Numerical stability of the predictor-corrector method in Monte Carlo burnup calculations of critical reactors

Monte Carlo burnup codes use various schemes to solve the coupled criticality and burnup equations. Previous studies have shown that the simplest methods, such as the beginning-of-step and middle-of-step constant flux approximations, are numerically unstable in fuel cycle calculations of critical reactors. Here we show that even the predictor-corrector methods that are implemented in established Monte Carlo burnup codes can be numerically unstable in cycle calculations of large systems. © 2013 Elsevier Ltd. All rights reserved.

Coupled neutronic thermo-hydraulic analysis of full PWR core with Monte-Carlo based BGCore system

BGCore reactor analysis system was recently developed at Ben-Gurion University for calculating in-core fuel composition and spent fuel emissions following discharge. It couples the Monte Carlo transport code MCNP with an independently developed burnup and decay module SARAF. Most of the existing MCNP based depletion codes (e.g. MOCUP, Monteburns, MCODE) tally directly the one-group fluxes and reaction rates in order to prepare one-group cross sections necessary for the fuel depletion analysis. BGCore, on the other hand, uses a multi-group (MG) approach for generation of one group cross-sections. This coupling approach significantly reduces the code execution time without compromising the accuracy of the results. Substantial reduction in the BGCore code execution time allows consideration of problems with much higher degree of complexity, such as introduction of thermal hydraulic (TH) feedback into the calculation scheme. Recently, a simplified TH feedback module, THERMO, was developed and integrated into the BGCore system. To demonstrate the capabilities of the upgraded BGCore system, a coupled neutronic TH analysis of a full PWR core was performed. The BGCore results were compared with those of the state of the art 3D deterministic nodal diffusion code DYN3D (Grundmann et al.; 2000). Very good agreement in major core operational parameters including k-eff eigenvalue, axial and radial power profiles, and temperature distributions between the BGCore and DYN3D results was observed. This agreement confirms the consistency of the implementation of the TH feedback module. Although the upgraded BGCore system is capable of performing both, depletion and TH analyses, the calculations in this study were performed for the beginning of cycle state with pre-generated fuel compositions. © 2011 Published by Elsevier B.V.

Efficient generation of one-group cross sections for coupled Monte Carlo depletion calculations

Coupled Monte Carlo depletion systems provide a versatile and an accurate tool for analyzing advanced thermal and fast reactor designs for a variety of fuel compositions and geometries. The main drawback of Monte Carlo-based systems is a long calculation time imposing significant restrictions on the complexity and amount of design-oriented calculations. This paper presents an alternative approach to interfacing the Monte Carlo and depletion modules aimed at addressing this problem. The main idea is to calculate the one-group cross sections for all relevant isotopes required by the depletion module in a separate module external to Monte Carlo calculations. Thus, the Monte Carlo module will produce the criticality and neutron spectrum only, without tallying of the individual isotope reaction rates. The onegroup cross section for all isotopes will be generated in a separate module by collapsing a universal multigroup (MG) cross-section library using the Monte Carlo calculated flux. Here, the term "universal" means that a single MG cross-section set will be applicable for all reactor systems and is independent of reactor characteristics such as a neutron spectrum; fuel composition; and fuel cell, assembly, and core geometries. This approach was originally proposed by Haeck et al. and implemented in the ALEPH code. Implementation of the proposed approach to Monte Carlo burnup interfacing was carried out through the BGCORE system. One-group cross sections generated by the BGCORE system were compared with those tallied directly by the MCNP code. Analysis of this comparison was carried out and led to the conclusion that in order to achieve the accuracy required for a reliable core and fuel cycle analysis, accounting for the background cross section (σ0) in the unresolved resonance energy region is essential. An extension of the one-group cross-section generation model was implemented and tested by tabulating and interpolating by a simplified σ0 model. A significant improvement of the one-group cross-section accuracy was demonstrated.

The stochastic implicit Euler method - A stable coupling scheme for Monte Carlo burnup calculations

Existing Monte Carlo burnup codes use various schemes to solve the coupled criticality and burnup equations. Previous studies have shown that the coupling schemes of the existing Monte Carlo burnup codes can be numerically unstable. Here we develop the Stochastic Implicit Euler method - a stable and efficient new coupling scheme. The implicit solution is obtained by the stochastic approximation at each time step. Our test calculations demonstrate that the Stochastic Implicit Euler method can provide an accurate solution to problems where the methods in the existing Monte Carlo burnup codes fail. © 2013 Elsevier Ltd. All rights reserved.

Numerically stable Monte Carlo-burnup-thermal hydraulic coupling schemes

This paper presents stochastic implicit coupling method intended for use in Monte-Carlo (MC) based reactor analysis systems that include burnup and thermal hydraulic (TH) feedbacks. Both feedbacks are essential for accurate modeling of advanced reactor designs and analyses of associated fuel cycles. In particular, we investigate the effect of different burnup-TH coupling schemes on the numerical stability and accuracy of coupled MC calculations. First, we present the beginning of time step method which is the most commonly used. The accuracy of this method depends on the time step length and it is only conditionally stable. This work demonstrates that even for relatively short time steps, this method can be numerically unstable. Namely, the spatial distribution of neutronic and thermal hydraulic parameters, such as nuclide densities and temperatures, exhibit oscillatory behavior. To address the numerical stability issue, new implicit stochastic methods are proposed. The methods solve the depletion and TH problems simultaneously and use under-relaxation to speed up convergence. These methods are numerically stable and accurate even for relatively large time steps and require less computation time than the existing methods. © 2013 Elsevier Ltd. All rights reserved.

Generation of SFR few-group constants using the Monte Carlo code serpent

In this study, the Serpent Monte Carlo code was used as a tool for preparation of homogenized few-group cross sections for the nodal diffusion analysis of Sodium cooled Fast Reactor (SFR) cores. Few-group constants for two reference SFR cores were generated by Serpent and then employed by nodal diffusion code DYN3D in 2D full core calculations. The DYN3D results were verified against the references full core Serpent Monte Carlo solutions. A good agreement between the reference Monte Carlo and nodal diffusion results was observed demonstrating the feasibility of using Serpent for generation of few-group constants for the deterministic SFR analysis.

Control of aircraft in the terminal manoeuvring area using parallelised sequential Monte Carlo

This paper reports on the use of a parallelised Model Predictive Control, Sequential Monte Carlo algorithm for solving the problem of conflict resolution and aircraft trajectory control in air traffic management specifically around the terminal manoeuvring area of an airport. The target problem is nonlinear, highly constrained, non-convex and uses a single decision-maker with multiple aircraft. The implementation includes a spatio-temporal wind model and rolling window simulations for realistic ongoing scenarios. The method is capable of handling arriving and departing aircraft simultaneously including some with very low fuel remaining. A novel flow field is proposed to smooth the approach trajectories for arriving aircraft and all trajectories are planned in three dimensions. Massive parallelisation of the algorithm allows solution speeds to approach those required for real-time use.

Monitoring and preventing numerical oscillations in 3D simulations with coupled Monte Carlo codes

Previous studies have reported that different schemes for coupling Monte Carlo (MC) neutron transport with burnup and thermal hydraulic feedbacks may potentially be numerically unstable. This issue can be resolved by application of implicit methods, such as the stochastic implicit mid-point (SIMP) methods. In order to assure numerical stability, the new methods do require additional computational effort. The instability issue however, is problem-dependent and does not necessarily occur in all cases. Therefore, blind application of the unconditionally stable coupling schemes, and thus incurring extra computational costs, may not always be necessary. In this paper, we attempt to develop an intelligent diagnostic mechanism, which will monitor numerical stability of the calculations and, if necessary, switch from simple and fast coupling scheme to more computationally expensive but unconditionally stable one. To illustrate this diagnostic mechanism, we performed a coupled burnup and TH analysis of a single BWR fuel assembly. The results indicate that the developed algorithm can be easily implemented in any MC based code for monitoring of numerical instabilities. The proposed monitoring method has negligible impact on the calculation time even for realistic 3D multi-region full core calculations. © 2014 Elsevier Ltd. All rights reserved.

高分子溶液中物理凝胶化的Monte Carlo模拟和小角中子散射研究

高分子凝胶广泛地存在于自然界以及日常生活中，按其形成作用力不同分为化学凝胶和物理凝胶两大类。由于高分子物理凝胶具有凝胶化的可逆性及其对环境条件强烈的响应性，因此，在近半个世纪的研究与应用中受到极大的关注。高分子溶液中的物理凝胶因其结构及形成机制复杂，在实验方面，除了散射技术及流变技术能够有效地揭示它的部分信息外，其它的实验手段很难用于这个领域的研究；在理论方面，化学凝胶的理论已经比较成熟，而物理凝胶的粘弹性质以及凝胶化是一个远离平衡态的松弛过程，除了一些特征的标度指数外，人们还没有得到适用于高分子物理凝胶的普适规律。当前，由于计算机模拟理论及模拟方法的发展，使得计算机模拟成为除了实验和理论研究方法之外的第三个重要的研究方法。但是，由于物理凝胶化行为的复杂性，用实验和理论获得的信息很难较好地描述凝胶化过程，而计算机模拟的高度透明性及反映信息的完整性，有助于理解这一复杂过程中所涉及的物理本质。因此，利用计算机模拟结合实验及理论方法深入研究高分子物理凝胶的形成机制、结构与性能关系已成为目前最有效的手段之一。 本论文主要运用Monte Carlo模拟方法，并结合小角中子散射（Small-Angle Neutron Scattering, SANS）和流变（Rheology）等实验手段从多个角度探讨了以下几类典型的高分子溶液物理凝胶化行为。 1. 温度对遥爪型三嵌段共聚物在选择性溶剂中的自组装及凝胶化行为影响的研究：采用二维简单方格子Monte Carlo模拟方法，结合逾渗（Percolation）理论，建立了溶胶－凝胶转变相图在统计热力学中的确定方法；甄别了具有特征构象的链，讨论了链及胶束的聚集，明晰了相互作用（体现为约化温度）、构象转变、聚集与凝胶化的一致的关联关系；提出了构象转变模型，进而明确了此体系的凝胶化过程，在微观尺度上表现为桥型链和环型链之间的竞争。 2. 模拟模型改进及其应用到持续长度对稀溶液中高分子链构象影响的研究：考虑到原始八位置键涨落模型效率低，实现复杂且不能应用到复杂的高分子体系，对该模型进行了改进，使其实现简单、效率高，并拓宽了该模型的应用范围。然后，以刚性对均聚物构象的影响为例，发现随着刚性增加，均聚物构象从球形椭球到棒状椭球的转变，并对比了自由连接链（Free Joint Chain, FJC）模型和蠕虫链（Wormlike Chain, WLC）模型在不同刚性范围内对高分子链末端距预测的偏差，首次给出了这两个经典模型的半定量的适用边界。 3. 溶剂尺寸对遥爪型三嵌段共聚物在选择性溶剂中的自组装及凝胶化行为影响的研究：用改进后的八位置键涨落Monte Carlo模型，研究了遥爪型三嵌段共聚物在选择性溶剂条件下的聚集和凝胶化对溶剂尺寸的依赖性，发现溶剂尺寸效应对凝胶化的作用是非单调的。由一个均聚物体系的对比模拟证明这种作用主要是由熵驱动的，并给出了中分子溶剂的半定量定义。在均聚物和嵌段共聚物溶液中，不同尺寸的溶剂分子可以使溶液由于高分子聚集不同而具有不同的微结构，并影响高分子链构象和溶液的性质。从多个角度研究了三嵌段共聚物在不同尺寸溶剂的溶液中所遵循的三种不同的凝胶化机理。 4. 聚氧化乙烯－氧化丙稀－氧化乙烯三嵌段共聚物（poly(ethylene oxide)-poly (propylene oxide)-poly-(ethylene oxide), PEO-PPO-PEO）重水溶液凝胶化的小角中子散射（SANS）和Monte Carlo研究：结合Pluronic F127（EO65PO99EO65）/D2O三嵌段共聚物溶液的特征，对照SANS数据，用改进后的八位置键涨落模型成功地从模拟中获得了F127/D2O的溶胶－凝胶转变相图。详细地考察了体系的微观结构，提出此类高分子溶液中形成的物理凝胶包含高分子逾渗网络的生成，以及被束缚溶剂（Bound Solvent）必须超过离散组分体系逾渗的临界体积分数的机理。着重研究了一定浓度的F127水溶液随温度升高引起的溶胶－凝胶转变以及凝胶－溶胶转变的Reentrant相行为，发现体系在低温区域的溶胶－凝胶转变遵循相同的机理，而在中等温度和较高温度以及不同浓度区域中的凝胶－溶胶转变遵循不同的机理。 5. 极性基团饱和度和溶剂条件对两亲性聚合物在溶液中的聚集行为和凝胶化影响的研究：用改进后的八位置键涨落模型，针对两亲性聚合物在不同溶剂条件的溶液建立了粗粒化模型，以两亲性聚合物中极性基团的饱和度，溶剂条件和高分子浓度为变量，考察了其对链构象、聚集及其凝胶化的影响。 6. 多糖水溶液凝胶化的流变和小角中子散射研究：用流变和SANS考察了两个多糖水溶液中物理凝胶化过程，针对由氢键主导的水基凝胶体系的典型特征进行了讨论，从分子链构象，聚集体结构及其关联以及流变特征等方面对聚强电解质角叉胶（Carrageenan）水溶液和聚弱电解质明胶（Pectin）水溶液进行了详细的讨论。考察了不同多糖的种类（聚合物链的电荷密度），盐的种类和浓度，溶液温度等对凝胶化和凝胶结构的影响，分析了不同多糖溶液的凝胶化机理。

聚烯烃官能化及其反应共混的Monte Carlo

针对马来酸酐(MAH)接枝聚丙烯(PP)和聚乙烯(PE)、不相容共混物的反应增容、聚丙烯共混物在剪切流动过程中形态结构演化等聚烯烃反应加工中典型的化学和物理问题开展了计算机模拟和实验研究。 首先，我们建立和完成了适用于Monte Carlo计算机模拟的MAH接枝PP和PE的反应动力学模型及模拟程序的编写与运行。对于MAH接枝PP，模拟结果表明接枝产物所占比例与MAH初始浓度有关。当MAH浓度较低时(小于2.5 wt%)，MAH主要接枝在由β裂解所产生PP链末端上；而在MAH浓度较大时(大于2.5 wt%)，大部分MAH接枝在PP链上的三级碳上。这一结论很好地解决了多年来存在的关于MAH接枝PP位置的争议。为了进一步验证模拟结果，在实验上我们制备了不同MAH(13C标记)初始浓度下接枝PP的系列样品。核磁共振(NMR)研究结果表明MAH的初始浓度明显影响不同接枝产物所占比例，具体表现为MAH接枝到PP三级碳的MAH的NMR共振峰（δ= 32.0 ppm）随MAH的初始浓度的增加而明显增强，而接枝到由β裂解所产生PP链末端的MAH的NMR共振峰（δ=30.3 ppm）随MAH的初始浓度的增加而明显减弱。这与模拟结果一致。对于MAH接枝线性PE，模拟结果表明MAH接在PE主链上形成枝状结构所占比例随MAH初始浓度的增加而增加，而接在两PE主链中间形成桥状结构所占比例随MAH初始浓度的增加而下降。当MAH浓度非常低的时候，两种结构所占比例相当接近。这一结果改变了人们对这一问题的传统认识，即在任何条件下桥状结构所占比例都远远低于枝状结构。以上结果为MAH接枝聚乙烯、聚丙烯分子结构的调控提供了科学依据。 其次，我们开展了伴有化学反应的不相容共混聚合物的增容，即反应增容的Monte Carlo模拟研究。模拟结果表明官能化聚合物A的加入有效改善了聚合物A与极性聚合物B的相容性。当聚合物A为分散相时，A-B和A-B-A嵌段共聚物的增容效果比B-A-B嵌段共聚物好。我们发现原位生成的A-B、A-B-A和B-A-B三种嵌段共聚物在共混体系中微观结构各不相同。所生成的A-B两嵌段共聚物分布在A/B两相界面上，其A、B嵌段分别嵌入A、B相区里；所生成的A-B-A三嵌段共聚物则通过“桥状结构”连接两个被分散的A相区；所生成的B-A-B三嵌段共聚物则以“折叠结构”存在于A/B两相界面。此外，我们还研究了接枝共聚物在A/B/接枝共聚物三元共混体系中的增容效果及其微观结构。模拟结果表明，接枝共聚物的结构和支链长度对其在共混物中的微观结构和分散相粒径影响很大。当添加A-g-B接枝共聚物作为增容剂时，如果其支链较短，部分接枝共聚物将会在连续相中形成胶束；当其支链较长时，它们会通过“桥状结构”连接分散相形成网络结构。当选择B-g-A接枝共聚物作为增容剂，如果其支链长度较短，部分共聚物会在分散相里形成胶束；如果其支链较长，大部分共聚物将存在于A/B两相界面上连接A、B两相。 最后，我们在线跟踪研究了聚丙烯共混物在剪切流动过程中形态结构演化过程，观察到了剪切流动下尼龙6(PA6)液滴在PP连续相中的破裂过程。结果表明PA6液滴同时存在fracture 和tip streaming 两种破裂模式。在该共混体中添加少量的SEBS或SEBS-g-MAH，发现在适当的剪切条件下PA6液滴可通过SEBS粘结形成非常有序的规则结构，即平行排列的线条结构。有趣的是这种平行排列的线条结构垂直于剪切流动方向。进一步研究结果表明该结构是一亚稳态，其最终要聚集成球状结构。有意义的是该亚稳态结构可以保持数十分钟以上，这使得人们有足够的时间降温将该结构“冻结”住。这一结果为通用高分子共混物有序结构的外场调控提供了成功范例。

多相多组分高分子体系相行为的Monte Carlo模拟

多相多组分高分子体系的相行为和形态结构一直是高分子科学研究的前沿领域。在这一领域的理论研究中，由于高分子间和高分子链段间相互作用的复杂性，理论结果往往是在一定的近似条件下得到的；另一方面，可用于实验研究的样品总是有限的，而改变体系中分子间的相互作用对实验来说又是极其困难的工作。而计算机模拟却可以避开这些条件的限制。目前计算机模拟不仅仅是实验和理论的补充，人们已经把它看成是"实验"和"理论"以外的第三个分支；由于Monte Carlo模拟能够在分子水平描述模型体系的信息、，进而对高分子体系的相行为和微观结构给出一些符合实际的预测，因而被广泛地应用于物理、数学、化学等一泊一二多领域中。本一沦文主要对二维的多相多组分高分子体系的相行为、形态结构等进行了Monte Carlo模拟研究。我们首先对刀B共聚物和均聚物C的共混体．系相行为进行了研究，其中链节B与C之间存在吸引的特殊相互作用。我们的模拟结果表明特殊相互作用的引入可以改善体系的共混相容性。在链二佩声B含量较低时，A/B无规共聚物和均聚物C的共混体系是相分离的，而在链节B的含量高于80％时，此共混体系是相容的；当链节B的含量较低时，链节B倾向于分布在共聚物相的界面上；另一方面，链节B的含量在O％到80%之间时共聚物的平均末端距小于均聚物的；并且，给出了温度一定时，B含量随均聚物组成变化关系的相图。在研究链结构对体系相形态的影响时，我们考虑了A/B交替共聚物、无规共聚物和嵌段共聚物分别与均聚物C的混合体系，其中，交替共聚物的链节B全部集中在均聚物相的界面上；无规共聚物的链节B大部分聚集在均聚物相的界面上；而嵌段共聚物更倾向于钻进均聚物聚集的区域。此外，我们研究了A/B二嵌段共聚物与均聚物C共混体系，在均聚物为连续相的体系中，出现了链段A聚集形成核链段B为壳的核壳结构。当链段B的含量较低时，共聚物的浓度越高，形成的核越大，核尺寸的分布也越来越宽，当链段B的含量很高时，核凡乎不随共聚物浓度的变化而变化。我们还研究了链节A、B和C之间是都是排斥作用的A／B共聚物和均聚物C共混体系的形态结构，并考虑了链结构对体系共混相容佩生佩狗影响（我们研究了交替共聚物、无规共聚物和嵌段共聚物三种链结构的共聚物）。随着共聚物分子内的链节A和B之间相互作用能的增加，共聚物和均聚物的共混相容性越好；当存在较强的分子内排斥作用时，共聚物的链结构强烈地影响此共混体系的相容性和均聚物的链构象。对于较强的分子内排斥作用的体系，交替共聚物与均聚物的共混相容性最好，并月＿形成了相容的体系，其次是无规共聚物和均聚物的共混体系，而嵌段共聚物与均聚物的共混相容性最差；当分子内的相互作用很弱时，共聚物的结构对共聚物与均聚物的共混相容性和均聚物的构象的影响均很刁、。此外，我们对高分子溶液的自组装进行了Monte Car to模拟。在我们的模拟中发现了：州B二嵌段共聚物在选择性溶剂中（溶剂是链段B的良溶剂）自组装成核壳结构的胶束，且随着高分子溶液浓度的增加，核不断增大。在A一B一A三嵌段共聚物溶液体系中，溶剂是链段B的良溶剂，是链段A的不良溶剂，此时我们得到了花型结构的胶束，其中花心是由链段A聚集而成，花瓣是由链段B形成的，而且在体系中也发现了有些胶束间存在着桥，并且在浓度很高时，体系出现了凝胶现象。此外，我们还研究了由特殊相互作用引起的结构变化。选取A／B嵌段共聚物和A／C无规共聚物的溶液体系，其中链节B和链节C之间存在吸引的特殊相互作用，而链节A与链节B和C之间都是排斥的，选择性溶剂是链节A的良溶剂，对于链节B和C是不良溶剂。在此条件刁（，体系形成了核壳结构，其中核主要由链节B和C以及少量的无规共聚物中的链节A组成。当链节C的含量较低时，壳由A／B嵌段共聚物的链段A和A/C无规共聚物的部分链节组成，当链节C的含量很高时，壳就几乎完全由A／B嵌段共聚物的链段A组成。此外，链节C在无规共聚物中的含量强烈影响核的密度极其分布。最后我们对一硬段一软段二嵌段共聚物进行了研究，发现随着硬段在共聚物中含量的增加和硬段的刚性程度的增加，嵌段共聚物更易形成有序结构。本论文中，我们充分考虑了高分子链结构及其分子内和分子间的相互作用对体系形态结构的影响，并且建立了核尺寸、核尺寸'分布、核密度、核密度分布以及平均末端距和平均木端距分布等计算机统计方法。所得到的结果将为高分子多相多组．分体系的相行为和形态结构以及高分子自组装超分子结构材料的研究和设计提供充分依据。

对称三嵌段共聚物薄膜有序-无序转变的Monte Carlo模拟

嵌段共聚物薄膜自组装的研究一直是高分子科学研究的前沿领域。嵌段共聚物受限于薄膜中，与边界之间的作用将影响介观相形貌和形貌的定向。由于聚合物体系的复杂性，理论结果往往是在各种近似条件下得到的；另一方面，实验研究中理想条件的实现极其困难，尤其是对薄膜和超薄膜的研究中。而计算机模拟却可以避开这些条件的限制。本文模拟结果表明：退火使聚合物链充分松弛，消除了淬火过程中可能存在的缺陷。与淬火比较，退火体系中分子链的均方回转半径变小，分子链的张角变大，桥健分子链的数量增加。与对称二嵌段共聚物相比，在对称三嵌段共聚物薄膜中，膜的中部先于表面区域形成有序结构。此外，、对薄膜中三嵌段共聚物的退火行为的模拟，得出了退火过程中静态性质、动态性质和分子链的构型等随温度和薄膜厚度的变化，并指出有序一无序转变温度随薄膜厚度增加而升高，直到接近本体，转变温度趋于平衡。通过对不同链长的三嵌段共聚物的模拟，得出了退火过程中分子链构型随链长的变化，并表明有序一无序转变温度＊与分子链长呈线性关系，随着分子链增加，T*升高，T*随分子链长变化的斜率（即：T*／N）为常数，即：不随链长变化而变化。