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ERROR: type should be string, got "https://doi.org/10.1051/epjconf/202124703002 https://doi.org/10.1051/epjconf/202124703002 EPJ Web of Conferences 247, 03002 (2021)\nPHYSOR2020 ABSTRACT Presently, APOLLO3 R⃝/MINARET solves the transport equation using the multigroup Sn\nmethod with discontinuous finite elements on triangular meshes with Lagrange polyno-\nmial bases. The goal of this work is to solve the spatial problem on hexagonal geometries\nin the context of honeycomb lattice reactors, without further refining the computational\nmesh. The idea here is to construct high-order basis functions on the hexagonal element\nin order to improve the trade-off between computational cost and accuracy, in particu-\nlar for multiphysics simulations where, often, thermalhydraulic modelling requires only\nassembly-average cross-sections to be defined (e.g. severe accident of fast breeder reac-\ntors) i.e. the assemblies are assumed homogeneous. One approach to achieve this goal\nis through the use of generalised barycentric functions such as the Wachspress rational\nfunctions. This research endeavour deals with the application of Wachspress rational\nfunctions to the neutron transport equation for hexagonal geometries up to order 3. With\nthis method, it is possible to decrease the number of spatial unknowns required for the\nsame accuracy, and thus the computational burden for complex geometries, such as hon-\neycomb lattices is reduced. KEYWORDS: Discontinuous Galerkin, Wachspress, Discrete ordinates, Polygons HIGH-ORDER FINITE ELEMENTS FOR THE NEUTRON\nTRANSPORT EQUATION ON HONEYCOMB MESHES\nAnsar Calloo1, Romain Le Tellier2 and David Labeurthre1\n1DEN - Service d’´etudes des r´eacteurs et de math´ematiques appliqu´ees (SERMA)\nCEA, Universit´e Paris-Saclay, F-91191 Gif-sur-Yvette, France 2DEN - DTN/STMA/LMAG\nCEA Cadarache, F-13108 Saint-Paul-lez-Durance, France ansar.calloo@cea.fr, romain.le-tellier@cea.fr, david.labeurthre@cea.fr © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0\n (http://creativecommons.org/licenses/by/4.0/). KEYWORDS: Discontinuous Galerkin, Wachspress, Discrete ordinates, Polygons DP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0\nses/by/4.0/). 1. INTRODUCTION The neutron transport equation [1] is solved increasingly in the framework of multiphysics cou-\npling schemes. It has been observed that high-fidelity calculations for such cases are still computa-\ntionally expensive for the neutronics part, especially in the case of parametric or accidental studies. One possible way of dealing with this situation is through the use of modern computing techniques\nto benefit from new computer architectures. A second means before optimising the code itself is\nby working on the discretisation scheme (in the present work, the spatial one). This work lies in the framework whereby the neutron transport equation is solved with discon-\ntinuous Galerkin finite elements, but on meshes that are not traditional triangles or quadrilater-\nals. Presently, APOLLO3 R⃝/MINARET [2,3] solves the transport equation using the multigroup Sn EPJ Web of Conferences 247, 03002 (2021)\nPHYSOR2020 https://doi.org/10.1051/epjconf/202124703002 method with discontinuous finite elements on triangular meshes with Lagrange polynomial bases. The goal of this work is to solve the spatial problem on hexagonal geometries in the context of\nhoneycomb lattice reactors, without further refining the computational mesh. Indeed, despite the\nlack of regularity of the angular flux, it has been shown in past works [4,5](especially on Adap-\ntive Mesh Refinement techniques) that increasing the discretization order is an effective means to\nsignificantly improve the accuracy of the solution. method with discontinuous finite elements on triangular meshes with Lagrange polynomial bases. The goal of this work is to solve the spatial problem on hexagonal geometries in the context of\nhoneycomb lattice reactors, without further refining the computational mesh. Indeed, despite the\nlack of regularity of the angular flux, it has been shown in past works [4,5](especially on Adap-\ntive Mesh Refinement techniques) that increasing the discretization order is an effective means to\nsignificantly improve the accuracy of the solution. The idea here is then to construct high-order basis functions on the hexagonal element in order to\nimprove the trade-off between computational cost and accuracy. One approach to achieve this goal\nis through the use of generalised barycentric functions such as the Wachspress rational functions. They have been employed in the past in neutron transport for quadrilaterals and arbitrary polygo-\nnals [6,7]. This research endeavour deals with the application of Wachspress rational functions [8]\nto the neutron transport equation for hexagonal geometries up to order 3. 1. INTRODUCTION With this method, it is\npossible to decrease the spatial unknowns required for the same accuracy, and thus the computa-\ntional burden for complex geometries, such as honeycomb lattices is reduced [9]. ∗It contains the external sources and the scattering source from other directions. 2.1. Discontinuous Galerkin form of the neutron transport equation Z\nκ\n((Ωn·∇ψn)vh + Σg\ntψnvh)dV +\nZ\n∂κ−\n(n · Ωn)ψ+\nn v+\nh dS Z\nκ\n((Ωn·∇ψn)vh + Σg\ntψnvh)dV +\nZ\n∂κ−\n(n · Ωn)ψ+\nn v+\nh dS\n=\nZ\nqnvhdV −\nZ\n∂\n(n · Ωn)ψ−\nn v+\nh dS\n(2) =\nZ\nκ\nqnvhdV −\nZ\n∂κ−\n(n · Ωn)ψ−\nn v+\nh dS\n(2) (2) Usually for meshes with triangular or quadrilateral elements, P is the space of polynomials of\ndegree k. However, in the general case of convex polygons, such as hexagons, it cannot be the case\nif P were the space of C0(κ) functions and such that at order k, the approximation space Pk ⊂P. Usually for meshes with triangular or quadrilateral elements, P is the space of polynomials of\ndegree k. However, in the general case of convex polygons, such as hexagons, it cannot be the case\nif P were the space of C0(κ) functions and such that at order k, the approximation space Pk ⊂P. 2.1. Discontinuous Galerkin form of the neutron transport equation Considering an open spatial domain D of R2 with boundary ∂D, meshed in a set Mh of Nκ\nhexagonal elements κ, the discrete-ordinates equation for the monoenergetic neutron transport\nequation in a given direction Ωn is expressed as: Ωn · ∇ψn(r) + Σt(r)ψn(r) = qn(r)∀r ∈D\n(1) (1) with Σt being the macroscopic total cross section and qn(r) being the neutron source in direction\nΩn ∗. Given n(r) the unit outward normal to ∂D at r ∈∂D, ∂D−is the inflow boundary defined\nas with Σt being the macroscopic total cross section and qn(r) being the neutron source in direction\nΩn ∗. Given n(r) the unit outward normal to ∂D at r ∈∂D, ∂D−is the inflow boundary defined\nas as ∂D−= {r ∈∂D | Ωn · n(r) < 0} The boundary conditions for r ∈∂D−are of Dirichlet type for ψn(r). Taking P(κ) as the space of\nreal-valued functions on κ, the set Vh is defined as The boundary conditions for r ∈∂D−are of Dirichlet type for ψn(r). Taking P(κ) as the space of\nreal-valued functions on κ, the set Vh is defined as Vh = {v ∈L2(D)|∀κ ∈Mh, v|κ = P(κ)} The unknown angular flux ψn(r) is thus expanded over the functions vh ∈Vh. These functions\nform a basis of the approximation space and are non-zero over κ and discontinuous across κ|κ′\ninterfaces. Defining ψ+\nn and ψ−\nn as the internal and external traces respectively, Eq. (2), the weak\nformulation for the transport equation, is obtained by multiplying Eq. (1) by vh, integrating by parts\nover element κ, applying the divergence theorem to the streaming term, and using the upwinding The unknown angular flux ψn(r) is thus expanded over the functions vh ∈Vh. These functions\nform a basis of the approximation space and are non-zero over κ and discontinuous across κ|κ′\ninterfaces. Defining ψ+\nn and ψ−\nn as the internal and external traces respectively, Eq. (2), the weak\nformulation for the transport equation, is obtained by multiplying Eq. (1) by vh, integrating by parts\nover element κ, applying the divergence theorem to the streaming term, and using the upwinding 2 2 EPJ Web of Conferences 247, 03002 (2021)\nPHYSOR2020 https://doi.org/10.1051/epjconf/202124703002 approximation for the numerical flux at the interfaces. More details can be found in [10]. 2.2. Wachspress rational functions on the regular hexagon Thus, the equation of the straight line\nd′\ni through ai−1,i and ai,i+1 is given as l′\ni(x) = 0. Using these geometrical constructions, and the\npreviously-defined ones, it is now possible to build the second-order Wachspress shape functions: Furthermore, the midpoint of ai and ai+1 is defined as ai,i+1. Thus, the equation of the straight line\nd′\ni through ai−1,i and ai,i+1 is given as l′\ni(x) = 0. Using these geometrical constructions, and the\npreviously-defined ones, it is now possible to build the second-order Wachspress shape functions: (\nwi\n= ci\nli+2li+3li+4li+5l′\ni\nq\nwi,i+1\n= ci,i+1\nli+2li+3li+4li+5li\nq\n(4) (\nwi\n= ci\nli+2li+3li+4li+5l′\ni\nq\nwi,i+1\n= ci,i+1\nli+2li+3li+4li+5li\nq\n(4)\nwhere ci and ci i+1 are chosen such that wi(ai) = 1 and wi i+1(ai i+1) = 1 (4) where ci and ci,i+1 are chosen such that wi(ai) = 1 and wi,i+1(ai,i+1) = 1. where ci and ci,i+1 are chosen such that wi(ai) = 1 and wi,i+1(ai,i+1) = 1. Besides, the nodes ai+1,i+1,i (respectively ai,i,i+1) are defined as 1\n3(2ai+1+ai) (respectively 1\n3(2ai+\nai+1)). The equation of the straight lines through ai+1,i+1,i and ai+3,i+3,i+4 is l′\ni+1,i+3(x) = 0. Finally, γ(x) = 0 is the equation of the circle Γ through the 12 points ai+1,i+1,i and ai,i,i+1. With\nthis last set of geometrical elements, the third-order Wachspress basis functions are defined: Besides, the nodes ai+1,i+1,i (respectively ai,i,i+1) are defined as 1\n3(2ai+1+ai) (respectively 1\n3(2ai+\nai+1)). The equation of the straight lines through ai+1,i+1,i and ai+3,i+3,i+4 is l′\ni+1,i+3(x) = 0. Finally, γ(x) = 0 is the equation of the circle Γ through the 12 points ai+1,i+1,i and ai,i,i+1. With\nthis last set of geometrical elements, the third-order Wachspress basis functions are defined: \n\n\n\n\n\n\nwi\n= ci\nli+2li+3li+4li+5γ\nq\nwi,i,i+1\n= ci,i,i+1\nli+2li+3li+4li+5lil′\ni+1,i+3\nq\nwi+1,i+1,i\n= ci+1,i+1,i\nli+2li+3li+4li+5lil′\ni,i+4\nq\n(5) (5) where ci, ci,i,i+1 and ci+1,i+1,i are chosen such that wi(ai) = 1, wi,i,i+1(ai,i,,i+1) = 1 and\nwi+1,i+1,i(ai+1,i+1,,i) = 1 respectively. Fig. 2 shows the typical shape functions for order 3 for\ni = 2. where ci, ci,i,i+1 and ci+1,i+1,i are chosen such that wi(ai) = 1, wi,i,i+1(ai,i,,i+1) = 1 and\nwi+1,i+1,i(ai+1,i+1,,i) = 1 respectively. Fig. 2 shows the typical shape functions for order 3 for\ni = 2. J. L. 3. NUMERICAL RESULTS In this section, we provide numerical results for a 2D one-group transport benchmark which has\nbeen set up in [12]. 2.2. Wachspress rational functions on the regular hexagon Gout proved that these rational functions form a basis for spanning space P, and such that at\norder k, Pk(κ) ⊂P and Pk+1(κ) ̸⊂P [8,11]. Besides, given that these functions are anchored at\nnodes on the sides only, they can be assimilated to serendipity functions [7]. 2.2. Wachspress rational functions on the regular hexagon J. L. Gout has defined Wachspress rational basis functions on the regular hexagon up to order 3 in\n[11]. For this particular case, the following geometrical elements are given as shown in Fig. 1 and\nfollowing the same notations as in [11]. x\ny\nO\na4\na5\na6\na1\na2\na3\nb4\nb5\nb6\nb1\nb2\nb3\na34\na665\na556\na332\na223\nFigure 1: Geometrical elements to define Wachspress shape functions on a regular hexagon. Figure 1: Geometrical elements to define Wachspress shape functions on a regular hexagon. Let us consider a regular “flat-top” hexagon with centre (0,0), a pitch (distance between two parallel\nsides) of p and thus, a sidelength of\np\n√\n3. The vertices ai of the hexagon are numbered as shown in Let us consider a regular “flat-top” hexagon with centre (0,0), a pitch (distance between two parallel\nsides) of p and thus, a sidelength of\np\n√\n3. The vertices ai of the hexagon are numbered as shown in 3 EPJ Web of Conferences 247, 03002 (2021)\nPHYSOR2020 https://doi.org/10.1051/epjconf/202124703002 Fig. 1, such that ai and ai+1 are consecutive. For any i in Z/6Z, li(x) = 0 is the equation of the\nstraight line di through ai−1 and ai. The intersection of straight lines di−1 and di+1 is the point bi,\nand q(x) = 0 is the equation of the circle Q through all the points bi (this is also called the adjoint\ncircle of the regular hexagon). These geometrical elements are sufficient to define the first-order\nWachspress shape functions wi which are anchored at the vertices of the hexagon: Fig. 1, such that ai and ai+1 are consecutive. For any i in Z/6Z, li(x) = 0 is the equation of the\nstraight line di through ai−1 and ai. The intersection of straight lines di−1 and di+1 is the point bi,\nand q(x) = 0 is the equation of the circle Q through all the points bi (this is also called the adjoint\ncircle of the regular hexagon). These geometrical elements are sufficient to define the first-order\nWachspress shape functions wi which are anchored at the vertices of the hexagon: wi = ci\nli+2li+3li+4li+5\nq\n(3) (3) where ci is a constant chosen such that wi(ai) = 1. where ci is a constant chosen such that wi(ai) = 1. Furthermore, the midpoint of ai and ai+1 is defined as ai,i+1. Table 1: Benchmark cross sections Table 1: Benchmark cross sections Table 1: Benchmark cross sections\nMixture\nΣt (cm−1)\nΣs0 (cm−1)\nΣs1 (cm−1)\nνΣf (cm−1)\n1\n0.025\n0.013\n0.0\n0.0155\n2\n0.025\n0.024\n0.006\n0.0\n3\n0.075\n0.0\n0.0\n0.0 ‡Throughout this analysis, the computational cost is not measured as time or arithmetic inten-\nsity. The WSF solver has not been implemented optimally with this goal. 3.2. Results analysis The one-group benchmark is analysed in two different settings: within the APOLLO3 R⃝/MINARET\nframework and using the Wachspress shape functions finite elements. Both cases use discrete\nordinates with product angular hexagonal quadratures shortened as HQ(n,m) in the results tables - n\nand m are the polar and azimuthal orders respectively, leading to 6nm directions in 2D geometries. APOLLO3 R⃝/MINARET[3] solves the neutron transport equations with a Discontinuous Galerkin\nformulation with the usual Lagrange shape functions on unstructured triangular meshes. Thus,\neach hexagonal cell within the meshed geometry is at least split in 6 equilateral triangles. In\nTable 2, splits of s (1, 2 and 3) are given and means that the side of a triangular mesh in the meshed\ngeometry should not exceed sidelength\ns\n, thereby leading to an unstructured grid of triangles. The\nsolutions are computed for spatial orders of 1 and 2. The reference case is the best-estimate case\ncomputed with HQ(4,5) angular quadrature, order 2, and a split of 3 - number of spatial unknowns\nNunk amounting to 38100. The best-estimate case with the Wachspress shape functions solver\n(WSF) is HQ(4,5) angular quadrature and spatial basis order of 3, equivalent to Nunk = 2286. For each case, the effective multiplication factor keff is given and compared to the best-estimate\nresult. Furthermore, the absorption rates Rabs are also analysed †. The maximum and average\nerrors are defined as in [12]. The results are summarised in Table 2 for APOLLO3 R⃝/MINARET\nand in Table 3 for WSF. The keff for the reference APOLLO3 R⃝/MINARET and WSF solutions\ndiffers by less than 1 pcm whereas the ϵmax and ϵ are both less than 0.1%. From Table 2, it can be deduced that APOLLO3 R⃝/MINARET satisfactorily predicts the keff for\nlow-order quadratures and coarser spatial discretisation, although the errors on the absorption rates\nare higher. It can also be observed that as the computational mesh is refined, the errors on the\nabsorption rates decreases, yet at the expense of more unknowns in the system, and thus a higher\ncomputational cost ‡. For a given computational mesh, increasing the order of the method is more\nefficient to decrease the error. Thus, these numerical results hint that increasing the spatial basis\norder would be a better option for solving the transport problem efficiently. †The absorption cross section is computed as Σt −Σs0.\n‡ 3.1. Benchmark description The benchmark consists of three mixtures with a radial distribution as given in Fig. 3. The cross\nsections for the mixtures are as given in Table 1. As stated by A. H´ebert, this benchmark is not\nrepresentative of a real-life problem but has been designed to magnify transport and anisotropy\neffects (as much as 2500 pcm!) within the domain and on the vacuum boundary condition. In this\nhoneycomb lattice, the length of the side of a hexagonal cell is 19 cm. 4 https://doi.org/10.1051/epjconf/202124703002 EPJ Web of Conferences 247, 03002 (2021)\nPHYSOR2020 (a) w2\n(b) w223\n(c) w332\nFigure 2: Example of Wachspress basis functions of order 3. (b) w223 (a) w2 (a) w2\n(b) w\n(c) w332 (c) w332 Figure 2: Example of Wachspress basis functions of order 3. Figure 3: 2D hexagonal benchmark problem proposed by [12]. Mixture 0 is here only to\ncomplete the hexagonal geometry description, and is equivalent to void. Figure 3: 2D hexagonal benchmark problem proposed by [12]. Mixture 0 is here only to\ncomplete the hexagonal geometry description, and is equivalent to void. 5 https://doi.org/10.1051/epjconf/202124703002 EPJ Web of Conferences 247, 03002 (2021)\nPHYSOR2020 3.2. Results analysis Increasing the order is cost effective in terms of Nunk to the\nquality of the result. For this problem, an order of 2 for a given angular quadrature is satisfactory\nwith respect to the solution obtained. Considering a purely computational aspect of using Wachs-\npress basis functions on unrefined hexagons, a coarser mesh is sufficient, and thus Nunk depends\nonly the order of the method, which increases linearly. Indeed, the matricial problem solved locally\ngrows as N 2\ndofs and increasing the order for better accuracy remains competitive so long the com-\nputation time remains reasonably low compared to refining meshes. Hence, the ideal framework\nwould be to increase the order locally to avoid increasing the global Nunk of the problem. All these results are consistent with TRIVAC and SNATCH results published in [12], and the new\ncapabilities of DRAGON for hexagonal geometries [13]. 3.2. Results analysis From Table 3, solutions with first-order Wachspress rational functions are satisfactory considering From Table 3, solutions with first-order Wachspress rational functions are satisfactory considering 6 6 https://doi.org/10.1051/epjconf/202124703002 EPJ Web of Conferences 247, 03002 (2021)\nPHYSOR2020 Table 2: Solutions from APOLLO3 R⃝/MINARET\nHQ(n,m)\norder\nsplit\nNunk = Nκ × Ndofs\nkeff\n∆keff (pcm)\nϵmax (%)\nϵ (%)\nHQ(2,3)\n1\n1\n762 × 3\n1.002160\n-16.9\n7.16\n0.60\nHQ(2,3)\n1\n2\n2540 × 3\n1.002235\n-9.4\n0.71\n0.11\nHQ(2,3)\n1\n3\n6350 × 3\n1.002251\n-7.8\n1.57\n0.13\nHQ(2,3)\n2\n1\n762 × 6\n1.002262\n-6.6\n2.55\n0.17\nHQ(2,3)\n2\n2\n2540 × 6\n1.002255\n-7.3\n2.12\n0.15\nHQ(2,3)\n2\n3\n6350 × 6\n1.002257\n-7.2\n1.83\n0.14\nHQ(4,5)\n1\n1\n762 × 3\n1.002224\n-10.4\n7.23\n0.53\nHQ(4,5)\n1\n2\n2540 × 3\n1.002304\n-2.4\n2.09\n0.13\nHQ(4,5)\n1\n3\n6350 × 3\n1.002324\n-0.5\n0.47\n0.03\nHQ(4,5)\n2\n1\n762 × 6\n1.002330\n0.1\n0.12\n0.02\nHQ(4,5)\n2\n2\n2540 × 6\n1.002330\n0.1\n0.13\n0.01\nHQ(4,5)\n2\n3\n6350 × 6\n1.002328\n–\n–\n– Table 2: Solutions from APOLLO3 R⃝/MINARET Table 3: Solutions with Wachspress shape functions\nHQ(n,m)\norder\nNunk = Nκ × Ndofs\nkeff\n∆keff (pcm)\nϵmax (%)\nϵ (%)\nHQ(2,3)\n1\n127 × 6\n1.001664\n-65.8\n3.94\n1.89\nHQ(2,3)\n2\n127 × 12\n1.002161\n-16.2\n0.32\n0.07\nHQ(2,3)\n3\n127 × 18\n1.002152\n-17.0\n0.20\n0.08\nHQ(4,5)\n1\n127 × 6\n1.001829\n-49.3\n3.50\n1.78\nHQ(4,5)\n2\n127 × 12\n1.002324\n0.2\n0.29\n0.10\nHQ(4,5)\n3\n127 × 18\n1.002323\n–\n–\n– Table 3: Solutions with Wachspress shape functions the Nunk involved in the problem. Increasing the order is cost effective in terms of Nunk to the\nquality of the result. For this problem, an order of 2 for a given angular quadrature is satisfactory\nwith respect to the solution obtained. Considering a purely computational aspect of using Wachs-\npress basis functions on unrefined hexagons, a coarser mesh is sufficient, and thus Nunk depends\nonly the order of the method, which increases linearly. Indeed, the matricial problem solved locally\ngrows as N 2\ndofs and increasing the order for better accuracy remains competitive so long the com-\nputation time remains reasonably low compared to refining meshes. Hence, the ideal framework\nwould be to increase the order locally to avoid increasing the global Nunk of the problem. the Nunk involved in the problem. 4. CONCLUSIONS In this work, a high-order discontinuous Galerkin formulation using Wachspress rational basis\nfunctions has been applied to the Sn neutron transport equation on 2D honeycomb lattices. In 7 7 EPJ Web of Conferences 247, 03002 (2021)\nPHYSOR2020 https://doi.org/10.1051/epjconf/202124703002 the context of precise solutions to the transport equation on coarse meshes as it is the case for\nmultiphysics calculations on hexagonal reactor geometries (e.g. fast-neutron reactor application),\nit is sufficient to use high-order finite element solutions on unrefined honeycomb meshes. This\nwork contributes to that effort by demonstrating that Wachspress rational basis functions of order\n2 or 3 on honeycomb lattices are well adapted. Further work on the use of these functions is to\nextend them in a polygonal framework and set up p-refinement strategies to reduce the number of\nunknowns. the context of precise solutions to the transport equation on coarse meshes as it is the case for\nmultiphysics calculations on hexagonal reactor geometries (e.g. fast-neutron reactor application),\nit is sufficient to use high-order finite element solutions on unrefined honeycomb meshes. This\nwork contributes to that effort by demonstrating that Wachspress rational basis functions of order\n2 or 3 on honeycomb lattices are well adapted. Further work on the use of these functions is to\nextend them in a polygonal framework and set up p-refinement strategies to reduce the number of\nunknowns. REFERENCES ] A. HEBERT. Applied Reactor Physics. Presses Internationales Polytechnique (2009). [2] D. SCHNEIDER, F. DOLCI, F. GABRIEL, and J.-M. P. et. al. “APOLLO3 R⃝: CEA/DEN\ndeterministic multipurpose platform for core physics analysis.” In Proc. of Int. Mtg. on the\nPhysics of Fuel Cycles and Advanced Nuclear Systems PHYSOR 2016. Sun Valley, USA\n(2016). [3] J. Y. MOLLER and J. J. LAUTARD. “MINARET, a deterministic neutron transport solver for\nnuclear core calculations.” In Proc. of Int. Conf. on Mathematics and Computational Methods\nApplied to Nuclear Science and Engineering. Rio de Janeiro, Brazil (2011). [4] D. FOURNIER and R. LE TELLIER and C. SUTEAU. “Analysis of an a posteriori Error\nEstimator for the Transport Equation with SN and Discontinuous Galerkin Discretizations.”\nAnnals of Nuclear Energy, volume 38, pp. 221–231 (2011). [5] Y. WANG and J. C. RAGUSA. “On the Convergence of DGFEM Applied to the Discrete\nOrdinates Transport Equation for Structured and Unstructured Triangular Meshes.” Nuclear\nScience and Engineering, volume 163, pp. 56–72 (2009). [6] G. G. DAVIDSON and T. S. PALMER. “Finite Element Transport Using Wachspress Ratio-\nnal Basis Functions on Quadrilaterals in Diffusive Regions.” Nuclear Science and Engineer-\ning, volume 159, pp. 242–255 (2008). [7] M. W. HACKEMACK and J. C. RAGUSA. “Quadratic serendipity discontinuous finite ele-\nment discretization for SN transport on arbitrary polygonal grids.” Journal of Computational\nPhysics, volume 374, pp. 188 – 212 (2018). [8] E. L. WACHSPRESS. A Rational Finite Element Basis. Academic Press (1975). [9] C. TALISCHI, G. H. PAULINO, and C. H. LE. “Honeycomb Wachspress finite elements\nfor structural topology optimization.” Structural and Multidisciplinary Optimization, vol-\nume 37(6), pp. 569–583 (2009). [10] L. GASTALDO, R. LE TELLIER, C. SUTEAU, D. FOURNIER, J. M. RUGGIERI. “High-\norder discrete ordinate transport in non-conforming 2D Cartesian meshes.” In Proceedings\nof International Conference on Mathematics, Computational Methods & Reactor Physics. M&C 2009, Saratoga Springs, New York (2009). [11] J. L. GOUT. “Rational Wachspress-type Finite Elements on Regular Hexagons.” IMA Jour-\nnal of Numerical Analysis, volume 5, pp. 59–77 (1985). [12] A. HEBERT. “A Raviart-Thomas-Schneider implementation of the Simplified Pn method in\n3D hexagonal geometry.” In Proc. of Int. Mtg. on Advances in Reactor Physics to Power the\nNuclear Renaissance PHYSOR 2010. ANS, Pittsburgh, USA (2010). [13] A. A. CALLOO and A. HEBERT. “Comparing the discontinuous Galerkin and high-order\ndiamond differencing methods for the discrete ordinates transport equation in hexagonal ge-\nometry.” In Proc. of this meeting (2020). REFERENCES 8 8"