Annals of Nuclear Energy 158 (2021) 108285

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Annals of Nuclear Energy

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Variational nodal method for three-dimensional multigroup neutrondiffusion equation based on arbitrary triangular prism

https://doi.org/10.1016/j.anucene.2021.1082850306-4549/� 2021 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: 29 Jiangjun Avenue, Nanjing 211106, PR China.E-mail address: kzhuang@nuaa.edu.cn (K. Zhuang).

Kun Zhuang a,b,⇑, Wen Shang a, Ting Li a, Jiangtao Yan a, Sipeng Wang a, Xiaobin Tang a,b, Yunzhao Li c

aDepartment of Nuclear Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, JiangSu, 211106, PR ChinabKey Laboratory of Nuclear Technology Application and Radiation Protection in Astronautics (Nanjing University of Aeronautics and Astronautics), Ministry of Industryand Information Technology, PR Chinac School of Nuclear Science and Technology, Xi’an Jiaotong University, ShaanXi, 710049, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 November 2020Received in revised form 30 January 2021Accepted 24 March 2021Available online 5 April 2021

Keywords:Variational nodal methodNeutron diffusionArbitrary triangular mesh

With the development of nuclear technology, new concepts of reactor core have been continuously pro-posed, and new types of geometric assembly and reactor core configuration would be adopted to achievebetter characteristics. Traditional nodal method based on rectangle or hexagonal nodal is no longer appli-cable for complex geometry modeling. Given that triangles have the advantage of constructing arbitrarilycomplex geometry, in this study, variational nodal method of three-dimensional multigroup neutron dif-fusion theory based on arbitrary triangular prism was developed. ANSYS code is used to divide calculationregion into arbitrary triangle mesh, which is then transformed into regular triangle mesh by coordinatetransformation, and a set of orthogonal polynomial basis function is built. Variables such as nodal neu-tron scalar flux and neutron current on nodal surface are expanded based on those polynomial basis func-tions. The response relationship between neutron partial neutron current and neutron flux can beobtained by variational principle, and traditional power method is employed to solve those responsematrix system. Based on above, TriVNM code was developed, and 4 types of typical benchmarks includingrectangle, hexagonal assembly reactor and reactor with curved boundary were employed to verifyTriVNM. Both power distributions and keff calculated by TriVNM code agree well with reference results,and the maximum relative error of all the benchmarks’ power doesn’t exceed 1% and error of keff doesn’texceed 50 pcm. TriVNM code achieves the same or higher level of calculation accuracy compared withother diffusion codes. Mesh sensitivity analysis shows that for large mesh size TriVNM code still has goodaccuracy.

� 2021 Elsevier Ltd. All rights reserved.

1. Introduction

The accurate calculation of neutron flux distributions in reactorcore is one of the most important tasks of nuclear reactor neutron-ics calculation, and the corresponding methods include ‘‘one-step”scheme and ‘‘two-step” scheme. ‘‘One-step” scheme includesMonte Carlo method such as OpenMC (Ruggieri et al., 2006), Ser-pent (Leppänen, 2013) etc. and deterministic methods such asNECP-X (Chen et al., 2018), MPACT code (Kochunas et al., 2013)etc. ‘‘One-step” scheme has the characteristics of high calculationaccuracy and strong geometric adaptability, it usually solves neu-tron transport equation. Due to the complexity of both neutrontransport equation and reactor core configuration, the calculationefficiency of ‘‘one-step” scheme is generally low, which limits its

further application in reactor core design and safety analysis.‘‘Two-step” scheme is an approximate method to solve the neutrontransport equation. For first step of ‘‘two-step” scheme, the few-group homogenized cross sections of assembly are generated bylattice code, which use PIJ method (Carlvik, 1964; Liu et al.,2015), SN method (Carlson and Bell, 1958; Hursin et al., 2014) orMOC method (Hursin et al., 2014; Li et al., 2015a) to solve neutrontransport equation to obtain the neutron flux required for homog-enization. Then those homogenized cross sections are employed insecond step of ‘‘two-step” scheme for whole core diffusion calcula-tion based on nodal method with coarse-mesh. In most cases, theaccuracy of results calculated by ‘‘two-step” scheme is close to thatby ‘‘one-step” scheme. However, the calculation efficiency of ‘‘two-step” scheme is several orders higher than that of ‘‘one-step”scheme.

As mentioned above, the reactor core neutron flux calculation of‘‘two-step” scheme usually adopts multigroup neutron diffusion

K. Zhuang, W. Shang, T. Li et al. Annals of Nuclear Energy 158 (2021) 108285

theory. This numerical method can be divided into finite differencemethod (Forsythe and Wasow, 1960; Lipnikov et al., 2014), finiteelement method (Rao, 2017; Wang and Ye, 2014; Zienkiewiczand Morice, 1971), and nodal method (Smith, 1979; Wang et al.,2019; Zhang et al., 2017) based on different processing procedureof spatial variables. Compared with finite difference method andfinite element method of usually using pin size mesh, the nodalmethod employs a coarser nodal mesh of usually fuel assemblysize, and the spatial variables in a nodal mesh are expanded byhigh-order polynomial functions/basis functions. The nodalmethod not only reaches the same calculation accuracy as finitedifference method but also increase largely the calculation effi-ciency. Thus, it makes nodal method the most widely used in cur-rent nuclear reactor design and simulation. Until now, manyresearchers from various countries have done a lot of work makenodal method more mature. Nodal method generally can bedivided into two categories. One is using polynomial functions toexpand variables of nodal, such as Nodal Expansion Method(NEM) (Christensen, 1985; Christoskov and Petkov, 2013; Singhet al., 2014), the Nodal Green’s Function Method (NGFM)(Lawrence and Dorning, 1980; Liska and Colonius, 2016), etc. Theother one is based on a set of analytical basis functions satisfyingthe neutron diffusion equation, such as Analytic Nodal Method(ANM) (Fu and Cho, 2002; Smith, 1979), Analytical Basis FunctionExpansion Method (AFEN) (Cho and Lee, 2006; Noh and Cho, 1994)etc. Besides, in order to avoid the iteration solving of coupled par-tial neutron current, K. Smith and Wagner (Smith, 1979) proposedthe advanced nodal method based on nonlinear iteration. The firstcategory such as NEM and NGFM etc. generally adopt transverseintegration technique to simplify three-dimensional (3D) diffusionequation to three one-dimensional (1D) coupled transverse inte-gration equations. The nodal method using transverse integrationtechnique has been well applied for rectangle mesh, however,some difficulties arises for hexagonal mesh. Singularity of trans-verse leakage term appears in transverse integration equation forhexagonal mesh, which results in complex relationship betweenneutron flux and surface neutron current. To overcome aboveshortcomings and improve the calculation accuracy, the secondcategory of nodal method based on analytical basis function wasdeveloped. Professor Cho of KAIST in Korea first proposed the AFEN(Noh and Cho, 1993), which employs a set of analytical basis func-tions satisfying the neutron diffusion equation to expand neutronflux, then constructs the response relationship between neutronflux and surface current. Subsequently, KASIT and Nuclear Engi-neering Computational Physics Lab (NECP) in Xi’an JiaotongUniversity conducted further researches to apply this method intorectangle (Kim and Cho, 2002; Noh and Cho, 1994) and hexagonalmesh (Chao, 1995; Cho and Lee, 2006; Christoskov and Petkov,2013; Jin et al., 1995). Even though AFEN has good calculation effi-ciency and accuracy, response matrix system may be ill-conditioned for diffusion equation of >2 neutron energy groupespecially when the nodal infinite multiplication factor kinf is equalto the reactor core effective multiplication factor.

Variational nodal method (VNM) (Dilber and Lewis, 1984) wasfirst proposed by Lewis of Northwestern University in 1980s. It isstill called the nodal method even without transverse integrationprocedure technique because variables’ expansion is performedwithin nodal of fuel assembly size. Compared to other nodalmethod, VNM has the following advantages: 1) 3D orthogonalbasis functions of nodal are directly employed to expand neutronflux instead of adopting transverse integration, and pin-to-pinpower distribution in a nodal can be directly calculated based onexpansion coefficient to avoid errors caused by pin-to-pin powerreconstruction; 2) the expansion order of neutron flux and surfacecurrent can be adjusted to adapt to the change of nodal size; 3) thenodal functional can be calculated by the sum of internal sub-nodal

2

functional. Until now, many codes have been developed based onVNM. VARIANT (Palmiotti et al., 1995) was the first code basedon VNM developed at the Argonne National Laboratory (ANL) inthe 1990s, and it performed nodal neutron transport calculationfor rectangle and hexagonal nodal. Then, REBUS code (Olson andSmith, 2019; Toppel, 1983) used to solve diffusion equation ortransport equation was developed by ANL and ERANOS (Le Tellieret al., 2010; Ruggieri et al., 2006) code including transport and dif-fusion solver for both rectangle and hexagonal nodal was devel-oped by Atomic Energy Commission (CEA) based on VNM fordesign of fast reactor. New version of VARIANT code named NODALas one of neutron transport solver in UNIC (Palmiotti et al., 2007)code package was developed by ANL in 2007. NODAL code can beapplicable for different geometrical nodal including rectangle,hexagonal and regular triangle. Idaho National Laboratory devel-oped INSTANT neutron transport code (Wang et al., 2011) basedon VNM, it can perform calculations on rectangle, hexagonal andirregular triangular nodal. In 2015, Yunzhao Li from NECP of Xi’anJiaotong University developed VIOLET code (Li et al., 2015b; Wanget al., 2016) based on VNM for rectangle nodal, then VIOLET wasintegrated in Bamboo-Core code (Yang et al., 2018) for PWR corecalculation. Furthermore, VIOLET code was extended to solve theneutron transport equation for hexagonal nodal and used as a sol-ver of the fast core calculation code system NECP-SARAX (Zhenget al., 2018). To further improve the efficiency of the VNM for solv-ing the neutron transport equation in hexagonal nodal, andenhance its application for high precision design of fast spectrumcore, Zhipeng Li from NECP of Xi’an Jiaotong University studiedhexagonal VNM (Li et al., 2017) based on group theory in 2017.Tengfei Zhang from Shanghai Jiaotong University proposed animproved VNM to solve 3D integral neutron transport equationfor rectangle and hexagonal nodal (Zhang et al., 2019, 2018). Thequasi-reflected interface condition (QRIC) method was employedto reduce the number of interfacial angular terms and save compu-tational costs.

It can be seen that VNM overcomes the disadvantages of nodalmethod based on transverse integration and analytical basis func-tion expansion, such as singularity of transverse leakage term andill-conditioned response matrix. A lot of studies have been carriedout by researchers around the world, but almost all of them focuson rectangle and hexagonal nodal. With the development ofnuclear technology, new concepts of reactor core have been con-tinuously proposed, such as fast reactor, space nuclear reactorand molten salt reactor. To achieve better characteristics inadvanced reactor, new types of geometric assembly and reactorcore configuration would be adopted. For example, space nuclearreactor generally consists of hexagonal fuel assembly and cylindri-cal control drum, of which the inner layer is coated with certainthickness of neutron absorption material of B4C to adjust reactivity.Traditional nodal method based on rectangle or hexagonal nodal isno longer applicable for control drummodeling. Since arbitrary tri-angle can construct a region with curved boundary, thus it can the-oretically be used to divide region of any shape geometry. Besides,local mesh encryption can be performed to improve the accuracyof calculation results. In this study, variational nodal method of3D multi-group neutron diffusion theory based on arbitrary trian-gle nodal was developed. ANSYS code is used to divide calculationregion into arbitrary triangle mesh and node information andnodal adjacent relationship are saved in files. The functional ofentire calculation region is obtained by Galerkin variational tech-nique (Lewis, 2013). Arbitrary triangular prism nodal is trans-formed into regular triangular prism nodal by coordinatetransformation, and a set of orthogonal polynomial basis functionis built. Then the neutron flux, the neutron source and the net neu-tron current on nodal surface are expanded based on those polyno-mial basis functions, which satisfy orthogonalization in regular

K. Zhuang, W. Shang, T. Li et al. Annals of Nuclear Energy 158 (2021) 108285

triangular prism nodal. The response relationship between neutronpartial neutron current and neutron flux can be obtained by varia-tion principle. Finally, traditional power method (Li et al., 2017; Xieand Deng, 2005) is employed to solve those response matrixsystem.

Finite element method and variational nodal method based onarbitrary triangular mesh are similar in aspect of employing Galer-kin variational technique, the concept of functional, and unstruc-tured mesh. However, there are also some differences. Firstly,finite element method generally adopts small mesh size and loworder trial functions, and the nodal method proposed in this studyemploys a set of high-order (>4) polynomial functions and a nodalmesh of fuel assembly size. Secondly, the expansion coefficients offinite element node are calculated in finite element method, andthe coupling between different mesh is performed through trans-ferring of finite element node value. However, for method of thisstudy, the spatial variables within an arbitrary triangular meshare directly expanded by high-order (>4) polynomial functions,and the coupling between different nodal mesh is realized by scan-ning all the nodal mesh and using the continuous condition of netneutron current and scalar neutron flux on the interface of thenodal mesh. Thirdly, for variational nodal method, pin-to-pinpower distribution of nodal can be calculated directly by theexpansion coefficients of neutron flux within a nodal. However,in finite element method, the pin-to-pin power distribution ofnodal need further calculation based on expansion coefficients offinite element node. Based on above theory, the code namedTriVNM was developed using FORTRAN-90 language object-oriented programming. 4 types of typical benchmarks includingrectangle (2D/3D-IAEA) (Michelsen, 1977), hexagonal assemblyreactor (4G-2D-VVER1000, 2G-3D-VVER1000) (Makai, 1982;Schulz, 1996) and unstructured geometry reactor (two-zone reac-tor with curved boundary) (Itagaki, 1985) were employed to verifyTriVNM’s accuracy and its applicability for complex geometricalreactor. Both power and keff calculated by TriVNM code agree wellwith reference results, and the maximum relative error of all thebenchmarks’ power doesn’t exceed 1% and the maximum error ofkeff doesn’t exceed 50 pcm. TriVNM code achieves the same orhigher level of calculation accuracy compared with other diffusioncodes. Besides, 2D IAEA benchmark was employed to analyze meshsensitivity of TriVNM code by using three kinds of different meshsize. It can be seen that the numerical model of variational nodalmethod based on arbitrary triangle mesh is correct, and TriVNMcode has good applicability and accuracy for reactor core withcomplex geometrical assembly.

The novelty of this study is that considering the drawbacks oftraditional nodal method based on rectangle or hexagonal nodalmesh in aspect of complex geometry reactor simulation, the insuf-ficiently accuracy pin-to-pin power distribution calculation fornodal method based on transverse integration procedure and lowcalculation efficiency for method based on fine mesh finite differ-ence method or finite element method, a variational nodal methodbased on arbitrary triangle mesh and three-dimensional directexpand procedure is developed. In Section 2, the numericalmethodology of VNM based on arbitrary triangle nodal is pre-sented. Section 3 discusses the numerical verification of TriVNMcode by 4 types of typical benchmarks including rectangle, hexag-onal assembly reactor and reactor with curved boundary. Besides,mesh sensitivity of TriVNM code is analyzed. Finally, some conclu-sions are summarized in Section 4.

2. Numerical method

In this section, the variational nodal method based on arbitrarytriangle mesh is introduced. Firstly, using Galerkin variational

3

technique to build the functional of neutron diffusion equation.Secondly, variables such as the neutron flux and neutron currentin nodal functional are discretized by orthogonal basis functions,which satisfy orthogonalization in regular triangular prism that istransformed from arbitrary triangular prism by coordinate trans-formation. The last part introduces the theory of coordinate trans-formation and orthogonal basis function generation.

2.1. Functional of neutron diffusion equation and Ritz discretization

Three-dimensional multigroup neutron diffusion equation iswritten as:

�r � DgðrÞrUgðrÞ þ Rr;gðrÞUgðrÞ

¼XG

g0¼1;g0–g

Rs;g0!gðrÞUg0 ðrÞ þ 1keff

vg

XGg0¼1

mRfg0 ðrÞUg0 ðrÞ ð1Þ

wheregis the energy group index; G is the total number of energygroup; r represents the spatial position (x,y,z); Ug is neutron scalarflux (cm�2�s�1); Dgis the diffusion coefficient (cm); Rr;g is themacroscopic removal cross section (cm�1); Rs;g0!g is the macro-scopic cross section of g group neutrons generated after scatteringg’ group neutron (cm�1); Rfg0 is the macroscopic fission cross section(cm�1); m is the average number of neutrons produced per fissionreaction; keff means effective multiplication factor; vg means fissionspectrum.

The right hand of Eq. (1) indicates the g-group neutron sourceterm, which contains scattering source and fission source term.Eq. (1) can be rewritten as the follows if the g-group neutronsource term is represented by Sg .

�r � DgðrÞrUg rð Þ þ Rr;gðrÞUg rð Þ ¼ Sg ð2Þwhere

Sg ¼XG

g0¼1;g0–g

Rs;g0!gðrÞUg0 ðrÞ þ 1keff

vg

XGg0¼1

mRfg0 ðrÞUg0 ðrÞ ¼ RsgUg

þ RfgUg

RsU ¼XG

g0¼1;g0–g

Rs;g0!gðrÞUg0 ðrÞ

RfU ¼ 1keff

vg

XGg0¼1

mRfg0 ðrÞUg0 ðrÞ

In the first place, the solution domain should be divided intoseveral arbitrary triangular nodal, named nodal, where the macro-scopic cross section is constant. In the following derivation, theneutron energy group g is omitted for simplification. Accordingto Galerkin variational technique, the functional of neutron diffu-sion equation for entire solution domain can be written as thesum of functional of each nodal:

F U; J½ � ¼Xv

Fv U; J½ � ð3Þ

where F U; J½ � is the functional of neutron flux U and net neutroncurrent J in entire region, and the functional of nodal v is repre-sented by:

Fv U; J½ � ¼ZvdVfD rUð Þ2 þ Rt � Rsð ÞU2 � 2USg þ 2

ZcUJcdC ð4Þ

Jc ¼ J � nc ð5Þ

J ¼ �DrU ð6Þ

K. Zhuang, W. Shang, T. Li et al. Annals of Nuclear Energy 158 (2021) 108285

where ncis the outer normal vector to the nodal boundary surface c,and Jcrepresents the net neutron current along the outer normalvector.

The neutron flux U, the neutron source term S and the net neu-tron current Jc at the nodal boundary surface are expanded asfollows:

U rð Þ ¼ Piuif i rð Þ

Jc rð Þ ¼ PkJkchkc rð Þ

S rð Þ ¼ Pisif i rð Þ

8>>>><>>>>:

ð7Þ

where i represents the number of expansions, which is determinedby user-defined the maximum expansion order. ui, Jkc, si areexpansion coefficients. f i and hkc are orthogonal polynomial basisfunctions for nodal volume and surface, respectively. And theysatisfy:R

v f i rð Þ � f j rð ÞdV ¼ dijRc hic rð Þ � hjc rð ÞdC ¼ dij

(ð8Þ

where dij represents Kronecker Delta, where the value is 1 for thesame i and j, and the value is 0 for otherwise.

2.2. The matrix form of variational nodal method

The expansions shown in Eq. (7) are substituted into the Eq. (4),then the matrix form of nodal functional is shown as follows,

Fv u; J½ � ¼ uTAu� 2uTsþ 2uTMJ ð9Þwhere u,s and J are vectors composed of expansion coefficients ofneutron flux density, neutron source, and net neutron current,respectively; The matrix A and M are calculated by:

Aij ¼ 3Rtrð Þ�1Pij þ dijVvRr ð10Þ

Pij ¼Z

rf i rð Þ � rf j rð ÞdV ð11Þ

M ¼ ½M1 M2 ::: Mc :::� ð12Þ

Mikc ¼Zcf i rð Þhkc rð ÞdC ð13Þ

Make the first order variation of the functional in Eq. (9) to bezero for uT and Jc, respectively. It can be got the followingequations.

u ¼ A�1s� A�1MJ ð14Þ

Wc ¼ MTcu ð15Þ

The partial neutron current j� is defined as Eq. (16), thenW, J areexpressed by Eqs. (17) and (18), respectively.

j� ¼ 14W� 1

2J ð16Þ

W ¼ 2 jþþj�� � ð17Þ

J ¼ jþ � j� ð18Þwhere jþ and j� denote the outgoing and incoming neutron currentvector at the nodal boundary, respectively. W means neutron flux atthe nodal boundary.

By substituting Eq. (17) into Eq. (14), the matrix form of neutronflux is shown as

4

u ¼ Hs� C jþ � j�� � ð19Þ

By substituting Eqs. (17) and (18) into Eq. (16), the matrix formof partial neutron current at boundary surface is shown as

jþ ¼ Bsþ Rj� ð20ÞwhereB, C, H and R are the response matrix, which are respect tonodal geometry and material macroscopic cross sections. Thoseresponse matrixes can be calculated before starting fission sourceiterations.

2.3. Spatial orthogonal basis function for arbitrary triangle prism

This section shows the procedure of establishing polynomialbasis functions. At the beginning of calculation, the region needsto be divided into arbitrary triangular meshes, which don’t overlapand are connected by nodes. As described in Eq. (7), the variableswithin a nodal are expanded by a set of basis functions, which sat-isfy orthogonality in triangular nodal. It should be noted that trian-gular meshes with different geometries and materials havedifferent basis functions, and the establishment of basis functionsis the main computational burden in variational nodal method.For the same material zone, arbitrary geometry triangles aftertransformation to equilateral triangles have the same basis func-tions, but the coordinate transformation relationship is different.Besides, establishing basis function within an equilateral triangleis easier than that within an arbitrary geometry triangle. Coordi-nate transformation is employed to transform arbitrary trianglenodal into regular triangle nodal, as shown in Fig. 1.

The coordinate relationship between arbitrary triangle in origi-nal coordinate system and regular triangle in new coordinate isexpressed as follows:

1x

y

264

375 ¼

1 0 0Fk21 Fk

22 Fk23

Fk31 Fk

32 Fk33

264

375

1x0

y0

264

375 ð21Þ

Fk21 ¼ 1

3 xk þ xn þ xp� �

; Fk22 ¼ �xk þ 1

2 xn þ xp� �

; Fk23 ¼

ffiffi3

p2 �xn þ xp� �

Fk31 ¼ 1

3 yk þ yn þ yp� �

; Fk32 ¼ �yk þ 1

2 yn þ yp� �

; Fk33 ¼

ffiffi3

p2 �yn þ yp� �

ð22Þwhere x; yð Þ represents the original coordinate, x0; y0ð Þ represents thenew coordinate.

The number of expanded basis functions is determined by thehighest order of basis functions given by user. For one-dimensional functionPmðzÞ, if the maximum expansion order is m,then the number of expansion moments is

f 1ðmÞ ¼ mþ 1 ð23ÞFor two-dimensional functionPnðx; yÞ, if the maximum expan-

sion order is n, then the number of expansion moments can beregarded as the sum of n + 1 one-dimensional case.

f 2ðnÞ ¼Xni¼0

ðiþ 1Þ ¼ ðnþ 1Þðnþ 2Þ2

ð24Þ

Similarly, for three-dimension. When the radial expansionorder is equal to, greater than and less than the axial expansionorder, the number of expansion basis functions are shown in Eqs.(25), (26) and (27), respectively.

f 3ðnÞ ¼Xni¼0

ðiþ 1Þðiþ 2Þ2

¼ ð6þ nÞnþ 11ð Þnþ 66

ð25Þ

f 3ðI;HÞ ¼ð6þ HÞH þ 11ð ÞH þ 6

6þ f 2ðIÞ � f 2ðHÞ ð26Þ

Fig. 1. Arbitrary triangle coordinate transformation.

K. Zhuang, W. Shang, T. Li et al. Annals of Nuclear Energy 158 (2021) 108285

f 3ðI;HÞ ¼ð6þ IÞI þ 11ð ÞI þ 6

6þ f 1ðHÞ � f 1ðIÞ ð27Þ

where n means the maximum expansion order. I, H are the maxi-mum expansion order in radial and axial direction.

Fig. 2. The calculation scheme of variational nod

5

After determining the number of expanded basis functions, theone set of basis functions is then constructed. The linearly indepen-dent function 1; x; y; z; x2; xy; xz; y2:::

� �constitutes a function vector

g!ðrÞ, which is not normalized orthogonally in the regular triangle

al method based on arbitrary triangle mesh.

Fig. 3. Configuration of 2D-IAEA benchmark (a) and its mesh generated by ANSYS code(b).

K. Zhuang, W. Shang, T. Li et al. Annals of Nuclear Energy 158 (2021) 108285

region V, that is, G in Eq. (29) is not equal to 1. The number of ele-

ments of vector g!ðrÞ is identified by Eqs. (24)–(27).

g*ðrÞ ¼ 1; x; y; z; x2; :::

� �T ð28Þ

G ¼ZVd r*

g*

g*T

ð29Þ

Obviously, the matrix G is a positive definite symmetric matrix,which must be similar to the identity matrix. Thus, there is a fullrank matrix Q satisfying Eq. (30). Eq. (31) can be obtained by sub-stituting Eq. (29) into Eq. (30), and by defining a new function vec-tor shown in Eq. (32).

QGQ T ¼ I ð30ÞZVdrf ðrÞf TðrÞ ¼ I ð31Þ

f*

¼ Q g* ð32Þ

The full rank matrix Q guarantees that the function spacesformed by the two sets of basis functions shown in Eqs. (28) and(32) are the same, as shown in Eq. (33).

span f 1; f 2; :::f g ¼ span g1; g2; :::f g ð33Þ

Table 1Macroscopic cross section of 2D-IAEA benchmark.

Material zone Group Dg=cm Rga=cm

�1 vRgf =cm

�1 R1�2s =cm�1

1 1 1.5 0.01 0.0 0.022 0.4 0.08 0.135

2 1 1.5 0.01 0.0 0.022 0.4 0.085 0.135

3 1 1.5 0.01 0.0 0.022 0.4 0.13 0.135

4 1 2.0 0.0 0.0 0.042 0.3 0.01 0.0

5 1 2.0 0.0 0.0 0.042 0.3 0.055 0.0

6

The orthogonal basis function {f1,f2,. . .} within a regular trianglecan be calculated by Eq. (32), but the full rank matrix Q need to becalculated first. The matrix Q can be solved by the Gram-Schmidtalgorithm (Giraud et al., 2005; Lyubashevsky and Prest, 2015),which is an typical orthogonal method. Orthogonal basis functionsare defined by linear combination of linearly independent func-tions {g1,g2,. . .}, and the linear combination coefficients are calcu-lated based on orthogonality. Those linear combinationcoefficients constitute the element of matrix Q. It should be notedthat the coordinate transformation of basis function as shown inEqs. (21) and (22) should be considered in calculation responsematrix shown in Eqs. (11) and (13).

2.4. Calculation scheme

The diffusion equation system after variational principle issolved via the traditional power method. Each iteration of powermethod contains three level iterations: fission source iteration,scattering source iteration and in-group iteration. Fig. 2 showsthe calculation scheme of variational nodal method based on arbi-trary triangle mesh.

First, arbitrary triangular meshes are generated by ANSYS codefor calculation region of interest, and node information and nodaladjacent relationship are saved in two files for the following calcu-lation. The expansion order of neutron flux, source term and netneutron current on boundary surface is determined by user. Thencalculations of four response matrixes B, C, H and R are performedfor all the triangle prism nodal. It is worth noting that the coordi-nate transformation needs to be considered for differential or inte-gral operation in calculation of response matrix since the basisfunctions are generated in the regular triangular mesh. Besides,two meshes have the same response matrixes if they have thesame geometry and material. Step 2, the effective multiplicationfactor, nodal neutron flux and partial neutron current on the nodalboundary surface are initialized before power iterations. Then thefission source of each neutron energy group in each nodal is calcu-lated based on new nodal neutron flux. Step 3, scattering sourceand fission source constitute the in-group source term, then in-group iteration starts. The Eq. (20) is continually adopted to updatethe outgoing partial neutron current for each nodal until it con-verges. Step 4, using Eq. (19) to update the nodal neutron flux

Fig. 4. Comparison of normalized power distribution between TriVNM and reference for 2D-IAEA.

Table 2Comparison of keff between TriVNM and reference for 2D-IAEA.

Code keff Dkeff =pcm Emax=%

QUANDRY 1.02962 3.399 0.94MEND 1.02961 3.998 0.88NODAN 1.02966 7.283 0.90TRAC/NEM 1.02950 �8.264 2.05NLNEM 1.02957 1.459 1.80NGFM 1.02960 1.071 0.72NLSANM 1.02962 3.823 0.59TriVNM 1.02959 �2.360 �0.20

Emax represents the maximum relative error of the normalized power of the nodal.

K. Zhuang, W. Shang, T. Li et al. Annals of Nuclear Energy 158 (2021) 108285

based on the above converged partial neutron current. Step 5, theiteration step from 3 to 5 need to scan all the neutron energy groupand continues until the neutron flux converges. Step 6, the con-verged neutron flux is used to calculate fission source, which thenis adopted to update multiplication factor. Step 7, the iteration stepfrom 3 to 6 continues until both multiplication factor and fissionsource converge. It can be seen that step 3 is namely inner itera-tion, step 5 means multi-group iteration and step 7 represents fis-sion source iteration.

3. Numerical results

Based on the above numerical model, the code named TriVNMwas developed based on FORTRAN-90 language object-orientedprogramming. In order to verify TriVNM and its applicability forcomplex geometry modeling, benchmarks including rectangle,hexagonal assembly reactor and reactor with curved boundarywere employed. In this section, the result comparisons betweenTriVNM and reference for 2D/3D-IAEA with rectangle assembly,2G/4G VVER1000 with hexagonal assembly and reactor withcurved edge are descripted. The following benchmarks are macro-scopic problem, and both TriVNM code and other reference codesemploy multi-group macroscopic cross sections, which are takenfrom literature. Finally, study on mesh sensitivity of TriVNM wasperformed based on 2D-IAEA benchmark. It should be noted thatacceleration calculation method is not the goal of this study, andthe current TriVNM code doesn’t integrate any acceleration calcu-lation method. Nevertheless, the calculation time of TriVNM is stilllisted in the following calculation, which is performed on the PCwith 2.3 GHz Intel Core i9. However, the reference results don’t list

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run time due to missing relevant data. In the following TriVNM cal-culation, the maximum expansion order of polynomial basis func-tion is respectively 5, 5 and 3 for variable of �, y and z.

3.1. 2D-IAEA benchmark

2D-IAEA benchmark problem (Michelsen, 1977) is a simplified2D PWR core problem, which is widely used to verify numericalcalculation of diffusion code. The reactor core contains 177 fuelassemblies and 9 control rods, and the size of assembly is20 cm � 20 cm. As shown in Fig. 3 (a), the reactor core is 1/8 sym-metrically arranged and a 20 cm thick water reflector stays outer ofactive reactor core and zero incoming neutron current occurs onthe outside boundary, namely vacuum boundary condition. Fivetypes of material with two-group macroscopic cross section arelisted in Table 1. It should note that 2D-IAEA benchmark only

Fig. 5. Radial and axial configuration of 3D-IAEA benchmark (Michelsen, 1977).

K. Zhuang, W. Shang, T. Li et al. Annals of Nuclear Energy 158 (2021) 108285

adopts first 4 types of material and the 5th material are used in thefollowing 3D-IAEA benchmark. TriVNM code employs arbitrary tri-angle mesh, and the meshing of reactor core shown in Fig. 3 (b) isperformed by ANSYS 14.0 code. Each rectangle assembly is dividedinto 4 triangle mesh.

Fig. 6. Comparison of normalized power distributio

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The normalized power of 1/8 core and effective multiplicationfactor keff calculated by TriVNM and reference results are shownin Fig. 4 and Table 2. And the run time of TriVNM code is 13.7 swithout any acceleration method. By comparison, the normalizedpower calculated by TriVNM agree well with reference, and the

n between TriVNM and reference for 3D-IAEA.

Table 3Comparison of keff between TriVNM and reference for 3D-IAEA.

Code keff Dkeff =pcm Emax=%

QUANDRY 1.02902 1.0 0.69MEND 1.02915 11.7 1.60NODAN 1.02916 12.6 1.70TRAC/NEM 1.02900 2.9 1.62NLNEM 1.02907 3.9 1.29NGFM 1.02909 5.8 0.67NLSANM 1.02912 8.3 0.89TriVNM 1.02908 4.7 �0.45

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relative error of most assembles stays below 0.2%, and the maxi-mum relative error of �0.20% occurs in the central assembly.Table 2 shows the relative error of keff between TriVNM and refer-ence, and the last column means the maximum relative error ofnormalized power. It can be seen that keff calculated by TriVNMis very close to results by other diffusion codes, and the error is2.36 pcm, which is better than results by most other codes. Fur-thermore, the maximum relative error of normalized power calcu-lated by TriVNM is less than that by all other codes. Overall,TriVNM code has applicability for 2D reactor core calculation andachieves the same or higher accuracy as other diffusion codes.

3.2. 3D-IAEA benchmark

3D-IAEA benchmark represents a typical 3D PWR reactor core,which is extended on the basis of 2D-IAEA benchmark. As shownin Figs. 5, 3D-IAEA benchmark has the same radial configurationwith 2D-IAEA shown in Section 3.1. The difference is that waterreflector of 20 cm thick is respectively located on the top and bot-tom reactor core, and the axial boundary condition is vacuum. Theaxial configuration is illustrated in Fig. 5 (Michelsen, 1977). 3D-IAEA benchmark contains 5 types of material, of which macro-scopic cross sections are listed in Table 1. In TriVNM calculation,the radial mesh is the same as that shown in Fig. 3 (b), and the axialdirection is divided into 19 layers, namely equal height size of20 cm.

The run time is 70.2 s, and the comparisons of normalizedpower of 1/8 core and effective multiplication factor keff are shownin Fig. 6 and Table 3, and the reference results come from literature(Michelsen, 1977). The relative error of normalized power betweenTriVNM and reference is very small, and the maximum value is

Table 4Macroscopic cross section of 4 energy group for the 3D-VVER1000 benchmark.

Material zone Group Rðg�1Þ�gs /cm�1 Dg /cm

1 1 0.0000E-0 2.1302 7.5487E-2 0.9193 8.9296E-2 0.6594 7.4945E-2 0.192

2 1 0.0000E-0 2.1312 7.5465E-2 0.9193 8.9163E-2 0.6564 7.3980E-2 0.191

3 1 0.0000E-0 2.1312 7.5457E-2 0.9193 8.9121E-2 0.6554 7.3696E-2 0.190

4 1 0.0000E-0 2.4602 1.0775E-1 0.8983 1.5590E-1 0.5894 1.2670E-1 0.139

where fission spectrum v is 0.75, 0.25, 0.0, 0.0 for 4 energy group; Rr is macroscopic re

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about �0.45%. The last column in Table 3 is the maximum relativeerror calculated by different other diffusion code. It can be seenthat the accuracy power calculated by TriVNM is better than thatby other diffusion code compared to the reference. As shown inTable 3, keff calculated by TriVNM is 1.02908, which has relativeerror of 4.7 pcm compared to reference and has the same accuracywith other similar diffusion codes.

3.3. 4G 2D-VVER1000 with reflector

To verify the multigroup (>2) calculation ability of TriVNMcode, a benchmark of 4 neutron energy group 2D-VVER1000 withreflector was employed in this section. This macroscopic problemhas 4 types of material with homogenized macroscopic cross sec-tion listed in Table 4. The reactor core is configured with hexagonalassemblies of 23.6 cm pitch in 1/6 reflective symmetry, and con-tains 8 rounds of fuel assembly and 1 round of reflector assembly.The outside boundary condition is vacuum, and more informationcan be found in reference (Makai, 1982). Each hexagonal assemblyis divided into 6 regular triangle mesh by ANSYS code for TriVNMcalculation.

The whole TriVNM calculation takes 19.2 s. Fig. 7 illustrates theelative error of fuel assembly normalized power of 1/12 reactorcore between TriVNM code and reference. Table 5 lists the keff com-parison between TriVNM and reference, and the maximum andaverage relative error of fuel assembly power over the whole core.The reference results are referred to the literature (Cho et al., 1997;Makai, 1982; Zimin and Baturin, 2002). The comparison shows thatTriVNM results agree well with the reference. The relative errors ofall fuel assembly normalized power are below 1%, and the maxi-mum value on outer fuel assembly is 0.92%. Table 5 shows keff errorof 16 pcm exists between TriVNM and reference. The good agree-ment of normalized power and keff shows that TriVNM is at thesame level of calculation accuracy compared with other diffusioncodes in aspect of multigroup (>2) calculation.

3.4. 3D-VVER1000 without reflector

Benchmark of 3D-VVER1000 without reflector is extended onthe basis of 2D-VVER1000. 8 rounds of hexagonal fuel assembliesof 23.6 cm pitch are arranged in the radial direction and the reactorcore is configured in 1/6 rotational symmetry and consists of 25control rods. Axial and radial outside boundaries adopt albedoboundary conditions of 0.15 and 0.125, respectively. Before

Rga/cm

�1 mRgf /cm

�1 Rr/cm�1

9 3.4210E-3 7.2825E-3 7.8908E-264 2.2480E-3 5.7354E-4 9.1544E-246 2.1423E-2 8.0392E-3 9.6368E-274 8.3783E-2 0.108070 8.3783E-28 3.4810E-3 7.4381E-3 7.8946E-264 2.3780E-3 8.6043E-4 9.1541E-230 2.3590E-2 1.1877E-2 9.7569E-220 1.0186E-1 0.147300 1.0186E-18 3.4980E-3 7.4850E-3 7.8955E-264 2.4220E-3 9.4610E-4 9.1543E-240 2.4257E-2 1.3010E-2 9.7953E-265 1.0679E-1 1.5780E-1 1.0679 E-10 4.5000E-4 0.0000E-0 1.082 E-17 0.0000E-0 0.0000E-0 1.559 E-16 1.0000E-3 0.0000E-0 1.277 E-13 1.6770E-2 0.0000E-0 1.6770E-2

moval cross section.

Fig. 7. Comparison of normalized power between TriVNM and reference for 2D-VVER1000 with reflector.

Table 5Comparison of keff between TriVNM and reference for 2D-VVER1000 with reflector.

Code keff Dkeff =pcm Emax=% Eavg=%

AFEN 1.11204 0.012 0.64 0.27SIXTUS-2 1.11193 0.001 2.42 0.69SKETCH-N 1.11190 �0.002 0.50 0.20FEMHEX 1.11207 0.015 0.57 0.28TriVNM 1.11208 0.016 0.92 0.52

where Eavg represents the average relative error of the normalized power of thenodal; The reference keff is 1.11192.

Table 6Macroscopic cross section for the 3D-VVER1000 benchmark.

Materialzone

Group Dg=cm Rga=cm

�1 vRgf =cm

�1 R1�2s =cm�1

1 1 1.3832 8.38590E-3 4.81619E-3 1.64977E-22 0.386277 6.73049E-2 8.46154E-2

2 1 1.38299 1.15550E-2 4.66953E-3 1.47315E-22 0.389403 8.10328E-2 8.52264E-2

3 1 1.39522 8.94430E-3 6.04889E-3 1.56219E-22 0.386225 8.44801E-2 1.19428E-1

4 1 1.39446 1.19932E-2 5.91507E-3 1.40185E-22 0.387723 9.89671E-2 1.20497E-1

5 1 1.39506 9.11600E-3 6.40256E-3 1.54981E-22 0.384492 8.93878E-2 1.29281E-1

K. Zhuang, W. Shang, T. Li et al. Annals of Nuclear Energy 158 (2021) 108285

TriVNM calculation, each hexagonal assembly is divided into 6 reg-ular triangle mesh by ANSYS code, and axial layer adopts 10.0 cmsize. 5 types of material with homogenized macroscopic cross sec-tion listed in Table 6 are employed in this benchmark. More infor-mation can be found in reference (Schulz, 1996).

The normalized power of 1/6 core calculated by TriVNM andreference results are compared in Fig. 8, and keff calculated byTriVNM and other diffusion codes are listed in Table 7, moreover,the maximum and average relative error of fuel assembly powerover the whole core are shown in Table 7. In this calculation, theTriVNM run time is 402 s, which is relative long for 2 group reactorcore calculation. This is because the calculation of whole corerather than 1/6 core is performed for TriVNM code since it doesn’thave the ability of processing rotational symmetry condition.Besides, the current TriVNM code doesn’t integrate any accelera-tion method. The calculation efficiency would be further improvedif TriVNM can use acceleration method and multiple boundarytypes. The reference results and results by other codes are referredto literature (Schulz, 1996). It can be seen that the results byTriVNM are in good agreement with reference, and most relativeerrors stay within 1%, and the maximum relative error of about0.3% is located in the outer fuel assembly. Compared to resultsby other diffusion code, results by TriVNM code have smaller max-imum and average relative error of fuel assembly. Table 7 showsthat error of 10.8 pcm can be found between keff calculated bythe TriVNM code and reference result. TriVNM code stays in thesame level of calculation accuracy compared with other diffusion

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codes. Therefore, TriVNM has good applicability and accuracy forreactor core with hexagonal assembly.

3.5. Two-zone core with curved boundary

As shown in Fig. 9, benchmark of two-zone core with curvedboundary (Itagaki, 1985) was employed to verify the applicabilityof TriVNM code for modeling of complicated geometry. This is asimplified 2D reactor model, and fuel of two-zones are located inan infinitely large pool. In actual simulation for TriVNM code andother reference code, a reflector of radius 45 cm is used to replacethe infinite reflector, and the outside boundary adopts reflectiveboundary condition. Three types of material are adopted in thisbenchmark, and its macroscopic cross sections are listed in Table 8.

Due to curved boundary, traditional nodal method diffusioncode based on rectangle or hexagonal geometry can’t be applicable.Thus, TriVNM code based on arbitrary triangle mesh has goodapplicability for modeling of complex geometry since the combina-tion of triangle meshes can theoretically form any geometries. Asdepicted in Fig. 10, curved boundary is simulated accurately byarbitrary triangular mesh. The normalized power and keff calcu-lated by TriVNM and results by other diffusion code are shown inTable 9, where FEM2D (Schmidt and Franke, 1975) and ABFEM-T(Lu and Wu, 2007) are both based on finite element method. By

Fig. 8. Comparison of normalized power distribution between TriVNM and reference for 3D-VVER1000 without reflector.

K. Zhuang, W. Shang, T. Li et al. Annals of Nuclear Energy 158 (2021) 108285

comparison, the errors of keff calculated by TriVNM are 20.1 pcm,and both keff and normalized power calculated by TriVNM are ingood agreement with other diffusion code. It can be concluded thatTriVNM code based on arbitrary triangle mesh can be applicablefor complex geometry reactor core and calculate the accurateresult.

Table 7Comparison of keff between TriVNM and reference for 3D-VVER1000 without reflector.

Code keff Dkeff =pcm Emax=% Eavg=%

AFEN 1.011460 11 1.70 0.81ANC-H 1.011480 13 0.90 –GTDIF-H 1.011980 63 3.09 –NLSANM 1.011500 15 1.57 0.62TriVNM 1.011242 �10.8 0.34 0.05

where Eavg represents the average relative error of the normalized power of thenodal; The reference keff is 1.011350.

Fig. 9. Configuration of two-zone core with curved boundary (Itagaki, 1985).

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Table 8Macroscopic cross section for two-zone core with curved boundary.

Material zone Group Dg=cm Rga=cm

�1 vRgf =cm

�1 R1�2s =cm�1

1 1 1.188 0.008179 0.005383 0.025032 0.2939 0.1008 0.1310

2 1 1.193 0.008801 0.006387 0.024222 0.2979 0.1079 0.1606

3 1 1.174 0.0007454 0.0 0.057032 0.1563 0.01885 0.0

Fig. 10. Mesh used in two-zone core with curved boundary.

Table 9Result comparison between TriVNM and reference code.

Code keff Dkeff =pcm

FEM2D 0.94465 82.6ABFEM-T 0.94371 �17.0TriVNM 0.94406 20.1

where the reference keff is 0.94387.

Fig. 11. Two meshing m

K. Zhuang, W. Shang, T. Li et al. Annals of Nuclear Energy 158 (2021) 108285

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3.6. Mesh sensitivity analysis

Due to using arbitrary triangle nodal method, TriVNM code canadopt larger mesh size compared to finite difference method. Thissection employs 2D IAEA benchmark to analyze mesh sensitivity ofTriVNM code by using three kinds of different mesh size as shownin Fig. 11. The first type of mesh is the same as that shown in Sec-tion 3.1, and each assembly is divided into 4 triangle meshes. Thesecond kind of mesh reaches further refinement, and each assem-bly contains about 10 triangle meshes. The third kind of meshrefine on the basis of the second kind of meshing, and each assem-bly is divided into about 40 triangle meshes. The power distribu-tions calculated by TriVNM for those three types of mesh size arerespectively drawn in Figs. 4, 12 and 13, and keff results and calcu-lation time are listed in Table 10. By comparison, the results by allthose three types of mesh size agree well with reference, and nomuch difference occurs between three cases. The maximum rela-tive error of assembly power is �0.2%, �0.6% and �0.6% for threekinds of mesh size, respectively. With the mesh size decreasing,

Power of core I Power of core II

0.80079 1.199210.80169 1.198310.80137 1.19863

ethods for 2D IAEA.

Fig. 12. Comparison of normalized power distribution for case 2.

K. Zhuang, W. Shang, T. Li et al. Annals of Nuclear Energy 158 (2021) 108285

the error keff become smaller but the improvement of from 2.36pcm to 0.76 pcm is small. However, the decreasing of mesh sizelead to the increasing of mesh number, thus the calculation timeincreases from 20 s to 500 s. In TriVNM calculation, it is need tobalance the relationship between calculation efficiency and accu-racy. A smaller mesh size slightly improves the calculation accu-racy, but greatly reduces the calculation efficiency. Generallyspeaking, the larger mesh size can guarantee calculation accuracyand efficiency at the same time for TriVNM code.

Overall, 4 types of typical benchmarks including rectangle,hexagonal assembly reactor and reactor core with curved bound-ary were employed to verify TriVNM and its applicability for com-plex geometry modeling. The comparisons of keff and assemblypower show that the numerical model of variational nodal methodbased on arbitrary triangle mesh is correct and TriVNM code can beapplied to reactor calculations with different assembly geometries.Mesh sensitivity analysis indicates that for large mesh size TriVNMcode still has good accuracy.

4. Conclusions

With the continuous development of new concept reactor, thecore configuration and assembly geometry have undergone greatchanges. Traditional core calculation codes based on rectangle orhexagonal nodal is no longer applicable for reactor design andanalysis. In this study, variational nodal method of diffusion theorybased on arbitrary triangle mesh was developed. Firstly, ANSYScode is used to divide calculation region into arbitrary triangle

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mesh, which is then transferred to regular triangle by coordinatetransformation theory, and orthogonal basis functions are gener-ated. After that step, a functional including nodal neutron balanceequations is established by Galerkin variational technique, andneutron flux, source term and net neutron current are expandedby orthogonal basis functions generated in step 1. Finally, the vari-ational principle is used to get the response relationship betweenthe neutron flux and surface partial neutron current, which is thensolved by power iteration method of three levels of iteration: fis-sion source iteration, scattering source iteration and in-group iter-ation. Based on above, TriVNM code was developed, and 4 types oftypical benchmarks including rectangle (2D/3D-IAEA), hexagonalassembly reactor (2G/4G VVER1000) and unstructured geometryreactor (two-zone reactor with curved boundary) were employedto verify TriVNM and its applicability for modeling complex geo-metrical reactor. Both power and keff calculated by TriVNM codeagree well with reference results, and the maximum relative errorof all the benchmarks doesn’t exceed 1% and error of keff doesn’texceed 50 pcm. TriVNM code achieves the same or higher levelof calculation accuracy compared with other diffusion codes.Besides, 2D IAEA benchmark was employed to analyze mesh sensi-tivity of TriVNM code by using three types of different mesh size. Itcan be seen that the numerical model of variational nodal methodbased on arbitrary triangle mesh is correct, and TriVNM code hasgood applicability and accuracy for reactor core with complex geo-metrical assembly. Furthermore, using large mesh size in TriVNMcalculation guarantees both good accuracy and reducing calcula-tion burden.

Fig. 13. Comparison of normalized power distribution for case 3.

Table 10Comparisons of keff and calculation time for three cases.

Case keff Dkeff =pcm Calculation time /s

Case 1 1.0295853 �2.360 20.875Case 2 1.0295930 0.777 301.516Case 3 1.0295929 0.767 532.391

K. Zhuang, W. Shang, T. Li et al. Annals of Nuclear Energy 158 (2021) 108285

CRediT authorship contribution statement

Kun Zhuang: Methodology, Software. Wen Shang: Software,Validation. Ting Li: Software, Validation. Jiangtao Yan: Writing -original draft. Sipeng Wang: Writing - review & editing. XiaobinTang: Writing - review & editing. Yunzhao Li: Investigation.

Declaration of Competing Interest

The authors declare that they have no known competing finan-cial interests or personal relationships that could have appearedto influence the work reported in this paper.

Acknowledgements

This work was supported by the Fundamental Research Fundsfor the Central Universities under Grant YAH17067 andNJ2020017-1, and China Postdoctoral Science Foundation fundedproject under Grant 2020 M681594, 2019TQ0148, and Jiangsu Pro-vince Postdoctoral Science Foundation funded project under Grant

14

2020Z231, and Science and Technology on Reactor System DesignTechnology Laboratory under Grant NBC20008.

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