胡澤華1)2)? 葉濤1)2) 劉雄國(guó)1)2) 王佳1)2)

1)(北京應(yīng)用物理與計(jì)算數(shù)學(xué)研究所,北京 100094)2)(中國(guó)工程物理研究院高性能數(shù)值模擬軟件中心,北京 100088)(2016年7月7日收到;2016年9月30日收到修改稿)

抽樣法與靈敏度法keff不確定度量化?

胡澤華1)2)? 葉濤1)2) 劉雄國(guó)1)2) 王佳1)2)

1)(北京應(yīng)用物理與計(jì)算數(shù)學(xué)研究所,北京 100094)2)(中國(guó)工程物理研究院高性能數(shù)值模擬軟件中心,北京 100088)(2016年7月7日收到;2016年9月30日收到修改稿)

核反應(yīng)堆的中子學(xué)模擬計(jì)算中,核數(shù)據(jù)的不確定度導(dǎo)致的積分量計(jì)算結(jié)果的不確定度,通常采用基于微擾理論的靈敏度與不確定度分析方法(簡(jiǎn)稱(chēng)靈敏度法)量化.靈敏度分析法原則上只適用于線(xiàn)性模型,且一般輸運(yùn)計(jì)算程序難以直接進(jìn)行靈敏度分析.而抽樣法直接抽樣核數(shù)據(jù)輸入中子學(xué)計(jì)算程序進(jìn)行計(jì)算,通過(guò)對(duì)計(jì)算結(jié)果的統(tǒng)計(jì)分析評(píng)估計(jì)算量的不確定度.抽樣法易于實(shí)現(xiàn)、計(jì)算精確、且適用性強(qiáng).在靈敏度分析與不確定度量化程序SURE中,增加了抽樣法不確定度的量化功能.為將抽樣法不確定度量化應(yīng)用于復(fù)雜問(wèn)題的模擬計(jì)算,需對(duì)其進(jìn)行細(xì)致的考核.為此,選取簡(jiǎn)單的臨界基準(zhǔn)實(shí)驗(yàn)?zāi)Ｐ?分別采用靈敏度分析法和抽樣法進(jìn)行不確定度量化,得到了各核素各反應(yīng)道核數(shù)據(jù)導(dǎo)致的keff計(jì)算不確定度.對(duì)比顯示,兩種方法的不確定度計(jì)算結(jié)果有很好的符合,驗(yàn)證了SURE程序抽樣法功能的正確性.抽樣法計(jì)算的keff符合正態(tài)分布,說(shuō)明在一般核數(shù)據(jù)的不確定度范圍內(nèi),keff與核數(shù)據(jù)近似成線(xiàn)性關(guān)系,利用靈敏度分析法評(píng)估keff計(jì)算值的不確定度是適用的.

不確定度量化,隨機(jī)抽樣法,靈敏度,核數(shù)據(jù)

1引 言

數(shù)值模擬是目前核反應(yīng)堆工程設(shè)計(jì)與安全分析的重要基礎(chǔ),而不確定度是計(jì)算結(jié)果可信度的量度.在核反應(yīng)堆工程領(lǐng)域,安全性是需要關(guān)注的首要問(wèn)題;而經(jīng)濟(jì)性則決定了商業(yè)核電的競(jìng)爭(zhēng)力.科學(xué)量化模擬計(jì)算的不確定度,對(duì)合理平衡核工程設(shè)計(jì)的安全性與經(jīng)濟(jì)性具有重要的意義,因而越來(lái)越受到關(guān)注.隨著新型反應(yīng)堆(GEN-IV)[1]研究與實(shí)驗(yàn)的發(fā)展,對(duì)反應(yīng)堆數(shù)值模擬不確定度量化的可靠性與精度提出了更高的要求[2].發(fā)展可靠的反應(yīng)堆數(shù)值模擬的不確定度分析方法,對(duì)提高數(shù)值模擬的可信度十分必要.

模擬計(jì)算的不確定度源自多方面因素,而輸入數(shù)據(jù)(如系統(tǒng)的物質(zhì)性質(zhì)、幾何尺寸)引入的不確定度是其中的重要部分.源自輸入數(shù)據(jù)的不確定度屬認(rèn)知不確定度,已經(jīng)發(fā)展了多種方法進(jìn)行不確定度量化(簡(jiǎn)稱(chēng)UQ).這些方法大致可分為兩類(lèi):確定論方法和隨機(jī)統(tǒng)計(jì)方法.兩者的主要區(qū)別在于:確定論方法基于對(duì)計(jì)算模型的簡(jiǎn)化近似進(jìn)行,只在一定范圍內(nèi)適用,計(jì)算速度快;而統(tǒng)計(jì)方法在理論上是精確的,但需要進(jìn)行大量的計(jì)算,基本上是普適的.

在反應(yīng)堆領(lǐng)域,常用的確定論UQ方法有:傳統(tǒng)的基于一階微擾近似的靈敏度與不確定度(S/U)分析方法[3]和最近引入的多項(xiàng)式混沌展開(kāi)(PCE)方法等[4].靈敏度分析方法在核反應(yīng)堆的中子學(xué)計(jì)算中有長(zhǎng)期的廣泛應(yīng)用[5?9].S/U分析方法基于一階微擾近似計(jì)算響應(yīng)量對(duì)核數(shù)據(jù)等輸入?yún)?shù)的靈敏度系數(shù)S,再利用線(xiàn)性近似的不確定度傳遞公式,由S結(jié)合核數(shù)據(jù)的協(xié)方差得到核數(shù)據(jù)導(dǎo)致的積分響應(yīng)量計(jì)算不確定度.靈敏度分析方法可用于分析中子有效增殖因數(shù)(keff)、反應(yīng)率等響應(yīng)量的不確定度,計(jì)算速度快,在一定范圍內(nèi)結(jié)果也比較可靠,但難以普遍適用于多物理復(fù)雜問(wèn)題計(jì)算的不確定度量化.近幾年,在流體力學(xué)和結(jié)構(gòu)力學(xué)等領(lǐng)域廣泛應(yīng)用的PCE方法也被引入核數(shù)據(jù)評(píng)價(jià)[10]以及反應(yīng)堆計(jì)算的不確定度分析[4,11,12].相比基于微擾的靈敏度分析法,PCE方法計(jì)算量大,但適用性較好.

隨機(jī)統(tǒng)計(jì)法不確定度量化,通常稱(chēng)為Monte Carlo(MC)法或抽樣方法,通過(guò)對(duì)問(wèn)題進(jìn)行一系列的模擬,統(tǒng)計(jì)得到計(jì)算結(jié)果的不確定度.抽樣方法UQ,除去統(tǒng)計(jì)不確定度(可通過(guò)增加抽樣次數(shù)降低)外是精確的,是普適的UQ方法,但是所需的計(jì)算量大.近來(lái),隨著計(jì)算能力的快速提升,多種基于隨機(jī)抽樣的UQ方法逐步發(fā)展,并得到較廣泛的應(yīng)用[13?16].隨機(jī)抽樣法在模擬計(jì)算的隨機(jī)輸入?yún)?shù)空間內(nèi)進(jìn)行抽樣,獲取系列隨機(jī)參數(shù)組,將各參數(shù)組代入計(jì)算程序中完成計(jì)算,得到系列計(jì)算結(jié)果;通過(guò)對(duì)計(jì)算結(jié)果的統(tǒng)計(jì)分析得到期望值與方差.抽樣法不需要了解模擬計(jì)算過(guò)程,可將計(jì)算過(guò)程視為“黑箱”,因此易于實(shí)現(xiàn);但抽樣法要進(jìn)行多次模擬計(jì)算以得到收斂的結(jié)果,需要大量的計(jì)算時(shí)間.

靈敏度分析與不確定度量化程序SURE[9],基于微擾理論發(fā)展了keff、反應(yīng)性系數(shù)等積分量對(duì)全套核數(shù)據(jù)的靈敏度分析功能;利用靈敏度系數(shù)結(jié)合協(xié)方差數(shù)據(jù),量化核數(shù)據(jù)導(dǎo)致的積分量計(jì)算不確定度.通過(guò)與直接法計(jì)算和成熟程序[8]的對(duì)比,SURE程序的靈敏度分析與不確定度量化功能,已得到比較充分的驗(yàn)證.為拓展SURE程序不確定度量化的適用范圍,發(fā)展了隨機(jī)抽樣法不確定度量化功能.在將抽樣法應(yīng)用于實(shí)際復(fù)雜問(wèn)題的不確定度量化前,為保證可靠性,需對(duì)SURE中實(shí)現(xiàn)的抽樣法功能的可靠性進(jìn)行比較充分的驗(yàn)證.由于靈敏度分析法不確定度量化是基于線(xiàn)性近似的,其適用范圍也值得細(xì)致考查.

本文通過(guò)與傳統(tǒng)的靈敏度法不確定度計(jì)算結(jié)果的對(duì)比,驗(yàn)證SURE中新發(fā)展的抽樣法不確定度量化功能的正確性.鑒于抽樣法是“精確”的UQ方法,通過(guò)兩種方法計(jì)算結(jié)果的對(duì)比,也考察了基于線(xiàn)性近似的靈敏度法不確定度量化的適用性范圍.本文的第2部分介紹核數(shù)據(jù)及其協(xié)防的基本概念,以及計(jì)算中所采用的核參數(shù)和程序;第3部分介紹靈敏度法和抽樣法不確定度量化理論;第4部分介紹計(jì)算分析采用的兩個(gè)基準(zhǔn)實(shí)驗(yàn)?zāi)Ｐ?第5部分給出靈敏度分析法和抽樣法的不確定度量化結(jié)果;最后給出結(jié)論.

2核數(shù)據(jù)與輸運(yùn)計(jì)算

2.1 核數(shù)據(jù)及其協(xié)方差

核(反應(yīng))數(shù)據(jù)主要描述粒子與原子核發(fā)生反應(yīng)的概率(以截面表述),以及反應(yīng)后出射粒子的能量和角度分布信息(以雙微分截面表述)等,是中子輸運(yùn)問(wèn)題計(jì)算的主要輸入?yún)?shù),也是計(jì)算不確定度的主要來(lái)源.核數(shù)據(jù)主要基于實(shí)驗(yàn)測(cè)量數(shù)據(jù)評(píng)價(jià)得到,其真值是未知的,應(yīng)視為服從正態(tài)分布的隨機(jī)量.最新的評(píng)價(jià)核數(shù)據(jù)庫(kù)[17],不僅包含核數(shù)據(jù)的期望值,還包含協(xié)方差.核數(shù)據(jù)的協(xié)方差描述核數(shù)據(jù)的不確定度及數(shù)據(jù)間的關(guān)聯(lián),是進(jìn)行不確定度量化的基礎(chǔ)數(shù)據(jù).核數(shù)據(jù)σi的期望值與協(xié)方差cov(σi,σj)構(gòu)建了核數(shù)據(jù)的聯(lián)合概率分布,可視為多維正態(tài)分布.

2.2 輸運(yùn)計(jì)算與核數(shù)據(jù)處理

本文對(duì)比考察靈敏度分析法與隨機(jī)抽樣法的keff不確定度量化,分別利用SURE程序的靈敏度分析模塊Sensitivity和抽樣模塊Sample進(jìn)行計(jì)算.靈敏度分析法需分別進(jìn)行一次輸運(yùn)計(jì)算和伴隨輸運(yùn)計(jì)算,得到通量和伴隨通量;抽樣法需進(jìn)行多次輸運(yùn)計(jì)算,以統(tǒng)計(jì)keff計(jì)算結(jié)果.兩方法中均采用一維多群SN輸運(yùn)程序ANISN[18]進(jìn)行輸運(yùn)計(jì)算.

輸運(yùn)計(jì)算所需的多群輸運(yùn)核數(shù)據(jù),靈敏度分析所需的多群核反應(yīng)截面與分反應(yīng)道群轉(zhuǎn)移矩陣,以及不確定度量化所需的多群協(xié)方差數(shù)據(jù),均采用NJOY[19]的172群結(jié)構(gòu),基于ENDF/B-VII.1評(píng)價(jià)庫(kù)[17]的中子核數(shù)據(jù)制作而成.圖1為235U的核數(shù)據(jù)的部分協(xié)方差,圖1(a)為非彈散射截面172群協(xié)方差;圖1(b)為輻射俘獲截面協(xié)方差;圖1(c)為彈散截面與非彈散射截面間的協(xié)方差;圖1(d)為裂變截面與俘獲截面間的協(xié)方差.

圖1 (網(wǎng)刊彩色)235U核數(shù)據(jù)的172群核數(shù)據(jù)協(xié)方差 (a)非彈散射;(b)輻射俘獲;(c)非彈散射-彈散;(d)裂變-輻射俘獲Fig.1.(color online)172group covariance data for235U nuclear data:(a)Inelastic scattering;(b)radiative capture;(c)inelastic scattering-elastic scattering;(d) fi ssion-radiative capture.

3keff的不確定度量化方法

分別簡(jiǎn)述傳統(tǒng)的基于微擾理論靈敏度分析的keff模擬計(jì)算不確定度量化方法和近來(lái)得到廣泛應(yīng)用的基于隨機(jī)抽樣的不確定度量化方法.

3.1 靈敏度分析法

靈敏度分析法不確定度量化通過(guò)一次輸運(yùn)計(jì)算和一次伴隨輸運(yùn)計(jì)算得到中子角通量和伴隨角通量,再基于微擾理論得到keff計(jì)算對(duì)全套核數(shù)據(jù)的靈敏度系數(shù);由靈敏度系數(shù)結(jié)合協(xié)方差,計(jì)算核數(shù)據(jù)不確定度導(dǎo)致keff計(jì)算的不確定度.

模擬計(jì)算中,靈敏度指輸入?yún)?shù)變化導(dǎo)致的輸出量(計(jì)算結(jié)果)的變化量,(相對(duì))靈敏度系數(shù)(sensitivity coefficient)SR,α定義為輸出量的相對(duì)變化量與導(dǎo)致其變化的輸入量相對(duì)變化量之比:

其中,R為模擬計(jì)算的輸出量(響應(yīng)量、積分量),α為計(jì)算的輸入?yún)?shù)(如核數(shù)據(jù)、核素豐度等).

記λ=1/keff,根據(jù)一階微擾理論,keff計(jì)算對(duì)核數(shù)據(jù)的靈敏度系數(shù)表示為

其中,?為?(r,E,?)的簡(jiǎn)寫(xiě),由keff本征值輸運(yùn)方程

計(jì)算得到;??為??(r,E,?)的簡(jiǎn)寫(xiě),由keff本征值伴隨輸運(yùn)方程

計(jì)算得到;B和F分別為輸運(yùn)算符和裂變算符,分別表示為

可見(jiàn),求得角通量和伴隨角通量,即可由(2)式得到keff對(duì)核數(shù)據(jù)的靈敏度系數(shù).一般利用確定論多群SN離散坐標(biāo)輸運(yùn)方法,進(jìn)行一次輸運(yùn)計(jì)算和一次伴隨輸運(yùn)計(jì)算,求解通量和伴隨通量.近來(lái)發(fā)展了基于連續(xù)能量MC計(jì)算的伴隨通量統(tǒng)計(jì)方法[20],進(jìn)行keff對(duì)連續(xù)能量核數(shù)據(jù)的靈敏度分析.本文采用多群SN方法進(jìn)行輸運(yùn)和伴隨輸運(yùn)計(jì)算.

假定keff計(jì)算值與核數(shù)據(jù)間是線(xiàn)性關(guān)系,利用靈敏度系數(shù)結(jié)合協(xié)方差數(shù)據(jù),由不確定度傳遞公式

即可量化keff計(jì)算的不確定度,其中,σi為第i個(gè)核數(shù)據(jù),Sk,σi為keff對(duì)核數(shù)據(jù)的靈敏度系數(shù),Cσi,σj為核數(shù)據(jù)間的協(xié)方差矩陣.

3.2 隨機(jī)抽樣法

一般可將多群SN輸運(yùn)計(jì)算中使用的N個(gè)核數(shù)據(jù)的隨機(jī)分布視為N維正態(tài)分布,利用核數(shù)據(jù)的期望值σi(i=1,N)及其協(xié)方差C(σi,σj)可建立核數(shù)據(jù)的N維正態(tài)分布空間.

若考慮其中的n(n≤N)個(gè)核數(shù)據(jù),核數(shù)據(jù)及其期望值組成的列矩陣分別記為

協(xié)方差矩陣記為C,則n個(gè)核數(shù)據(jù)的概率密度分布為

為在n維正態(tài)分布空間內(nèi)進(jìn)行隨機(jī)抽樣,首先對(duì)協(xié)方差矩陣C進(jìn)行奇異值分解,得到T矩陣,使得

再?gòu)臉?biāo)準(zhǔn)正態(tài)分布中抽樣n個(gè)隨機(jī)數(shù),記為Z,則的核數(shù)據(jù)抽樣為

在核數(shù)據(jù)的隨機(jī)分布空間進(jìn)行抽樣,每次抽樣得到一套核數(shù)據(jù),將抽樣核數(shù)據(jù)輸入計(jì)算程序,得到一個(gè)keff計(jì)算值.進(jìn)行M次抽樣,計(jì)算得到M個(gè)keff值.通過(guò)對(duì)keff計(jì)算結(jié)果進(jìn)行統(tǒng)計(jì)分析,得到keff計(jì)算的期望值和方差:

與靈敏度分析法不同,抽樣法不需要假定keff與核數(shù)據(jù)間成線(xiàn)性關(guān)系.由于抽樣法不需線(xiàn)性假定,且可將計(jì)算程序視為“黑箱”使用,因而抽樣法不但計(jì)算精確,而且具有更廣泛的適用性.

抽樣法中,在量化某一反應(yīng)道數(shù)據(jù)導(dǎo)致的不確定度時(shí),在該反應(yīng)道數(shù)據(jù)隨機(jī)分布內(nèi)抽樣,統(tǒng)計(jì)計(jì)算結(jié)果的標(biāo)準(zhǔn)差;在量化兩反應(yīng)道關(guān)聯(lián)導(dǎo)致的不確定度Uc時(shí),首先在兩反應(yīng)道核數(shù)據(jù)的聯(lián)合隨機(jī)分布內(nèi)進(jìn)行抽樣,得到keff不確定度量化值U0,再分別獨(dú)立量化兩反應(yīng)道數(shù)據(jù)導(dǎo)致的不確定度U1和U2,利用下式

得到反應(yīng)道關(guān)聯(lián)導(dǎo)致的不確定度.

4基準(zhǔn)實(shí)驗(yàn)?zāi)Ｐ?/h2>

由于抽樣法不確定度量化,需進(jìn)行大量的模擬計(jì)算,計(jì)算時(shí)間消耗大.為節(jié)約時(shí)間,選取了簡(jiǎn)單的基準(zhǔn)實(shí)驗(yàn)?zāi)Ｐ虶odiva和Jezebel[21]進(jìn)行計(jì)算.Godiva和Jezebel都是美國(guó)洛斯阿拉莫斯國(guó)家實(shí)驗(yàn)室開(kāi)展的臨界積分實(shí)驗(yàn).Godiva為球形高濃鈾(235U)裸臨界基準(zhǔn)裝置,半徑為8.7407cm,keff的基準(zhǔn)實(shí)驗(yàn)值為1.0±0.001;Jezebel-239Pu為球形239Pu裸基準(zhǔn)裝置,半徑為6.38493cm,keff的基準(zhǔn)實(shí)驗(yàn)值為1.0±0.002.兩裝置的核素成分見(jiàn)表1.

表1 Godiva和Jezebel的核子數(shù)密度Table 1.Atom densities for the Godiva and Jezebel Benchmark.

5計(jì)算分析

利用SURE程序的靈敏度分析模塊和抽樣模塊,分別對(duì)基準(zhǔn)模型Godiva和Jezebel,計(jì)算了各核素的彈散(n,n)、非彈(n,n′),n 2n(n,2n)、裂變(n,f)和輻射俘獲截面(n,gamma),以及平均裂變中子數(shù)(nubar)導(dǎo)致的keff計(jì)算不確定度.

5.1 Godiva基準(zhǔn)實(shí)驗(yàn)?zāi)Ｐ?/p>

通常的輸運(yùn)計(jì)算,直接采用核數(shù)據(jù)的期望值.采用制作的172群核數(shù)據(jù)(期望值),計(jì)算的Godiva基準(zhǔn)實(shí)驗(yàn)的keff為1.0006;采用隨機(jī)抽樣法,抽樣10000組核數(shù)據(jù)(這里主要是為了考察keff計(jì)算的分布規(guī)律進(jìn)行了10000次抽樣,實(shí)際的抽樣法量化中,并不必要進(jìn)行這么多次抽樣計(jì)算),計(jì)算得到keff期望值(平均值)為1.0010,兩者相差僅為4×10?4,符合很好.采用靈敏度法量化的全部核數(shù)據(jù)導(dǎo)致的keff計(jì)算不確定度為1.206%,采用抽樣量化為1.212%,兩者相差6×10?5,符合得很好.

圖2 (網(wǎng)刊彩色)Godiva基準(zhǔn)模型抽樣法不確定度量化I(PDF為概率密度函數(shù)) (a)彈散截面導(dǎo)致的keff計(jì)算值分布;(b)非彈截面導(dǎo)致的keff計(jì)算值分布;(c)彈散與非彈截面導(dǎo)致的keff計(jì)算值分布;(d)三者對(duì)比Fig.2.(color online)Uncertainty quanti fi cation using sampling method for Godiva Benchmark I(PDF is probability density function):(a)keffdistribution due to elastic cross sections;(b)keffdistribution due to inelastic cross sections;(c)keffdistribution due to elastic and inelastic cross sections;(d)comparison of the three cases.

進(jìn)一步采用抽樣法分別量化各核素各反應(yīng)道截面數(shù)據(jù)導(dǎo)致的Godiva基準(zhǔn)實(shí)驗(yàn)的keff計(jì)算不確定度,每次計(jì)算抽樣10000次.如圖2所示,圖2(a)—(c)為keff抽樣計(jì)算值分布直方圖與keff抽樣計(jì)算的平均值與標(biāo)準(zhǔn)差u為參數(shù)的正態(tài)分布曲線(xiàn).圖2(a)和圖2(b)分別給出了235U的彈性散射截面與非彈性散射截面的隨機(jī)分布導(dǎo)致的keff計(jì)算值分布;圖2(c)給出了235U的彈性散射截面與非彈性散射截面的聯(lián)合隨機(jī)分布導(dǎo)致的keff計(jì)算值分布.可見(jiàn),三種情況下,keff計(jì)算值的隨機(jī)分布均很好地符合正態(tài)分布.由于截面由正態(tài)分布抽樣得到,keff分布符合正態(tài)分布表明在三種情況下截面的抽樣擾動(dòng)值與keff計(jì)算擾動(dòng)值基本符合線(xiàn)性關(guān)系.圖2(d)給出了三種情況下keff分布擬合正態(tài)分布曲線(xiàn).可見(jiàn),彈散截面導(dǎo)致的keff計(jì)算不確定度最小,非彈導(dǎo)致的不確定度最大,彈散與非彈共同導(dǎo)致的不確定度介于兩者之間.這是由于彈散與非彈截面間是負(fù)相關(guān)(如圖1(c)),而彈散與非彈截面主要有正的靈敏度系數(shù),因而其對(duì)不確定度的貢獻(xiàn)為負(fù)值.

圖3對(duì)比了235U的裂變截面、輻射俘獲截面以及裂變-輻射俘獲截面聯(lián)合分布下的keff抽樣計(jì)算結(jié)果.圖3(a)—(c)分別為裂變、輻射俘獲和裂變-輻射俘獲截面隨機(jī)分布下keff抽樣計(jì)算值分布直方圖與keff抽樣計(jì)算的平均值與標(biāo)準(zhǔn)差u為參數(shù)的正態(tài)分布曲線(xiàn),圖3(d)給出了三種情況下keff分布擬合正態(tài)分布曲線(xiàn).可見(jiàn),裂變與俘獲截面共同導(dǎo)致的keff計(jì)算不確定度最大,其次為俘獲截面.裂變截面與俘獲截面基本上是負(fù)相關(guān)(如圖1(d)),但裂變截面有正的靈敏度,俘獲截面為負(fù)的靈敏度,兩者的關(guān)聯(lián)對(duì)不確定度的貢獻(xiàn)為正.俘獲截面的抽樣擾動(dòng)值仍與keff計(jì)算擾動(dòng)值也基本符合線(xiàn)性關(guān)系.

利用MATLAB軟件的normplot命令分別繪制了彈散截面和俘獲截面抽樣的keff計(jì)算值的正態(tài)概率分布圖,分布點(diǎn)越接近直線(xiàn)越符合正態(tài)分布.如圖4所示,兩種情況下keff分布均基本符合正態(tài)分布,只是不確定度較大的俘獲截面抽樣的keff分布略偏離正態(tài)分布一些.

采用靈敏度分析法,先計(jì)算keff對(duì)各核數(shù)據(jù)的靈敏度系數(shù),再結(jié)合協(xié)方差數(shù)據(jù),利用不確定度傳遞(7)式,計(jì)算了各核素各反應(yīng)道截面不確定度導(dǎo)致的keff的不確定度.抽樣法與靈敏度法量化的keff不確定度的對(duì)比見(jiàn)表2.

圖3 (網(wǎng)刊彩色)Godiva基準(zhǔn)模型抽樣法不確定度量化II (a)裂變截面導(dǎo)致的keff計(jì)算值分布;(b)俘獲截面導(dǎo)致的keff計(jì)算值分布;(c)裂變與俘獲截面導(dǎo)致的keff計(jì)算值分布;(d)三者對(duì)比Fig.3.(color online)Uncertainty quanti fi cation using sampling method for Godiva Benchmark II(PDF is Probability Density Function):(a)keffdistribution due to fi ssion cross sections;(b)keffdistribution due to capture cross sections;(c)keffdistribution due to fi ssion and capture cross sections;(d)comparison of the three cases.

圖4 (網(wǎng)刊彩色)正態(tài)概率分布 (a)彈散截面抽樣的keff分布;(b)俘獲截面抽樣的keff分布Fig.4.(color online)Normal probability distribution:(a)keffdistribution due to elastic cross sections;(b)keffdistribution due to capture cross sections.

表2 各核素各反應(yīng)道核數(shù)據(jù)導(dǎo)致的Godiva模型keff計(jì)算不確定度Table 2.Uncertainties of kefffrom every nuclides and every reaction types for Godiva.

可見(jiàn),對(duì)各核素各反應(yīng)道,采用靈敏度分析法和抽樣法得到的keff計(jì)算不確定度量化結(jié)果,均有很好的符合,最大偏差＜6%.SURE程序靈敏度分析法不確定度量化結(jié)果已通過(guò)與成熟程序(TSUNAMI,SUSD等[5])結(jié)果比較得到充分的驗(yàn)證.抽樣法不確定度量化結(jié)果與靈敏度法結(jié)果符合很好,一方面表明SURE抽樣法不確定度量化結(jié)果的可靠性,另一方面也進(jìn)一步印證了靈敏度法量化的可靠性.

鑒于靈敏度法中采用的不確定度傳遞公式是基于輸入數(shù)據(jù)與計(jì)算結(jié)果間是線(xiàn)性關(guān)系的假定,靈敏度法與抽樣法的結(jié)果符合很好也顯示對(duì)Godiva模型的計(jì)算,在核數(shù)據(jù)協(xié)方差的擾動(dòng)范圍內(nèi),keff計(jì)算值與核數(shù)據(jù)成線(xiàn)性關(guān)系.

5.2 Jezebel基準(zhǔn)實(shí)驗(yàn)?zāi)Ｐ?/p>

采用核數(shù)據(jù)期望值,計(jì)算的Jezebel基準(zhǔn)實(shí)驗(yàn)的keff為0.99996,采用隨機(jī)抽樣法計(jì)算得到keff期望值為1.0001,兩者相差僅為1.4×10?4,符合得很好.進(jìn)一步,采用靈敏度分析法和抽樣法分別量化各核素各反應(yīng)道截面數(shù)據(jù)導(dǎo)致的Jezebel基準(zhǔn)實(shí)驗(yàn)的keff計(jì)算不確定度,結(jié)果見(jiàn)表3.

可見(jiàn),與Godiva模型計(jì)算相似,對(duì)各核素各反應(yīng)道,采用靈敏度分析法和抽樣法得到的keff計(jì)算不確定度量化結(jié)果均有很好的符合,最大偏差僅為2.3%.

表3 各核素各反應(yīng)道核數(shù)據(jù)導(dǎo)致的Jezebel模型keff計(jì)算不確定度Table 3.Uncertainties of kefffrom every nuclides and every reaction types for Jezebel.

6結(jié) 論

分別采用SURE程序的靈敏度分析方法和隨機(jī)抽樣方法,對(duì)兩個(gè)典型臨界基準(zhǔn)實(shí)驗(yàn)Godiva和Jezebel,計(jì)算了各核素各反應(yīng)道核數(shù)據(jù)不確定度導(dǎo)致的keff計(jì)算的不確定度.對(duì)比顯示,兩方法得到的不確定度量化結(jié)果符合很好.在多維正態(tài)分布空間內(nèi),對(duì)核數(shù)據(jù)進(jìn)行抽樣,代入輸運(yùn)程序得到keff計(jì)算值.統(tǒng)計(jì)分析表明,keff計(jì)算值很好地符合了正態(tài)分布,驗(yàn)證了在一般核數(shù)據(jù)的不確定度范圍內(nèi)keff與核數(shù)據(jù)近似成線(xiàn)性關(guān)系.這表明在一般的核數(shù)據(jù)不確定度范圍內(nèi),基于線(xiàn)性模型的靈敏度法不確定度量化方法是可靠的.但對(duì)不確定度很大的核數(shù)據(jù),靈敏度法不確定度量化的可靠性需進(jìn)一步考察.對(duì)各核素各反應(yīng)道截面,抽樣法不確定度量化結(jié)果與靈敏度法結(jié)果均有很好的符合,驗(yàn)證了SURE程序抽樣法模塊的可靠性,在進(jìn)一步檢驗(yàn)后將應(yīng)用于復(fù)雜非線(xiàn)性問(wèn)題模擬計(jì)算的不確定度量化.為減少計(jì)算時(shí)間,將在抽樣法的基礎(chǔ)上發(fā)展多項(xiàng)式混沌展開(kāi)方法用于復(fù)雜的多物理問(wèn)題的靈敏度分析與不確定度量化.

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PACS:28.20.—v,28.20.Gd,28.41.AkDOI:10.7498/aps.66.012801

*Project supported by the Key Laboratory of Neutron Physics of China Academy of Engineering Physics(Grant No.2013AA02),Sub-item of Special Project of the National Energy Bureau,China(Grant No.2015ZX06002008),National Magnetic Con fi nement Fusion Energy Research Project,China(Grant No.2015GB108002).

?Corresponding author.E-mail:hu_zehua@iapcm.ac.cn

Uncertainty quanti fi cation in the calculation of keffusing sensitity and stochastic sampling method?

Hu Ze-Hua1)2)?Ye Tao1)2)Liu Xiong-Guo1)2)Wang Jia1)2)

1)(Institute of Applied Physics and Computational Mathematics,Beijing 100094,China)2)(Software Center for High Performance Numerical Simulation,China Academy of Engineering Physics,Beijing 100088,China)(Received 7 July 2016;revised manuscript received 30 September 2016)

The sensitivity and uncertainty analysis(S/U)method based on the fi rst order perturbation theory is commonly employed to calculate the uncertainties in-nuclear reactor’s integral parameters,such as the neutron e ff ective multiplication factor(keff),due to uncertainties in nuclear data.However,this method is only theoretically suitable for the linear model because of its fi rst order approximation.Moreover,S/U method is difficult to incorporate into a neutronics code,because the adjoint angular fl ux is needed to obtain the sensitivity coefficient of an integral parameter to nuclear data.Meanwhile,the sampling approach based on parametric random sampling of input parameters,an easy implemented method,evaluates the uncertainties in the integral parameters by performing a set of neutronics simulations inputted with a set of stochastic nuclear data sampled from a multinomial normal distribution with nuclear cross section mean values and covariance data.The sampling approach is considered as a more exact method,as linear approximation is not needed.With the increase of computational power,the sampling methods with consuming more time are now possible.The sampling approach is incorporated into SURE,a sensitivity and uncertainty analysis code developed in IAPCM,as a functional module.A careful veri fi cation of the new function is necessary before it is used to analyze complicated problems,such as multi-physical coupling calculations of nuclear reactor.Two simple fast criticality benchmark experiments,namely Godiva(HEU-MET-FAST-001)and Jezebel(PU-MET-FAST-001),are selected to verify the sampling module of SURE.The uncertainties in nuclear data are given by multigroup covariance matrices processed from ENDF/B-VII.1data.The uncertainties in the computed value of keffresulting from uncertainties in the nuclear data are calculated with both S/U and sampling methods.The uncertainties due to reaction cross sections for each nuclide in two benchmarks given by two methods with the multigroup covariance matrices are in good agreement.Since the S/U module of SURE code is veri fi ed extensively,the correctness of the sampling function of the code is con fi rmed as well.The distribution of the kefffrom the sampling approach obeys the normal distribution pretty well,which indicates that keffvaries linearly with the nuclear data under its uncertainty range,since the nuclear data used in calculations are assumed to be normal distribution in the sampling method.The results from the sampling method also support the S/U method with linear approximation as a suitable uncertainty quanti fi cation method for keffcalculation.

uncertainty quanti fi cation,stochastic sampling method,sensitivity,nuclear data

10.7498/aps.66.012801

?中國(guó)物理研究院中子物理學(xué)重點(diǎn)實(shí)驗(yàn)室基金(批準(zhǔn)號(hào):2013AA02)、能源局06重大專(zhuān)項(xiàng)(批準(zhǔn)號(hào):2015ZX06002008)和國(guó)家磁約束

核聚變能研究專(zhuān)項(xiàng)(批準(zhǔn)號(hào):2015GB108002)資助的課題.

?通信作者.E-mail:hu_zehua@iapcm.ac.cn