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This paper describes an hp-angular adaptivity algorithm in the discrete ordinates method for Boltzmann transport applications with strong angular effects. This adaptivity uses discontinuous finite element quadrature sets with different degrees, which updates both angular mesh and the degree of the underlying discontinuous finite element basis functions, allowing different angular local refinement to be applied in space. The regular and goal-based error metrics are considered in this algorithm to locate some regions to be refined. A mapping algorithm derived by moment conservation is developed to pass the angular solution between spatial regions with different quadrature sets. The proposed method is applied to some test problems that demonstrate the ability of this hp-angular adaptivity to resolve complex fluxes with relatively few angular unknowns. Results illustrate that a reduction to approximately 1/50 in quadrature ordinates for a given accuracy compared with uniform angular discretization. This method therefore offers a highly efficient angular adaptivity for investigating difficult particle transport problems.

The discrete ordinates method (SN) is one of the major shielding calculation method, which is suitable for solving deep-penetration transport problems. Our objective is to explore the available quadrature sets and to improve the accuracy in shielding problems involving strong anisotropy. The linear discontinuous finite-element (LDFE) quadrature sets based on the icosahedron (in short, ICLDFE quadrature sets) are developed by defining projected points on the surfaces of the icosahedron. Weights are then introduced in the integration of the discontinuous finite-element basis functions in the relevant angular regions. The multivariate secant method is used to optimize the discrete directions and their corresponding weights. The numerical integration of polynomials in the direction cosines and the Kobayashi benchmark are used to analyze and verify the properties of these new quadrature sets. Results show that the ICLDFE quadrature sets can exactly integrate the zero-order and first-order of the spherical harmonic functions over one-twentieth of the spherical surface. As for the Kobayashi benchmark problem, the maximum relative error between the fifth-order ICLDFE quadrature sets and references is only −0.55%. The ICLDFE quadrature sets provide better integration precision of the spherical harmonic functions in local discrete angle domains and higher accuracy for simple shielding problems.

This paper proposes a family of high-degree discontinuous finite element (DFEM) quadrature sets designed to improve accuracy of integrals over local angular domain in the discrete ordinates method. The angular domain is divided into three types of spherical meshes by inscribing the octahedron, icosahedron and cube, respectively. High-degree discontinuous finite element basis functions are generated over different spherical meshes to resolve associated quadrature weights, producing high-degree DFEM quadrature sets. The performance and accuracy of the DFEM quadrature family was tested in the integration of spherical harmonics functions and some transport problems. Results demonstrate that k th-degree DFEM quadrature sets with any quadrature order can exactly integrate up to all k th-degree spherical harmonics functions in the direction cosines over local angular domain. Results also suggest that the performance of high-degree DFEM quadrature sets is comparable or better than that of traditional quadrature sets for transport problems with unsmooth or nearly discontinuous angular flux solutions. Locally-refined DFEM quadrature sets can obtain the same accuracy with less expense compared with uniform quadrature sets. DFEM quadrature sets with different degrees can offer two types of local refinement by increasing the quadrature order and changing the degree of discontinuous finite element basis functions, which can be further expected for use in an angular adaptive algorithm. [ABSTRACT FROM AUTHOR]