• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.15 no.1
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• pp.19-30
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• 2011

In [21], we compared the Newton-Krylov method and some high-order methods to solve nonlinear systems. In this paper, we propose high-order Newton-Krylov methods combining the Newton-Krylov method with some high-order iterative methods to solve systems of nonlinear equations. We provide some numerical experiments including comparisons of CPU time and iteration numbers of the proposed high-order Newton-Krylov methods for several nonlinear systems.

• Nuclear Engineering and Technology
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• v.54 no.8
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• pp.3059-3072
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• 2022

A Jacobian-Free Newton Krylov Two-Nodal Coarse Mesh Finite Difference algorithm based on Nodal Expansion Method (NEM_TNCMFD_JFNK) is successfully developed and proposed to solve the three-dimensional (3D) and multi-group reactor physics models. In the NEM_TNCMFD_JFNK method, the efficient JFNK method with the Modified Incomplete LU (MILU) preconditioner is integrated and applied into the discrete systems of the NEM-based two-node CMFD method by constructing the residual functions of only the nodal average fluxes and the eigenvalue. All the nonlinear corrective nodal coupling coefficients are updated on the basis of two-nodal NEM formulation including the discontinuity factor in every few newton steps. All the expansion coefficients and interface currents of the two-node NEM need not be chosen as the solution variables to evaluate the residual functions of the NEM_TNCMFD_JFNK method, therefore, the NEM_TNCMFD_JFNK method can greatly reduce the number of solution variables and the computational cost compared with the JFNK based on the conventional NEM. Finally the NEM_TNCMFD_JFNK code is developed and then analyzed by simulating the representative PWR MOX/UO₂ core benchmark, the popular NEACRP 3D core benchmark and the complicated full-core pin-by-pin homogenous core model. Numerical solutions show that the proposed NEM_TNCMFD_JFNK method with the MILU preconditioner has the good numerical accuracy and can obtain higher computational efficiency than the NEM-based two-node CMFD algorithm with the power method in the outer iteration and the Krylov method using the MILU preconditioner in the inner iteration, which indicates the NEM_TNCMFD_JFNK method can serve as a potential and efficient numerical tool for reactor neutron diffusion analysis module in the JFNK-based multiphysics coupling application.

• 한국전산유체공학회:학술대회논문집
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• 1998.11a
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• pp.153-159
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• 1998

The Newton-Krylov method on the unstructured grid flow solver using the cell-centered spatial discretization oi compressible Euler equations is presented. This flow solver uses the reconstructed primitive variables to get the higher order solutions. To get the quadratic convergence of Newton method with this solver, the careful linearization of face flux is performed with the reconstructed flow variables. The GMRES method is used to solve large sparse matrix and to improve the performance ILU preconditioner is adopted and vectorized with level scheduling algorithm. To get the quadratic convergence with the higher order schemes and to reduce the memory storage. the matrix-free implementation and Barth's matrix-vector method are implemented and compared with the traditional matrix-vector method. The convergence and computing times are compared with each other.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.26 no.4
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• pp.223-231
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• 2013

This paper presents a time-domain Gauss-Newton full-waveform inversion method for the material profile reconstruction in heterogeneous semi-infinite solid media. To implement the inverse problem in a finite computational domain, perfectly-matchedlayers( PMLs) are introduced as wave-absorbing boundaries within which the domain's wave velocity profile is to be reconstructed. The inverse problem is formulated in a partial-differential-equations(PDE)-constrained optimization framework, where a least-squares misfit between measured and calculated surface responses is minimized under the constraint of PML-endowed wave equations. A Gauss-Newton-Krylov optimization algorithm is utilized to iteratively update the unknown wave velocity profile with the aid of a specialized regularization scheme. Through a series of one-dimensional examples, the solution of the Gauss-Newton inversion was close enough to the target profile, and showed superior convergence behavior with reduced wall-clock time of implementation compared to a conventional inversion using Fletcher-Reeves optimization algorithm.

• Nuclear Engineering and Technology
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• v.52 no.2
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• pp.258-270
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• 2020

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.

• Nuclear Engineering and Technology
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• v.49 no.6
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• pp.1181-1188
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• 2017

This paper proposes the adoption of Monte Carlo perturbation theory to approximate the Jacobian matrix of coupled neutronics/thermal-hydraulics problems. The projected Jacobian is obtained from the eigenvalue decomposition of the fission matrix, and it is adopted to solve the coupled problem via the Newton method. This avoids numerical differentiations commonly adopted in Jacobian-free Newton-Krylov methods that tend to become expensive and inaccurate in the presence of Monte Carlo statistical errors in the residual. The proposed approach is presented and preliminarily demonstrated for a simple two-dimensional pressurized water reactor case study.

• Nuclear Engineering and Technology
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• v.55 no.1
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• pp.310-323
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• 2023

The k-eigenvalue neutronics/thermal-hydraulics coupling calculation is a key issue for reactor design and analysis. Jacobian-free Newton-Krylov (JFNK) method, featured with super-linear convergence rate and high efficiency, has been attracting more and more attention to solve the multi-physics coupling problem. However, it may converge to the high-order eigenmode because of the multiple solutions nature of the k-eigenvalue form of multi-physics coupling issue. Based on our previous work, a modified JFNK with a line search method is proposed in this work, which can find the fundamental eigenmode together with thermal-hydraulics feedback in a wide range of initial values. In detail, the existing modified JFNK method is combined with the line search strategy, so that the intermediate iterative solution can avoid a sudden divergence and be adjusted into a convergence basin smoothly. Two simplified 2-D homogeneous reactor models, a PWR model, and an HTR model, are utilized to evaluate the performance of the newly proposed JFNK method. The results show that the performance of this proposed JFNK is more robust than the existing JFNK-based methods.

• Nuclear Engineering and Technology
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• v.45 no.6
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• pp.707-720
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• 2013

We analyze piecewise homogenization with flux-weighted cross sections and preservation of averaged currents at the boundary of the homogenized domain. Introduction of a set of flux discontinuity ratios (FDR) that preserve reference interface currents leads to preservation of averaged region reaction rates and fluxes. We consider the class of numerical discretizations with one degree of freedom per volume and per surface and prove that when the homogenization and computing meshes are equal there is a unique solution for the FDRs which exactly preserve interface currents. For diffusion submeshing we introduce a Jacobian-Free Newton-Krylov method and for all cases considered obtain an 'exact' numerical solution (eight digits for the interface currents). The homogenization is completed by extending the familiar full assembly homogenization via flux discontinuity factors to the sides of regions laying on the boundary of the piecewise homogenized domain. Finally, for the familiar nodal discretization we numerically find that the FDRs obtained with no submesh (nearly at no cost) can be effectively used for whole-core diffusion calculations with submesh. This is not the case, however, for cell-centered finite differences.