 transmission probability, 49–51 Transmission probability of surface, 49–51, 55 Transport equation discretization, 61 Transport operator, 110 Triangular prism meshes, SN nodal method for, 187–205 Triangular unit, Jacobian transformation matrix of, 179 Two-dimensional (2D) C5G7 problem, CCCP method, 68–70 Two-dimensional (2D) modular ray tracing technique, 81, 82f Two-dimensional (2D) planar equation, 95 Two-dimensional angular geometry, quadrant of, 172, 172f Two-dimensional calculation, transmission probability method, 46–51 approximation, 46–47 boundary conditions and unstructured-mesh sweeping, 51 leakage probability, 48–49 probability, 47–48 transmission probability, 49–51 Two-dimensional four-group (2D4G) problem, 228 Two-dimensional method of characteristics (MOC), 75–80 2D/1D fusion method. See Method of characteristics (MOC) technique, 2D/1D fusion method Two-dimensional pressurized water reactor (PWR) benchmark problem, 228 2D-TWIGL benchmark problem, 233–237 Two-dimensional-two-group (2D2G) problem, 225–228 Typical tritium breeding blanket (TBB) module, of CFETR, 153–156 U Unified expression, of stabilized finite element method, 126–127, 130 V Vacuum boundary condition, 120–121, 181 VARIANT code, 254

Index

Variational nodal method (VNM), 253–254 eigenvalue system, 256 formulations, 254–256 source equation, 255 within-group current equation, 255 within-group flux equation, 255 Vector, 6 VNM. See Variational nodal method (VNM) Void zone cross-sections, SN method, 207t W Wavelet basis function, 243–246 Wavelet expansion method, 243, 246–251

279

in neutron transport numerical results, 251–253 wavelet basis function, 243–246 wavelet expansion method, 246–251 Weak formulation, 111–112, 116–117, 121–122, 124 Weighted residual method, weak formulation from, 111–112 WG RBGS algorithm, 257–258 White boundary, 16 X XData data type, 86