Comparison between correlated sampling and the perturbation technique of MCNP5 for fixed-source problems

Correlated sampling and the differential operator perturbation technique are two methods that enable MCNP (Monte Carlo N-Particle) to simulate small response change between an original system and a perturbed system. In this work the performance of the MCNP differential operator perturbation technique is compared with that of the MCNP correlated sampling method for three types of fixed-source problems. In terms of precision of predicted response changes, the MCNP perturbation technique outperforms correlated sampling for the problem involving variation of nuclide concentrations in the same direction but performs comparably with or even underperforms correlated sampling for the other two types of problems that involve void or variation of nuclide concentrations in opposite directions. In terms of accuracy, the MCNP differential operator perturbation calculations may predict inaccurate results that deviate from the benchmarks well beyond their uncertainty ranges for some of the test problems. However, the accuracy of the MCNP differential operator perturbation can be improved if the midpoint correction technique is used.     

Multi-Assembly Analysis with an Advanced Correlated Sampling Method

The Advanced Correlated Sampling (ACS) method, which can greatly enhance the calculation efficiency of Monte Carlo (MC) methods using a perturbation technique and a pseudo-scattering technique, is improved so as to be able to handle multi-assembly analysis. Two boundary connection methods—the neutron path connection and neutron current connection methods—are developed, and their accuracies for a 2×2 multi-assembly system are confirmed. The speedup factor against the corresponding direct MC calculation is a maximum of about 100. These boundary connection methods are also applied in pin-wise 2-D quarter core calculations as an example of a large-scale system. A newly developed perturbed neutron source iteration technique allows for very smooth source convergence. Furthermore, by incorporating the Direct Response Matrix (DRM) method as the acceleration means for the source convergence, the number of source iteration cycles is significantly shortened. The results confirm that this improved ACS method has the ability to perform pin-wise whole core calculations very accurately and efficiently.     

Improvement of fission source distribution by correlated sampling method in Monte Carlo perturbation calculations

To increase the accuracy of Monte Carlo perturbation calculations, the main issue is to estimate the variations of fission source distribution in perturbed systems. For estimating the difference between effective multiplication factors in unperturbed and perturbed systems, this study proposes combining the fission matrix method and the correlated sampling method, and applying the weight window technique for stabilizing the weight fluctuation of fission sources in perturbed systems. By applying the proposed method to the Smart and User-frIendly Monte Carlo Particle Transport Code (SUIT) Monte Carlo code, perturbation calculations are carried out for GODIVA and for simplified STACY problems. The results thus estimated show good agreement compared with those of reference calculations, demonstrating that the proposed method can effectively estimate variations of fission source distribution in perturbed systems for improving the accuracy of Monte Carlo correlated sampling method, especially for large variations.     

Development of a hybrid method to improve the sensitivity and uncertainty analysis for homogenized few-group cross sections

In the framework of two-step method of reactor core calculation, few-group homogenized cross sections generated by lattice-physics calculations are key input parameters for the three-dimensional full-core calculation. Conventional method for few-group cross-sections sensitivity and uncertainty (S&U) analysis related to the nuclear data was performed based on the effective self-shielding cross sections instead of the continuous-energy cross sections, which means resonance self-shielding effect (implicit effect) is neglected. Furthermore, the multi-group covariance data is generated from the continuous-energy cross sections. Therefore, in order to perform S&U analysis with respect to the continuous-energy cross sections for both accuracy and consistency, a hybrid method is proposed in this paper. The subgroup-parameter sensitivity-coefficients are calculated based on the direct perturbation (DP) method. The sensitivity-coefficients of the effective self-shielding cross sections and the responses (k_eff and few-group homogenized cross sections) are calculated based on the generalized perturbation theory (GPT). A boiling water reactor (BWR) pin-cell problem under different power conditions is calculated and analyzed. The numerical results reveal that the proposed hybrid method improves the sensitivity-coefficients of eigenvalue and few-group homogenized cross sections. The temperature effects on the sensitivity-coefficients are demonstrated and the uncertainties are analyzed.     

Improvement of Monte Carlo Lattice Burnup Calculation Performance with the Correlated Sampling Method

For next generation reactor designs, which are attempting wide variations of assembly configurations, the flexibility Monte Carlo method holds is attractive, but still costly for repetitive design study works. This paper presents an advanced correlated sampling (ACS) method which was developed to speed up Monte Carlo lattice burnup calculations. The ACS method is the combination of the correlated sampling method and a pseudo-scattering technique. All burnup steps are considered as consecutive perturbed problems using the same neutron collision history, which is pre-calculated based on a selected unperturbed problem. Since neutron weights can be adjusted on every collision point, rather than along paths between them, the perturbed calculation is very fast and the neutron collision history is light enough to be stored in memory or physical storage, which is an indispensable feature for consecutive perturbed calculations. The presented theory shows that the ACS method has good potential to work for a wide range of neutron absorption variations, the dominant perturbation in the lattice burnup. In an example calculation on a BWR lattice, the ACS calculation results of 600,000 neutrons/step agree well with the independent Monte Carlo runs of 20,000,000 neutrons/step within 0.1%dk/k in terms of k_? throughout 95 steps (~50GWd/t). Average calculation time of neutron tracking with the former method is 3.4 s/step with 600,000 neutron histories on a single processor of an Alpha21164-600 MHz, and the speed-up factor against the Monte Carlo calculation turns out to be about 100.     

核活化法反应计算中的蒙特卡罗模拟

用MCNP/4B程序,对核活化法反应计算中的截断能量、通量计数方式、数据库、阈探测器之间的扰动等因素的影响作了分析.截断能量可选为有效反应阈能,不影响计算结果,但可减少计算时间.栅元通量计数方式稳定可靠,效率高,计算值略高.不同的数据库的计算结果可能有些差别.阈探测器之间的扰动对计算结果的影响很小.     

基于经典微扰理论的特征值灵敏度和不确定度分析

核数据不确定度作为组件/栅元计算不确定度的重要来源,备受重视和研究。本文采用经典微扰理论,推导输运计算中keff对于核数据的灵敏度系数和不确定度的计算方法。基于ENDF/B-Ⅶ.1制作多群协方差数据库,并根据所采用的组件输运求解程序的截面模型对分反应道协方差矩阵进行归并。开发灵敏度和不确定度分析程序COLEUS,对传统压水堆燃料栅元进行计算分析。数值结果表明,栅元计算的keff对235 U每次裂变中子产额的扰动最为敏感,238 U俘获截面对keff不确定度的贡献最大。目前的核数据的不确定度会给keff带来0.4%~0.5%的不确定度。     

用激光干涉法制备平面调制靶的表面起伏图形

表面起伏靶是惯性约束聚变（ＩＣＦ）分解实验中的重要实验用靶。采用激光干涉方法制备初始微扰振幅和波长分别在几和几十微米范围内的正弦调制形状，摸索了相应的工艺条件和工艺过程。用台阶仪及光学显微轮廓仪观测微加工后的形貌。探讨了调制波长的精确控制与干涉工艺之间的关系，并用电镀工艺转移图形得到用于压制的模具。     

Combination of sensitivity-based and random sampling-based methodologies for efficient uncertainty quantification calculations with control variates method

A combined method of the sensitivity-based and random sampling-based methodologies is proposed for efficient uncertainty quantification calculations. The proposed method is based on the control variates (CV) method, in which a mean value of a target parameter can be estimated efficiently with a help of a mockup parameter whose mean value is well known. Standard deviations can be also efficiently estimated from two mean values of stochastic parameters; a target parameter itself and its square. In the present work, the CV method is applied to a toy problem, in which a linear approximation to a target parameter is regarded as a mockup parameter. This case corresponds to our proposed method to combine the sensitivity-based and random sampling-based methodologies. Numerical results reveal that the proposed method efficiently works. As a preliminary test of application of our proposed method to realistic problems, nuclear fuel burnup calculations are considered, and uncertainties of nuclides number densities after burnup are calculated. Uncertainties of number densities of cesium-134 and europium-151 are calculated by the proposed method, and it is demonstrated that we can carry out uncertainty quantification calculations more efficiently with our proposed method than with the normal random sampling method.     

Perturbation theory for nuclear fuel depletion calculations with predictor–corrector method

The perturbation theory for nuclear fuel depletion calculations with the predictor–corrector method is derived. This theory is implemented to a reactor physics code system CBZ, and the theory itself and its implementation are numerically verified. Sensitivities of nuclide number densities after fuel depletion with respect to nuclear data calculated with this theory are compared with reference sensitivities calculated by numerical differentiation, and good agreements are obtained. Importance of accurate angle integration on product of neutron flux and generalized adjoint neutron flux is also pointed out. Sensitivities in a 3×3 multi-cell system including a gadolinium-bearing fuel pin are calculated, and it is demonstrated that the derived theory yields accurate sensitivities even if coarse depletion time step division is adopted. The present work drastically increases the applicability of the depletion perturbation theory to actual problems.     

Physical design of a new set of high poloidal mode number coils in the EAST tokamak

An external resonant magnetic perturbation (RMP) field, which is an effective method to mitigate or suppress the edge localized mode (ELM), has been planned to be applied on the ELM control issue in ITER. A new set of magnetic perturbation coils, named as high m coils, has been developed for the EAST tokamak. The magnetic perturbation field of the high m coils is localized in the midplane of the low field side, with the spectral characteristic of high m and wide n, where m and n are the poloidal and toroidal mode numbers, respectively. The high m coils generate a strong localized perturbation field. Edge magnetic topology under the application of high m coils should have either a small or no stochastic region. With the combination of the high m coils and the current RMP coils in the EAST, flexible working scenarios of the magnetic perturbation field are available, which is beneficial for ELM control exploration on EAST. Numerical simulations have been carried out to characterize the high m coil system, including the magnetic spectrum and magnetic topology, which shows a great flexibility of magnetic perturbation variation as a tool to investigate the interaction between ELM and external magnetic perturbation.     

Three-Dimensional First-Order Perturbation Calculation Method for Reactivity Changes Due to Core Deformations of Fast Breeder Reactors

A three-dimensional diffusion calculation method has been proposed to rapidly and accurately calculate reactivity changes of LMFBRs caused by assembly displacements in accidental events. The method requires shorter computation times and provides almost the same accuracy as a conventional direct eigenvalue calculation method. In this method, changes in macroscopic neutron cross-sections and diffusion coefficient are defined so that changes in both region volume and material composition can be treated in a mesh-centered finite-difference program under the same coarse mesh division as used for the normal, non-deformed core. Reactivity changes are calculated from the above-mentioned changes by the first-order perturbation method using normal and adjoint neutron fluxes calculated beforehand for the normal core.

The method was applied to deformations of a 1,000-MWe LMFBR core. Reactivity changes calculated by the method agreed within 0.4% with those by a conventional direct eigenvalue calculation method, while computation time was less than 1/35.     

Range shift and dose perturbation with high-density materials in proton beam therapy

Radiotherapy with proton beams requires accurate knowledge of the proton range. When materials with high atomic numbers (Z) and densities (e.g. prostheses or implants) are present in the patient, they give rise to pronounced uncertainties in computed tomography data and to large errors in proton range and dose calculations.A modified analytical expression is proposed for the observed range shift in water in the presence of a high-density material of known thickness and density. The expression was verified experimentally in a clinical beam with various thicknesses and materials in a water phantom, at several beam ranges and at different depths. Measurements were also made behind the medium-to-water interface to evaluate dose perturbation using a thin window parallel plate ion chamber. Primary particle fluence variations due to the range shift were studied in a separate experiment.The measured range shift was in good agreement (±0.3 mm) with the analytical expression for most of the materials studied. A small, but consistent dependence of range shift on the energy of impinging protons was found. Dose perturbation factor in water downstream of the material is less than +5% for thicknesses up to 8 g/cm².The proposed analytical expression can be used in clinical situations to determine the range shift in patient caused by an implanted material. Dose perturbation in the presence of an implant is due to the changes in primary particle fluence resulting from several physical processes.     

用一阶扰动改进低阶谐波展开法

针对文献提出的用于换料方案快速评价的低阶谐波展开法,提出用一阶微扰方法来替代原先经验性的扰动基函数产生方法,从而使低阶谐波展开法的理论更完善、适用性更广、精度更高。     

用于换料方案快速评价的低阶谐波结合线性扰动展开法——理论模型

为解决核电厂装料方案优化搜索过程计算最大和耗时的难题,提出了用于装料方案快速评价的谐波结合线性扰动法.在该方法中,由核燃料倒换所引起的堆芯中子注量率空间分布变化,被区分为局部扰动和全局宏观倾斜两种效应,并分别采用扰动基函数和参考堆芯装载方案的低阶谐波基函数来近似表达.再通过剩余权重方法,将原本大规模矩阵特征值问题的求解转换成有关展开系数的小规模矩阵特征值问题求解,从而实现对堆芯装载方案的快速评价.     

Experiments and Analyses on Fuel Expansion and Bowing Reactivity Worth in Mock-up Cores of Metallic Fueled Fast Reactors at FCA

Axial fuel expansion and radial fuel bowing were simulated in mock-up cores of metallic fueled fast reactors at the Fast Critical Assembly (FCA). Reactivity worth caused by the simulation was measured and compared with calculations. Based on these experiments and calculations, the applicability of current calculation methods was discussed for both the first order perturbation theory (FOP) and the exact perturbation theory (EP).

For the axial fuel expansion reactivity worth, both FOP and EP showed 10 to 20% smaller values than the experiment. This underestimation was consistent to a C/E trend of axial distributions of plutonium sample worth. No significant difference was observed between FOP and EP, when transport correction was applied.

For the radial fuel bowing reactivity worth, the FOP showed about 10% larger values than the EP. Near the core central plane, the EP with transport correction showed good agreement with the experiment, while FOP showed overestimation by 14%. At the core axial edge, however, both FOP and EP underestimated the reactivity worth by more than 10%.     

针对裂变产额和半衰期的燃耗计算灵敏度和不确定度分析方法

基于广义微扰理论推导了裂变产额和半衰期的燃耗灵敏度系数理论模型,该模型考虑了原子核密度和中子通量的相互影响,并开发了燃耗计算中有效增殖因数和原子核密度等响应参数对核数据的灵敏度和不确定度分析程序。基于评价核数据中裂变产物独立产额的标准差数据,产生了针对压缩燃耗数据库的裂变产额协方差矩阵,以提高不确定度的计算精度。基于ENDF/B-Ⅶ.1数据库量化了UAM基准题TMI-1栅元无限增殖因数及重要裂变产物和重核的原子核密度由裂变产额和半衰期引入的不确定度。数值结果表明,对于栅元无限增殖因数,裂变产额和半衰期引入的不确定度很小;对于部分裂变产物的原子核密度,裂变产额和半衰期会引入较大的不确定度。     

基于统计抽样的敏感性系数计算方法

分析核系统的不确定性和敏感性,对于减小核设计的设计余量、提高核系统的经济性具有重要意义。基于统计抽样的不确定度分析方法,由于算法简单、可考虑高阶效应且对响应量没有特殊要求等,越来越受到重视。但之前认为基于统计抽样法很难进行敏感性系数分析,其原因主要是响应量的变化是由多变量同时变化引起,很难把单独一个变量的变化导致的响应量的变化确定出来。本文首先推导了利用统计抽样法进行敏感性系数分析的理论公式,然后利用裸堆双群近似的临界公式和复杂的压水堆单栅元问题进行了验证,验证了统计抽样法的可行性。针对实际问题协方差矩阵求逆困难的问题,本文提出了两种替代解决方法,即采用简化协方差矩阵或统一微扰量的方法,利用²³⁵U裂变截面对上述方法进行了验证分析,证明了方法的可行性和正确性;同时分析了不同敏感性系数对不确定度计算的影响。     

High-order cross-section homogenization method

A high-order cross-section homogenization method based on boundary condition perturbation theory is developed to improve the accuracy of nodal methods for coarse-mesh eigenvalue calculations. The method expands the homogenized parameters such as the cross-sections and the neutron flux discontinuity factor in terms of the node surface current-to-flux ratio. The expansion coefficients are evaluated during the nodal calculations using additional precomputed homogenization parameters. As a result, it is possible to correct (update) the homogenized parameters to arbitrary order of accuracy for the effect of reactor core environment (fuel assembly neutron leakage) with very little computational effort in the core calculation. The reconstructed fine-mesh flux (fuel-pin power) is a natural byproduct of the new method. A benchmark problem typical of a BWR core is analyzed in one dimension, monoenergetic diffusion theory by modifying a nodal method based on a bilinear, flat as well as a fine-mesh intranodal flux shape. The homogenized parameters are first computed using exact (fine-mesh) albedos and compared to those determined from a fine-mesh core calculation. Two nodal (coarse-mesh) examples are given to show how well this approach works as a higher-order perturbation method is utilized. The paper concludes by showing that this method succeeds in giving excellent results for cores that may be difficult to model using standard nodal methods.     

Burnup Calculation Sensitivity and Uncertainty Analysis Method of Fission Yield and Decay Half-life

The calculation model of sensitivity coefficient for decay half-life and fission product yield in burnup calculation was derived based on generalized perturbation theory, which considered the interaction between nuclear concentration and neutron flux. A code was developed to calculate sensitivity and uncertainty of effective neutron multiplication factors and nuclide concentration caused by nuclear data. Covariance matrix of fission yield for a simplified burnup library was generated based on standard deviation data of independent fission yield in evaluated nuclear data library to improve the accuracy of uncertainty quantification. Uncertainties induced by decay half-life and fission yield on infinite neutron multiplication factors and nuclide concentration for TMI-1 pin-cell in the UAM burnup benchmark were quantified based on ENDF/B-Ⅶ.1. The numerical results show that the uncertainty of infinite neutron multiplication factors induced by decay half-lives and fission yields is low, while the uncertainty of concentration of some fission product nuclide is high.