A Multilevel in Space and Energy Solver for Multigroup Diffusion and Coarse Mesh Finite Difference Eigenvalue Problems

In reactor physics, the efficient solution of the multigroup neutron diffusion eigenvalue problem is desired for various applications. The diffusion problem is a lower-order but reasonably accurate approximation to the higher-fidelity multigroup neutron transport eigenvalue problem. In cases where the full-fidelity of the transport solution is needed, the solution of the diffusion problem can be used to accelerate the convergence of transport solvers via methods such as Coarse Mesh Finite Difference (CMFD). The diffusion problem can have O(108) unknowns, and, despite being orders of magnitude smaller than a typical transport problem, obtaining its solution is still not a trivial task. In the Michigan Parallel Characteristics Transport (MPACT) code, the lack of an efficient CMFD solver has resulted in a computational bottleneck at the CMFD step. Solving the CMFD system can comprise 50% or more of the overall runtime in MPACT when the de facto default CMFD solver is used; addressing this bottleneck is the motivation for our work. The primary focus of this thesis is the theory, development, implementation, and testing of a new Multilevel-in-Space-and-Energy Diffusion (MSED) method for efficiently solving multigroup diffusion and CMFD eigenvalue problems. As its name suggests, MSED efficiently converges multigroup diffusion and CMFD problems by leveraging lower-order systems with coarsened energy and/or spatial grids. The efficiency of MSED is verified via various Fourier analyses of its components and via testing in a 1-D diffusion code. In the later chapters of this thesis, the MSED method is tested on a variety of reactor problems in MPACT. Compared to the default CMFD solver, our implementation of MSED in MPACT has resulted in an ~8-12x reduction in the CMFD runtime required by MPACT for single statepoint calculations on 3-D, full-core, 51-group reactor models. The number of transport sweeps is also typically reduced by the use of MSED, which is able to better converge the CMFD system than the default CMFD solver. This leads to a further savings in overall runtime that is not captured by the differences in CMFD runtime.PHDNuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/146075/1/bcyee_1.pd

Multigroup diffusion preconditioners for multiplying fixed-source transport problems

Several preconditioners based on multigroup di usion are developed for application to multiplying fi xed-source transport problems using the discrete ordinates method. By starting from standard, one-group, diff usion synthetic acceleration (DSA), a multigroup diff usion preconditioner is constructed that shares the same fi ne mesh as the transport problem. As a cheaper but effective alternative, a two-grid, coarse-mesh, multigroup diff usion preconditioner is examined, for which a variety of homogenization schemes are studied to generate the coarse mesh operator. Finally, a transport-corrected diff usion preconditioner based on application of the Newton-Shulz algorithm is developed. The results of several numerical studies indicate the coarse-mesh, diff usion preconditioners work very well. In particular, a coarse-mesh, transport-corrected, diff usion preconditioner reduced the computational time of multigroup GMRES by up to a factor of 17 and outperformed best-case Gauss-Seidel results by over an order of magnitude for all problems studied

Adaptive tree multigrids and simplified spherical harmonics approximation in deterministic neutral and charged particle transport

A new deterministic three-dimensional neutral and charged particle transport code, MultiTrans, has been developed. In the novel approach, the adaptive tree multigrid technique is used in conjunction with simplified spherical harmonics approximation of the Boltzmann transport equation. The development of the new radiation transport code started in the framework of the Finnish boron neutron capture therapy (BNCT) project. Since the application of the MultiTrans code to BNCT dose planning problems, the testing and development of the MultiTrans code has continued in conventional radiotherapy and reactor physics applications. In this thesis, an overview of different numerical radiation transport methods is first given. Special features of the simplified spherical harmonics method and the adaptive tree multigrid technique are then reviewed. The usefulness of the new MultiTrans code has been indicated by verifying and validating the code performance for different types of neutral and charged particle transport problems, reported in separate publications.Väitöstutkimuksen tuloksena on kehitetty uusi tietokoneohjelma varauksettomien ja varauksellisten hiukkasten kuten neutronien, fotonien ja elektronien etenemisen mallinnukseen. MultiTrans-ohjelma mahdollistaa säteilyn etenemisen mallinnuksen mielivaltaisessa 3D-geometriassa. Laskentageometria generoidaan suoraan CAD-mallista, jolloin voidaan käyttää moderneja suunnittelutyökaluja. Laskentaverkko on puumaisesti itsetarkentuva materiaalien rajapinnoilla, joissa hila muodostuu automaattisesti hienojakoisimmaksi. Näin monimutkainenkin geometria voidaan kuvata yksityiskohtaisesti merkittävästi pienemmällä hilapisteiden määrällä verrattuna tasajakoiseen hilaan. Laskentaverkon puumaisuudesta seuraa että ongelmalle löytyy aina myös karkeammat hilaesitykset. Tällöin kuljetusyhtälön iteratiivisessa ratkaisussa voidaan käyttää ns. moniverkkotekniikkaa jossa ongelma ratkaistaan ensin hyvin karkeassa esityksessä ja tätä ratkaisua käytetään alkuarvauksena yhä hienojakoisemmissa hiloissa. Näin nopeutetaan iteratiivisen ratkaisun löytymistä huomattavasti. Myös laskentaverkon puumaisuus ja sen myötä hilapisteiden vähäisempi määrä nopeuttaa iteratiivista ratkaisua. Kyseessä on tiettävästä ensimmäinen puumoniverkkotekniikan sovellutus säteilyn etenemisen mallinnukseen. MultiTransia on testattu erilaisten sädehoitojen (esimerkiksi VTT:n Otaniemen ydintutkimusreaktorilla annettavan boorineutronikaappaushoidon) sekä reaktorifysiikan laskentaongelmiin. Ongelmaksi on jossain määrin osoittautunut säteilyn kulkeutumisyhtälölle käytetty yksinkertaistettu palloharmoninen kehitelmä, jonka tarkkuus ei kaikissa tapauksissa vastaa asetettuja vaatimuksia

CASTRO: A New Compressible Astrophysical Solver. III. Multigroup Radiation Hydrodynamics

We present a formulation for multigroup radiation hydrodynamics that is correct to order $O(v/c)$ using the comoving-frame approach and the flux-limited diffusion approximation. We describe a numerical algorithm for solving the system, implemented in the compressible astrophysics code, CASTRO. CASTRO uses an Eulerian grid with block-structured adaptive mesh refinement based on a nested hierarchy of logically-rectangular variable-sized grids with simultaneous refinement in both space and time. In our multigroup radiation solver, the system is split into three parts, one part that couples the radiation and fluid in a hyperbolic subsystem, another part that advects the radiation in frequency space, and a parabolic part that evolves radiation diffusion and source-sink terms. The hyperbolic subsystem and the frequency space advection are solved explicitly with high-order Godunov schemes, whereas the parabolic part is solved implicitly with a first-order backward Euler method. Our multigroup radiation solver works for both neutrino and photon radiation.Comment: accepted by ApJS, 27 pages, 20 figures, high-resolution version available at https://ccse.lbl.gov/Publications/wqzhang/castro3.pd

Anisotropic Adaptivity and Subgrid Scale Modelling for the Solution of the Neutron Transport Equation with an Emphasis on Shielding Applications

This thesis demonstrates advanced new discretisation and adaptive meshing technologies that improve the accuracy and stability of using finite element discretisations applied to the Boltzmann transport equation (BTE). This equation describes the advective transport of neutral particles such as neutrons and photons within a domain. The BTE is difficult to solve, due to its large phase space (three dimensions of space, two of angle and one each of energy and time) and the presence of non-physical oscillations in many situations. This work explores the use of a finite element method that combines the advantages of the two schemes: the discontinuous and continuous Galerkin methods. The new discretisation uses multiscale (subgrid) finite elements that work locally within each element in the finite element mesh in addition to a global, continuous, formulation. The use of higher order functions that describe the variation of the angular flux over each element is also explored using these subgrid finite element schemes. In addition to the spatial discretisation, methods have also been developed to optimise the finite element mesh in order to reduce resulting errors in the solution over the domain, or locally in situations where there is a goal of specific interest (such as a dose in a detector region). The chapters of this thesis have been structured to be submitted individually for journal publication, and are arranged as follows. Chapter 1 introduces the reader to motivation behind the research contained within this thesis. Chapter 2 introduces the forms of the BTE that are used within this thesis. Chapter 3 provides the methods that are used, together with examples, of the validation and verification of the software that was developed as a result of this work, the transport code RADIANT. Chapter 4 introduces the inner element subgrid scale finite element discretisation of the BTE that forms the basis of the discretisations within RADIANT and explores its convergence and computational times on a set of benchmark problems. Chapter 5 develops the error metrics that are used to optimise the mesh in order to reduce the discretisation error within a finite element mesh using anisotropic adaptivity that can use elongated elements that accurately resolves computational demanding regions, such as in the presence of shocks. The work of this chapter is then extended in Chapter 6 that forms error metrics for goal based adaptivity to minimise the error in a detector response. Finally, conclusions from this thesis and suggestions for future work that may be explored are discussed in Chapter 7.Open Acces

Multilevel matrix-free preconditioner to solve linear systems associated with a the time-dependent SPN equations

[EN] The evolution of the neutronic power inside of a nuclear reactor core can be approximated by means of the diffusive time-dependent simplified spherical harmonics equations (SPN). For the spatial discretization of these equations, a continuous Galerkin high order finite element method is applied to obtain a semi-discrete system of equations that is usually stiff. A semi-implicit time scheme is used for the time discretization and many linear systems are needed to be solved and previously, preconditioned. The aim of this work is to speed up the convergence of the linear systems solver with a multilevel preconditioner that uses different degrees of the polynomials used in the finite element method. Furthermore, as the matrices that appear in this type of system are very large and sparse, a matrix-free implementation of the preconditioner is developed to avoid the full assembly of the matrices. A benchmark transient tests this methodology. Numerical results show, in comparison with the block Gauss-Seidel preconditioner, an improvement in terms of number of iterations and the necessity of computational resources.This work has been partially supported by Spanish Ministerio de Economía y Competitividad under projects ENE2017-89029-P and MTM2017-85669-P. Furthermore, this work has been financed by the Generalitat Valenciana under the project PROMETEO/2018/035.Carreño, A.; Vidal-Ferràndiz, A.; Ginestar, D.; Verdú, G. (2022). Multilevel matrix-free preconditioner to solve linear systems associated with a the time-dependent SPN equations. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 68-77. https://doi.org/10.4995/YIC2021.2021.12510OCS687