Jacobian-free Newton Krylov Research Articles

On the performance of harmonic balance method for unsteady flow with oscillating shocks

The harmonic balance method (HBM) is a promising alternative for simulating time-periodic unsteady flows; however, its performance in strong nonlinear problems has not been fully investigated. In this paper, we focus on the unsteady flow with oscillating shocks through which several unique aspects regarding the accuracy and efficiency of the HBM are addressed. First, different from most existing studies, influences of the number of harmonics and the number of time samples on the solution are separately analyzed. It is found that the mode factor plays a role in accurately describing the unsteady physics, while the sampling factor is associated with the numerical oscillations in shock oscillation regions. Second, based on the recently developed Jacobian-free Newton–Krylov method with a simplified preconditioner and comparison with other methods, the efficiency of the present HBM and its relationship with the two factors are studied. The computational cost is found to increase almost linearly with the increase of the number of harmonics, but nonlinearly with the number of time samples. Finally, noticing the fact that fewer harmonics (time samples) can ensure accurate solutions in smooth regions, while many more are required in shock oscillation regions, a non-uniform HBM is developed by adopting various numbers of harmonics (time samples) throughout the whole flow domain. The superiority of this method compared with the traditional uniform HBM is demonstrated, showing a good application prospect in large-scale unsteady flows with strong nonlinearities.

A New Precursor Integral Method for Solving Space-Dependent Kinetic Equations in Neutronic and Thermal-Hydraulic Coupling System

The accurate prediction of the neutronic and thermal-hydraulic coupling system transient behavior is important in nuclear reactor safety analysis, where a large-scale nonlinear coupling system with strong stiffness should be solved efficiently. In order to reduce the stiffness and huge computational cost in the coupling system, the high-performance numerical techniques for solving delayed neutron precursor equation are a key issue. In this work, a new precursor integral method with an exponential approximation is proposed and compared with widely used Taylor approximation-based precursor integral methods. The truncation errors of exponential approximation and Taylor approximation are analyzed and compared. Moreover, a time control technique is put forward which is based on flux exponential approximation. The procedure is tested in a 2D neutron kinetic benchmark and a simplified high-temperature gas-cooled reactor-pebble bed module (HTR-PM) multiphysics problem utilizing the efficient Jacobian-free Newton–Krylov method. Results show that selecting appropriate flux approximation in the precursor integral method can improve the efficiency and precision compared with the traditional method. The computation time is reduced to one-ninth in the HTR-PM model under the same accuracy when applying the exponential integral method with the time adaptive technique.

A comparison of three algorithms applied in thermal-hydraulics and neutronics codes coupling for lbe-cooled fast reactor

In this research, the Operator-Splitting semi-implicit (OSSI) method, fixed-point implicit (FPI) method and Approximate Block Newton (ABN) method have been applied to the coupling of thermal-hydraulics code COBRA and neutronics code SKETCH-N. In the OSSI method, there’s only one data exchange within a time step. In the FPI coupling, an iteration convergence has been added to the OSSI coupling framework to ensure the coupling results converged. And in the ABN method, a variant of Jacobian-free Newton Krylov method has been applied, which removes the Jacobian matrix construction and storage. Three temporal schemes have been validated through a PWR control rod bank ejection benchmark to show their applicability. Moreover, a LBE-cooled fast reactor control rods withdrawal accident applying two time steps has been simulated. And the results indicate that the ABN method applied in the mild transient generally outperforms the other two methods in both the convergence speed and the computational cost.

The Efficient 3D Gravity Focusing Density Inversion Based on Preconditioned JFNK Method under Undulating Terrain: A Case Study from Huayangchuan, Shaanxi Province, China

Since polymetallic ores show higher anomalies in gravity exploration methods, we usually obtain the position and range of ore bodies by density inversion of gravity data. The three-dimensional (3D) gravity focusing density inversion is a common interpretation method in mineral exploration, which can directly and quantitatively obtain the density distribution of subsurface targets. However, in actual cases, it is computation inefficient. We proposed the preconditioned Jacobian-free Newton-Krylov (JFNK) method to accomplish the focusing inversion. The JFNK method is an efficient algorithm in solving large sparse systems of nonlinear equations, and we further accelerate the inversion process by the preconditioned technique. In the actual area, the gravity anomalies are distributed on the naturally undulating surface. Nowadays, the gravity inversion under undulating terrain was mainly achieved by discretizing the ground into unstructured meshes, but it is complicated and time-consuming. To improve the practicality, we presented an equivalent-dimensional method that incorporates unstructured meshes with structured meshes in gravity inversion, and the horizontal size is determined by the gradient of observed gravity and terrain data. The small size meshes are adopted at the position where the terrain or gravity gradient is large. We used synthetic data with undulating-terrain to test our new method. The results indicated that the recovered model obtained by this method was similar to the inversion method of unstructured meshes, and the new method computes faster. We also applied the method to field data in Huayangchuan, Shaanxi Province. The survey area has complicated terrain conditions and contains multiple polymetallic ores. Based on the high-density characteristics of polymetallic ore bodies in the area, we calculate the field data into 3D density models of the subsurface by the preconditioned JFNK method and infer six polymetallic ores.

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.

Finite difference Jacobian based Newton-Krylov coupling method for solving multi-physics nonlinear system of nuclear reactor

Newton-Krylov (NK) method with the explicit Jacobian matrix is a promising method to solve the nonlinear multi-physics coupling system. However, how to form the Jacobian matrix efficiently is a key issue that is the most cumbersome step in NK method. In this work, a new solution with the finite difference Jacobian based NK (DJNK) method is proposed, where the Jacobian matrix is calculated by finite difference approximation instead of the analytical form. Moreover, DJNK is further improved to form Jacobian by calculating the diagonals of the submatrixes, leading to significant reduction of the time cost. Two benchmarks and a complicated transient neutronics/thermal-hydraulics coupling problem are conducted. Results show that DJNK outperforms widely used Jacobian-free NK (JFNK) method, because of the higher performance in matrix-vector product calculation and the more efficient preconditioning in Krylov iteration. A speedup ratio of DJNK over JFNK is observed to be 1.7.

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.

Identifying disruptive contingencies for catastrophic cascading failures in power systems

Due to the evolving nature and the complicated coupling relationship of power system components, it has been a great challenge to identify the disruptive contingencies that can trigger cascading blackouts. This paper aims to develop a generic approach for identifying the initial disruptive contingencies that can result in the catastrophic cascading failures of power systems. The problem of contingency identification is formulated in the mathematical framework of hybrid differential-algebraic system, and it can be solved by the Jacobian-Free Newton-Krylov method, which allows to circumvent the Jacobian matrix and relieve the computational burden. Moreover, an efficient numerical algorithm is developed to search for the disruptive contingencies that lead to catastrophic cascading failures with the guaranteed convergence accuracy in theory. Finally, case studies are presented to demonstrate the efficacy of the proposed identification approach on the IEEE test systems by using different cascade models.

Nektar++: Design and implementation of an implicit, spectral/[formula omitted] element, compressible flow solver using a Jacobian-free Newton Krylov approach

At high Reynolds numbers the use of explicit in time compressible flow simulations with spectral/hp element discretization can become significantly limited by time step. To alleviate this limitation we extend the capability of the spectral/hp element open-source software framework, Nektar++, to include an implicit discontinuous Galerkin compressible flow solver. The integration in time is carried out by a singly diagonally implicit Runge–Kutta method. The non-linear system arising from the implicit time integration is iteratively solved by the Jacobian-free Newton Krylov (JFNK) method. A favorable feature of the JFNK approach is its extensive use of the explicit operators available from the previous explicit in time implementation. The functionalities of different building blocks of the implicit solver are analyzed from the point of view of software design and placed in appropriate hierarchical levels in the C++ libraries. In the detailed implementation, the contributions of different parts of the solver to computational cost, memory consumption and programming complexity are also analyzed. A combination of analytical and numerical methods is adopted to simplify the programming complexity in forming the preconditioning matrix. The solver is verified and tested using cases such as manufactured compressible Poiseuille flow, Taylor–Green vortex, turbulent flow over a circular cylinder at Re=3900 and shock wave boundary-layer interaction. The results show that the implicit solver can speed-up the simulations while maintaining good simulation accuracy.

Finite particle method for static deformation problems solved using JFNK method

Abstract In this work, the Finite Particle Method (FPM) which is a typical meshfree particle method is formulated for static deformation and plastic flow problems. The novel formulation utilizes the Lagrangian-type kernel, which guarantees the method being free from the tension instability. A pseudo-viscosity is introduced into the particle system to prevent the hourglass phenomenon in the numerical procedure. The nonlinear system of governing equations is solved with implicit time integration scheme. For numerical efficiency the Jacobian-Free Newton-Krylov (JFNK) method is applied to obtain the solution for each time step. Numerical examples are provided to demonstrate the validity and the effectiveness of the method in capturing finite deformation with moderate distortion.

A hyperbolic Poisson solver for tetrahedral grids

This paper presents a robust and efficient Poisson solver that can produce accurate solutions and gradients (e.g., heat flux) on unstructured tetrahedral grids. The solver is constructed based on the hyperbolic method for diffusion, where the Laplacian operator is discretized in the form of a hyperbolic system with solution gradients introduced as additional unknown variables. A practical formula for defining a reference length is proposed, which is needed to properly scale the relaxation length associated with the hyperbolic formulation for scale-invariant computations. The resulting system of residual equations is efficiently solved by a Jacobian-Free Newton-Krylov solver with an implicit defect-correction solver used as a variable-preconditioner. Robustness and superior gradient accuracy are demonstrated for linear and nonlinear Poisson equations through a series of numerical tests with unstructured tetrahedral grids.

Preliminary verification of incompressible Navier-Stokes equations solved by The Newton method

The thermal-hydraulics module for reactor multi-physics coupling is studied. Multi-physics coupling for high temperature gas cooling reactor is very important due to the interaction between different physics, especially neutronics and thermal-hydraulics. This paper demonstrates an Incompressible Navier-Stokes (INS) module in the Multiphysics Object-Oriented Simulation Environment (MOOSE) to validate the accuracy and efficiency of the module, and the research on INS module is the significant basis of the future coupling work. MOOSE is an open-source finite element platform developed by Idaho National Lab (INL), which is for solving nonlinear equations from different physics simultaneously and can also couple different codes implicitly. The Jacobian-free Newton Krylov (JFNK) method adopted in MOOSE uses the finite difference method to avoid explicitly solving the Jacobian matrix so as to effectively save memory, and different preconditioning methods are also applied to accelerate the Krylov iteration. Besides, the Newton method is also supplied in MOOSE to solve nonlinear equations. Incompressible Navier-Stokes equations have been solved by the Newton method through the finite element method in MOOSE, and different numerical solution methods can be used to solve different cases. MOOSE regards problem equations as kernels and boundary conditions, and solves the discretized equations using the embedded Petsc or Trilinos solver. A convective heat transfer lid driven cavity problem is calculated using the INS module and the CFD tool Fluent to compare simulation results, and the temperature average error of 0.31% validates the module accuracy. In addition, JFNK and Newton methods are adopted to solve the same problem, and the results reveal that Newton method can save nearly 23% of the calculation time and has higher efficiency. The simulation results prove the module capability of applying to the reactor thermal-hydraulics, multi-physics coupling and safety analysis.

Gas-Kinetic Harmonic Balance Method for Unsteady Supersonic/Hypersonic Flows with Oscillating Motions

In this paper, a novel gas-kinetic harmonic balance method (HBM) is developed for unsteady flows with oscillating motions. Particularly, the emphasis here is on supersonic and hypersonic flows. Because of the presence of an additional source term in the harmonic balance equations, usually very small time steps are needed for explicit algorithms, and some factorization-based implicit algorithms may fail to converge in the case of higher frequencies. To achieve better convergence performance and robustness, a Jacobian-free Newton–Krylov method is adopted. By constructing a simplified preconditioning matrix, the lower–upper symmetric Gauss–Seidel scheme is employed as a preconditioner, so that the computational cost and memory requirement can be reduced. Considering that conventional spatial discretization schemes may encounter numerical difficulties and low robustness in high-Mach-number flows, a more general gas-kinetic scheme (GKS) is extended for the harmonic balance equations. As the HBM solves a mathematically steady problem, the present GKS focuses on evaluation of the time-independent fluxes at each cell interface and each subtime level. The gas-kinetic HBM is applied to a supersonic oscillating flat plate cascade and an oscillating panel in hypersonic flow. By comparison with other methods, the advantages of the developed method are demonstrated.

Multi-physics coupling simulation in virtual reactors

Nuclear power stations involve a range of complex and interacting multi-physical processes. With the rapid development of high-performance computing technology, accurate multi-physics simulation in virtual reactors has drawn more and more attention in industry and academia. Great efforts have been made toward the simulation of multi-physics coupling processes in reactors. The interpretations of many terms that describe multi-physics simulation vary in different literatures. We organize and discuss some important terms relevant to the multi-physics coupling simulation. We compare the three most frequently used multi-physics coupling strategies: the operator splitting, Picard iteration, and Jacobian-Free Newton–Krylov methods. We summarize three main viewpoints on the degree of coupling of the three strategies (loose, tight, or full coupling). Then we review the coupling software and corresponding coupling strategies in some representative virtual reactor projects. We present the research focuses of Spider coupling platform. The Spider is developed in the China Virtual Reactor (CVR) project. The multi-physics phenomena are considered in the CVR project from three scales: fuel scale, reactor core scale, and system scale. Both loose and tight coupling strategies are supported in the Spider platform.

Subcell finite volume multigrid preconditioning for high-order discontinuous Galerkin methods

ABSTRACTWe suggest a new multigrid preconditioning strategy for use in Jacobian-free Newton–Krylov (JFNK) methods for the solution of algebraic equation systems arising from implicit Discontinuous Galerkin (DG) discretisations. To define the new preconditioner, use is made of an auxiliary first-order finite volume discretisation that refines the original DG mesh, but can still be implemented algebraically. As smoother, we consider the pseudo-time iteration W3 with a symmetric Gauss–Seidel-type approximation of the Jacobian. As a proof of concept numerical tests are presented for the one-dimensional Euler equations, demonstrating the potential of the new approach.

A spectral deferred correction strategy for low Mach number reacting flows subject to electric fields

We propose an algorithm for low Mach number reacting flows subjected to electric field that includes the chemical production and transport of charged species. This work is an extension of a multi-implicit spectral deferred correction (MISDC) algorithm designed to advance the conservation equations in time at scales associated with advective transport. The fast and nontrivial interactions of electrons with the electric field are treated implicitly using a Jacobian-Free Newton Krylov approach for which a preconditioning strategy is developed. Within the MISDC framework, this enables a close and stable coupling of diffusion, reactions and dielectric relaxation terms with advective transport and is shown to exhibit second-order convergence in space and time. The algorithm is then applied to a series of steady and unsteady problems to demonstrate its capability and stability. Although developed in a one-dimensional case, the algorithmic ingredients are carefully designed to be amenable to multi-dimensional applications.

Predictor–Corrector Nodal Integral Method for simulation of high Reynolds number fluid flow using larger time steps in Burgers ’ equation

Nodal Integral Methods (NIM), although frequently used for solving neutron transport equations, have not found wider acceptance for solving fluid flow problems. One of the pivotal reasons behind this is the lack of efficient non-linear solvers for the system of coupled algebraic equations obtained by these methods, thus making such methods prohibitively expensive for highly nonlinear (higher Reynolds number) fluid flow problems. Successive improvements in earlier attempts resulted in an improved version called Modified NIM (MNIM) scheme; following which, a further modified version appeared (called as M2NIM), that used the concept of the delayed coefficients. Upon comparing the solutions obtained using MNIM and M2NIM schemes, respectively, it was observed that although the modified scheme with delayed coefficients (M2NIM) has significantly faster convergence, it is less accurate. The convergence and accuracy are very likely to suffer to a significant extent in the high Reynolds number fluid flows, especially when a large time-step is to be considered. In order to resolve these difficulties, a new type of physics-based predictor–corrector algorithm is proposed in the present study. In the proposed computational algorithm, the linearized M2NIM scheme is used to predict the solution, whereas the MNIM scheme is utilized to improve the predictive guess. The novelty of such a hybrid numerical algorithm comes from the fact that it combines the advantage of the faster convergence of M2NIM with the accuracy of the MNIM scheme. The algorithm also benefits from the unique inclusion of Jacobian-free Newton-Krylov (JFNK) method, which helps in getting rid of the formation of large Jacobian matrices, thereby reducing unnecessary computational overhead. The proposed methodology has been applied to solve a non-linear convection–diffusion problem, represented by the Burgers’ equation. The computational results for bothone-dimensional and two-dimensional Burgers’ equation are presented to demonstrate the effectiveness of the developed novel algorithm, as well as, the advantage it offers over the existing numerical methods.

The JFNK method for the PWR's transient simulation considering neutronics, thermal hydraulics and mechanics

Abstract A new task of using the Jacobian-Free-Newton-Krylov (JFNK) method for the PWR core transient simulations involving neutronics, thermal hydraulics and mechanics is conducted. For the transient scenario of PWR, normally the Picard iteration of the coupled coarse-mesh nodal equations and parallel channel TH equations is performed to get the transient solution. In order to solve the coupled equations faster and more stable, the Newton Krylov (NK) method based on the explicit matrix was studied. However, the NK method is hard to be extended to the cases with more physics phenomenon coupled, thus the JFNK based iteration scheme is developed for the nodal method and parallel-channel TH method. The local gap conductance is sensitive to the gap width and will influence the temperature distribution in the fuel rod significantly. To further consider the local gap conductance during the transient scenario, a 1D mechanics model is coupled into the JFNK scheme to account for the fuel thermal expansion effect. To improve the efficiency, the physics-based precondition and scaling technique are developed for the JFNK iteration. Numerical tests show good convergence behavior of the iterations and demonstrate the influence of the fuel thermal expansion effect during the rod ejection problems.

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.

Low-order multiphysics coupling techniques for nuclear reactor applications

The accurate modeling and simulation of nuclear reactor designs depends greatly on the ability to couple differing sets of physics together. Current coupling techniques most often use a fixed-point, or Picard, iteration scheme in which each set of physics is solved separately, and the resulting solutions are passed between each solver. In the work presented here, two different coupling techniques are investigated: a Jacobian-Free Newton-Krylov (JFNK) approach and a new methodology called Coarse Mesh Finite Difference Coupling (CMFD-Coupling). In this work, both techniques were applied to the low-order CMFD system of equations. This allows for the multiphysics feedback effects to be captured on the low-order system without having to perform a neutron transport solve.The JFNK and CMFD-Coupling approaches were implemented in the MPACT neutron transport code, which is being developed for the Consortium for Advanced Simulation of Light Water Reactors (CASL). These methods were tested on a wide range of practical reactor physics problems, from a single 3D fuel pin to a massively parallel 3D full core problem. When coupled neutronics-thermal hydraulics problems were investigated with both JFNK and CMFD-Coupling, it was concluded that CMFD-Coupling outperformed JFNK in terms of both accuracy and runtime for every problem. When applied to large full core problems with multiple sources of strong feedback enabled, CMFD-Coupling reduced the overall number of transport sweeps required for convergence.