Newton-Krylov Algorithm Research Articles

A method to regularize optimization problems governed by chaotic dynamical systems

Long-time averages of outputs from chaotic dynamical systems exhibit high-frequency oscillations in parameter space, which makes gradient-based optimization impractical. We show that these oscillations can be eliminated by coupling the initial and final states of the trajectory to produce discontinuous periodic orbits (DPOs). This approach can be regarded as a generalization of (continuous) unstable periodic orbits, which have been shown to regularize time-averages from chaotic problems. To solve the DPO governing equations, we present a Newton–Krylov algorithm that is globalized using a line search and parameter continuation. We study the accuracy of DPO-based outputs using the Lorenz dynamical system and the Kuramoto–Sivashinsky partial differential equation, and we demonstrate the effectiveness of DPO-based optimization on two simple examples. We conclude the paper with a discussion of open questions for future research.

An aneurysm-specific preconditioning technique for the acceleration of Newton-Krylov method with application in the simulation of blood flows.

In this paper, we develop an algorithm to simulate blood flows in aneurysmal arteries and focus on the construction of robust and efficient multilevel preconditioners to speed up the convergence of both linear and nonlinear solvers. The work is motivated by the observation that in the local aneurysmal region, the flow is often quite complicated with one or more vortices, but in the healthy section of the artery, the principal component of blood flows along the centerline of the artery. Based on this observation, we introduce a novel two-level additive Schwarz method with a mixed-dimensional coarse preconditioner. The key components of the preconditioner include (1) a three-dimensional coarse preconditioner covering the aneurysm; (2) a one-dimensional coarse preconditioner covering the central line of the healthy section of the artery; (3) a collection of three-dimensional overlapping subdomain preconditioners covering the fine meshes of the entire artery; (4) extension/restriction operators constructed by radial basis functions. The blood flow is modeled by the unsteady incompressible Navier-Stokes equations with resistance outflow boundary conditions discretized by a stabilized finite element method on fully unstructured meshes and the second-order backward differentiation formula in time. The resulting large nonlinear algebraic systems are solved by a Newton-Krylov algorithm accelerated by the new preconditioner in two ways: (1) the initial guess of Newton is obtained by solving a linear system defined by the coarse preconditioner; (2) the Krylov solver of the Jacobian system is preconditioned by the new preconditioner. Numerical experiments indicate that the proposed preconditioner is highly effective and robust for complex flows in a patient-specific artery with aneurysm, and it significantly reduces the numbers of linear and nonlinear iterations.

Recycling Newton–Krylov algorithm for efficient solution of large scale power systems

Power flow calculations are crucial for the study of power systems, as they can be used to calculate bus voltage magnitudes and phase angles, as well as active and reactive power flows on lines. In this paper, a new approach, the Recycling Newton–Krylov (ReNK) algorithm, is proposed to solve the linear systems of equations in Newton–Raphson iterations. The proposed method uses the Generalized Conjugate Residuals with inner orthogonalization and deflated restarting (GCRO-DR) method within the Newton–Raphson algorithm and reuses the Krylov subspace information generated in previous Newton runs. We evaluate the performance of the proposed method over the traditional direct solver (LU) and iterative solvers (Generalized Minimal Residual Method (GMRES), the Biconjugate Gradient Stabilized Method (Bi-CGSTAB) and Quasi-Minimal Residual Method (QMR)) as the inner linear solver of the Newton–Raphson method. We use different test systems with a number of busses ranging from 300 to 70000 and compare the number of iterations of the inner linear solver (for iterative solvers) and the CPU times (for both direct and iterative solvers). We also test the performance of the ReNK algorithm for contingency analysis and for different load conditions to simulate optimization problems and observe possible performance gains.

Implicit and Conservative Mixing-Plane Method for Multistage Turbomachinery Aerodynamic Analysis

A fully implicit Reynolds-averaged Navier–Stokes (RANS) flow solver and a corresponding adjoint solver using a Newton–Krylov algorithm has recently been developed and demonstrated to exhibit good solution robustness and efficiency for flow and adjoint aerodynamic analyses of single-row turbomachines at design and off-design conditions. To extend the algorithm to multistage applications, a robust mixing-plane technique and its fully implicit treatment are crucial. In this work, a mixing-plane approach featuring 1) flux conservation, 2) implicitness, 3) one-dimensional nonreflectiveness, and 4) robustness in handling reserve flow is proposed and incorporated into the existing Newton–Krylov turbomachinery computational fluid dynamics solver. Following a flux-based approach that ensures exact flux conservation, the algorithm of the proposed mixing-plane method is first rigorously derived, and implementation details into the existing single-row RANS solver are then discussed, with emphasis on the fully implicit treatment of the mixing-plane fluxes. Finally, the fully implicit multistage flow solution capability is extended to the corresponding adjoint solver. The resulting algorithms are tested on a subsonic and a transonic compressor to validate the flow solution accuracy, verify the adjoint sensitivity, and demonstrate its robustness in handling reverse flows.

A multi-level nonlinear elimination-based JFNK method for multi-scale multi-physics coupling problem in pebble-bed HTR

In the pebble-bed high-temperature gas-cooled reactor, various physical phenomena occur at different spatial scales and are coupled together tightly in a nonlinear manner. More specifically, the local temperature profiles inside the fuel sphere and TRISO particle, which are essential in reactor design and safety analysis, are coupled with the global physical fields, such as the pebble bed temperature, helium temperature, as well as the fission heat source. Therefore, the multi-physics calculation combined with multi-scale model should be employed to describe the reactor behavior. The computational efficiency is the critical issue for the numerical solution of such global and local coupled systems. Jacobian-free Newton Krylov (JFNK) algorithm is an excellent candidate for multi-physics coupling issues due to its high convergence rate and superior computational performance, but it doesn’t consider the physical features at different scales. In this work, a multi-level nonlinear elimination is proposed within the JFNK framework to deal with the multi-scale coupled system in high-temperature gas-cooled reactor. The numerous local multi-scale temperature variables, including the fuel sphere and TRISO particle temperatures, are eliminated efficiently based on the Schur complement factorization due to its well sparse matrix structure. As a result, the number of unknowns is significantly reduced and the computational performance is improved. A steady-state pebble-bed HTR model is used in this work, and the numerical results show that multi-level nonlinear elimination-based JFNK method has high computational efficiency and numerical robustness, whose performance is 4 ∼ 6 times higher than that of the original JFNK algorithm.

A movable boundary model for helical coiled once-through steam generator using preconditioned JFNK method

Helical coiled once-through steam generator (H-OTSG) is the largest and the most complicated heat exchanger in the high-temperature gas-cooled reactor (HTGR), whose performance plays an important role in the nuclear power plant design and dynamic behavior analysis. A mathematical model of H-OTSG is developed based on the movable boundary method and solved by the advanced fully implicit Jacobian-free Newton-Krylov algorithm. The physical-based preconditioners are proposed and analyzed in this work, which achieves high-performance in solving the nonlinear steam generator system. In order to evaluate the performance of the movable boundary model, the fixed fine mesh model is also considered. The comparison is made between these two models from the perspective of the computational efficiency and accuracy. The simulation results are validated by the HTR-10 steam generator design data, showing that both models agree well with the design data, but have their own features. The fixed fine mesh model has its advantages in the accuracy, while the movable boundary model can realize the high computational efficiency.

Path integral approach unveils role of complex energy landscape for activated dynamics of glassy systems

The complex dynamics of an increasing number of systems is attributed to the emergence of a rugged energy landscape with an exponential number of metastable states. To develop this picture into a predictive dynamical theory, I discuss how to compute the exponentially small probability of a jump from one metastable state to another. This is expressed as a path integral that can be evaluated by saddle-point methods in mean-field models, leading to a boundary value problem. The resulting dynamical equations are solved numerically by means of a Newton-Krylov algorithm in the paradigmatic spherical $p$-spin glass model that is invoked in diverse contexts from supercooled liquids to machine-learning algorithms. I discuss the solutions in the asymptotic regime of large times and the physical implications on the nature of the ergodicity-restoring processes.

Numerical methods for solving the equivalent inclusion equation in semi-analytical models

Equivalent inclusion method is the basis for semi-analytical models in tackling inhomogeneity problems. Equivalent eigenstrains are obtained by solving the consistency equation system of the equivalent inclusion method and then stress disturbances caused by inhomogeneities are determined. The equivalent inclusion method equation system can only be solved numerically, but the current fixed-point iteration method may not be able to achieve deep convergence when the Young's modulus of inhomogeneity is lower than that of the matrix material. The most significant innovation of this paper is to reveal the non-convergence mechanism of the current method. Considering the limitation, the Jacobian-free Newton Krylov algorithm is selected to solve the equivalent inclusion method equation. Results indicate that the new algorithm has significant advantages of computing accuracy and efficiency compared with the classic method.

Smooth Local Correlation-Based Transition Model for the Spalart–Allmaras Turbulence Model

A smooth version of the local correlation-based transition model (LM2015) for the Spalart–Allmaras (SA) turbulence model is presented. The LM2015 model with helicity-based crossflow correlations is modified and coupled to the SA turbulence model, designated SA-LM2015. The LM2015 and SA-LM2015 transition models include source terms that contain stiff and nonsmooth functions. Approximations to these functions are introduced to eliminate discontinuities and improve the numerical behavior of the model, with the smooth model designated SA-sLM2015. Deep convergence is achieved using a fully coupled, implicit Newton–Krylov algorithm globalized using an efficient pseudotransient continuation strategy. Modifications to the Newton–Krylov algorithm are introduced, including a source-term time stepping strategy, to address the large sources introduced by the turbulence and transition model equations. Two- and three-dimensional transition test cases demonstrate that both models, SA-LM2015 and SA-sLM2015, are able to predict transition due to a variety of mechanisms accurately, with the smooth variant displaying significantly improved numerical behavior. Several of the strategies presented, including the smoothing techniques and source-term time stepping, could be useful in the context of other transition models as well.

Fully implicit hybrid two-level domain decomposition algorithms for two-phase flows in porous media on 3D unstructured grids

Simulation of subsurface flows in porous media is difficult due to the nonlinearity of the operators and the high heterogeneity of material coefficients. In this paper, we present a scalable fully implicit solver for incompressible two-phase flows based on overlapping domain decomposition methods. Specifically, an inexact Newton-Krylov algorithm with analytic Jacobian is used to solve the nonlinear systems arising from the discontinuous Galerkin discretization of the governing equations on 3D unstructured grids. The linear Jacobian system is preconditioned by additive Schwarz algorithms, which are naturally suitable for parallel computing. We propose a hybrid two-level version of the additive Schwarz preconditioner consisting of a nested coarse space to improve the robustness and scalability of the classical one-level version. On the coarse level, a smaller linear system arising from the same discretization of the problem on a coarse grid is solved by using GMRES with a one-level preconditioner until a relative tolerance is reached. Numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed solver for 3D heterogeneous medium problems. We also report the parallel scalability of the proposed algorithms on a supercomputer with up to 8,192 processor cores.

Newton–Krylov Solver for Robust Turbomachinery Aerodynamic Analysis

Steady computational fluid dynamics solvers based on the Reynolds-averaged Navier–Stokes equations are the primary workhorses for turbomachinery aerodynamic analysis due to their good engineering accuracy at a low computational cost. However, even state-of-the-art steady solvers suffer from convergence slowdown or failure when applied to challenging off-design conditions. This severely limits the reliable nonlinear and linearized turbomachinery aerodynamic analysis over a wide operating range. To alleviate the convergence difficulties, a nonlinear flow solver using the Newton–Krylov method is developed. This is the first time the Newton–Krylov algorithm is used for achieving robust analysis of turbomachinery aerodynamics in the open literature. The proposed solution algorithm features 1) the exact Jacobian matrix forming, 2) straightforward parallelization, and 3) a reliable globalization strategy; and it aims to achieve fast machine-zero convergence. The solver accuracy is validated using four test cases: an airfoil, a linear turbine cascade, a centrifugal compressor, and an axial compressor. Machine-zero convergence is achieved for all cases over a wide range of operating conditions without manual intervention. The method shows great potential for enabling an automated and reliable whole-map turbomachinery aerodynamic analysis, and it paves the way for a robust and efficient linearized aerodynamic analysis, such as adjoint, time-linearized, and eigenvalue analyses.

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.

A Jacobian-free approximate Newton–Krylov startup strategy for RANS simulations

The favorable convergence rates of Newton–Krylov-based solution algorithms have increased their popularity for computational fluid dynamics applications. Unfortunately, these methods perform poorly during the initial stages of convergence, particularly for three-dimensional Reynolds-averaged Navier–Stokes simulations. Addressing this problem requires the use of a globalization method such as pseudo-transient continuation, along with an approximate Newton–Krylov startup stage. This class of methods marches the solution in pseudo-time with a matrix-based approximate Jacobian that has a lower bandwidth and better conditioning properties compared with the exact Jacobian. However, this matrix-based approach also has shortcomings, including the large cost of computing and storing the approximate Jacobian and its preconditioner, along with the need for an updated Jacobian for every nonlinear iteration. To rectify these shortcomings, we use approximate residual formulations in a Jacobian-free approximate Newton–Krylov algorithm. With the approximate Jacobian, we compute the vector products by using the approximate residual computations in a matrix-free manner while forming a preconditioner based on the matrix-based approximate Jacobian. This approach keeps the approximate Jacobian up to date and mitigates the cost of forming a matrix-based Jacobian and its preconditioner at each iteration by lagging the preconditioner between nonlinear iterations. We use varying levels of approximations with the matrix-free approach and thereby demonstrate the trade-off between rate of convergence and the cost of each nonlinear iteration. The proposed implementation uses only the exact and approximate residual formulations and can therefore be generalized with minimal additional implementation effort to a range of solvers and discretizations. The code is available under an open-source license.

Investigation into Aerodynamic Shape Optimization of Planar and Nonplanar Wings

Problems in three-dimensional aerodynamic shape optimization can produce complex design spaces due to the nonlinear physics of the Navier–Stokes equations and the large number of design variables used. In this paper, a Newton–Krylov optimization algorithm is applied to a set of complex aerodynamic optimization problems in order to investigate its behaviour and performance. The methodology solves the Reynolds-averaged Navier–Stokes equations with a parallel Newton–Krylov algorithm. Aerodynamic geometries are meshed using structured multiblock grids, which are then fitted with B-spline control volumes for mesh deformation and geometry control. A gradient-based optimization method is used, with adjoint variables calculated using a Krylov method. The optimization of the Common Research Model (CRM) wing is revisited, with a focus on the effect of varying geometric constraints and on the possibility of multimodality. In addition, several cases are presented that involve a high degree of shape change: two planform optimizations starting from a rectangular wing, and investigation of various wingtip treatments. The results characterize the methodology, demonstrating its robustness and ability to address optimization problems with substantial geometric freedom.

A new numerical method for solution of boiling flow using combination of SIMPLE and Jacobian-free Newton-Krylov algorithms

In this paper, an efficient and stable numerical approach is developed in which a different combination of Jacobian-free Newton-Krylov (JFNK) and SIMPLE methods are introduced for solution of two-phase flow in both steady state and transient conditions. It is shown that, combination of Krylov subspace methods and SIMPLE type preconditioner give a fast convergence in the solution of the Drift Flux Model (DFM). It is demonstrated that the stability problem with SIMPLE approach, for two-phase flow problems, can be avoided by combining with JFNK. RELAP5 code and experimental data are considered for verification study. For all benchmarks, the proposed methods predictions are in good agreement with experiment and RELAP5 results. The accuracy of the proposed schemes were studied and compared with the classical SIMPLE method. It was found that the present approach to two-phase flow simulation predicts accurate results over a wide pressure range; whereas the classical SIMPLE algorithm underestimated the void fraction at low pressures. Finally, a typical power transient is considered to demonstrate how the problem fastest time scale can change during the course of a transient. The present assessment of the numerical method showed that the time step control is not based on a stability time step restriction, like the material Courant limit.

Globalization technique for projected Newton–Krylov methods

SummaryLarge‐scale systems of nonlinear equations appear in many applications. In various applications, the solution of the nonlinear equations should also be in a certain interval. A typical application is a discretized system of reaction diffusion equations. It is well known that chemical species should be positive otherwise the solution is not physical and in general blow up occurs. Recently, a projected Newton method has been developed, which can be used to solve this type of problems. A drawback is that the projected Newton method is not globally convergent. This motivates us to develop a new feasible projected Newton–Krylov algorithm for solving a constrained system of nonlinear equations. Combined with a projected gradient direction, our feasible projected Newton–Krylov algorithm circumvents the non‐descent drawback of search directions which appear in the classical projected Newton methods. Global and local superlinear convergence of our approach is established under some standard assumptions. Numerical experiments are used to illustrate that the new projected Newton method is globally convergent and is a significate complementarity for Newton–Krylov algorithms known in the literature. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd.

Computational Algorithms for Solving Spectral/$hp$ Stabilized Incompressible Flow Problems

In this paper we implement the element-by-element preconditioner and inexact Newton-Krylov methods (developed in the past) for solving stabilized computational fluid dynamics (CFD) problems with spectral methods. Two different approaches are implemented for speeding up the process of solving both steady and unsteady incompressible Navier-Stokes equations. The first approach concerns the application of a scalable preconditioner namely the element by element LU preconditioner, while the second concerns the application of Newton-Krylov (NK) methods for solving non-linear problems. We obtain good agreement with benchmark results on standard CFD problems for various Reynolds numbers. We solve the Kovasznay flow and flow past a cylinder at Re-$100$ with this approach. We also utilize the Newton-Krylov algorithm to solve (in parallel) important model problems such as flow past a circular obstacle in a Newtonian flow field, three dimensional driven cavity, flow past a three dimensional cylinder with different immersion lengths. We explore the scalability and robustness of the formulations for both approaches and obtain very good speedup. Effective implementations of these procedures demonstrate for relatively coarse macro-meshes<br />the power of higher order methods in obtaining highly accurate results in CFD. While the procedures adopted in the paper have been explored in the past the novelty lies with applications with higher order methods which have been known to be computationally intensive.

Solving quasi‐static equations with the material‐point method

SummaryThe material‐point method models continua by following a set of unconnected material points throughout the deformation of a body. This set of points provides a Lagrangian description of the material and geometry. Information from the material points is projected onto a background grid where equations of motion are solved. The grid solution is then used to update the material points. This paper describes how to use this method to solve quasi‐static problems. The resulting discrete equations are a coupled set of nonlinear equations that are then solved with a Jacobian‐free, Newton–Krylov algorithm. The technique is illustrated by examining two problems. The first problem simulates a compact tension test and includes a model of material failure. The second problem computes effective, macroscopic properties of a polycrystalline thin film. Copyright © 2015 John Wiley & Sons, Ltd.

Drag Minimization Based on the Navier–Stokes Equations Using a Newton–Krylov Approach

A methodology is presented for performing numerical aerodynamic shape optimization based on the three-dimensional Reynolds-averaged Navier–Stokes (RANS) equations. An initial multiblock structured mesh is first fit with B-spline volumes that form the basis for a hybrid mesh movement scheme that is tightly integrated with the geometry parameterization based on B-spline surfaces. The RANS equations and the one-equation Spalart–Allmaras turbulence model are solved in a fully coupled manner using an efficient parallel Newton–Krylov algorithm with approximate-Schur preconditioning. Gradient evaluations are performed using the discrete-adjoint approach with analytical differentiation of the discrete flow and mesh movement equations. The overall methodology remains robust even in the presence of large shape changes. Several examples of lift-constrained drag minimization are provided, including a study of the common research model wing geometry, a wing–body–tail geometry with a prescribed spanwise load distribution, and a blended-wing–body configuration. An example is provided that demonstrates that a wing optimized based on the Euler equations exhibits substantially inferior performance when subsequently analyzed based on the RANS equations relative to a wing optimized based on the RANS equations.

A High-Order Central ENO Finite-Volume Scheme for Three-Dimensional Low-Speed Viscous Flows on Unstructured Mesh

AbstractHigh-order discretization techniques offer the potential to significantly reduce the computational costs necessary to obtain accurate predictions when compared to lower-order methods. However, efficient and universally-applicable high-order discretizations remain somewhat illusive, especially for more arbitrary unstructured meshes and for incompressible/low-speed flows. A novel, high-order, central essentially non-oscillatory (CENO), cell-centered, finite-volume scheme is proposed for the solution of the conservation equations of viscous, incompressible flows on three-dimensional unstructured meshes. Similar to finite element methods, coordinate transformations are used to maintain the scheme’s order of accuracy even when dealing with arbitrarily-shaped cells having non-planar faces. The proposed scheme is applied to the pseudo-compressibility formulation of the steady and unsteady Navier-Stokes equations and the resulting discretized equations are solved with a parallel implicit Newton-Krylov algorithm. For unsteady flows, a dual-time stepping approach is adopted and the resulting temporal derivatives are discretized using the family of high-order backward difference formulas (BDF). The proposed finite-volume scheme for fully unstructured mesh is demonstrated to provide both fast and accurate solutions for steady and unsteady viscous flows.