HyeongKae Park

Application of the Jacobian-Free Newton-Krylov Method to Nonlinear Acceleration of Transport Source Iteration in Slab Geometry

Abstract The use of the Jacobian-free Newton-Krylov (JFNK) method within the context of nonlinear diffusion acceleration (NDA) of source iteration is explored. The JFNK method is a synergistic combination of Newton’s method as the nonlinear solver and Krylov methods as the linear solver. JFNK methods do not form or store the Jacobian matrix, and Newton’s method is executed via probing the nonlinear discrete function to approximate the required matrix-vector products. Current application of NDA relies upon a fixed-point, or Picard, iteration to resolve the nonlinearity. We show that the JFNK method can be used to replace this Picard iteration with a Newton iteration. The Picard linearization is retained as a preconditioner. We show that the resulting JFNK-NDA capability provides benefit in some regimes. Furthermore, we study the effects of a two-grid approach, and the required intergrid transfers when the higher-order transport method is solved on a fine mesh compared to the low-order acceleration problem.

Tightly Coupled Multiphysics Algorithms for Pebble Bed Reactors

Abstract We have developed a tightly coupled multiphysics simulation tool for the pebble bed reactor (PBR) concept, a specific type of very high temperature gas-cooled reactor. The simulation tool PRONGHORN takes advantage of the Multiphysics Object-Oriented Simulation Environment library and is capable of solving multidimensional thermal-fluid and neutronics problems implicitly with a Newton-based approach. Expensive Jacobian matrix formation is alleviated via the Jacobian-free Newton-Krylov method, and physics-based preconditioning is applied to minimize Krylov iterations. Motivation for the work is provided via analysis and numerical experiments on simpler multiphysics reactor models. We then provide detail of the physical models and numerical methods in PRONGHORN. Finally, PRONGHORN’s algorithmic capability is demonstrated on a number of PBR test cases.

Acceleration of k-Eigenvalue / Criticality Calculations using the Jacobian-Free Newton-Krylov Method

Abstract We present a new approach for the k-eigenvalue problem using a combination of classical power iteration and the Jacobian-free Newton-Krylov (JFNK) method. The method poses the k-eigenvalue problem as a fully coupled nonlinear system, which is solved by JFNK with an effective block preconditioning consisting of the power iteration and algebraic multigrid. We demonstrate effectiveness and algorithmic scalability of the method on a one-dimensional, one-group problem and two two-dimensional two-group problems and provide comparison to other efforts using similar algorithmic approaches.

An Efficient and Time Accurate, Moment-Based Scale-Bridging Algorithm for Thermal Radiative Transfer Problems

We present physics-based preconditioning and a time-stepping strategy for a moment-based scale-bridging algorithm applied to the thermal radiative transfer equation. Our goal is to obtain (asymptotically) second-order time accurate and consistent solutions without nonlinear iterations between the high-order (HO) transport equation and the low-order (LO) continuum system within a time step. Modified equation analysis shows that this can be achieved via a simple predictor-corrector time stepping that requires one inversion of the transport operator per time step. We propose a physics-based preconditioning based on a combination of the nonlinear elimination technique and Fleck--Cummings linearization. As a result, the LO system can be solved efficiently via a multigrid preconditioned Jacobian-free Newton--Krylov method. For a set of numerical test problems, the physics-based preconditioner reduces the number of GMRES iterations by a factor of 3

On physics-based preconditioning of the Navier-Stokes equations

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Parallel multiphysics algorithms and software for computational nuclear engineering

4 as compared to a standard preconditioner for advection-di...

Hybrid Deterministic/Monte Carlo Neutronics

We develop a fully implicit scheme for the Navier-Stokes equations, in conservative form, for low to intermediate Mach number flows. Simulations in this range of flow regime produce stiff wave systems in which slow dynamical (advective) modes coexist with fast acoustic modes. Viscous and thermal diffusion effects in refined boundary layers can also produce stiffness. Implicit schemes allow one to step over the fast wave phenomena (or unresolved viscous time scales), while resolving advective time scales. In this study we employ the Jacobian-free Newton-Krylov (JFNK) method and develop a new physics-based preconditioner. To aid in overcoming numerical stiffness caused by the disparity between acoustic and advective modes, the governing equations are transformed into the primitive-variable form in a preconditioning step. The physics-based preconditioning incorporates traditional semi-implicit and physics-based splitting approaches without a loss of consistency between the original and preconditioned systems. The resulting algorithm is capable of solving low-speed natural circulation problems (M~10^-^4) with significant heat flux as well as intermediate speed (M~1) flows efficiently by following dynamical (advective) time scales of the problem.

Comparison of multimesh hp-FEM to interpolation and projection methods for spatial coupling of thermal and neutron diffusion calculations

There is a growing trend in nuclear reactor simulation to consider multiphysics problems. This can be seen in reactor analysis where analysts are interested in coupled flow, heat transfer and neutronics, and in fuel performance simulation where analysts are interested in thermomechanics with contact coupled to species transport and chemistry. These more ambitious simulations usually motivate some level of parallel computing. Many of the coupling efforts to date utilize simple code coupling or first-order operator splitting, often referred to as loose coupling. While these approaches can produce answers, they usually leave questions of accuracy and stability unanswered. Additionally, the different physics often reside on separate grids which are coupled via simple interpolation, again leaving open questions of stability and accuracy. Utilizing state of the art mathematics and software development techniques we are deploying next generation tools for nuclear engineering applications. The Jacobian-free Newton-Krylov (JFNK) method combined with physics-based preconditioning provide the underlying mathematical structure for our tools. JFNK is understood to be a modern multiphysics algorithm, but we are also utilizing its unique properties as a scale bridging algorithm. To facilitate rapid development of multiphysics applications we have developed the Multiphysics Object-Oriented Simulation Environment (MOOSE). Examples from two MOOSE-based applications: PRONGHORN, our multiphysics gas cooled reactor simulation tool and BISON, our multiphysics, multiscale fuel performance simulation tool will be presented.

Direct Numerical Simulation of Interfacial Flows: Implicit Sharp-Interface Method (I-SIM)

In this paper we describe a hybrid deterministic/Monte Carlo algorithm for neutron transport simulation. The algorithm is based on nonlinear accelerators for source iteration, using Monte Carlo methods for the purely absorbing high-order problem and a Jacobian-free Newton--Krylov iteration for the low-order problem. We couple the Monte Carlo solution with the low-order problem using filtering to smooth the flux and current from the Monte Carlo solver and an analytic Jacobian-vector product to avoid numerical differentiation of the Monte Carlo results. We use a continuous energy deposition tally for the Monte Carlo simulation. We conclude the paper with numerical results which illustrate the effectiveness of the new algorithm.

Recovery Discontinuous Galerkin Jacobian-free Newton-Krylov Method for all-speed flows

Multiphysics solution challenges are legion within the ?eld of nuclear reactor design and analysis. One major issue concerns the coupling between heat and neutron ?ow (neutronics) within the reactor assembly. These phenomena are usually very tightly interdependent, as large amounts of heat are quickly produced with an increase in ?ssion events within the fuel, which raises the temperature that a?ects the neutron cross section of the fuel. Furthermore, there typically is a large diversity of time and spatial scales between mathematical models of heat and neutronics. Indeed, the di?erent spatial resolution requirements often lead to the use of very di?erent meshes for the two phenomena. As the equations are coupled, one must take care in exchanging solution data between them, or signi?cant error can be introduced into the coupled problem. We propose a novel approach to the discretization of the coupled problem on di?erent meshes based on an adaptive multimesh higher-order ?nite element method (hp-FEM), and compare it to popular interpolation and projection methods. We show that the multimesh hp-FEM method is signi?cantly more accurate than the interpolation and projection approaches considered in this study.