Preconditioned Jacobian-free Newton Krylov Research Articles

An effective method for modeling 1D two-phase two-fluid six-equation model with automatic differentiation approach

The modeling of the two-fluid model with plenty of realistic closure correlations has been a tough problem because of the difficulty of forming the Jacobian matrix manually. To predict the phase appearance/disappearance in two-phase flow problems, the high-order spatial and fully implicit temporal discretization typically should be adopted. However, considering the requirement of a fairly accurate evaluation of the Jacobian matrix, the implementation of high-order schemes is a challenging task. In this study, the automatic differentiation (AD) method was employed to overcome these difficulties, which seems the first demonstration of the high-order scheme and the AD method in the two-fluid model with realistic relations. With the AD method, the high-order fully implicit scheme, and the preconditioned Jacobian-free Newton-Krylov (PJFNK) method, the two-fluid code was developed to solve the two-phase flow and heat transfer problems in this study. Through simulations of a series of hydrodynamics two-phase flow problems, this code was verified and validated in both transients and boiling conditions. Meanwhile, code-to-code comparisons with well-validated code, RELAP5, were presented. The results demonstrate the promising performance of the developed code for solving 1D two-phase flow problems.

Preconditioned Jacobian-free Newton–Krylov fully implicit high order WENO schemes and flux limiter methods for two-phase flow models

Motivated by the high-resolution properties of high-order Weighted Essentially Non-Oscillatory (WENO) and flux limiter (FL) for steep-gradient problems and the robust convergence of Jacobian-free Newton-Krylov (JFNK) methods for nonlinear systems, the preconditioned JFNK fully implicit high-order WENO and FL schemes are proposed to solve the transient two-phase two-fluid models. Specially, the second-order fully-implicit BDF2 is used for the temporal operator and then the third-order WENO schemes and various flux limiters can be adopted to discrete the spatial operator. For the sake of the generalization of the finite-difference-based preconditioning acceleration methods and the excellent convergence to solve the complicated and various operational conditions, the random vector instead of the initial condition is skillfully chosen as the solving variables to obtain better sparsity pattern or more positions of non-zero elements in this paper. Finally, the WENO_JFNK and FL_JFNK codes are developed and then the two-phase steep-gradient problem, phase appearance/disappearance problem, U-tube problem and linear advection problem are tested to analyze the convergence, computational cost and efficiency in detailed. Numerical results show that WENO_JFNK and FL_JFNK can significantly reduce numerical diffusion and obtain better solutions than traditional methods. WENO_JFNK gives more stable and accurate solutions than FL_JFNK for the test problems and the proposed finite-difference-based preconditioning acceleration methods based on the random vector can significantly improve the convergence speed and efficiency.

A diffusion synthetic acceleration approach to k-eigenvalue neutron transport using PJFNK

This paper investigates a diffusion synthetic acceleration (DSA) approach to the k-eigenvalue problem of the neutron transport equation. The novelties of this study reside in 1) casting the low-order diffusion equation directly as a nonlinear problem, which is solved with the PJFNK (preconditioned Jacobian-free Newton Krylov) method, circumventing the need for power iteration; 2) showing the appealing features on solving the diffusion equation with DSA compared with nonlinear diffusion acceleration (NDA) with a simplified model of ATR (Advanced Test Reactor); 3) proposing the DSA weak form for the discretization scheme with SAAF (self-adjoint angular flux) formulation that is stable with anisotropic scatterings; 4) showing the impact of the vacuum boundary coefficient of the low-order equation on the convergence of DSA. Both DSA and NDA are implemented in and tested with Rattlesnake code, the INL MOOSE-based radiation transport application for multi-physics modeling and simulations.

Subchannel analysis of annular fuel assembly using the preconditioned Jacobian-free Newton Krylov methods

In this paper, a numerical procedure for thermal–hydraulic subchannel analysis of the annular fuel assembly in a PWR using the drift-flux model (DFM) is presented. The system of nonlinear equations was solved with a fully-implicit solution scheme using the Jacobian-free Newton-Krylov (JFNK) method. To improve the computational efficiency and numerical stability of the JFNK method, a semi-implicit physics-based preconditioner was implemented. To validate the present method, various annular fuel arrays with different operating conditions were modeled, and the results were compared with the results of other codes. The key thermal–hydraulic parameters were in good agreement and the difference in total pressure drop values obtained by different methods was less than 2%. Also, it was found that the departure from nucleate boiling ratio (DNBR) of the outer channel was significantly larger than the inner channel, and the heat split ratio of the inner surface to the outer surface was approximately 1.4.

Physics-based preconditioning of Jacobian free Newton Krylov for Burgers’ equation using modified nodal integral method

The application of Nodal Integral Method (NIM) is mostly limited to the low Reynolds number problems; due to lack of suitable nonlinear solvers. This necessitates the development of advanced nonlinear solvers for higher nonlinearity. Newton’s method is a well-established solver for such equations. However, the rate of convergence in Newton methods is dependent on the initial guess. Additionally, the condition number of Jacobian matrix dictates the time required for matrix inversion. In this work, for the first time a preconditioner is developed for fluid flow problem using Modified-NIM (MNIM). This preconditioned Jacobian free Newton Krylov algorithm uses the linearized Modified-MNIM (M2NIM) as the preconditioner, resulting in better eigenvalue clustering, reducing Krylov iterations. The effectiveness of proposed method is demonstrated using 1-D Burgers' equation for larger time steps and higher Reynolds number up to 2500. The proposed preconditioner drastically reduces the spectral radius, CPU run-time and Krylov iterations.

Coupling Methods for HTR-PM Multicircuit Problem

The High-Temperature Gas-Cooled Reactor–Pebble Bed Module (HTR-PM) is a large-scale complex system that includes reactor core, steam generator, helium circulator, and other important components. When integrating these components, coupling problems such as multiphysics problem, multicircuit problem, multiscale problem, and multimodule problem arise in the numerical simulation. The HTR-PM multicircuit system comprises the primary circuit and secondary circuit, which are simulated by two independent codes and coupled by the interface in the once-through steam generator. Although time-consuming, Picard iteration is a feasible and convenient coupling method to integrate two components because oversolving in the early stages of the iteration causes strong fluctuation between circuits. To address this problem, optimization of the maximum subiteration number and convergence precision have been implemented to improve the efficiency and numerical stability of the Picard iteration. The Dynamic Residual Balance method, an improved version of the Residual Balance method, is proposed to prevent oversolving inside the subiterations. It takes into consideration fluctuation between circuits, and this method is robust in a wide range of cases. Moreover, the nonlinear preconditioned Jacobian-Free Newton-Krylov method, which has less fluctuation between circuits than Picard iteration, is a coupling scheme that updates all the solution variables from the primary circuit and the secondary circuit simultaneously. Outstanding convergence and efficiency can be obtained by implementing the proper preconditioner in this HTR-PM multicircuit problem. The downside is that it requires significant modification to the legacy codes.

A Newton solution for the Superhomogenization method: The PJFNK-SPH

This work presents two novel topics regarding the Superhomogenization method: 1) the formalism for the implementation of the method with the linear Boltzmann Transport Equation, and 2) a Newton algorithm for the solution of the nonlinear problem that arises from the method. These new ideas have been implemented in a continuous finite element discretization in the MAMMOTH reactor physics application. The traditional solution strategy for this nonlinear problem uses a Picard, fixed-point iterative process whereas the new implementation relies on MOOSE’s Preconditioned Jacobian-Free Newton Krylov method to allow for a direct solution. The PJFNK-SPH can converge problems that were either intractable or very difficult to converge with the traditional iterative approach, including geometries with reflectors and vacuum boundary conditions. This is partly due to the underlying Scalable Nonlinear Equations Solvers in PETSc, which are integral to MOOSE and offer Newton damping, line search and trust region methods. The PJFNK-SPH has been implemented and tested for various discretizations of the transport equation included in the Rattlesnake transport solver. Speedups of five times for diffusion and ten to fifteen times for transport were obtained when compared to the traditional Picard approach. The three test problems cover a wide range of applications including a standard Pressurized Water Reactor lattice with control rods, a Transient Reactor Test facility control rod supercell and a prototype fast-thermal reactor. The reference solutions and initial cross sections were obtained from the Serpent 2 Monte Carlo code. The SPH-corrected cross sections yield eigenvalues that are near exact, relative to reference solutions, for reflected geometries, even with reflector regions. In geometries with vacuum boundary conditions the accuracy is problem dependent and solutions can be within a few to a few hundred pcm. The root-mean-square error in the power distribution is below 0.8% of the reference Monte Carlo. There is little benefit from SPH-corrected transport in typical scoping calculations, but for more detailed analyses it can yield superior convergence of the solution in some of the test problems. This PJFNK-SPH approach is currently being used in the modeling of the Transient Test Reactor at Idaho National Laboratory, where full reactor core SPH-corrected cross sections are employed to reduce the homogenization errors in transient or multi-physics calculations. This base implementation of the PJFNK-SPH provides an extremely robust solver and a springboard to further improve the Superhomogenization method in order to better preserve neutron currents, one of the primary deficiencies of the method.

On the plastic driving force of grain boundary migration: A fully coupled phase field and crystal plasticity model

Dislocations stored in heavily deformed materials play an important role in driving microstructure evolution. Here, we developed a full coupling model that concurrently couples the phase field method with crystal plasticity finite element analysis to study grain boundary (GB) migration under a plastic driving force. In our model, we describe multiple active grains in GB regions with crystal plasticity theory and use a weighted sum of their properties (i.e., stress and elastic/plastic potentials, etc.) to evaluate the plastic driving force for GB migration. The model can qualitatively capture the absorption of dislocations by mobile GBs through re-initialization of slip system resistances of newly active grains. A finite element based preconditioned Jacobian-free Newton-Krylov approach is used to simultaneously solve all the nonlinear partial differential equations for the coupled physics models. Determining model parameters and validation of the model are accomplished by simulating copper bicrystals and comparing the results to available experiments. This model provides a useful tool for effectively simulating GB migration in metals undergoing large plastic deformation. All the developments have been implemented in the MOOSE/MARMOT simulation package.

An assessment of coupling algorithms for nuclear reactor core physics simulations

This paper evaluates the performance of multiphysics coupling algorithms applied to a light water nuclear reactor core simulation. The simulation couples the k -eigenvalue form of the neutron transport equation with heat conduction and subchannel flow equations. We compare Picard iteration (block Gauss–Seidel) to Anderson acceleration and multiple variants of preconditioned Jacobian-free Newton–Krylov (JFNK). The performance of the methods are evaluated over a range of energy group structures and core power levels. A novel physics-based approximation to a Jacobian-vector product has been developed to mitigate the impact of expensive on-line cross section processing steps. Numerical simulations demonstrating the efficiency of JFNK and Anderson acceleration relative to standard Picard iteration are performed on a 3D model of a nuclear fuel assembly. Both criticality ( k -eigenvalue) and critical boron search problems are considered.

A communication-avoiding implicit–explicit method for a free-surface ocean model

We examine a nonlinear elimination method for the free-surface ocean equations based on barotropic–baroclinic decomposition. The two dimensional scalar continuity equation is treated implicitly with a preconditioned Jacobian-free Newton–Krylov method (JFNK). The remaining three dimensional equations are subcycled explicitly within the JFNK residual evaluation with a method known as nonlinear elimination. In this approach, the memory footprint of the underlying Krylov vector is greatly reduced over that required by fully coupled implicit methods. The method is second-order accurate and scales algorithmically, with allowed timesteps much larger than fully explicit methods. Moreover, the hierarchical nature of the algorithm lends itself readily to emerging architectures. In particular, we introduce a communication staging strategy for the three dimensional explicit system that greatly reduces the communication costs of the algorithm and provides a key advantage as communication costs continue to dominate relative to floating point costs in emerging architectures.

Multi-dimensional, fully-implicit, spectral method for the Vlasov–Maxwell equations with exact conservation laws in discrete form

A spectral method for the numerical solution of the multi-dimensional Vlasov–Maxwell equations is presented. The plasma distribution function is expanded in Fourier (for the spatial part) and Hermite (for the velocity part) basis functions, leading to a truncated system of ordinary differential equations for the expansion coefficients (moments) that is discretized with an implicit, second order accurate Crank–Nicolson time discretization. The discrete non-linear system is solved with a preconditioned Jacobian–Free Newton–Krylov method.It is shown analytically that the Fourier–Hermite method features exact conservation laws for total mass, momentum and energy in discrete form. Standard tests involving plasma waves and the whistler instability confirm the validity of the conservation laws numerically. The whistler instability test also shows that we can step over the fastest time scale in the system without incurring in numerical instabilities. Some preconditioning strategies are presented, showing that the number of linear iterations of the Krylov solver can be drastically reduced and a significant gain in performance can be obtained.

The preconditioned Jacobian-free Newton-Krylov methods for nonequilibrium radiation diffusion equations

In general, it is difficult to use the Newton-Krylov methods to solve the large-scale multi-variable nonequilibrium reaction-diffusion systems. In this paper, by employing two new semi-implicit dis...

The preconditioned Jacobian-free Newton–Krylov methods for nonequilibrium radiation diffusion equations

In general, it is difficult to use the Newton–Krylov methods to solve the large-scale multi-variable nonequilibrium reaction–diffusion systems. In this paper, by employing two new semi-implicit discretization schemes to construct the preconditioners, the preconditioned Newton–Krylov methods are presented to solve the multidimensional problems. These methods cannot only improve the number of iterations, but also speed up the convergence of solutions. Numerical results are given to illustrate the effectiveness.

A parallel, fully coupled, fully implicit solution to reactive transport in porous media using the preconditioned Jacobian-Free Newton-Krylov Method

Modeling large multicomponent reactive transport systems in porous media is particularly challenging when the governing partial differential algebraic equations (PDAEs) are highly nonlinear and tightly coupled due to complex nonlinear reactions and strong solution-media interactions. Here we present a preconditioned Jacobian-Free Newton-Krylov (JFNK) solution approach to solve the governing PDAEs in a fully coupled and fully implicit manner. A well-known advantage of the JFNK method is that it does not require explicitly computing and storing the Jacobian matrix during Newton nonlinear iterations. Our approach further enhances the JFNK method by utilizing physics-based, block preconditioning and a multigrid algorithm for efficient inversion of the preconditioner. This preconditioning strategy accounts for self- and optionally, cross-coupling between primary variables using diagonal and off-diagonal blocks of an approximate Jacobian, respectively. Numerical results are presented demonstrating the efficiency and massive scalability of the solution strategy for reactive transport problems involving strong solution-mineral interactions and fast kinetics. We found that the physics-based, block preconditioner significantly decreases the number of linear iterations, directly reducing computational cost; and the strongly scalable algebraic multigrid algorithm for approximate inversion of the preconditioner leads to excellent parallel scaling performance.

Improving parallel scalability for edge plasma transport simulations with neutral gas species

Simulating the transport of multi-species plasma and neutral species in the edge region of a tokamak magnetic fusion energy device is computationally intensive and difficult due to coupling among various components, strong nonlinearities and a broad range of temporal scales. In addition to providing boundary conditions for the core plasma, such models aid in the understanding and control of the associated plasma/material–wall interactions, a topic that is essential for the development of a viable fusion power plant. The governing partial differential equations are discretized to form a large nonlinear system that typically must be evolved in time to obtain steady-state solutions. Fully implicit techniques using preconditioned Jacobian-free Newton–Krylov methods with parallel domain-based preconditioners are shown to be robust and efficient for the plasma components. Inclusion of neutral gas components, however, increases the condition number of the system to the point where improved parallel preconditioning is needed. Standard algebraic preconditioners that provide sufficient coupling throughout the global domain to handle the neutrals are not generally scalable. We present a new preconditioner, termed FieldSplit, which exploits the character of the neutral equations to improve the scalability of the combined plasma/neutral system.

Three dimensional coupled simulation of thermomechanics, heat, and oxygen diffusion in [formula omitted] nuclear fuel rods

The simulation of nuclear reactor fuel performance involves complex thermomechanical processes between fuel pellets, made of fissile material, and the protective cladding barrier that surrounds the pellets. This paper examines a subset of phenomena that are important in the development of an analysis capability for fuel performance calculations, focusing on thermomechanics and diffusion within UO 2 fuel pellets. In this study, correlations from the literature are used for thermal conductivity, specific heat, and oxygen diffusion. This study develops a three dimensional thermomechanical model fully-coupled to an oxygen diffusion model. Both steady state and transient results are examined to compare this three dimensional model with the literature. Further, this equation system is solved in a parallel, fully-coupled, fully-implicit manner using a preconditioned Jacobian-free Newton Krylov method. Numerical results are presented to explore the efficacy of this approach for examining selected fuel performance problems. INL’s BISON fuels performance code is used to perform this analysis.