Transport Calculations. Tseelmaa Byambaakhuu, Dean Wang*, and Sicong Xiao

Transcription

1 A Locl hp Adptive Diffusion Synthetic Accelertion Method for Neutron Trnsport Clcultions Tseelm Bymbkhuu, Den Wng*, nd Sicong Xio University of Msschusetts Lowell, 1 University Ave, Lowell, MA USA *Emil: Den_Wng@uml.edu Number of pges: 21 Number of figures: 7 Number of tbles: 1 1

2 ABSTRACT We present locl hp dptive diffusion synthetic ccelertion (DSA) method for neutron trnsport clcultions. This new DSA method, clled DG-DSA, solves the diffusion eqution on corse mesh using the interior penlty discontinuous Glerkin methods. We investigte vrious numericl spects of the DG-DSA method such s convergence, locl hp dpttion, nd penlty size. We demonstrte tht our DG-DSA method cn effectively nd efficiently ccelerte trnsport source itertions. Keywords: Accelertion methods, DSA, DG-DSA 2

3 I. INTRODUCTION There hs been considerble reserch on diffusion synthetic ccelertion (DSA) of trnsport source itertions. 1-4 The DSA method is bsed on the use of diffusion clcultion for pproximting the itertive error of the trnsport source itertion. Most work ws focused on the development of so-clled consistent (or prtilly consistent) diffusion discretiztions in order to obtin the optiml convergence performnce. While the use of consistent discretiztions mkes the DSA method effective, it does not necessrily mke it efficient becuse the numericl solution of discretized elliptic diffusion problem itself cn be costly, prticulrly when the problem size becomes lrge. Adms nd Mrtin developed DSA scheme bsed on discontinuous finite element (DFE) diffusion discretiztion to ccelerte DFE trnsport itertions. 5 They noted tht their DFE bsed DSA scheme ws very effective, however, the efficient solution of the DFE diffusion equtions in 2D geometries remined n open question. In this pper, we present new discontinuous Glerkin (DG) discretiztion of the diffusion eqution, clled DG-DSA, which cn effectively nd efficiently ccelerte the S N trnsport itertions. The novelty of our method is tht the diffusion eqution is solved on corse mesh grid using the DG methods insted on the trnsport fine mesh, nd the DG diffusion discretiztion incorportes locl hp dpttion, i.e., locl dpttion of mesh size nd/or polynomil degree, bsed on locl totl cross section (or opticl thickness). Therefore, the resulting number of degrees of freedom of the DG 3

4 discretiztion is much less thn the conventionl consistent DSA discretiztions, nd thus DG-DSA cn chieve significnt improvement in computtionl efficiency. The remining pper is orgnized s follows. In Sec. II, we present in detil the formultion nd lgorithm of the DG-DSA method. A numericl study of DG-DSA is crried out in Sec. III, focusing on vrious numericl spects such s convergence performnce, locl hp dpttion, nd penlty. Sec. IV concludes the pper with brief summry nd discussion. II. DG-DSA FORMULATION AND ALGORITHM In this pper, we introduce the DG-DSA method bsed on monoenergetic S N neutron trnsport fixed source problems on 2D Crtesin geometry. The scttering nd neutron source re ssumed isotropic. The flowchrt of the DG-DSA lgorithm is shown in Fig. 1. 4

5 Fig. 1. Flowchrt of the DG-DSA lgorithm. The l &' itertion cycle begins with the S N trnsport eqution with itertion indices is expressed s μ ) )* ψ,-. / x, y, μ, η + η ) )5 ψ,-. / x, y, μ, η + Σ 7 ψ,-. / x, y, μ, η = : ; < φ, x, y + > < Q x, y, (1) where φ nd ψ re the sclr flux nd ngulr flux, respectively. Σ & nd re the totl cross section nd scttering cross section. μ nd η re the neutron ngulr directions. x 5

6 nd y re the sptil positions. Q is the externl neutron source. l is the source itertion index nd l + 1/2 is the intermedite step. During ech source itertion, the diffusion is utilized to pproximte the trnsport itertive flux error s ) )* > ) E: F )* δφ,->/h x, y ) )5 > ) E: F )5 δφ,->/h x, y + Σ I δφ,->/h x, y = Σ J [φ,->/h x, y φ, x, y ], (2) where Σ I is the bsorption cross section. We solve the bove diffusion eqution with priml discontinuous Glerkin method (DG) 6, which is commonly used method for solving elliptic nd prbolic problems. D δφ x, y + Σ I δφ x, y = f(x, y), in Ω, (3) Reflective BC: D δφ x, y n = 0, on Γ XYZ, (3b) Vcuum BC: D δφ x, y n = > δφ x, y, H on Γ [I\, (3c) where Ω is polygonl domin in R H, Γ XYZ nd Γ [I\ re the disjoint sets tht prtition the domin boundry. n is the unit norml vector to the boundry exterior to Ω. The functions D nd f re defined s D = > E: F, (4) f x, y = Σ J [φ,->/h x, y φ, x, y ]. (4b) Multiplied by test function υ nd integrted over one element E gives υ D δφ + Σ I δφ υ = f υ, E E, (5) 6

7 where E is the discretiztion of Ω, i.e. the mesh. Appling the Green s theorem on the first term υ D δφ = D δφ υ D δφ n E υ ), E E, (6) where n E denotes n outwrd norml vector to, E, the boundry of element E. Summing over ll the elements gives E (D δφ υ + Σ I δφ υ) Y h ijf D δφ n e υ D δφ Y Y h klm h opq Y n e υ = E fυ, (7) where Γ rs7 is set contining only interior edges nd n e is n outwrd unit norml vector of edge e. Two new opertors re introduced: jump nd verge Interior edge: υ = > H υ ḷ + υ / l, (8) υ = υ ḷ υ / l, e = E > Y E H Y, (8b) Boundry edge: υ = υ = υ ḷ, e = E > Y Ω, (8c) where Ω is the boundry of the domin Ω. For continuous nd second order differentible functions, the jump nd verge opertors re simpler δφ = 0, e Γ rs7, (9) D δφ n e = 0, e Γ rs7, (9b) D δφ n e = D δφ n e, e Γ rs7. (9c) 7

8 Using Eqs. (8) nd (9) on Eq. (7), it is strightforwrd to obtin the vritionl formultion (or wek formultion) of Eq. (3). E (D δφ υ + Σ I δφ υ) D δφ n e υ Y h ijf + Y > H Y h opq [δφ][υ] = fυ Y E. (10) Using the smoothness of the solution δφ expressed in Eq. 9, we cn dd the following two terms to the wek formultion ε Y h ijf D υ n e [δφ] + Y x l Y Y Y h ijf δφ [ υ] = 0. (11) where e is the edge length, σ Y is nonnegtive rel penlty number nd ε is nother prmeter tht my tke the vlue of {-1, 0, 1}. Thus, the vritionl formultion cn be rewritten s { δφ, υ = L(υ), (12) where { δφ, υ is biliner form nd L(υ) is liner form which re defined s: { δφ, υ = E (D δφ υ + Σ I δφ υ) D δφ n e υ Y h ijf + Y > H Y h opq δφ υ + ε D υ n e [δφ] Y Y h ijf + Y x l Y Y Y h ijf δφ [ υ], (13) L υ = E fυ. (13b) Depending on the choice of prmeter ε, the methods cn be clled differently. A detiled discussion of these three types of DG methods cn be in Reference [6]. ε = 1, Symmetric interior penlty Glerkin SIPG +1, Nonsymmetric interior penlty Glerkin NIPG 0, Incomplete interior penlty Glerkin IIPG. (14) 8

9 DG-DSA utilizes finite element spce D E, the spce of discontinuous polynomils. The globl bsis functions of D E hve support contined in ech element nd D E = spn{p r : 1 i N, E E}, (15) with P r x, y = p r x, y, (x, y) E 0, (x, y) E, (15b) where {p r } is set of locl bsis functions tht re chosen to be the monomil bsis functions, trnslted from the intervl (-1,1) for qudrilterl mesh: p r x, y = * * š.// H(* š. * š ) œ 5 5.// H(5. 5 ) ž, I + J = i, 0 i k, (15c) nd (x ->/H, y ->/H ) is the midpoint of the element E bounded by (x, x -> ) (y, y -> ). This yields the locl dimension N = ( ->)( -H) H where k is the highest polynomil degree of n element E., (15d) Finlly, the DG-DSA method is to find δφ,->/h (x, y) in D E such tht { δφ,->/h (x, y), υ = L υ, υ D E. (16) The solution δφ,->/h (x, y) cn be n expnsion of locl bsis functions s δφ,->/h x, y = rª> α r E P r (x, y), (17) where α r is the unknown rel numbers to be solved. Substituting Eq. (17) into Eq. (16) gives Aα = b, (18) 9

10 where α is the vector with components α r, b is the vector with components L P, nd A is sprse mtrix expressed s rª> A = { P r, P E, E E, 1 j N ±. (18b) At the end of the l &' source itertion, the sclr flux cn be updted in the next trnsport itertion s φ,-> x, y = φ,->/h x, y + δφ,->/h (x, y) (19) The trnsport source itertion will continue until the convergence criterion is stisfied. The DG-DSA method is locl dptive, which mens tht both the mesh size nd the polynomil degree re loclly djustble bsed on locl totl cross section. The penlty number, σ Y, is chosen to give the optimum performnce bsed on scoping nlysis, which will be discussed lter. III. NUMERICAL RESULTS III.A. Numericl Convergence Study A numericl study of the DG-DSA ccelertion performnce ws crried out bsed on 2D S N fixed source model problem, which is homogeneous 6cm 6cm squre with the reflective boundry on the left nd bottom sides nd the vcuum boundry on the top nd right sides. The domin is discretized into 5 5 uniform corse-mesh cells. The fine-mesh number in ech corse-mesh cell is The numericl solution for 10

11 the S N trnsport ws obtined on the fine mesh grid (60 60) using the Guss-Legendre S 12 qudrture set for ngulr discretiztion nd the dimond difference (DD) method for sptil discretiztion. The DG-DSA results were obtined on vrious corse mesh grids, which ws determined bsed on the totl cross sections. The corse mesh (5 5) ws used for smll cross sections (Σ 7 1 cm > ), nd the fine mesh (60 60) ws used for lrge cross sections (Σ 7 > 6 cm > ). For medium cross sections (1 < Σ 7 < 6 cm > ), the DG-DSA mesh size ws determined by mintining the opticl thickness (i.e., Σ & Δ, where Δ is the corse-mesh size) round 1.2. The symmetric interior penlty Glerkin method (SIPG) ws used with piecewise liner polynomils nd the optimized penlty number. The S 12 with DG-DSA ws implemented in MATLAB. In order to chrcterize the convergence behvior, we estimte the spectrl rdius numericlly s defined by ρ = lim, º»¼.»¼» ¼» ¼½.. (20) Note tht the convergence is rpid when ρ 1, nd it slows down when ρ increses. When ρ 1 the scheme fils to converge. Fig. 2 presents the numericl spectrl rdius of DG-DSA s function of totl cross section for vrious scttering rtios, c. It shows tht the DG-DSA method is very effective nd stble for wide rnge of totl cross sections (or opticl thickness). The generl trend is tht the convergence rte decreses with the increse in totl cross sections up to Σ cm > (i.e., the opticl thickness of 20), therefter the spectrl 11

12 rdius decreses becuse the S N solution tends to the diffusion limit. In ddition, the spectrl rdius increses with the scttering rtio in generl. 1 x = 1.2 cm x = 1.2/Σ 7 x = 0.1 cm Spectrl Rdius c = 0.6 c = 0.8 c = 0.9 c = S t (cm -1 ) Fig. 2. Spectrl rdius vs. Σ 7. The bove convergence nlysis cn be used to develop the locl mesh refinement strtegy for DG-DSA. For exmple, in typicl light wter rector the totl cross section of wter in fst neutron groups is less thn 1 cm >, nd it is lrger thn 1 cm > in therml groups. The totl cross section of the fuel is typiclly less thn 1 cm >. Therefore, for fst group trnsport clcultions the DG-DSA mesh cn be s lrge s fuel pin size (~1.2 cm) or even lrger. Only for therml groups, it requires reltively fine mesh. It is interesting to note tht in most neutronics trnsport codes the conventionl consistent DSA discretiztion uses the sme mesh s the trnsport discretiztion, which typiclly hs more thn 100 cells in fuel pin. However, our DG-DSA discretiztion cn hve less thn 20 cells to chieve the sme convergence performnce. An exmple of the conventionl DSA nd our DG-DSA mesh structures is shown Fig. 12

13 3. It shows tht the number of degrees of freedom of the DG-DSA method is much less thn those of the conventionl DSA methods. This is why our DG-DSA scheme cn be very efficient. A detiled comprison is given in Sec. III.C. Consistent DSA Mesh DG-DSA Mesh Fig. 3. Consistent DSA mesh vs. DG-DSA mesh. III.B. Locl p Adpttion We study the locl p (polynomil degree) dptivity of the DG-DSA method bsed on 2D monoenergetic trnsport fixed source problem with homogeneous cross section. Similr to the problem in Sec. III.A, the model problem is 6cm 6cm squre with the reflective boundry on the left nd bottom sides nd the vcuum boundry on the top nd right sides. The domin is divided into 5 5 uniform corse-mesh cells. The finemesh number in ech corse-mesh is The numericl S N trnsport solutions were obtined on the fine mesh (60 60) using DD for sptil discretiztion nd the Guss- Legendre S >H qudrture set for ngulr discretiztion. The DG-DSA solutions were obtined on the corse mesh (5 5), using piecewise constnt (P0), liner (P1), nd qudrtic (P2) polynomils, respectively. 13

14 The numericl results re presented in Fig. 4. The converged flux is shown in Fig. 4, nd the flux reltive error s function of trnsport itertion is shown in Fig. 4b. It shows tht the DG-DSA ccelertion scheme is more effective with higher polynomils, but the computtionl sving decreses with incresing polynomil degree (e.g., P1 vs. P2). The use of higher polynomils is more expensive becuse of lrger number of degrees of freedom. For nucler neutron trnsport problems, we recommend P1 for the region of lrge totl cross sections (Σ 7 > 1 cm > ), nd P1 or P0 for smll cross sections (Σ 7 < 1 cm > ). () Converged sclr flux 14

15 Flux Reltive Error 1.0E E E E E-08 SI P0 P1 P2 1.0E Trnsport Sweep # (b) Flux reltive error vs. Itertion number Fig. 4. Numericl results of locl p dpttion. III.C. Locl h Adpttion In this section, we solve the sme problem s bove with inhomogeneous cross sections s shown in Fig. 5. The problem hs 5 5 uniform corse-mesh cells, nd ech corse cell consists of fine-mesh cells. This cse is mimic of mini fuel ssembly, but there is dditionlly highly bsorbing region (in blue) t the loction (2, 2). The numericl solutions for the S N trnsport were obtined on the fine mesh (60 60) using the Guss-Legendre S 12 qudrture set for ngulr discretiztion nd the dimond difference (DD) method for sptil discretiztion. The DG-DSA solutions were obtined on both the fine-mesh (FM) nd corse-mesh (CM) grids. Piecewise liner polynomil functions were used for the DG-DSA solutions. It should be noted tht locl mesh refinement, 6 6, ws pplied to the bsorbing region (in blue), where the opticl thickness is lrge. The resulting mesh is nonconforming mesh with some 15

16 hnging nodes. The totl number of cells in the DG-DSA mesh is 252, while the S N mesh hs 3600 cells. The numericl results re shown in Fig. 6. The converged sclr flux is plotted in Fig. 6. The convergence performnce, i.e., the flux reltive error vs. trnsport sweep number, is illustrted in Fig. 6b. The results of the unccelerted S N source itertion re shown for comprison. It shows tht DG-DSA cn effectively converge the S N itertions on the corse mesh s on the fine mesh. Fig. 5. Specifictions of 2D problem. 16

17 () Converged sclr flux Flux Reltive Error 1.0E E E E E-08 SI FM DG-DSA CM DG-DSA 1.0E Trnsport Sweep # (b) Flux Reltive Error vs. Itertion number Fig. 6. Numericl results of locl h dpttion. The comprison of computing time is summrized in TABLE I for the bove problem. It shows tht DG-DSA is very effective when solved on consistent fine mesh, but computtionlly inefficient becuse the discretized liner diffusion system is very lrge with the number of degrees of freedom of (only 756 for the corse- 17

18 mesh solution). This cse demonstrtes tht the DG-DSA method cn effectively nd efficiently ccelerte the trnsport itertion by using corse mesh grid. In ddition, s compred with continuous Glerkin (CG) methods, n dvntge of the DG methods is tht they cn be discretized on nonconforming mesh which enbles flexible implementtion of locl hp dpttion, while the CG implementtion on the noncomforming mesh is much more involving nd complicted. It should be noted tht in this study the liner system of DG-DSA, i.e., Eq. (18), ws solved simply using the MATLAB built-in bckslsh function. TABLE I. Computtionl Performnce Comprison* SI FM DG-DSA CM DG-DSA Number of degrees of freedom Globl mtrix size b Trnsport itertion number Trnsport time (s) Globl mtrix ssembly time (s) c DG-DSA time (s) Totl clcultion time (s) Speedup *Computtionl results were obtined with MATLAB R2017 on McBook Pro with Processor 2.9 GHz Intel Core i7. SI: totl number of ngulr flux unknowns; DG-DSA: totl number of polynomil coefficients. b Mtrix A of Eq. (18b). c Computing time for clculting the entries of the globl mtrix. III.D. Penlty Number Our DG-DSA method is bsed on the primry DG pproch which employs interior penlty to stbilize the numericl solution. It is found in our model problems tht the 18

19 ccelertion performnce of DG-DSA is sensitive to the penlty number, σ Y, lthough it is numericlly stble for wide rnge of penlty. To chieve the optiml convergence performnce, the suggested penlty number for SIPG s function of totl cross section is depicted in Fig. 7. It shows tht for smll cross section (< 0.8 cm > ), the convergence of DG-DSA is not much sensitive to penlty, for which wide rnge of vlues cn be chosen. For 0.8 < Σ 7 < 1 cm >, the penlty cn be determined bsed on liner interpoltion between 1 nd 0.3. For Σ 7 > 1 cm >, constnt penlty of 0.3 cn be used. Nevertheless, for different implementtion it is suggested tht similr sensitive study be conducted to find the optiml rnge of the penlty number se St Fig. 7. Penlty number vs. totl cross section. IV. CONCLUSIONS In this pper, we hve presented our newly developed diffusion synthetic ccelertion method, DG-DSA, for speeding up the convergence of neutron trnsport clcultions. This new DSA method cn gretly improve the computtionl efficiency of the convectionl DSA methods by using the DG methods. The novelty of DG-DSA is 19

20 tht it reduces the number of degrees of freedom by discretizing the diffusion eqution on corse mesh with locl hp dption. Our numericl results hve demonstrted its rpid convergence performnce nd efficiency. In ddition, it is worth mentioning tht for LWR pplictions DG-DSA cn further employ mesh dpttion for different neutron energy group, i.e., the mesh for fst group clcultions cn be corser thn tht for therml groups (e.g., pin or qurter ssembly size) since the totl cross sections of wter nd fuel in fst groups re smller thn therml group cross sections. Finlly, the ppliction of DG-DSA for other unstructured mesh nd trnsport solvers will be studied in the future. ACKNOWLEDGEMENTS This reserch ws supported by the Deprtment of Energy Nucler Energy University Progrm(NEUP). REFERENCES 1. R. E. ALCOUFFE, Diffusion Synthetic Accelertion Methods for the Dimond- Differenced Discrete-Ordintes Equtions, Nucl. Sci. Eng., 64, 344 (1977). 2. E. W. LARSEN, Unconditionlly Stble Diffusion- Synthetic Accelertion Methods for the Slb Geometry Discrete Ordintes Equtions. Prt I: Theory, Nucl. Sci. Eng., 82, 47 (1982). 3. M. L. ADAMS nd E. W. LARSEN., Fst Itertive Methods for Discrete Ordintes Prticle Trnsport Clcultions, Prog. Nucl. Energy., 40, 3, (2002). 4. J. S. WARSA, T. A. WAREING, nd J. E. MOREL, Krylov Itertive Methods nd 20

21 the Degrded Effectiveness of Diffusion Synthetic Accelertion for Multidimensionl S N Clcultions in Problems with Mteril Discontinuities, Nucl. Sci. Eng., 147, 218 (2004). 5. M. L. ADAMS nd W. R. MARTIN, Diffusion Synthetic Accelertion of Discontinuous Finite Element Trnsport Itertions, Nucl. Sci. Eng., 111, 145 (1992). 6. B. Riviere, Discontinuous Glerkin Methods for Solving Elliptic nd Prbolic Equtions: Theory nd Implementtion, Society for Industril nd Applied Mthemtics, (2008). 21