Bruno Turcksin

The deal.II Library, Version 8.4

Abstract This paper provides an overview of the new features of the finite element library deal.II version 8.5.

Goal-Oriented h-Adaptivity for the Multigroup SPN Equations

Abstract We investigate application of goal-oriented mesh adaptivity to the SPN multigroup equations. This technique utilizes knowledge of the computational goal and combines it with mesh adaptivity to accurately and rapidly compute quantities of interest. Specifically, the local error is weighted by the importance of a given cell toward the computational goal, resulting in appropriate goal-oriented error estimates. Even though this approach requires the solution of an adjoint (dual) problem, driven by a specific source term for a given quantity of interest, the work reported here clearly shows the benefits of such a method. We demonstrate the level of accuracy this method can achieve using two-dimensional and three-dimensional numerical test cases for one-group and two-group models and compare results with more traditional mesh refinement and uniformly refined meshes. The test cases consider situations in which the radiative flux of a source is shielded and are designed to prototypically explore the range of conditions under which our methods improve on other refinement algorithms. In particular, they model strong contrasts in material properties, a situation ubiquitous in nuclear engineering.

Angular Multigrid Preconditioner for Krylov-Based Solution Techniques Applied to the Sn Equations with Highly Forward-Peaked Scattering

It is well known that the diffusion synthetic acceleration (DSA) methods for the Sn equations become ineffective in the Fokker-Planck forward-peaked scattering limit. In response to this deficiency, Morel and Manteuffel (1991) developed an angular multigrid method for the 1-D Sn equations. This method is very effective, costing roughly twice as much as DSA per source iteration, and yielding a maximum spectral radius of approximately 0.6 in the Fokker-Planck limit. Pautz, Adams, and Morel (PAM) (1999) later generalized the angular multigrid to 2-D, but it was found that the method was unstable with sufficiently forward-peaked mappings between the angular grids. The method was stabilized via a filtering technique based on diffusion operators, but this filtering also degraded the effectiveness of the overall scheme. The spectral radius was not bounded away from unity in the Fokker-Planck limit, although the method remained more effective than DSA. The purpose of this article is to recast the multidimensional PAM angular multigrid method without the filtering as an Sn preconditioner and use it in conjunction with the Generalized Minimal RESidual (GMRES) Krylov method. The approach ensures stability and our computational results demonstrate that it is also significantly more efficient than an analogous DSA-preconditioned Krylov method.

WorkStream -- A Design Pattern for Multicore-Enabled Finite Element Computations

Many operations that need to be performed in modern finite element codes can be described as an operation that needs to be done independently on every cell, followed by a reduction of these local results into a global data structure. For example, matrix assembly, estimating discretization errors, or converting nodal values into data structures that can be output in visualization file formats all fall into this class of operations. Using this realization, we identify a software design pattern that we call WorkStream and that can be used to model such operations and enables the use of multicore shared memory parallel processing. We also describe in detail how this design pattern can be efficiently implemented, and we provide numerical scalability results from its use in the deal.II software library.