Year: 2011
Type: article
Abstract: The least-squares finite-elements spherical-harmonics (Pn) and discrete-ordinates (Sn) solvers in the SCEPTRE radiation transport code are described. These solvers interface seamlessly with the first-order transport sweeps-base algorithm and the other second-order transport solvers (even-odd parity and self-adjoint angular flux) in SCEPTRE, using the same unstructured-mesh, angular quadrature and cross section framework, designed for massively-parallel computations. The leastsquared methods are derived as a weighted-residual method, using the first-order transport operator to form weighting functions. The result is a symmetric positive definite linear system that can be solved efficiently with a parallel, preconditioned conjugate-gradients algorithm. Unlike the even-odd parity and self-adjoint angular flux implementations, the least-squares formulation does not break down in void or perfectly scattering regions. For the least-squares Pn algorithm, two alternative methods for imposing boundary conditions are described: one using a penalty method to minimize a residual on the boundary, and the other making use of the divergence theorem to transform volume integrals to terms involving boundary integrals. The moments integrals needed for the Pn algorithms are generated using recurrence relations, and the Lebedev angular quadrature is used to provide accurate half-range boundary integral terms on element surfaces up to order 130, needed for imposing the Pn boundary conditions. In particular, a Lebedev/Galerkin quadrature is developed using a Gram-Schmidt procedure for mapping between the discrete angular space and moments space, without loss of accuracy.
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W187415528