A two-dimensional modal method for spatial rehomogenization of nodal cross sections and discontinuity-factor correction

Article history: Received 12 August 2018 Received in revised form 29 September 2018 Accepted 3 October 2018


Introduction
The accuracy of two-group, coarse-mesh nodal diffusion calculations is hindered by spatial homogenization and energy condensation errors in the infinite-medium cross sections and discontinuity factors (Smith, 1994;Palmtag and Smith, 1998). This aspect is a major drawback of the conventional two-step procedure, which combines off-line single-assembly heterogeneous transport calculations with the on-line simulation of the assembly-homogenized core representation (Smith, 1986;Sanchez, 2009). In previous work (Gamarino et al., 2018a;Gamarino et al., 2018b), spectral rehomogenization was applied successfully to improve the accuracy of nodal cross sections by computing the variation in the neutron flux spectrum between the core environment and the infinite-lattice approximation. In this paper, we aim to correct the spatial component of singleassembly cross-section defects.
Spatial homogenization errors are significant when highly absorbing elements (such as control rods and burnable-poison rods) are located at the assembly periphery, thus experiencing steep flux gradients, as well as in the presence of strongly hetero-geneous fuel composition (such as in MOX assemblies). Several methods have been proposed to incorporate spatial effects into the nodal cross sections. Spatial heterogeneity due to nonuniform intranodal depletion (i.e., gradients in the burn-up and nuclide concentrations) and fuel temperature is commonly modeled with a separable (along each coordinate axis) quadratic expansion of the nodal cross sections (Wagner et al., 1981;Forslund et al., 2001). A similar approach has been applied to represent design heterogeneity (Shatilla et al., 1996). However, these methods do not correct the spatial homogenization error due to deviations in the flux distribution between the environment and the infinite lattice. One of the first spatial rehomogenization techniques was proposed by Smith (1994). In his approach, the transverse-integrated intranodal cross sections are spatially (re) homogenized at each power iteration or thermal-feedback update with the computed one-dimensional (1-D) heterogeneous flux shape. This shape is obtained via superposition of the heterogeneous infinite-lattice transverse-integrated flux form function and the homogeneous flux distribution from the transverseintegrated nodal calculation. This method uses single-assemblygenerated rehomogenization coefficients and can be easily incorporated into existing codes. However, it does not correct the assembly discontinuity factors. A variant of it was later proposed by Palmtag (1997). In Dall'Osso (2014), the variation in the 1-D transverse-integrated flux between the real environment and the infinite lattice is evaluated with a modal expansion. This approach also corrects the discontinuity factors.
Another method for spatial rehomogenization is the 2-D submesh model that has been implemented in Studsvik's nodal code SIMULATE (Bahadir et al., 2005;Bahadir and Lindahl, 2009). In this strategy, each radial node (i.e., fuel assembly or assembly quarter) is subdivided into N Â N homogeneous rectangular subnodes (typically with N = 5). The 2-D diffusion equation is solved in every subregion with the corresponding infinite-medium homogenization parameters. The axial leakage (known from the global 3-D solution) is converted into an equivalent absorption cross section. The resulting intranodal flux distribution is used to rehomogenize the single-assembly cross sections over the submeshes and to estimate the environmental discontinuity factors at the assembly outer edges. The 2-D submesh calculation is combined with an axial homogenization model to handle axial heterogeneity.
Radial zoning of fuel-assembly homogenization can also be used to reduce spatial errors. For example, EDF's core simulator COCAGNE (Guillo et al., 2017) features a discretization of 2Â2 subregions in each assembly quarter. Non-uniform nodes are used to model separately (i) the four fuel pins in every external corner of the assembly, (ii) the outer and first inboard pin rows (at the assembly periphery), and (iii) the remainder of the fuel bundle. This meshing strategy has been chosen to minimize the level of heterogeneity in each subregion that is homogenized. In spite of their enhanced accuracy, refined-mesh approaches entail a somewhat higher computational effort than conventional nodal methods, which only use one or four nodes per fuel assembly.
Other methods do not specifically focus on spatial effects and try to model the global environmental effect. For example, in Rahnema and Nichita (1997), the corrections on the nodal cross sections and discontinuity factors are tabulated in the standard parameterized libraries versus the current-to-flux ratio (or other albedo parameters) at the node surfaces. These corrections are computed during the lattice calculation via parametric assembly simulations with varying albedo boundary conditions. They are interpolated during the nodal calculation just as the infinitelattice group constants. Recently, a variant of this approach has been investigated (Kim et al., 2017). The drawback of this kind of method is that it demands multiple single-assembly calculations for each lattice state. Rahnema et al. (2002) developed high-order cross-section homogenization, which does not require additional lattice simulations. Using high-order boundary-condition perturbation theory (McKinley and Rahnema, 2000), the environmental homogenization parameters are expanded in terms of the surface current-to-flux ratio. The expansion coefficients are evaluated using the known unperturbed solution of the zero-leakage problem. This method requires two infinite-medium adjoint functions, which are precomputed and stored as additional parameters in the cross-section tables. Clarno and Adams (2005) proposed to capture neighbor effects during the single-assembly calculation via spatial superposition of typical four-assembly configurations. Recently, Groenewald et al. (2017) developed a semi-heterogeneous transport-embedded approach, in which the embedded transport calculations are performed with a simplified handling of spatial heterogeneity, energy discretization and solution operator. The computational burden of transport-nodal iterations is thus mitigated.
In this paper, we propose an extension of the spatial rehomogenization technique described in Dall'Osso (2014). This method aims to compute on the fly the change in the intranodal flux shape that is used for cross-section spatial homogenization when the assembly is in the core environment. The rehomogenization problem is solved locally (namely, independently in each node) after each power iteration, using as boundary conditions the estimates of the volume-averaged fluxes, the surface-averaged fluxes and currents, and the multiplication factor from the global (i.e., corewide) nodal calculation. Only radial heterogeneity is addressed. The effect of axial heterogeneity is taken into account via axial homogenization or control-rod cusping models (Dall'Osso, 2002). Two significant approximations are made in the formulation presented in Dall'Osso (2014): The 2-D rehomogenization problem is simplified via transverse integration into two 1-D problems, which are solved sequentially in the x and y directions. The transverse-integrated fine-mesh cross sections (to be weighted with the computed 1-D flux change) are obtained by collapsing the 2-D pin-by-pin cross-section distributions with the infinite-medium flux form function, instead of the environmental flux shape.
In this work, the above assumptions are relaxed by developing a full 2-D rehomogenization model, which retains the nonseparability of the flux variation in the x and y directions. In this way, the 2-D distributions of the environmental flux and directional net currents can be estimated at the nodal flux iteration level. Moving to a 2-D model is also motivated by the possibility to use the so obtained 2-D information (i) to compute the transverse-leakage distributions for the transverse-integrated nodal equations, and (ii) to reconstruct the pin-by-pin flux and power distributions directly (i.e., without the dehomogenization phase).
This paper is structured as follows. The 2-D rehomogenization method is described in Section 2. Section 3 shows the numerical results of several Pressurized-Water-Reactor (PWR) multiassembly configurations, in which the spatial effects of the environment are important. The accuracy of the method is assessed on both nodal and pin-by-pin quantities. A comparison with the simplified 1-D approach is made. In Section 4, we discuss various aspects of interest of spatial rehomogenization. These include the correction of assembly discontinuity factors, the calculation of the transverseleakage shape with the information from the 2-D rehomogenization, and the contributions of various environmental effects (spatial, spectral, and mixed energy-space) to the deviations of nodal cross sections from the single-assembly values. Concluding remarks follow in Section 5.

Description of the method
We introduce a non-dimensional coordinate u d , where d stands for x or y. The following change of variable is applied: We define the 2-D intranodal distribution of the environmental neutron flux density in the coarse energy group G as where U G is the volume-averaged flux from the nodal calculation, u 1;G u x ; u y À Á is the infinite-medium flux distribution (normalized to unity) used for cross-section spatial homogenization in the lattice calculation, and dU G u x ; u y À Á is the flux spatial variation between the environmental and infinite-medium conditions. The distributions U env;G u x ; u y À Á and u 1;G u x ; u y À Á are heterogeneous quantities, whereas dU G u x ; u y À Á is assumed to be a smoothly varying function.
The node-averaged value of dU G u x ; u y À Á is zero to satisfy the normalization condition The flux change is approximated with a 2-D modal expansion: X Ny cy¼1 cxþcy6Nc a G;xy;cx;cy P G;cx ðu x ÞP G;cy ðu y Þ: The non-separability of the flux variation is modeled with the cross terms a G;xy;cx;cy . In this work, for each coarse group we consider directional expansions with four basis functions (i.e., N x ¼ N y ¼ N ¼ 4) and we retain four cross terms (N cross ¼ 4). The basis functions P G;i d u d ð Þ are the conventional polynomial and hyperbolic functions used for the 1-D transverse-integrated flux expansion in the semi-analytic Nodal Expansion Method (NEM), with the only exception of P 2;4 u d ð Þ: In Eqs. (5c) and (5d), the coefficient g is chosen so that the hyperbolic functions sinh gu d ð Þ and cosh gu d ð Þ are particular solutions of the 1-D homogeneous diffusion equation in the thermal group: The basis functions of Eq. (5) proved to accurately reconstruct the 1-D transverse-integrated flux variation (Dall'Osso, 2014) and have been also used in this work for their generality. The above modes have zero average value in the interval À1=2; 1=2 ½ , thus satisfying Eq. (3). The four cross terms in Eq. (4) only have polynomial components, with global order up to 4 (i.e., N c = 4) and directional order up to 2.
The spatial distribution of the directional net neutron current J env;G;d u x ; u y À Á is defined as In Eq. (7), scaling to the volume-averaged flux U G ensues from the application of Fick's law to Eq. (2). In Eq. (8), D G u x ; u y À Á is the spatially dependent diffusion coefficient (in units of cm). We define the environmental discontinuity factors f env G;dAE at the node surfaces as where f 1 G;dAE denotes the single-assembly discontinuity factor, and the signs AE refer to the interfaces along the positive and negative directions of the d axis.
The following nodal unknowns must be found for each coarse group [Eqs. (4) and (9)]: the directional modal coefficients a G;d;i d (4 unknowns per direction), the cross modal coefficients a G;xy;cx;cy (4 unknowns), and the discontinuity-factor corrections df G;dAE (2 unknowns per direction). With the aforementioned choice of N and N c , the number of unknowns per coarse group is 16. In order to solve for them, we identify a set of equations for the following quantities: the environmental surface-averaged fluxes (2 equations per direction, per group); the environmental surface-averaged net currents (2 equations per direction, per group); the environmental corner-point fluxes (4 equations per group).
The remaining four (or N x þ N y þ N cross À Á À 8 Â Ã in a more general framework) equations per group are found applying a standard weighted-residual technique to the 2-D few-group balance equation in the environmental conditions. The procedure used to derive the aforementioned equations is explained below.

Equations for the environmental surface-averaged flux
The surface-averaged heterogeneous (i.e., continuous) flux in the environmental conditions U het G;dAE can be written, for the x direction, as U het where U hom G;xAE denotes the homogeneous (i.e., discontinuous) surfaceaveraged flux from the nodal calculation. Introducing Eqs. (2) and (9) into Eq. (10) yields Using the definition of single-assembly discontinuity factor and taking into account that the volume-averaged value of u 1;G u x ; u y À Á is unity, we rewrite the first term on the left-hand side of Eq. (11) as After substituting Eqs. (4) and (12) into Eq. (11), we obtain X Nx ix¼1 a G;x;ix P G;ix AE In Eq. (13), the transverse-direction (i.e., y-directional) and cross components of the surface-averaged flux variation vanish because, by definition, Z 1=2 À1=2 du y P G;iy u y À Á ¼ 0: An equation analogous to Eq. (13) holds for the y direction.

Equations for the environmental surface-averaged current
We consider the simple case with homogeneous diffusion coefficient [i.e., D G u x ; u y À Á ¼ D G ]. Using Eqs. (7) and (8), the surfaceaveraged directional net current reads (for the x axis) In Eq. (15), we have used the fact that the infinite-medium current at the assembly surface is zero, namely Eq. (16) is valid at the assembly outer edges and, when considering four nodes per fuel assembly, at the internal surfaces of fully symmetrically-loaded fuel bundles (i.e., with octant or quadrant symmetry). The sought equation is found equating Eq. (15) to the surfaceaveraged directional net current J G;xAE from the nodal calculation:

Equations for the environmental corner-point flux
The heterogeneous (i.e., continuous) flux at a nodal corner point is where u c x and u c y denote the corner-point coordinates within the node. Introducing the single-assembly corner discontinuity factor f An estimate of U het G;c must be found to use Eq. (19) in the spatial rehomogenization algorithm. For this purpose, we use two different approaches: a combination of Smith's method (Rempe et al., 1989) and the Method of Successive Smoothing (MSS) (Böer and Finnemann, 1992); a Corner Point Balance (CPB) condition (Chang et al., 1989).
The above methods are commonly used in the context of fuelassembly dehomogenization (Joo et al., 2009). Their application in the framework of our work is briefly explained in the following.

Smith's method and the Method of Successive Smoothing
In Smith's method, the intranodal flux distribution is considered as separable in the x and y directions. The homogeneous flux at a given node vertex is approximated using the volume-averaged flux in the node and the surface-averaged flux at the nodal edges crossing the corner. For instance, using the nodal coordinate system of The MSS is based on the assumption that the flux varies linearly in the neighborhood of a corner point. The homogeneous flux at a nodal vertex is estimated with a linear extrapolation in terms of the surface-averaged and volume-averaged fluxes: The heterogeneous corner flux in the environment is related to its homogeneous counterpart U hom G;c via the corner discontinuity factor: Using Eqs. (20) or (21) and Eq. (22), four estimates of U het G;c are available for a given corner point, one from each node surrounding the corner. The heterogeneous corner flux is thus approximated with the arithmetic average of the available estimates: where we have denoted with i and j the coordinates (along the x and y axes, respectively) of the four nodes sharing the corner c.
In the real environment, the corner discontinuity factors also change compared to their infinite-lattice estimates. An environmental correction on this parameter (df G;c ) should therefore be introduced for each node vertex. In order not to increase the number of unknowns and equations of the rehomogenization problem, we use an approximate relation to evaluate this correction. We assume that df G;c is proportional to the corrections on the discontinuity factors of the x-and y-directional surfaces crossing the corner (df G;x ; df G;y ), namely where f env G;x and f env G;y are the environmental discontinuity factors estimated with Eq. (9) at the previous rehomogenization iteration.
Previous work showed that the MSS generally performs better than Smith's method, in which spatial cross effects are neglected (F. Khoshahval et al., 2014). On the other hand, Eq. (21) may lead to a negative corner flux (especially at the first nodal iterations, when the surface-averaged flux estimates are not yet accurate enough). For this reason, we use Eq. (21) as a default option and switch to Eq. (20) in case of negative corner-flux values from the MSS. Despite the ease of implementation, this approach may fail to provide accurate estimates of the corner fluxes if the flux distribution exhibits strong spatial gradients (Rempe et al., 1989;Joo et al., 2009).

The Corner Point Balance approach
The Corner Point Balance (Chang et al., 1989) is a neutron balance equation imposing the absence of net source conditions at the corner points (i.e., no accumulation of neutrons within an infinitesimally small volume). With this method, continuity of the neutron current at the vertices of a node is preserved.
We briefly review this approach with an analogy to the simple case of 1-D problem (Joo et al., 2009). From diffusion theory, the heterogeneous surface-averaged flux at the interface s between two adjacents node k and l is related to the inlet (J in G;s ) and outlet (J out G;s ) partial currents as If no continuity of the net current J G;s is imposed, the estimates of J G;s computed for the nodes k and l will be different (i.e., J k G;s -J l G;s ).
Eq. (27) defines an iterative algorithm, in which a new estimate of the surface flux U Ã G;s is obtained from the previous one and the difference in the computed net currents, which will eventually converge to zero.
A similar scheme can be applied in the neighborhood of a corner point. As illustrated in Fig. 2, four sets of directional net currents exist at a given nodal vertex (one for each adjacent node): J 1 x;c and J 1 y;c for node 1, J 2 x;c and J 2 y;c for node 2, etc. The currents along a certain direction must have the same value (i.e., to ensure a zero-netleakage balance in an infinitesimally small volume containing the corner point. However, since no condition on the corner current continuity is imposed in the nodal calculation, they may be different. Using Eq. (27), four new estimates of the heterogeneous corner flux are obtained (one for each node). Taking the average of them yields This iterative scheme converges as the sum of the differences in the net currents (i.e., the term within brackets on the righthand side of Eq. (28)) vanishes. Assuming that Eq. (2) is the exact solution of the neutron balance equation in the nodes sharing the corner, an estimate of the net currents at the corner itself is obtained from Eq. (7). For instance, the following expression is derived for J 1 x;c in node 1 (we use the nodal coordinate system of Fig. 1): Eq. (29) ensues from the fact that the infinite-medium net current in a corner point is zero.
With the CPB approach, a new estimate U Ã G;c of the heterogeneous corner flux U het G;c is found at the end of each rehomogenization iteration (namely, after sweeping all the nodes) with the computed flux-variation modal coefficients. This new estimate is to be used as a known term on the right-hand side of Eq. (19) in the following rehomogenization update. At the first iteration, in which no estimate of the a coefficients is available, we use Eq.
Compared to Smith's method and the MSS, the CPB approach has a physical foundation. Moreover, it does not involve the corner discontinuity factors, so an environmental correction on these parameters is not necessary. No further approximation is introduced when solving the rehomogenization problem. On the other hand, as it will be shown in Section 3, this strategy causes a slower convergence rate of the rehomogenization algorithm due to its non-linearity. This is because the heterogeneous corner fluxes depend on the flux-variation modal coefficients, which are the unknowns of the rehomogenization linear system.

Weighted-residual balance equations
The 2-D multigroup neutron balance equation in the real environment is 1 Dx The symbols in Eq. (30) correspond to conventional notation in reactor physics literature (Stacey, 2007). Following the example of spectral rehomogenization (Gamarino et al., 2018a;Gamarino et al., 2018b), we project Eq. (30) over a set of weighting functions W G;j u x ; u y À Á (with j = 1,. . .,4) and integrate it in the two directions. The cross-section projection term for the generic reaction rate r is where the rehomogenization coefficients are defined as In Eq. (32b), the subscript t refers to the direction transverse to d.
The projection of the x-directional current divergence term yields (we assume again that the diffusion coefficient is spatially constant within the node) Similar equations can be written for the divergence of the ydirectional current. The rehomogenization parameter detailed in Eq. (34a) is determined for the two directions from the projection of the 2-D balance equation in the infinite medium: where the spectral ratio is computed solving the single-assembly zero-dimensional (i.e., node-averaged) neutron balance equation. In a two-group framework and with neutron emission from fission only in the fast group where B 2 denotes a buckling coefficient. We set B 2 ¼ B 2 crit if the nodal cross sections are generated with the critical-spectrum correction (Hebert, 2009) To summarize, with the above definitions the j-th weightedresidual equation is X Ny cy ¼1 cxþcy Nc a G;xy;cx;cy h V;t;G;xy;cx;cy;j À D G X d¼x;y h div d ;G;xy;cx;cy;j Using Eq. (37), we make the assumption that the nodal estimates U G and k eff satisfy the neutron balance in space [Eq. (30)] in a weighted-integral sense.
The choice of the weighting functions W G;j u x ; u y À Á is, in principle, arbitrary. However, these modes must be selected carefully to avoid ill-conditioning of the rehomogenization linear system that ensues from the set of equations defined above [Eqs. (13), (17), (19), and (37)]. Using Galerkin projection, the lowest condition number of the solving matrix has been achieved with the following set of test functions in one variable: With this choice, the weighting operators do not depend on the energy group.

Summary of the procedure
At the end of a non-linear flux iteration, the nodes are swept to solve the spatial rehomogenization problem. The rehomogenization linear system is set up independently for each node with Eqs. (13), (17), (19) and (37). In Eq. (19), the corner flux is determined using (i) Eqs. (20)-(24), or (ii) Eqs. (28) and (29) with the flux-variation modal coefficients from the previous rehomogenization update. After solving the system, the new estimates of the environmental discontinuity factors at the node surfaces are determined with Eq. (9). The spatial cross-section correction for reaction r is computed as X Ny cy¼1 cxþcy6Nc a G;xy;cx;cy h V;r;G;xy;cx;cy;0 ; ð39Þ where the index j ¼ 0 refers to the unitary weighting function [W G;0 u d ð Þ ¼ 1]. When the CPB condition is applied, as the corner fluxes converge, the corrections df G;dAE and dR r;G also converge, and so do the effective multiplication factor and the nodal variables U G ; U hom G;dAE , and J G;dAE (which are the input quantities of the rehomogenization algorithm).
We exploit the information on the 2-D directional net current distributions [Eq. (7)] to compute the few-group transverseleakage shape for the NEM (or the Analytic Nodal Method) equations. This approach replaces the approximation commonly adopted in industrial nodal codes, which is based on a three-node quadratic fit along the direction of interest (Lawrence, 1986). This approximation lacks a theoretical justification and introduces an inconsistency in the solution of the nodal equations, because it uses information from the adjacent nodes in the direction under consideration to determine the internodal leakage in the transverse direction. For the 1-D transverse-integrated problem along the x direction, the transverse leakage is formulated as Introducing Eqs. (4), (7) and (8) into Eq. (40), after some algebraic manipulation we obtain where L G;y is the node-averaged leakage in the y direction (i.e., the transverse direction), and b G;cy is a constant term defined as Setting N c ¼ 4 and limiting the order of the directional polynomial basis functions in the cross terms to 2, we obtain b G; The transverse leakage detailed in Eq. (43) is still a quadratic polynomial. However, its shape retains some information on the 2-D node-to-node leakage distribution (i.e., information from the transverse direction) via the cross coefficients a G;xy;cx;2 . In this way, the unphysical feature of the standard approximation discussed above is removed.

Numerical results
We validate the method on the same colorset (i.e., fourassembly) configurations considered in Gamarino et al. (2018b): (i) a UO 2 colorset with burnable-absorber (Pyrex) rods, (ii) a UO 2 colorset with AIC-type control rods, (iii) a UO 2 colorset with gadolinium-bearing fuel pins, and (iv) a UO 2 /MOX colorset. Reflective boundary conditions are set at the assembly centerlines. Two-group nodal simulations are performed with the TU Delft inhouse-developed BRISINGR nodal code (Gamarino et al., 2018b), which combines the conventional Coarse Mesh Finite Difference (CMFD) and NEM solution strategies. The infinite-medium homogenization parameters are generated with the APOLLO2-A deterministic lattice transport code (Martinolli et al., 2010). Since in this work we do not apply spectral rehomogenization, the critical-buckling correction (which is a default option in advanced nodal codes) is applied to the nodal cross sections (Hebert, 2009;Gamarino et al., 2018a). All the test cases are simulated in critical conditions (k eff =1), which are determined with a critical-boron search, and at zero burn-up. Isothermal conditions (i.e., no thermal-hydraulic feedback) during nominal operation are considered. We use a spatial discretization of one node per assembly quarter. We present the results of the calculations with: single-assembly cross sections and discontinuity factors (calculation a); 2-D spatial rehomogenization of cross sections and discontinuity factors, using the CPB approach (calculation b) and the MSS/ Smith's method (calculation c) to determine the corner fluxes; 1-D spatial rehomogenization of cross sections and discontinuity factors (calculation d).
For the sake of convenience, from now on we will refer to the hybrid MSS/Smith's method simply as MSS. In calculations b and c, the spatial rehomogenization coefficients [Eqs. (32) and (34)] are computed with the infinite-medium cell-homogenized cross sections and flux distribution.
The reference calculations are performed with APOLLO2-A. We test the accuracy of spatial rehomogenization on the main nodal parameters and on the pin-by-pin flux and fission-power distributions. In all the calculations, the 2-D heterogeneous intranodal flux is reconstructed with the dehomogenization method described in Joo et al. (2009). This approach is based on a 2-D, fourth-order Legendre-polynomial expansion of the neutron source distribution, which results in a semi-analytic solution of the 2-D, groupdecoupled neutron diffusion equation. In calculations b and c, the corner fluxes computed with the 2-D rehomogenization are used as boundary conditions for the pin-flux reconstruction. The pinpower Root-Mean-Square (RMS) deviations are expressed in terms of the relative error (RMS P;r ) and of the power-weighted absolute error (RMS P;wa ), namely where N fc is the number of fuel cells, P calc i and P ref i are the computed and reference (i.e., from APOLLO2-A) values of the fission power in the i-th cell, and P ref CS is the reference average power in the colorset. With Eq. (44b), the errors in the hot spots (i.e., the fuel cells having the highest thermal load) have more weight in the accuracy assessment. We also use Eq. (44a) to compute the RMS deviation in the two-group flux distribution (RMS U G ;r ).

Example 1: UO 2 colorset with burnable-absorber (Pyrex) rods
The first colorset is made of four 17 Â 17 PWR fuel assemblies of fresh UO 2 . Two of these have 1.8% enrichment, whereas the other two have 3.1% enrichment and host 16 burnable-poison rods made of borosilicate glass (Pyrex). The colorset layout is depicted in Fig. 3.
The critical boron concentration is 1465 ppm. The reference values of the normalized fission power are 0.92 in the 1.8%-enriched assembly and 1.08 in the 3.1%-enriched poisoned assembly. For the sake of completeness, we report here the reference absorption and production cross sections (condensed and homogenized in the colorset environment) computed in APOLLO2-A: R a;1 ¼ 0:00873 cm À1 ;R a;2 ¼ 0:0686 cm À1 ;mR f ;1 ¼ 0:00483 cm À1 , and mR f ;2 ¼ 0:0811 cm À1 in the 1.8%-enriched assembly; R a;1 ¼ 0:0101 cm À1 ; R a;2 ¼ 0:104 cm À1 ; mR f ;1 ¼ 0:00657 cm À1 , and mR f ;2 ¼ 0:131 cm À1 in the 3.1%-enriched assembly with Pyrex. Fig. 4 depicts the reference variation in the 2-D flux distribution between the real environment and the infinite lattice in the two fuel assemblies. The percent values are computed with respect to the node-averaged flux. Fig. 5 shows the flux variation computed with the 2-D spatial rehomogenization, combined with the CPB approach for the corner-point fluxes. The result of the MSS approach is not presented because the differences are small. Rehomogenization captures the overall flux variation reasonably well. The main differences between the reference and computed distributions are observed in the outer pin rows of both assemblies. In the 1.8%-enriched bundle, the magnitude of the thermal flux change is overestimated at the assembly edges and underestimated in the external-corner fuel cell. The opposite is found in the assembly with Pyrex rods. Fig. 6 shows the variation in the transverse-integrated flux U G;x x ð Þ computed with the 1-D rehomogenization. For the sake of comparison, we also plot the 1-D curves obtained by transverse integration of the 2-D distributions of Fig. 5. The deviations between the 1-D and 2-D approaches are not significant. They are more evident in the fast group, especially in the assembly with Pyrex. Table 1 reports the number of non-linear flux iterations (N iter ) and the errors in the integral parameters and main nodal cross sections (the errors in the discontinuity factors will be addressed in Section 4.2). Compared to the calculation with infinitemedium cross sections and discontinuity factors, both the 2-D and 1-D models give a significantly more accurate prediction of the multiplication factor and fission power. Much of this    improvement comes from the correction of the thermal absorption cross section in the assembly with burnable poison, even if the corresponding error becomes higher than that of the singleassembly estimate. The increase in the error in this cross section is because of the exclusion of spectral effects, which go in the opposite direction to spatial ones (see Section 4.4). The improvement in the power is more apparent with the 2-D rehomogenization. The differences between the CPB and MSS strategies are negligible. Table 2 compares the errors in the cross sections rehomogenized with (i) the reference 2-D flux change (Fig. 4) and (ii) a least-squares best fit of it. The best fit has been computed with the basis functions defined in Eq. (5), using the same number of directional and cross terms as in the modal reconstruction [Eq.
(4)]. Since these two sets of cross sections do not account for the spectral effects of the environment, they can be considered as a reference to assess the accuracy of spatial rehomogenization. Comparing Tables 1 and 2, it turns out that the cross-section corrections computed with the 2-D rehomogenization go in the right direction and are very close to those obtained with a best fit of the 2-D reference. The inaccuracy in the flux-change reconstruction observed in Fig. 5 is therefore ascribable to the limited fitting capability of the 1-D basis functions [Eq. (5)] rather than to a deficiency of the method. We have verified that increasing the number of cross terms in the best fit from four to six or eight improves the computed corrections only slightly. Rehomogenization underestimates the corrections on R a;2 and mR f ;2 in the 1.8%enriched assembly and on mR f ;2 in the assembly with Pyrex (see Tables 1(b) and 2). It overcorrects R a;2 in the latter. This overcorrection is due to an overly negative estimate of the flux variation in the center of the assembly, where the Pyrex rods are located and thermal absorption is higher. This mismatch is smaller with the 1-D model. The flux spatial variation has no effect on the fastgroup cross sections. The same has been observed in fast-to-thermal scattering and in the diffusion coefficients of both energy groups, which are not reported in the previous tables. For these quantities, both the reference and computed corrections are negligible. Fig. 7 compares the relative errors in the thermal-flux pin-bypin distribution of the 1.8%-enriched assembly, as computed with calculations a and b. A general improvement is observed, especially in the fuel cells surrounding the assembly external vertex. The reference fission-power distribution and the corresponding relative errors are in Figs. 8 and 9, respectively. Table 3 shows the RMS errors in the two-group flux and total power. The reduction in the RMS deviations is apparent in the poison-free bundle, whereas it is less evident in the assembly with Pyrex. The error in the fuel cell with the highest power (i.e., the external-corner cell in the assembly with Pyrex) increases from À0.64% to À1.71% when the 2-D rehomogenization is applied (Fig. 9).

Example 2: UO 2 colorset with AIC-type control rods
In this test case, two banks of twenty-four black control rods each are inserted into two of the four fuel assemblies. The type of the control elements is AIC (silver-indium-cadmium), with the following mass percent composition: 80% 47 Ag, 15% 48 Cd, and 5% 49 In. The 235 U enrichment is 1.8% in the unrodded assembly and 2.4% in the rodded one. The colorset layout is shown in Fig. 10. The critical configuration is achieved with a boron concentration of 222 ppm. The reference values of the normalized fission power are 1.12 in the unrodded assembly and 0.88 in the rodded one.
The reference cross sections are: R a;1 ¼ 0:00833 cm À1 ; R a;2 ¼ 0:0573 cm À1 ; mR f ;1 ¼ 0:00483 cm À1 , and mR f ;2 ¼ 0:0828 cm À1 in the unrodded assembly; R a;1 ¼ 0:0119 cm À1 ; R a;2 ¼ 0:0942 cm À1 ; mR f ;1 ¼ 0:00554 cm À1 , and mR f ;2 ¼ 0:109 cm À1 in the rodded assembly.  The reference and computed 2-D flux variations are in Figs. 11 and 12. The prediction is quite accurate in the unrodded assembly. In the rodded bundle, the magnitude of the flux change is underestimated around the external corner, where it is positive, and overestimated in the control-rod cells closer to the assembly center (i.e., at the coordinates [À3.79, 0], [À3.79, 3.79] and [0, 3.79] cm), where it is negative. The decrease in thermal absorption in the environment is therefore overestimated (see Tables 4 and 5). The variation in the transverse-integrated flux is shown in Fig. 13. In the thermal group of the rodded assembly, the 1-D curve from the 2-D model approximates the reference better in the neighborhood of the coordinate À3.79 cm, which spans three rodded cells along the y direction (see Figs. 10(c) and 11). However, it provides a less accurate estimate at the periphery.
Tables 4 and 5 report the errors in the nodal quantities. Also in this case, an improvement in k eff and in the fission power is observed, particularly with the 1-D rehomogenization. Calculations b, c and d mainly differ in the prediction of the correction dR a;2 in the rodded assembly, which is overestimated to a different extent in the three cases (see Tables 4(b) and 5). As in the example with Pyrex rods, the 2-D model reproduces faithfully the corrections computed with a best fit of the reference flux change. Therefore, the overestimation of dR a;2 is mostly because the basis functions cannot reconstruct the dip in dU 2 x; y ð Þ in the inner rodded cells.
The relative errors in the pin-by-pin thermal flux in the unrodded assembly and in the power distribution are depicted in Figs. 14 and 16. The reference fission power is in Fig. 15. Table 6 summarizes the RMS deviations. With the 2-D model, a general improvement is only found in the unrodded bundle, even though not in the cells with higher power. Since in this assembly the 2-D rehomogenization provides a better intranodal flux distribution than calculation a (see Fig. 14), the observed increase in RMS P;wa is probably due in part to the inaccuracy in the infinite-medium pin-by-pin fission cross sections. With the 1-D approach, the improvement is negligible in both assemblies. The highest power (126% of the colorset-averaged value) is found in the external-corner cell of the rodded assembly and in the two cells adjacent to the empty instrumentation tube at the center of the unrodded bundle (i.e., at [0,0] cm). With the 2-D rehomogenization, the power error    decreases (in absolute value) from 3.38% to À1.0% in the former and increases (in absolute value) from À0.70% to À0.76% in the latter.
The reference and computed flux changes are shown in Figs. 18-20. The nodal errors are in Tables 7 and 8. With the 2-D model, the computed dR a;2 in the 3.9%-enriched assembly is higher than the expected value. This is because the thermal-flux variation is overestimated in the gadolinium-bearing fuel pins located in the outer rows (at the coordinates [À2.53, 10.78] and [À10.78, 2.53] cm). In these cells the reference flux change is 8.12%, whereas the computed one is 13.4%. In the same fuel bundle, the correction on mR f ;2 goes in the wrong direction because of the combined underestimation and overestimation of the magnitude of dU 2 x; y ð Þin the neighborhood of the assembly vertex and in the assembly center, respectively. Despite the above mismatches, the 2-D model significantly improves the k eff and nodal-power estimates compared to the calculation without rehomogenization. These parameters improve less with the 1-D model. Figs. 21 and 22 show the reference fission-power distribution and the corresponding relative errors. Table 9 summarizes the RMS deviations. Rehomogenization better predicts the fission power in the center of the gadoliniumbearing assembly quarter (where the power is higher), which results in a reduction in RMS P;wa .

Example 4: UO 2 /MOX colorset
This colorset is made of two 18 Â 18 UO 2 assemblies and two 18 Â 18 MOX assemblies. The 235 U enrichment in the UO 2 assemblies is 2.1%. The MOX bundle contains three different types of fuel pins: with low plutonium content (1.78% 239 Pu, 0.22% 235 U) at the assembly corners, with intermediate plutonium content (2.53% 239 Pu, 0.21% 235 U) along the assembly outer edges, and with high plutonium content (3.86% 239 Pu, 0.20% 235 U) in the remainder of the fuel bundle. The colorset and assembly layouts are shown in Fig. 23. The concentration of diluted boron in the moderator is 2907 ppm. The reference normalized fission power and cross sections are: P fiss ¼ 0:85, R a;1 ¼ 0:00924 cm À1 ; R a;2 ¼ 0:0887 cm À1 ; mR f ;1 ¼ 0:00543 cm À1 , and mR f ;2 ¼ 0:0974 cm À1 in the UO 2 assembly; P fiss ¼ 1:15, R a;1 ¼ 0:0143cm À1 ;R a;2 ¼ 0:258 cm À1 ; mR f ;1 ¼ 0:00994 cm À1 , and mR f ;2 ¼ 0:372 cm À1 in the MOX assembly.     Table 5 Example 2: errors in the absorption and production cross sections rehomogenized with the reference 2-D flux variation and with a least-squares best fit of it. nodal errors. The 2-D model overestimates the absolute value of the flux change (which is negative) in the center of the MOX assembly. Here the plutonium content is higher (see Fig. 23(c)), and so are the thermal-group absorption and production pin-bypin cross sections. The corrections on the thermal cross sections are thus considerably higher than those computed with the reference flux variation and, in this case, also with a least-squares best fit of it. The error in k eff increases, whereas the errors in the fission power still decrease significantly. The pin-by-pin errors are shown in Figs. 27-30 and in Table 12. In spite of the limited accuracy in the prediction of the flux varia-tion, the flux and power distributions in the MOX assembly improve with the 2-D rehomogenization. An increase in the error is only found around the four nodal vertices, particularly in the two cells surrouding the central water-carrying instrumentation tube (in the SW corner of the quadrant). These observations are confirmed by the RMS deviations, which also decrease appreciably. The improvement is more significant in the UO 2 assembly with the 2-D model.
In this test case, the CPB-based 2-D rehomogenization has a considerably slower convergence rate than the MSS approach and the 1-D rehomogenization (Table 10).

On the transverse-leakage approximation
As mentioned in Section 2.5, the conventional three-node quadratic fit for the approximation of the transverse-leakage distribution is a non-consistent feature of nodal calculations (Lawrence, 1986). Various methods can be found in the literature to relax this approximation (Prinsloo et al., 2014). In this section, we compare the transverse-leakage distributions computed with the aforementioned standard approach and with the quadratic approximation ensuing from the 2-D rehomogenization [Eq. (43)]. The impact of these two strategies on the nodal simulations is also addressed. Fig. 31 shows the transverse-leakage shape L G;y x ð Þ in the 1.8%enriched UO 2 assembly of Example 1. The curves of calculations a and d have been computed with the three-node parabolic fit, whereas the curves of calculation b have been determined with Eq. (43). The distributions obtained with the 2-D rehomogenization differ significantly from those of the standard approximation, especially in the thermal group.
We have found that, when spatial rehomogenization is applied, the nodal calculation becomes insensitive to the transverse-    leakage approximation. We have observed convergence to the same solution (in terms of k eff , the nodal-flux distribution U G , and the cross-section corrections dR x;G ) irrespective of the intranodal shape assumed for L G;y x ð Þ and L G;x y ð Þ. This finding is related to the corrections on the assembly-surface discontinuity factors df G;dAE and can be explained as follows. The transverse-leakage approximation affects the nodal estimates of U hom G;dAE and J G;dAE , which are input quantities to the spatial rehomogenization algorithm (see Section 2). However, the corrections df G;dAE vary in such a way that the modal coefficients of the flux spatial perturbation [Eq. (4)] converge to the same values obtained with other transverse-leakage approximations. Eventually, also the heterogeneous quantities G;dAE and J G;dAE converge to the same values. The discontinuity-factor correction acts thus as a free parameter that enables convergence to the unique solution of the coupled nodal (i.e., CMFD + NEM) and rehomogenization fields.
For Example 1, Table 13 shows the corrections df G;dAE determined with calculation b and three different transverse-leakage distributions: the flat-leakage approximation (in which the intranodal shape is considered as uniform and equal to the node-averaged value of the transverse leakage), the standard three-node quadratic fit, and the quadratic shape from Eq. (43). The flat-leakage approximation leads to the highest variations in df G;dAE , whereas Eq. (43) leads to the smallest.
The insensitivity to the transverse-leakage approach has also been observed with the 1-D rehomogenization. More investigation is needed to find a mathematical justification of this behavior. Table 7 Example 3: (a) number of non-linear flux iterations and errors in the integral parameters, and (b) errors in the absorption and production nodal cross sections.  Table 8 Example 3: errors in the absorption and production cross sections rehomogenized with the reference 2-D flux variation and with a least-squares best fit of it.

Analysis on the discontinuity factors
In Section 3, we have only assessed the accuracy of the crosssection corrections. We now make a similar analysis on the discontinuity factors.
We evaluate the reference environmental discontinuity factors with a nodal-equivalence approach applied to the colorset transport calculation. We define the homogeneous transverseintegrated neutron flux in the real environment with a quartic expansion, as in the conventional NEM equations: We exploit the symmetry of the four-assembly configurations considered in this work and determine the reference surfaceaveraged currents with the two-group nodal balance in the colorset: where U ref G;wg denotes the spatially averaged value of the water-gap flux from the reference calculation. Table 14 reports the relative differences (in percentage) between the discontinuity-factor estimates of calculations a, b and d and the reference values determined with Eq. (49). The improvement in the discontinuity factors is less apparent than that observed in the cross sections in Section 3. In particular, in the UO 2 and MOX assemblies of Example 4 (Table 14(d)) the corrected values have significantly higher errors than the infinite-medium ones. We remark that the reference quantities in Eqs. (46)-(49) (cross sections, nodeaveraged fluxes, surface-averaged fluxes and currents) come from the solution of the 281-group transport equation in the real environment, with successive collapsing to two groups. These quantities incorporate not only the spatial effects of the environment but also the spectral ones, which are not taken into account by spatial rehomogenization. The discontinuity factors computed with Eq. (49), which we consider here as reference values, are therefore not fully consistent with our spatial rehomogenization approach. A more rigorous assessment should be made computing them with a reference solution that excludes the spectral effects (namely, solving the transport equation in the colorset environment directly in a twogroup structure, without energy condensation).
To conclude the analysis on the discontinuity factors, we assess the impact of the reference df G;dAE on the nodal estimates of the   Table 15. Whilst the discontinuityfactor corrections have a mild impact on the nodal simulations of Examples 1 and 4, their effect is substantial in Examples 2 and 3. Comparing the errors in k eff and P fiss shown in Table 15 with those resulting from rehomogenization (Tables 1, 4, 7 and 10), the following conclusions can be drawn for the various test cases: In Example 1, both the 1-D and 2-D models overestimate the corrections on k eff and on the fission power (especially the 2-D one). In Example 2, similar considerations apply to the 2-D model, whereas the 1-D approach provides a very accurate result. In Example 3, the correction on k eff is overestimated with the 2-D model and underestimated with the 1-D one; both approaches underestimate the corrections on P fiss (particularly the 1-D model).
In Example 4, the computed corrections on k eff and P fiss go in the wrong direction. The spatial effects have negligible impact on the assembly-averaged fission power.

Numerical features and implementation
As spectral rehomogenization, spatial rehomogenization also translates into an additional level of feedback in the global core calculation (Gamarino et al., 2018a). In a two-group framework with four basis functions per direction and four cross terms (see Section 2), the rank of the 2-D rehomogenization linear system is 32. Since the rehomogenization problem is solved independently in each node after updating the corner-point flux distribution [Eq. (23) or (28)], the corresponding computational cost can be mitigated via parallelization of the algorithm.
In the present work, the rehomogenization coefficients [Eqs. (32 and (34)] have been computed post-processing the APOLLO2-A single-assembly cell-homogenized cross sections and flux distribution. Better modeling may be achieved incorporating their calculation into the lattice-code routines, thus accounting for within-cell heterogeneity. As in the case of spectral rehomogenization, an   Table 16 shows the errors of calculation b with these improved rehomogenization parameters. Comparing Tables 16(a) and 4, we observe that the differences in the results are negligible in the colorset with control rods. In the UO 2 /MOX colorset (Tables 16(b) and 10), the differences are more tangible. Whilst the error in k eff increases of about 40 pcm compared to the case with standard rehomogenization coefficients (i.e., computed with the infinite-medium pin-bypin cross sections), the errors in the fission power are halved. However, in both cases the power errors are significantly smaller than those of calculation a. Moreover, the improvement in the nodal cross sections of the MOX assembly observed with the environmental coefficients is small if compared to the reference estimates (see Table 11). We have verified that the impact of the aforementioned approximation on the rehomogenization parame-ters is marginal also in the test cases with Pyrex rods and gadolinium-bearing fuel pins. It can therefore be concluded that using the infinite-medium fine-mesh cross sections in Eq. (32) does not significantly affect the performance of the method. Intranodal depletion effects can be easily incorporated into the rehomogenization coefficients via the conventional within-assembly crosssection polynomial expansions (Wagner et al., 1981;Forslund et al., 2001).
The memory requirement for the storage of the rehomogenization parameters in the cross-section libraries is minimized exploiting the symmetry of the fuel-assembly internal layout. These additional coefficients are only computed and stored for an assembly quarter. During the nodal calculation, their sign is determined based on the orientation of the node within the fuel assembly.
As observed for spectral rehomogenization (Gamarino et al., 2018a), convergence of the algorithm benefits from underrelaxation, which dampens numerical oscillations. Convergence difficulties have never been encountered. Except for the CPB approach in the UO 2 /MOX benchmark problem (Table 10), the increase in the number of power iterations of the eigenvalue   calculation is below a factor of 1.8 and, therefore, acceptable. In coupled calculations (namely, with thermal feedback), the additional level of non-linearity introduced by the cross-section and discontinuity-factor updates would be nested in the thermalhydraulic iterations, thus alleviating the observed increase in the number of power iterations and computing time.
To conclude, the 2-D rehomogenization method presented in this work can also be viewed as an original dehomogenization approach. If Eq. (2) is used, the assumption is made that the 2-D flux variation is a smooth function of the x and y coordinates and is not affected by intranodal heterogeneity. The impact of this approximation on the reconstruction of the pin-by-pin flux and power distributions should be assessed. A successful application of the 2-D rehomogenization for assembly dehomogenization as well would increase the attractiveness of the method and largely compensate for its higher computational cost compared to the 1-D model.

Complementarity between the spatial and spectral effects of the environment
We briefly address the complementarity of the various environmental effects on the nodal cross sections. For the benchmark problems analyzed in Section 3, Table 17 shows: (i) the reference spectral corrections dR spec;ref , computed with the reference spectrum variation (Gamarino et al., 2018a;Gamarino et al., 2018b); (ii) the reference spatial corrections dR spat;ref , computed with the reference 2-D flux spatial change; and (iii) the cross (i.e., mixed energy-space) corrections dR cross r;G , which have been estimated as dR cross where R ref r;G is the cross section condensed and homogenized in the colorset configuration, thus taking into account the global environmental effect. Even if the cross corrections are generally smaller than the spectral and spatial ones, they are not negligible, especially in the assemblies with the strongest heterogeneity. In particular, in the MOX assembly of Example 4 [Table 17(d)], the thermal absorption and production cross terms have magnitude comparable to or larger than the spectral and spatial ones, which have opposite sign and roughly cancel each other. One of the reasons of this behavior is that the flux spectrum deformation varies significantly with the distance from the assembly outer edge, whereas the correction is computed with an average deformation in the assembly (Gamarino et al., 2018a). Taking into account the cross corrections is therefore important for an accurate prediction of the global cross-section variation.
From Table 17, some general features of the cross-section corrections can be observed: The spatial effects are only relevant in the thermal group, whereas the spectral ones are significant in both energy groups. There is no clear pattern for cross effects. In the heterogeneous assemblies, the sign and magnitude of the spatial corrections depend on the position of the main sources of heterogeneity within the fuel bundle.
Separating the various effects on the discontinuity factors is not equally straightforward for the reasons discussed in Section 4.2.

Conclusions
In this paper, we have developed a first-principle modal method to estimate the variation in the 2-D few-group flux distribution between the core environment and the infinite lattice. This method relaxes the main approximations of the original 1-D transverseintegrated rehomogenization.
Numerical results show that the errors in the multiplication factor and assembly-averaged fission power significantly decrease compared to the calculation with infinite-medium homogenization parameters. The cross-section corrections computed by the 2-D model always go in the right direction. In most cases they match the corrections determined with a least-squares best fit of the reference flux change, but they are higher than the reference values. One of the reasons of this outcome is that the polynomial and hyperbolic basis functions used for the modal expansion cannot reproduce to a high degree of accuracy the flux variation dip in the cells hosting absorbing elements (burnable-poison rods and control rods). In the 1-D approach, the flux change dip in these cells is smoothed out by the transverse integration and is therefore better fitted by the basis functions. When applying spatial and spectral rehomogenization sequentially, the overestimation of the spatial corrections in the thermal group may affect the accuracy of the global results. The differences between the CPB-based and MSS-based approaches are small, with the former usually performing slightly better. An apparent superiority of the 2-D model over the 1-D model has not been observed. Except for the test case with gadolinium-bearing fuel pins in which the 2-D method is more accurate, the discrepancies between the two approaches are not significant. In general, the 1-D model overestimates the crosssection corrections to a lesser extent.
At the pin-by-pin level, on average better results are found when rehomogenization is applied. Nevertheless, the improvement in the heterogeneous flux and fission-power distributions is often not fully generalized, but limited to certain subregions of the fuel assemblies. Therefore, the accuracy of the computed fission power in the hot spots (that is, the fuel cells with the highest thermal load) is tied to their location within the fuel bundles.
We have observed that when 1-D or 2-D spatial rehomogenization is applied, the type of the transverse-leakage approximation does not affect the accuracy of the nodal calculation. An analysis on the complementarity of the various environmental corrections revealed that mixed energy-space effects cannot be neglected, especially in UO 2 /MOX configurations.
The accuracy of the 2-D spatial rehomogenization might be improved by searching basis functions that better fit the intranodal flux spatial variation. Based on the satisfactory outcome of Proper Orthogonal Decomposition (POD) in the framework of spectral rehomogenization (Gamarino et al., 2018a;Gamarino et al., 2018b), this modal approach could be extended to the spatial problem as well. This strategy would allow the direct calculation of two-dimensional basis functions in the domain of space, thus discarding the separation of the directional and cross components of the modal expansion with one-dimensional modes.
In core configurations with strong heterogeneity in the axial direction (such as enrichment and gadolinium-concentration zoning in boiling water reactors), the 2-D rehomogenization in the x and y coordinates may be combined with the 1-D rehomogenization along the z axis. With this approach, modeling of axial heterogeneity could be improved while retaining coarse axial meshes for both 3-D assembly homogenization (Schneider et al., 2016), when applicable, and the on-line nodal calculation.

Acknowledgment
This work was supported by the Framatome Ph.D. program.

Table 17
Cross-section changes due to the various types of environmental effects: spectral, spatial, and cross (i.e., mixed energy-space). The variations have been computed with respect to the infinite-medium values by using the reference data from APOLLO2-A.
(a) Example