Comparison of multimes p-fem to interpolation and projection metods for spatial coupling of reactor termal and neutron diffusion calculations Lenka Dubcova a, Pavel Solin a,b,, Glen Hansen c, HyeongKae Park c a Institute of Termomecanics, Dolejskova 5, Prague, CZ 18200. b Department of Matematics and Statistics, University of Nevada - Reno, 1664 N. Virginia St., Reno, NV 89557-0208. c Multipysics Metods Group, Idao National Laboratory, P.O. Box 1625, Idao Falls, ID 83415-3840. Abstract Multipysics solution callenges are legion witin te field of nuclear reactor design and analysis. One major issue concerns te coupling between eat and neutron flow (neutronics) witin te reactor assembly. Tese penomena are usually very tigtly interdependent, as large amounts of eat are quickly produced wit an increase in fission events witin te fuel, wic raises te temperature tat affects te neutron cross section of te fuel. Furtermore, tere typically is a large diversity of time and spatial scales between matematical models of eat and neutronics. Indeed, te different spatial resolution requirements often lead to te use of very different meses for te two penomena. As te equations are coupled, one must take care in excanging solution data between tem, or significant error can be introduced into te coupled problem. We propose a novel approac to te discretization of te coupled problem on different meses based on an adaptive multimes iger-order finite element metod (p-fem), and compare it to popular interpolation and projection metods. We sow tat te multimes p-fem metod is significantly more accurate tan te interpolation and projection approaces considered in tis study. Keywords: reactor multipysics simulation, coupled diffusion, multipysics error 2000 MSC: 65Z05, 65N30, 65N22, 80M10 1. Introduction Nuclear reactor modeling and simulation is filled wit multipysics modeling callenges. Inside of a ligt water reactor, coolant flows troug fuel bundles to remove eat tat is used to generate electric power. Tis coolant flow can cause vibration and wear of te fuel rods wic reduce teir life. Furter, te coolant moderates te speed of neutrons in te reactor, wic affects te fission efficiency witin te fuel, wic canges te amount of eat produced tat needs to be removed by te coolant. As suc, calculations tat describe te flow of neutrons (neutronics), coolant (fluid mecanics), and eat are coupled witin te reactor. Indeed, te coupling may be tigt, wic requires tat care be taken in te solution of te equation systems tat describe te penomena. Furter, one must be cognizant of numerical error tat may be introduced into te solution of te coupled system, especially if a large number of time steps are needed for te solution. Generally, eac of te penomena to be modeled inside of te reactor ave different spatial discretization requirements [7]. For example, te coolant mes requires a refined anisotropic mes near te fuel to resolve te turbulent boundary layer to correctly compute te eat flow from te fuel into te coolant, as seen in te gas-cooled reactor sown in Fig. 1. Likewise, te neutronics equations require resolution to capture effects at material interfaces and discontinuities. If a single mes spanning te entire domain and all pysics penomena were used, it would need to be fine enoug at every location to meet te discretization needs of Corresponding autor Email addresses: dubcova@centrum.cz (Lenka Dubcova), solin@unr.edu (Pavel Solin), Glen.Hansen@inl.gov (Glen Hansen), Ryosuke.Park@inl.gov (HyeongKae Park) Preprint submitted to Journal of Computational Pysics April 7, 2010
te most demanding pysics at tat location. Furter, as te lengt and time scales of te various pysics components may vary greatly during a transient simulation, tis mes would only be appropriate for a certain period in time. Tis leads one to conclude tat to capture individual beavior of te solution components more efficiently, different pysics sould be discretized on teir individual meses, and an adaptive metod sould be used to refine eac one based on te needs of te pysics it supports. Grapite Moderator Fuel Coolant Figure 1: Triangular mes generated on a cross-section of a gas-cooled reactor core. Te gas coolant flows between te sections of te grapite moderator tat contain te reactor fuel. Note tat graded anisotropic mes is used were te coolant meets te surface of te moderator. Substantial refinement is needed in tis area as te correct resolution of te coolant flow in te boundary layer is crucial for te accurate calculation of bot te eat flux and forces between te coolant and te moderator. Traditionally, reactor multipysics problems ave been analyzed by dividing tem into several distinct processes, one for eac pysical penomena of interest, and addressing eac penomena independently using existing monopysics codes. In reality, eac component of te solution is often strongly coupled to te oters, necessitating interaction between eac of te models (and codes). Traditional coupling paradigms rely on solving te different pysics in a loosely coupled fasion, a tecnique matematically described as operator splitting (OS). Tis is often implemented in a simple black-box fasion, were te output of one code serves as input of anoter code, using various metods of data transfer between te non-matcing meses. Iteration is used between te codes to converge te overall problem. Tis approac, wile attractive because it employs existing codes and user experience, may exibit accuracy and stability issues. Te different codes employed typically use different meses tat necessitate a data excange at eac iteration. Tese data excanges may be quite callenging to perform accurately, as it is often necessary to conduct an expensive geometric 2
intersection of te meses to ost te solution of one component on te oter s mes, and vice versa [12]. For tree-dimensional problems on complex reactor geometry, te generation of individual meses can be very time consuming and expensive [7]. Secondly, it is often callenging to perform te data transfer so tat conservation of important quantities (mass, momentum, energy) of te solution is acieved, originating from te fact tat coupling terms between te various pysics components are dealt wit in an inconsistent fasion. Tese issues are discussed in some detail in te study by Jiao and Heat [8]. Tis paper proposes a multimes p-fem metod tat employs an assembly process to discretize multipysics coupled problems in a monolitic way witout employing operator splitting or data transfer between meses. Eac solution component is approximated on a different mes tat is obtained adaptively to best fit te specific pysics requirements. Te multimes p-fem as been previously employed to coupled termomecanical models [6], coupled moisture and eat transfer models [14], and coupled electromagnetics-termal models [15]. Tis study explores te use of adaptive multimes p-fem for computational nuclear engineering problems. It begins wit Section 2 tat describes te algoritm for automatic adaptation of p-meses wit arbitrary-level anging nodes and demonstrates advantages of p-fem over low-order FEM using an IAEA bencmark problem and an axisymmetric problem involving te multigroup neutron diffusion equation. Section 3 describes te multimes discretization tecnology and compares it to selected interpolation and projection data transfer metods. Conclusions are formulated in Section 6. 2. Te adaptive iger-order finite element metod (p-fem) Te p-fem is a modern version of te finite element metod (FEM) tat attains very fast (exponential) convergence rates by combining optimally finite elements of variable size () and polynomial degree (p) [16]. Te main principles of exponential convergence are tat (a) very smoot, polynomial-like functions are approximated using large ig-order elements and (b) non-analytic (non-polynomial-like) functions suc as singularities are approximated via small low-order ones. Te superiority of te p-fem over standard (low-order) FEM as been demonstrated by many independent researcers [1, 2, 3, 10, 9, 16]. Altoug te implementation of te p-fem is involved, te metod is becoming increasingly popular among practitioners. Te p-fem differs from standard FEM in a number of aspects, and in particular te complexity of spatial adaptation due to te large number of ways an element can be refined. One can increase te polynomial degree of an element witout subdividing it in space or one can split te element in space and distribute te polynomial degrees in te subelements in multiple ways. Te number of refinement candidates considered ere is approximately 100 in 2D and several undred in 3D. An example for cubic triangles and cubic quadrilaterals is sown in Fig. 2. Figure 2: Examples of different p refinement options for a triangular and quadrilateral element. Note tat te ability to refine elements using subdivision and elevate te order of te element leads to many different possibilities. Te number sown at te center of te element is te polynomial degree of te element. In low-order FEM, te number of refinement candidates is small and tus it is sufficient to guide adaptation using element-wise constant error estimators. However, suc error estimators do not provide enoug 3
information to guide p-adaptation. For tis, one needs to ave muc better information about te sape of te error function e,p = u u,p in addition to its magnitude. In principle, it is possible to obtain tis information from estimates of iger derivatives of te solution, but tis approac is not usually practical. One way to circumvent tis problem is to employ a reference solution; an approximation u ref tat is at least one order more accurate tan te coarse solution u,p. Te p-adaptation algoritm is ten guided by an a-posteriori error estimate of te form e,p = u ref u,p. More details on automatic p-adaptation on meses wit arbitrary level anging nodes may be found in [13]. 2.1. Example: IAEA bencmark EIR-2 Tis example was cosen due to te excellent work on tis problem by Wang and Ragusa ([17], p. 46). Tis paper poses a multiple mes p-fem metod and demonstrates it on several examples of multigroup diffusion in one dimension, using a modified version of te 1Dp90 code [4]. Tey ten present a single mes, single group neutron diffusion solution of EIR-2, te Sapir bencmark. Tis classical nuclear engineering bencmark [11] describing an external-force-driven configuration witout fissile materials present will be used to compare te performance of adaptive p-fem and adaptive -FEM wit linear and quadratic quadrilateral elements 1. Te domain is a 96 86 cm rectangle comprising five regions as illustrated in Fig. 3. Solved is te one-group neutron diffusion equation (D(x, y) φ) + Σ a (x, y)φ = Q ext (x, y) for te unknown neutron flux φ(x, y). Zero flux boundary conditions are assumed on te entire outer boundary. Te values of te diffusion coefficient D(x, y), absorption cross-section Σ a (x, y) and te source term Q ext (x, y) are constant in te subdomains. Te total cross-section Σ t is 0.60 in Ω 1, 0.48 in Ω 2, 0.70 in Ω 3, 0.85 in Ω 4, and 0.90 cm 1 in Ω 5. Te scattering cross-section Σ s is 0.53 in Ω 1, 0.20 in Ω 2, 0.66 in Ω 3, 0.50 in Ω 4, and 0.89 cm 1 in Ω 5. Te absorption cross-section Σ a = Σ t Σ s. Te diffusion coefficient D(x, y) is calculated from Σ t using te standard relation D = 1/(3Σ t ). Te source Q ext = 1 cm 3 t 1 in areas Ω 1 and Ω 3 and zero elsewere. Figure 3: Geometry of IAEA bencmark EIR-2. 2.1.1. Results Te results presented below were obtained using Hermes2D 2, an open-source modular C++ library for solving various types of partial differential equations (PDE) as well as multipysics PDE systems using adaptive iger-order finite element metods (p-fem) and Discontinuous Galerkin metods (p-dg). It provides various capabilities for automatic adaptation of p-meses, including dynamical meses for timedependent problems, and supports te solution of coupled multipysics problems in a monolitic way witout operator splitting. Te solution φ(x, y) is sown in Fig. 4. Figures 5 and 6 sow te resulting meses (adaptive -FEM wit linear and quadratic elements and adaptive p-fem, respectively). 1 Note: it would be more correct to call tese elements bilinear and biquadratic, respectively, but we will stay wit te sorter notation. 2 ttp://pfem.org/ermes2d/ 4
Figure 4: Solution Φ(x, y). Figure 5: Resulting meses for adaptive -FEM wit linear and quadratic elements. In Fig. 6, colors are used to distinguis between different polynomial degrees of elements. In te p-fem, edges and element interiors can ave different polynomial degrees. Polynomial degrees of edges are depicted using tin color belts along tem. If te interior of an element is split diagonally and contains two colors, different directional (anisotropic) polynomial degrees are used. Figure 6: Resulting mes for adaptive p-fem. 5
A comparison of te corresponding convergence rates in te H 1 -norm, bot in terms of te discrete problem size (degrees of freedom, DOF) and CPU time, is sown in Figs. 7 and 8. Figure 7: Convergence comparison in terms of discrete problem size (DOF) and CPU time. In tese results, adaptive p-fem is superior to bot adaptive -FEM wit linear elements and adaptive -FEM wit quadratic elements. One may also see tat te teoretically-predicted asymptotic slope 1/2 for -FEM wit linear elements as well as te slope 1 for -FEM wit quadratic elements is attained wit approximately 5000 degrees of freedom. Tis section closes by comparing te performance of te adaptive p-fem algoritm in Hermes2D wit te performance of te p-fem algoritm presented by Wang and Ragusa for te same bencmark problem (see Fig. 25 in [17]). Te code used by Wang and Ragusa is based on an adaptive p-fem code developed by Demkowicz [5]. Bot codes use te same type of reference solution and measure te error in te same norm. Figure 8: Convergence comparison of adaptive -FEM wit linear elements (left) and adaptive p-fem (rigt) performed using Hermes2D in comparison wit te results of Wang and Ragusa [17]. Figure 8 sows tat Hermes2D provides substantially better convergence for bot -FEM (wit linear elements) and p-fem. To explore tis result, consider te p-fem results. Using Wang s notation, te teoretical convergence rate of p-fem is proportional to exp(γn 1/3 ), were γ < 0. Note tat te slope of 6
te data is very similar, but te constant of proportionality is smaller wit Hermes2D, resulting in a constant advantage of requiring rougly alf te DOFs for te same level of accuracy. Te advantage is amplified wen te results for -FEM are considered. Te better performance of Hermes2D may be related to te fact tat it employs a different adaptivity algoritm, bot spatially and polynomially anisotropic p-refinements, and arbitrary-level anging nodes. 3. Multimes p-fem Many of te pysical penomena occurring in nuclear reactors are strongly coupled. Te lengt scales of te various pysics components typically vary significantly in time and space, and te discretization requirements of te components are often quite different. To address tese requirements, different meses are typically used to ost te solution of te different components, and tese meses may evolve independently over time. Tis study proposes a novel monolitic approac tat preserves te coupling structure of multipysics problems from te continuous to te discrete level. Te metod is based on an adaptive multimes p-fem discretization tecnique [14, 6, 15], wic preserves te consistent discretization of te problem even as components of te solution are discretized on different meses. In standard p-fem, te domain Ω is covered wit a finite element mes T consisting of non-overlapping convex elements K 1, K 2,..., K M (typically triangles or quadrilatera) osting polynomial bases 1 p 1, p 2,..., p M. In general, te N solution components exibit different beavior requiring tat tey be approximated on different meses T n = {K1 n, K2 n,..., KM n }, n = 1,..., N. Te elements in tese meses possess nonuniform polynomial degrees p(ki n) = pn i. In order to limit te algoritm to a reasonable level of complexity, tis approac establises a geometric relationsip between te meses. Tis relationsip begins wit te definition of a coarse master mes T m, tat serves as te common basis of refinement for te coupled solution. Given tis basis T m, a set of refinement operations are applied to develop an appropriate mes T n for te n t solution component of interest. Te refinement procedure applied for eac T n may be mutually independent for eac component. Use of te master mes as a common basis of refinement significantly simplifies te approac, as it eliminates te need to perform geometric intersection operations to relate te solution between meses. Furter, te ierarcical relationsip between te discrete spaces provides an effective mecanism for te representation and assembly of te finite element problem. In tis case, te stiffness matrix is assembled on a virtual union mes T u, wic is te geometrical union of te meses T n. To visualize tis union of mes representations, imagine printing te meses on transparencies and overlaying tem one above te oter, as illustrated in Figure 3). Figure 9: Diagram A) sows te master mes T m tat is te common basis of refinement of te multiple multipysics meses. B) troug D) are meses T 1, T 2, T 3 obtained by refinement of te master mes to meet te requirements of pysics packages osted upon tem. Te multipysics problem is actually assembled on te union mes T u, tat is te intersection of B) troug D). Te union mes T u is not explicitly constructed; its virtual elements are traversed by te element-byelement assembling procedure employed in standard p-fem [16]. Tis approac does not incur interpolation or projection error, as information is not actually transferred between te meses; eac element on eac mes is used to integrate te solution tat te element osts. In oter words, all integrals in te weak formulation of eac coupled problem are evaluated exactly (up to te error in numerical quadrature). 7
4. Comparison to interpolation- and projection-based intermes data transfer metods In tis section we use a problem wit known exact solution to compare te multimes discretization approac wit two standard tecniques to transfer solution data between different meses: linear interpolation of vertex values and L 2 projection. We will solve a eat transfer equation wit a spatially-dependent termal conductivity k = k(x, y), (k u) = f in Ω = (0.01, 1.0) 2, (1) u = u D on Ω. Te weak problem is to find u H 1 (Ω) satisfying u = u D on Ω suc tat k u v dxdy = f v dxdy, for all v H0 1 (Ω). (2) Wit Ω ( ) y k(x, y) = x 3 + y 3 3, f(x, y) = 2 2 x 3 + x3 y 3 te exact solution is given by Ω u(x, y) = 1 x + 1 y. and u D (x, y) = 1 x + 1 y, For comparison purposes, two different meses T A and T B were generated as follows: A piecewise-linear mes T A is obtained by solving equation (2) using adaptive -FEM. Tis mes contains 189 DOF and it is virtually optimal to represent te exact solution u (see left part of Fig. 10). Anoter mes T B is a uniform piecewise-linear 256 256 subdivision of te computational domain containing 65025 DOF, as sown in te rigt part of Fig. 10. Te termal conductivity k(x, y) is represented on te mes T B via its continuous, piecewise-linear vertex interpolant k (x, y). Figure 10: Left: non-uniform piecewise-linear mes T A wit 189 DOF were equation (2) is solved. Rigt: uniform piecewiselinear mes T B wit 65025 DOF were te termal conductivity k(x, y) is represented via its continuous, piecewise-linear vertex interpolant k (x, y). In te following we solve equation (2) in four different ways and compare te results to te exact solution u: 8
(1) Transfering k (x, y) from T B to T A via interpolation and solving on T A Let us begin wit linear interpolation of vertex values, wic probably is te most widely used metod for transferring solution data between different meses. Te piecewise-linear function k (x, y) defined on te mes T B is approximated on te mes T A by means of a continuous function k (1) (x, y) tat matces k (x, y) exactly at all grid vertices of te mes T A and is bilinear in eac element of T A. In oter words, k (1) = i c iv i (x, y) were v i V A are te vertex basis functions on te mes T A, and te coefficients c i are te values of te function k(x, y) at te corresponding grid points of T A. (For igerorder ierarcic finite elements, te natural extension of linear interpolation is te so-called projection-based interpolation [16].) Te corresponding approximation u (1), computed on te mes T A using linear finite elements and te, is sown in part (a) of Fig. 11. Te approximation error e(1) (x, y) = (x, y) is sown in part (a) of Fig. 12. approximate termal conductivity k (1) u(x, y) u (1) (2) Transfering k (x, y) from T B to T A via L 2 -projection and solving on T A Let us denote by V A te finite element subspace of H1 (Ω) on te piecewise-linear mes T A. Te projection problem k (2) k L 2 0 can be rewritten in weak sense as (k (2) k, v i ) L 2 = 0 or for all basis functions v i of te space V A (k (2), v i) L 2 = (k, v i ) L 2 (3). Te unknown projection k(2) k (2) N = y j v j j=1 is written as a linear combination were y j are unknown coefficients. Substituting tis sum into te variational identity (3), one obtains a system of linear algebraic equations. Te mass matrix integrals of te form Ω v j v i dxdy are computed over elements of te mes T A. Following te common refinement algoritm proposed by [8], te rigt-and side integrals Ω k v i dxdy are computed using elements of te geometrical union of te meses T A and T B (tat is T B in our case). Te corresponding piecewise-linear approximation u (2) on te mes T A is sown in part (b) of Fig. 11. Te approximation error e (2) (x, y) = u(x, y) u(2) (x, y) is sown in part (b) of Fig. 12. (3) Using te multimes discretization metod (no data transfer between meses) Next we compute a piecewise-linear approximation u (3) using te multimes discretization metod tat was described in Section 3. Note tat in tis case, te union mes of te meses T A and T B is T B. Te approximation is sown in part (c) of of Fig. 11, and te approximation error e (3) (x, y) = u(x, y) u(3) (x, y) is sown in part (c) of Fig. 12. (4) Solving on T A wit te exact termal conductivity k(x, y) For comparison purposes we also compute a piecewise-linear approximation u (4) on te mes T A using te original function k(x, y). Tis is te best finite element approximation one can obtain on te mes T A. Te result is sown in part (d) of Fig. 11 and te corresponding approximation error e (4) (x, y) = u(x, y) u(4) (x, y) is sown in part (d) of Fig. 12. Te results of te four computations are summarized in Table 1 wic sows H 1 -norms of te approximation errors on te mes T A calculated wit respect to te exact solution u. Approaces (1) and (2) resulted in a large error in te solution. In particular, notice te extreme error in case (2). As discussed in [8], one must typically use a iger order integration on te target mes. Tus, a second calculation was performed using quadratic elements on T A to yield te tird row of Table 1. Note tat te multimes discretization provides sligtly better results tan te approximation calculated using te exact data k(x, y) on te mes T A. Tis is due to te fact tat te multimes metod integrates over elements of te union mes (in tis case te mes T B ) wic is muc finer tan te mes T A used for assembling in case (4). 9
Table 1: Comparison of relative H 1 -norm errors for te four metods described above. Metod Relative H 1 error Interpolation of k (1) on mes T A 17.766% L 2 -projection of k (1) on mes T A (lin.) 31.808% L 2 -projection of k (1) on mes T A (quadr.) 6.734% Multimes discretization metod 6.727% Using exact data k(x, y) on mes T A 6.729% (a) Approximation u (1) k (1) on te mes T A obtained using te interpolant (b) Approximation u (2) k (2) on te mes T A obtained using te projection (c) Approximation u (3) discretization metod obtained using te multimes (d) Approximation u (4) obtained using te exact function k(x, y) on te mes T A Figure 11: Finite element solutions to (2). Note tat te two metods tat transfer te termal conductivity between te meses T B and T A (a and b) possess significant error close to te origin, wile te multimes discretization yields results tat are quite close to te exact solution. 10
(a) Error e (1) due to vertex interpolation of k (x, y) on (b) Error e (2) te mes T A mes T A due to L 2 -projection of k (x, y) on te obtained using te multimes discretiza- (c) Error e (3) tion metod (d) Error e (4) obtained using k(x, y) on te mes T A Figure 12: Approximation errors on te mes T A calculated wit respect to te exact solution u. Note tat te scale of te error magnitude is significantly different between te plots. 5. Automatic adaptivity in multimes p-fem Automatic adaptivity in multimes p-fem [6] is muc more involved tan standard adaptive p-fem [5, 16] were only one mes is used. In te multimes case, one needs to define a global error norm tat includes error contributions from all meses considered as a group, and refine te meses for eac pysical model to minimize tat global error norm. As discussed in [17], Fig. 4, one may take eac pysical model in turn and adapt it (using te standard p-fem approac) until te error for tat model is below some tolerance. At tis point, one moves to te next model and repeats te process. Unfortunately, refinement on eac mes may effect te refinement needed on te oters at particular spatial locations. Figure 5 in [17] presents a second algoritm tat selects elements to be refined based on te maximum error across all models and across all elements. Wile tis approac requires tat te error magnitudes be normalized across 11
all pysical models, it is employed for tese results. Tus, in te multimes p-fem presented ere, every element competes directly wit all elements on all meses in te system. 5.1. Example: Multigroup neutron diffusion Tis section compares te performance of adaptive p-fem and -FEM by solving a multigroup neutron diffusion problem on a two dimensional multimes. Te reactor core sown in Fig. 13 is modeled in two dimensions using cylindrical coordinates. Te computational domain is divided into 5 zones; center, outer, upper, and lower reflector regions, and a fissile core. Detailed geometry witin eac zone is omogenized into te zone for simplification. Figure 13: Diagram of te simplified prismatic VHTR core model. Te reactor neutronics is given by te following eigenproblem, Σ g g s φ g = χ g k eff wit boundary conditions D g φ g + Σ Rg φ g g g g ν g Σ fg φ g, g = 1,..., 4, (4) φ g n = 0 on Γ SYM, and D g φ g n = 1 2 φ g on Γ V. (5) Here, φ g is te neutron flux, D g is te diffusion coefficient, Σ Rg = Σ ag +Σ g g+1 s is te removal cross section, Σ fg is te fission cross section, Σ S is te scattering cross section, ν is te average number of neutrons born per fission event, and te subscript g denotes te group index in te multigroup neutron diffusion equation. Lastly, k eff is te eigenvalue. Tis eigenproblem is numerically solved using te power metod (or power iteration), as outlined by te following algoritm: 1. Make an initial estimate of φ (0) g and k (0) 12
2. For n = 1, 2,... (a) solve for φ (n) g using D g φ (n) g + Σ Rg φ (n) g g g Σ g g s φ (n) g = χ g k (n 1) g ν g Σ fg φ (n 1) g (6) (b) solve for k (n) using 3. Finally, stop iterating wen k (n) = k (n 1) 4 Ω C g=1 4 Ω C g=1 ν g Σ fg φ (n) g ν g Σ fg φ (n 1) g k (n) k (n 1) < ɛ (8) k (n) (7) 5.1.1. Results Te calculations tat follow are using te material properties sown in Tab. 2: Table 2: Material properties for te geometry sown in Fig. 13. Active core properties 1 2 3 4 D g( m) 0.0235 0.00121 0.0119 0.0116 Σ Rg ( m 1 ) 0.00977 0.162 0.156 0.535 Σ fg ( m 1 ) 0.00395 0.0262 0.0718 0.346 ν g 2.49 2.43 2.42 2.42 χ g 0.9675 0.03250 0.0 0.0 Σ 1 2 s ( m 1 ) 1.23 0 0 0 Σ 2 3 s ( m 1 ) 0 0.367 0 0 Σ 3 4 s ( m 1 ) 0 0 2.28 0 Reflector properties 1 2 3 4 D g( m) 0.0164 0.0085 0.00832 0.00821 Σ Rg ( m 1 ) 0.00139 0.000218 0.00197 0.0106 Σ fg ( m 1 ) 0 0 0 0 ν g 0 0 0 0 χ g 0 0 0 0 Σ 1 2 s ( m 1 ) 1.77 0 0 0 Σ 2 3 s ( m 1 ) 0 0.533 0 0 Σ 3 4 s ( m 1 ) 0 0 3.31 0 Te problem was solved using bot p-adaptation and standard -adaptation on low-order elements (wit bilinear and biquadratic quadrilateral elements). Te relative error between te coarse mes solution φ g and te reference solution φ g was calculated using φ g φ g 2 H 1 E 2 rel = g φ. g 2 H 1 g Figure 14 sows te neutron flux for eac of te four energy groups (eigenvectors), were flux intensity is sown in color (blue is te lowest flux and red is te igest). Te eigenvalue in tis problem was k eff = 1.14077. Using p adaptation, a relative error of 0.0164% was acieved using 27903 degrees of freedom (DOF). Te refined results provided a relative error of 3.87% using 30112 DOF. Figures 15 and 16 sow four optimal p-meses and corresponding -meses, respectively. Figure 15 sows p adapted solutions to eac group of te four group problem. It is clear from tis figure tat te coupled problem is being solved wit a mes of a different refinement and elements of different order in eac group. Note tat te solutions for groups 1 3 sown in Fig. 14 are similar, as are teir corresponding p meses. However, wit detailed examination, bot te solutions and meses differ between groups. Note tat bot te solution and mes for group 4 differs significantly from te oters. 13
(a) Group 1 (b) Group 2 (c) Group 3 (d) Group 4 Figure 14: Neutron flux for te four group neutron diffusion calculation on te VHTR geometry. Color indicates te intensity of te flux in eac group. (a) Group 1 (b) Group 2 (c) Group 3 (d) Group 4 Figure 15: p-meses for tat correspond to te above neutron flux results. Colors ave te same meaning as in Fig. 6. Note tat different levels of subdivision and different polynomial orders are used for eac energy group. Tis solution contains 27903 DOF, and as a relative error of 0.0164%. Figure 17 compares convergence curves on tis problem using p-fem and standard FEM using bot bilinear and biquadratic elements. It is apparent tat in regions were te solutions are smoot, te algoritm tends to use larger elements wit iger polynomial degree. In areas near singularities or geometric corners, 14
(a) Group 1 (b) Group 2 (c) Group 3 (d) Group 4 Figure 16: -meses (linear elements, p = 1) for te four group neutron flux problem. refinement. Solution contains 30112 DOF, and as a relative error of 3.87%. Again, eac group uses a different Figure 17: Comparison of te spatial convergence of p- vs. -adaptation on te multigroup multimes neutron diffusion problem sown in Fig. 13. Tree curves are sown, results wit bot linear (p = 1) and quadratic (p = 2) polynomial order, and p. Note te significant accuracy advantage of te p metod for a given number of DOFs. 15
smaller elements wit lower polynomial degree yield te best results. Tis is in good correspondence wit te teory of p-fem. On te oter and, -adaptation can refine only spatially, wic results in more degrees of freedom and a larger relative error as illustrated in Fig. 17. Figure 18 sows two solutions of te multigroup problem, te first using a single mes for eac of te pysics models, and te second using te multimes approac to solve eac group on its own mes (Figs. 14 and 15). It is apparent tat more degrees of freedom are needed to solve te four group problem using a single mes, to obtain comparable accuracy. On tis problem, te savings in total DOF obtained by using a multiple mes strategy is not striking. Future work includes more study of te error metric used to adapt eac mes in turn, as a more discriminating metric may yield a greater performance difference wen compared to a single mes approac. Secondly, wile tere is some diversity in te spatial resolution requirements of te pysics solutions osted on eac mes, it is fairly small across te groups on tis multigroup problem. A muc greater diversity is expected between multigroup neutronics and models for fluid (coolant) flow, especially in areas of te turbulent boundary layer of te fluids calculation. Figure 18: Comparison of p-fem spatial convergence for solving eac group on a separate mes using te multimes approac, vs. using te same mes for all te groups. 6. Conclusion Tis paper began by introducing some of te multipysics callenges witin te field of computational nuclear engineering and motivating te need for integration metods tat andle a wide diversity of spatial resolution needs across multipysics models. Te p-fem metod was introduced, and solutions were presented tat compared te metod vs. -FEM on te IAEA bencmark EIR-2. Te study ten considered te use of p-fem as a metodology to spatially integrate and couple multipysics problems using multiple meses. Here, a ierarcy of meses were used to define te metod to 16
simplify te implementation and to eliminate te callenges of performing geometric intersection operations between topologically-distinct meses. Te proposed approac was compared to commonly used tecniques for data transfers between meses, on a problem wit a known solution. It was sown tat by using te proposed multimes assembly process no additional error arises due to te use of distinct meses in te problem considered. Te final portion of tis study explored te adaptive, iger order finite element metod (p-fem) and potential advantages of te approac over typical low-order FEM for te four group neutron diffusion equations. In tis case, a multiple mes problem was developed tat osted te solution for eac group on a independent (but ierarcically related) mes. An example problem was considered tat exibited an error of 3.87% using 30112 DOF, employing adaptation and linear elements, but using a different mes for eac group. Te second example used te proposed p metod, and produced an error of 0.0164% using 27903 DOF, again wit a separate mes per group. Tese results may be interpreted two ways; for a given level of accuracy te p metod will require considerable fewer DOF, or for a similar number of DOF, te p metod will provide over 200 times te accuracy (on tis problem). Finally, a multimes vs. single mes calculation was compared. Here, te reduction in te number of DOFs needed for te same level was limited, likely in part due to te closeness of te multigroup solutions on eac mes. Future work includes study of tis approac on problems wit greater diversity suc as Navier-Stokes coupled to neutronics, tree dimensional problems, and more sopisticated error indicators and refinement strategies to refine eac individual mes separately. Acknowledgments Te submitted manuscript as been autored by a contractor of te U.S. Government under Contract No. DEAC07-05ID14517 (INL/JOU-10-18340). Accordingly, te U.S. Government retains a non-exclusive, royalty-free license to publis or reproduce te publised form of tis contribution, or allow oters to do so, for U.S. Government purposes. Te autors also acknowledge te financial support of te DOE NEUP program under BEA Contract No. 00089911. Tis work was also supported financially by te Grant Agency of te Academy of Sciences of te Czec Republic under Grant No. IAA100760702. Te autors wis to tank Yaqi Wang for saring te convergence data from is paper [17]. References [1] I. Babuska and B. Q. Guo. Te, p, and -p version of te finite element metod; basis teory and applications. Advances in Engineering Software, 15:159 174, 1992. [2] I. Babuška and M. Suri. Te p and p versions of te finite element metod, basic principles and properties. SIAM Rev., 36:578 632, 1994. [3] L. Demkowicz. Computing wit p-adaptive Finite Elements, Volume 1: One and Two Dimensional Elliptic and Maxwell Problems. Capman & Hall/CRC, Boca Raton, FL, 2006. [4] L. Demkowicz and C.-W. Kim. 1D p-adaptive finite element package, Fortran-90 implementation (1Dp90). Tecnical Report TICAM Report 99-38, Texas Institute of Computational and Applied Matematics, University of Texas, Austin, 1999. [5] L. Demkowicz, W. Racowicz, and P. Devloo. A fully automatic p-adaptivity. J. Sci. Comput., 17(1 3):127, 2002. [6] L. Dubcova, P. Solin, J. Cerveny, and P. Kus. Space and time adaptive two-mes p-fem for transient microwave eating problems. Electromagnetics, 2010. In press. [7] Glen Hansen and Steve Owen. Mes generation tecnology for nuclear reactor simulation; barriers and opportunities. Nucl. Engrg. Design, 238(10):2590 2605, 2008. 17
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