Three-dimensional aerodynamic computations on unstructured grids using a Newton–Krylov approach

A Newton–Krylov algorithm is presented for the compressible Navier–Stokes equations in three dimensions on unstructured grids. The algorithm uses a preconditioned matrix-free Krylov method to solve the linear system that arises in the Newton iterations. Incomplete factorization is used as the preconditioner, based on an approximate Jacobian matrix after the reverse Cuthill–McKee reordering of the unknowns. Several approximate viscous operators that involve only the nearest neighboring terms are studied to reduce the cost of preconditioning. The performance of the algorithm is demonstrated through numerical studies of the ONERA M6 wing and the DLR-F6 wing-body configuration. A ten-order-of-magnitude residual reduction for the wing and wing-body configurations can be obtained with a computing cost equivalent to 5500 and 8000 function evaluations, respectively, on grids with a half million nodes.

Efficient nonlinear solvers for Laplace–Beltrami smoothing of three-dimensional unstructured grids

This paper describes a component‐based nonlinear physical system simulation prototyping package written entirely in Java using object‐oriented design. The package provides scientists and engineers with a ‘developer‐friendly’ software environment for large‐scale computational algorithm and physical model development. The software design centers on the Jacobian‐free Newton–Krylov solution method surrounding a finite‐volume treatment of conservation equations. This enables a clean component‐like implementation. We first provide motivation for the development of the software and then discuss software structure. The discussion includes a description of the use of Java's built‐in thread facility that enables parallel, shared‐memory computations on a wide variety of unstructured grids with triangular, quadrilateral, tetrahedral and hexahedral elements. We also discuss the use of Java's inheritance mechanism in the construction of a hierarchy of physics systems objects and linear and nonlinear solver objects that simplify development and foster software re‐use. We provide a brief review of the Jacobian‐free Newton–Krylov nonlinear system solution method and discuss how it fits into our design. Following this, we show results from example calculations and then discuss plans including the extension of the software to distributed‐memory computer systems. Copyright © 2003 John Wiley & Sons, Ltd.

3D minimum-structure inversion of controlled-source EM data using unstructured grids

In this article, the results of aeroacoustic cases obtained by high-order methods are presented. One internal and one external case are considered. The internal case is a transonic cavity. It has been computed by three parters, namely FOI-LiU (Linköping University), University of Stuttgart (USTUTT) and Dassault Aviation (DASSAV). FOI-LiU has computed with their higher order accurate finite difference solver using a block structured grid. Comparisons are made to reference calculations using an unstructured grid with Edge. USTUTT uses a high order discontinuous Galerkin spectral element code for the final high-order computations and a mixed modal/nodal DG code for the baseline computations. Dassault Aviation uses their in-house stabilized finite element code Aether, both for the reference and the higher-order computations. In both instances unstructured tetrehedral grids are used. The external case is a quasi-two dimensional generic wing and flap configuration defined in the VALIANT FP7 project. Two institutions were involved in the high-order simulation of this low-Mach number test case, M = 0.15, Central Institute of Aero- and Hydrodynamics, TsAGI, and the von Karman Institute, VKI. In the following, the acoustic predictions corresponding to the two test cases will be described.KeywordsCFDCAAhybrid methodhigher order

During the last decades, research efforts have been focused on the derivation of effective preconditioned iterative methods. The preconditioned iterative methods are mainly categorized into implicit preconditioned methods and explicit preconditioned methods. In this manuscript we review implicit preconditioned methods, based on incomplete and approximate factorization, and explicit preconditioned methods, based on sparse approximate inverses and explicit approximate inverses. Modified Moore-Penrose conditions are presented and theoretical estimates for the sensitivity of the explicit approximate inverse matrix of the explicit preconditioned method are derived. Finally, the performance of the preconditioned iterative methods is illustrated by solving characteristic 2D elliptic problem and numerical results are given indicating a qualitative agreement with the theoretical estimates.

We present a study of the effectiveness of asynchronous incomplete LU factorization preconditioners for the time-implicit solution of compressible flow problems while exploiting thread-parallelism within a compute node. A block variant of the asynchronous fine-grained parallel preconditioner adapted to a finite volume discretization of the compressible Navier--Stokes equations on unstructured grids is presented, and convergence theory is extended to the new variant. Experimental (numerical) results on the performance of these preconditioners on inviscid and viscous laminar two-dimensional steady-state test cases are reported. It is found, for these compressible flow problems, that the block variant performs much better in terms of convergence, parallel scalability, and reliability than the original scalar asynchronous ILU preconditioner. For viscous flow, it is found that the ordering of unknowns may determine the success or failure of asynchronous block-ILU preconditioning, and an ordering of grid cells suitable for solving viscous problems is presented.

A fast, matrix-free implicit method has been developed to solve the three-dimensional compressible Euler and Navier?Stokes equations on unstructured meshes. An approximate system of linear equations arising from the Newton linearization is solved by the GMRES (generalized minimum residual) algorithm with a LU-SGS (lower?upper symmetric Gauss?Seidel) preconditioner. A remarkable feature of the present GMRES+LU-SGS method is that the storage of the Jacobian matrix can be completely eliminated by approximating the Jacobian with numerical fluxes, resulting in a matrix-free implicit method. The method developed has been used to compute the compressible flows around 3D complex aerodynamic configurations for a wide range of flow conditions, from subsonic to supersonic. The numerical results obtained indicate that the use of the GMRES+LU-SGS method leads to a significant increase in performance over the best current implicit methods, GMRES+ILU and LU-SGS, while maintaining memory requirements similar to its explicit counterpart. An overall speedup factor from eight to more than one order of magnitude for all test cases in comparison with the explicit method is demonstrated.

Considering the drag-divergence performance as a constraint, this paper presents a multi-point aero-structural design optimization of wings on a wing-body-tail-engine configuration of long-range dual-aisle civil aircraft by using a gradient-based method based on the discrete adjoint method. Firstly, the accuracy of the coupled aero-structural analysis method used in this paper was validated with DLR-F6 wing-body configuration. Then a wing aero-structural design optimization based on a dual-aisle civil aircraft is offered. The drag coefficient of the optimized configuration in every state decreased, which was reduced by 13.67 counts at the cruise condition, the difference in drag coefficient from Mach 0.85 to Mach 0.87 was decreased from 28.52 counts to 18.98 counts, indicating its drag-divergence performance has been improved undoubtedly. Finally, we compared the performance among the multi-point aero-structural optimized configuration, the single-point aerodynamic optimized configuration and the single-point structural optimized configuration. The results show that the multi-point aero-structural design optimization considering drag-divergence performance has great potential to gain a design configuration with better comprehensive and practical performance compared with a single-point optimization in a single discipline.

Several implementations of Newton-like iteration schemes based on Krylov subspace projection methods for solving nonlinear equations are considered. The simplest such class of methods is Newton’s algorithm in which a (linear) Krylov method is used to solve the Jacobian system approximately. A method in this class is referred to as a Newton–Krylov algorithm. To improve the global convergence properties of these basic algorithms, hybrid methods based on Powell’s dogleg strategy are proposed, as well as linesearch backtracking procedures. The main advantage of the class of methods considered in this paper is that the Jacobian matrix is never needed explicitly.

Coupled dynamics in the physics of molten salt reactors

The controlled Cholesky factorization (CCF) has been shown to be a robust preconditioner for the conjugate gradient (CG) method. In this scheme the amount of fill-in is defined in terms of a parameter $\eta$, the number of extra nonzero elements allowed per column. This parameter $\eta$ can be chosen in such a way that the preconditioning factor will require more or less storage than the coefficient matrix. The selection of nonzero elements kept for the preconditioner is made by value instead of by position, so the sparsity pattern of the original matrix is never considered by the proposed factorization. This preconditioned method is termed the CCCG($\eta$) method and is a generalization of a method developed by Jones and Plassmann [\textit{ACM Trans.\ Math.\ Software}, 21 (1995), pp.\ 5--17]. It is demonstrated how an optimum value of $\eta$ can be automatically determined when solving systems with multiple right-hand sides. Typically such systems occur when using time-stepping procedures. Numerical examples are presented for some time-dependent parabolic partial differential equations discretized using linear triangular finite elements on unstructured grids. Comparisons made between CCCG($\eta$), standard ICCG, and the Jones and Plassmann method show CCCG($\eta$) to be an efficient solver.

At very high concentrations phosphorus diffusion in silicon exhibits marked nonlinearities. The hierarchy of physical models that attempt to explain this anomalous diffusion are reviewed. An eight-species kinetic model is derived that yields a quasilinear, partly-dissipative system of reaction-diffusion partial differential equations. The numerical method of lines is used to solve the system for a simplified five-species model in three dimensions. The linear system in the Newton iteration is solved using several matrix-free methods. In all cases the dimension of the Krylov subspace must be quite large to insure convergence. This suggests that preconditioning will be more important for efficiency than choice of an accelerator.

An unstructured grid method is presented to compute three-dimensional compressible turbulent flows for complex geometries. The Navier-Stokes equations, along with the one-equation turbulence model of Spalart-Allmaras are solved by the use of a parallel, matrix-free implicit method on unstructured tetrahedral grids. The developed method has been used to predict drags in the transonic regime for both DLR-F4 and DLR-F6 configurations to assess the accuracy and efficiency of the method

SummaryThis study explores the performance and scaling of a GMRES Krylov method employed as a smoother for an algebraic multigrid preconditioned Newton‐Krylov solution approach applied to a fully implicit variational multiscale finite element resistive magnetohydrodynamics formulation. In this context, a Newton iteration is used for the nonlinear system and a parallel MPI‐based Krylov method is employed for the linear subsystems. The efficiency of this approach is critically dependent on the scalability and performance of the parallel algebraic multigrid preconditioner for the linear solutions and the performance of the multigrid smoothers play a critical role. Krylov multigrid smoothers are considered in an attempt to reduce the time and memory requirements of existing robust smoothers based on additive Schwarz domain decomposition with incomplete LU factorization solves on each subdomain. Three time‐dependent resistive magnetohydrodynamics test cases are considered to evaluate the method. Compared with a domain decomposition incomplete LU smoother, the GMRES smoother can reduce the solve time due to a significant decrease in the preconditioner setup time and often a reduction in outer Krylov solver iterations, and requires less memory, typically 35% less memory.