Finite Element Continuous Vs Discontinuous Galerkin Ppt

A Discontinuous Galerkin Finite Element Method based axial SN for the 2D/1D transport method

Abstract

The 2D/1D transport method is a prominent solver for the 3D Boltzmann equation due to its strong geometric capability. To pursue high accuracy, the discrete ordinate (SN) method for transport equation is a straightforward choice as the axial 1D solver instead of the traditional nodal expansion method (NEM) for diffusion equation. In practice, the axial SN solver is usually combined with the transverse leakage splitting method to deal with the negative source issue, which strongly affects the convergence of the 2D/1D method. However, transverse leakage splitting breaks the consistency of scalar fluxes between the 2D MOC and 1D SN equations. To solve this problem, a discontinuous Galerkin finite element method (DGFEM) for axial SN together with an improved 2D/1D transport approach is proposed in the present study to ensure that the scalar flux from the 1D axial solver is the same as that from the 2D solver. The accuracy, consistency and computational efficiency of the new scheme are evaluated using various well-known numerical problems, including heterogeneous assembly problem, KUCA benchmark, and C5G5-3D benchmark.

Introduction

High-fidelity three-dimensional transport calculations have attracted more and more attention to pursue the pin-by-pin power resolution. In particular, the method of characteristics (MOC) is recognized as one of the most common approaches due to its excellent geometric capability and easy implementation. However, using the direct 3D MOC calculations in large-scale problems is still a challenge in terms of the computational cost and storage burden. Therefore, preference is given to the 2D/1D transport method that has been successfully applied to various engineering issues, such as the well-known codes CRX-3D (Cho et al., 2002), MPACT (Shen et al., 2018), NECP-X (Wang et al., 2018), and KUAFU (Zhao et al., 2021).

The basic idea of the 2D/1D transport method is to decompose the 3D neutron transport equation into the coupling of a 2D equation and a 1D equation. In detail, the 3D equation is integrated axially to obtain the radial 2D equation, which is usually solved by MOC to resolve the complicated geometry in the radial plane. The 3D equation is also integrated along the radial direction to derive the axial 1D equation to be solved by nodal expansion method (NEM) or discrete ordinate method (SN). To achieve high accuracy, SN is often chosen as the 1D solver rather than NEM. The 2D and 1D equations are coupled with each other by the transverse leakage. Meanwhile, this may result in the negative source and lead to the divergence of the whole 2D/1D calculation scheme (Jarrett, 2017). To solve the problem, transverse leakage splitting approximation has been proposed (Kelly et al., 2013). However, this approach can cause the inconsistency in scalar fluxes between 2D MOC and 1D SN calculations, which makes it difficult for the scalar fluxes in the flat source region to converge thoroughly (Zhao et al., 2021).

In this work, the axial SN approximation based on a discontinuous Galerkin finite element method (DGFEM) (Chauchat et al., 2021, Kanevsky et al., 2007) together with an improved 2D/1D transport scheme is proposed. In particular, the approach keeps the zero-order flux moment consistent with that of the 2D MOC solver. To evaluate the accuracy, consistency, and computational efficiency of the developed scheme, a heterogeneous assembly problem, a KUCA benchmark problem, and a C5G7 benchmark problem are used.

The remainder of this paper is organized as follows. In section 2, the original 2D/1D transport method and its combination with DGFEM-based SN approach are described, respectively. In section 3, a heterogeneous assembly problem, a KUCA benchmark problem, and a C5G5 benchmark problem are utilized to evaluate the excellent consistency, accuracy, and computational efficiency. The conclusions and discussions are given in section 4.

Section snippets

2D/1D transport method

The 2D/1D method starts from the multigroup discrete ordinates 3D neutron transport equation as follows: $\left(\sqrt{1-{\mu }_m^2}\left(\cos \left({\alpha }_m\right)\frac{\partial }{\partial x}+\sin \left({\alpha }_m\right)\frac{\partial }{\partial y}\right)+{\mu }_m\frac{\partial }{\partial z}\right){\phi }_{g,m}\left(x,y,z\right)+{\Sigma }_{t,g}\left(x,y,z\right){\phi }_{g,m}\left(x,y,z\right)=Q_{g,m}\left(x,y,z\right)$ $Q_{g,m}\left(x,y,z\right)=\frac{{\chi }_g}{4\pi k_{eff}}\sum _{g^{\prime }=1}^{G}v{\Sigma }_{f,{\mathrm{g}}^{\prime }}\left(x,y,z\right){\phi }_{g^{\prime }}\left(x,y,z\right)+\frac{1}{4\pi }\sum _{g^{\prime }=1}^G{\Sigma }_{s0,g^{\prime }\to g}\left(x,y,z\right){\phi }_{g^{\prime }}\left(x,y,z\right)$ where $\mu$ is the polar angle, $\alpha$ is the azimuthal angle, ${\Sigma }_t$ , ${\Sigma }_f$ , ${\Sigma }_s$ are the total cross-section, the fission cross-section, and the scattering cross-section, respectively, $\phi$ is the angular flux, $\phi$ is the scalar flux, $v$ is the number of

Numerical results

In this section, the numerical results for a heterogeneous assembly problem, a KUCA benchmark problem, and a C5G7 benchmark problem are presented. The value of theta is set as 0.126, the order of the DGFEM-based SN is fixed at 3, and the number of sweepings per SN calculation is 3. A Gauss-Legendre quadrature set is applied. The number of azimuthal angles is 64, and S₈ is used for polar angles. The order of Fourier moment expansion of the transverse leakage term ranges from 1 to 3. There are 5

Conclusions

Compared with the NEM, the SN-based 1D axial solver could capture the anisotropic angular information. Transverse leakage splitting is an important method to tackle the negative source problems in the 2D/1D scheme. However, this method breaks the consistency between the 2D-MOC/1D-SN solvers and the original 3D equations. The scalar fluxes calculated using the 2D MOC and 1D SN are not consistent as well, which may lead to the convergence issue of the 2D/1D method. To solve this problem, a

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work is supported by Chinese National Natural Science Foundation Project 12075067, 11505102 and 11375099, National Key R&D Program of China 2018YFE0180900, Chinese National S&T Major Project 2018ZX06 529 902013, and IAEA CRP I31020.

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Source: https://www.sciencedirect.com/science/article/pii/S0149197022002657

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