Spelling suggestions: "subject:"diffusion synthetic acceleration"" "subject:"diiffusion synthetic acceleration""
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Domain decomposition methods for nuclear reactor modelling with diffusion accelerationBlake, Jack January 2016 (has links)
In this thesis we study methods for solving the neutron transport equation (or linear Boltzmann equation). This is an integro-differential equation that describes the behaviour of neutrons during a nuclear fission reaction. Applications of this equation include modelling behaviour within nuclear reactors and the design of shielding around x-ray facilities in hospitals. Improvements in existing modelling techniques are an important way to address environmental and safety concerns of nuclear reactors, and also the safety of people working with or near radiation. The neutron transport equation typically has seven independent variables, however to facilitate rigorous mathematical analysis we consider the monoenergetic, steady-state equation without fission, and with isotropic interactions and isotropic source. Due to its high dimension, the equation is usually solved iteratively and we begin by considering a fundamental iterative method known as source iteration. We prove that the method converges assuming piecewise smooth material data, a result that is not present in the literature. We also improve upon known bounds on the rate of convergence assuming constant material data. We conclude by numerically verifying this new theory. We move on to consider the use of a specific, well-known diffusion equation to approximate the solution to the neutron transport equation. We provide a thorough presentation of its derivation (along with suitable boundary conditions) using an asymptotic expansion and matching procedure, a method originally presented by Habetler and Matkowsky in 1975. Next we state the method of diffusion synthetic acceleration (DSA) for which the diffusion approximation is instrumental. From there we move on to explore a new method of seeing the link between the diffusion and transport equations through the use of a block operator argument. Finally we consider domain decomposition algorithms for solving the neutron transport equation. Such methods have great potential for parallelisation and for the local application of different solution methods. A motivation for this work was to build an algorithm applying DSA only to regions of the domain where it is required. We give two very different domain decomposed source iteration algorithms, and we prove the convergence of both of these algorithms. This work provides a rigorous mathematical foundation for further development and exploration in this area. We conclude with numerical results to illustrate the new convergence theory, but also solve a physically-motivated problem using hybrid source iteration/ DSA algorithms and see significant reductions in the required computation time.
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DSA Preconditioning for the S_N Equations with Strictly Positive Spatial DiscretizationBruss, Donald 2012 May 1900 (has links)
Preconditioners based upon sweeps and diffusion-synthetic acceleration (DSA) have been constructed and applied to the zeroth and first spatial moments of the 1-D transport equation using SN angular discretization and a strictly positive nonlinear spatial closure (the CSZ method). The sweep preconditioner was applied using the linear discontinuous Galerkin (LD) sweep operator and the nonlinear CSZ sweep operator. DSA preconditioning was applied using the linear LD S2 equations and the nonlinear CSZ S2 equations. These preconditioners were applied in conjunction with a Jacobian-free Newton Krylov (JFNK) method utilizing Flexible GMRES.
The action of the Jacobian on the Krylov vector was difficult to evaluate numerically with a finite difference approximation because the angular flux spanned many orders of magnitude. The evaluation of the perturbed residual required constructing the nonlinear CSZ operators based upon the angular flux plus some perturbation. For cases in which the magnitude of the perturbation was comparable to the local angular flux, these nonlinear operators were very sensitive to the perturbation and were significantly different than the unperturbed operators. To resolve this shortcoming in the finite difference approximation, in these cases the residual evaluation was performed using nonlinear operators "frozen" at the unperturbed local psi. This was a Newton method with a perturbation fixup. Alternatively, an entirely frozen method always performed the Jacobian evaluation using the unperturbed nonlinear operators. This frozen JFNK method was actually a Picard iteration scheme. The perturbed Newton's method proved to be slightly less expensive than the Picard iteration scheme.
The CSZ sweep preconditioner was significantly more effective than preconditioning with the LD sweep. Furthermore, the LD sweep is always more expensive to apply than the CSZ sweep. The CSZ sweep is superior to the LD sweep as a preconditioner. The DSA preconditioners were applied in conjunction with the CSZ sweep. The nonlinear CSZ DSA preconditioner did not form a more effective preconditioner than the linear DSA preconditioner in this 1-D analysis. As it is very difficult to construct a CSZ diffusion equation in more than one dimension, it will be very beneficial if the results regarding the effectiveness of the LD DSA preconditioner are applicable to multi-dimensional problems.
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Um método sintético de difusão para aceleração do esquema de fonte de espalhamento em cálculos SN unidimensionais de fonte fixa / A diffusion synthetic acceleration method for the scattering source iteration scheme in fixed source slab-geometry SN calculationsFrederico Pereira Santos 09 September 2011 (has links)
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O esquema iterativo de fonte de espalhamento (SI) é tradicionalmente aplicado para a
convergência da solução numérica de malha fina para problemas de transporte de nêutrons
monoenergéticos na formulação de ordenadas discretas com fonte fixa. O esquema SI é muito
simples de se implementar sob o ponto de vista computacional; porém, o esquema SI pode
apresentar taxa de convergência muito lenta, principalmente para meios difusivos (baixa
absorção) com vários livres caminhos médios de extensão. Nesta dissertação descrevemos
uma técnica de aceleração baseada na melhoria da estimativa inicial para a distribuição da
fonte de espalhamento no interior do domínio de solução. Em outras palavras, usamos como
estimativa inicial para o fluxo escalar médio na grade de discretização de malha fina,
presentes nos termos da fonte de espalhamento das equações discretizadas SN usadas nas
varreduras de transporte, a solução numérica da equação da difusão de nêutrons em grade
espacial de malha grossa com condições de contorno especiais, que aproximam as condições
de contorno prescritas que são clássicas em cálculos SN, incluindo condições de contorno do
tipo vácuo. Para aplicarmos esta solução gerada pela equação da difusão em grade de
discretização de malha grossa nas equações discretizadas SN de transporte na grade de
discretização de malha fina, primeiro implementamos uma reconstrução espacial dentro de
cada nodo de discretização, e então determinamos o fluxo escalar médio em grade de
discretização de malha fina para usá-lo nos termos da fonte de espalhamento. Consideramos
um número de experimentos numéricos para ilustrar a eficiência oferecida pela presente
técnica (DSA) de aceleração sintética de difusão. / The scattering source iterative (SI) scheme is traditionally applied to converge finemesh
numerical solutions to fixed-source discrete ordinates neutron transport problems.
The SI scheme is very simple to implement under a computational viewpoint. However, the
SI scheme may show very slow convergence rate, mainly for diffusive media (low absorption)
with several mean free paths in extent. In this work we describe an acceleration technique
based on an improved initial guess for the scattering source distribution within the slab. In
other words, we use as initial guess for the fine-mesh average scalar flux in the scattering
source terms of the SN discretized equations used in the transport sweeps, the coarse-mesh
solution of the neutron diffusion equation with special boundary conditions to account for the
classical SN prescribed boundary conditions, including vacuum boundary conditions. To
apply this coarse-mesh diffusion solution into the fine-mesh SN transport sweep discretized
equations, we first perform within-node spatial reconstruction, and then we determine the
fine-mesh average scalar flux for use in the scattering source terms. We consider a number of
numerical experiments to illustrate the efficiency of the offered diffusion synthetic
acceleration (DSA) technique.
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Um método sintético de difusão para aceleração do esquema de fonte de espalhamento em cálculos SN unidimensionais de fonte fixa / A diffusion synthetic acceleration method for the scattering source iteration scheme in fixed source slab-geometry SN calculationsFrederico Pereira Santos 09 September 2011 (has links)
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O esquema iterativo de fonte de espalhamento (SI) é tradicionalmente aplicado para a
convergência da solução numérica de malha fina para problemas de transporte de nêutrons
monoenergéticos na formulação de ordenadas discretas com fonte fixa. O esquema SI é muito
simples de se implementar sob o ponto de vista computacional; porém, o esquema SI pode
apresentar taxa de convergência muito lenta, principalmente para meios difusivos (baixa
absorção) com vários livres caminhos médios de extensão. Nesta dissertação descrevemos
uma técnica de aceleração baseada na melhoria da estimativa inicial para a distribuição da
fonte de espalhamento no interior do domínio de solução. Em outras palavras, usamos como
estimativa inicial para o fluxo escalar médio na grade de discretização de malha fina,
presentes nos termos da fonte de espalhamento das equações discretizadas SN usadas nas
varreduras de transporte, a solução numérica da equação da difusão de nêutrons em grade
espacial de malha grossa com condições de contorno especiais, que aproximam as condições
de contorno prescritas que são clássicas em cálculos SN, incluindo condições de contorno do
tipo vácuo. Para aplicarmos esta solução gerada pela equação da difusão em grade de
discretização de malha grossa nas equações discretizadas SN de transporte na grade de
discretização de malha fina, primeiro implementamos uma reconstrução espacial dentro de
cada nodo de discretização, e então determinamos o fluxo escalar médio em grade de
discretização de malha fina para usá-lo nos termos da fonte de espalhamento. Consideramos
um número de experimentos numéricos para ilustrar a eficiência oferecida pela presente
técnica (DSA) de aceleração sintética de difusão. / The scattering source iterative (SI) scheme is traditionally applied to converge finemesh
numerical solutions to fixed-source discrete ordinates neutron transport problems.
The SI scheme is very simple to implement under a computational viewpoint. However, the
SI scheme may show very slow convergence rate, mainly for diffusive media (low absorption)
with several mean free paths in extent. In this work we describe an acceleration technique
based on an improved initial guess for the scattering source distribution within the slab. In
other words, we use as initial guess for the fine-mesh average scalar flux in the scattering
source terms of the SN discretized equations used in the transport sweeps, the coarse-mesh
solution of the neutron diffusion equation with special boundary conditions to account for the
classical SN prescribed boundary conditions, including vacuum boundary conditions. To
apply this coarse-mesh diffusion solution into the fine-mesh SN transport sweep discretized
equations, we first perform within-node spatial reconstruction, and then we determine the
fine-mesh average scalar flux for use in the scattering source terms. We consider a number of
numerical experiments to illustrate the efficiency of the offered diffusion synthetic
acceleration (DSA) technique.
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Adaptive Mesh Refinement Solution Techniques for the Multigroup SN Transport Equation Using a Higher-Order Discontinuous Finite Element MethodWang, Yaqi 16 January 2010 (has links)
In this dissertation, we develop Adaptive Mesh Refinement (AMR) techniques
for the steady-state multigroup SN neutron transport equation using a higher-order
Discontinuous Galerkin Finite Element Method (DGFEM). We propose two error estimations,
a projection-based estimator and a jump-based indicator, both of which
are shown to reliably drive the spatial discretization error down using h-type AMR.
Algorithms to treat the mesh irregularity resulting from the local refinement are
implemented in a matrix-free fashion. The DGFEM spatial discretization scheme
employed in this research allows the easy use of adapted meshes and can, therefore,
follow the physics tightly by generating group-dependent adapted meshes. Indeed,
the spatial discretization error is controlled with AMR for the entire multigroup SNtransport
simulation, resulting in group-dependent AMR meshes. The computing
efforts, both in memory and CPU-time, are significantly reduced. While the convergence
rates obtained using uniform mesh refinement are limited by the singularity
index of transport solution (3/2 when the solution is continuous, 1/2 when it is discontinuous),
the convergence rates achieved with mesh adaptivity are superior. The
accuracy in the AMR solution reaches a level where the solution angular error (or ray
effects) are highlighted by the mesh adaptivity process. The superiority of higherorder
calculations based on a matrix-free scheme is verified on modern computing architectures.
A stable symmetric positive definite Diffusion Synthetic Acceleration (DSA)
scheme is devised for the DGFEM-discretized transport equation using a variational
argument. The Modified Interior Penalty (MIP) diffusion form used to accelerate the
SN transport solves has been obtained directly from the DGFEM variational form of
the SN equations. This MIP form is stable and compatible with AMR meshes. Because
this MIP form is based on a DGFEM formulation as well, it avoids the costly
continuity requirements of continuous finite elements. It has been used as a preconditioner
for both the standard source iteration and the GMRes solution technique
employed when solving the transport equation. The variational argument used in
devising transport acceleration schemes is a powerful tool for obtaining transportconforming
diffusion schemes.
xuthus, a 2-D AMR transport code implementing these findings, has been developed
for unstructured triangular meshes.
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