Three-Dimensional Aerodynamic Computations on Unstructured Grids Using a Newton-Krylov Approach

Abstract

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. Approximate viscous operators that involve only the nearest neighboring terms are studied to construct an efficient and effective preconditioner. Two incomplete factorization techniques are examined: one based on a level-of-fill approach while the other utilizes a threshold strategy. The performance of the algorithm is demonstrated with numerical studies over the ONERA M6 wing and the DLR-F6 wing-body configuration. A ten-order-of-magnitude residual reduction can be obtained in 20-25 hours on a single processor for a mesh with roughly 500,000 nodes.

Fast unsteady flow computations with a Jacobian-free Newton–Krylov algorithm

“…The Jacobian is approximately factored with an incomplete lower upper factorization [30] based on the footprint of the Jacobian (ILU(k)), because incomplete multilevel ILU factorizations [31] and incomplete dual thresholds factorizations [31] were found to be much less effective, as also reported and observed by others [25,28,32]. ILU(k) implies that a zero entry in the Jacobian may only become a nonzero in the ILU when it is up to k positions away from an existing nonzero element in the Jacobian.…”

A Jacobian-free Newton–Krylov algorithm for compressible turbulent fluid flows

“…In the threshold strategy, ILUTðP; sÞ, elements are dropped if they are smaller than s, and at most the P largest elements are kept in each row. Although the ILUTðP; sÞ strategy allows more precise control, it is much more expensive to form, and both Pueyo and Zingg [13] and Wong and Zingg [19] have demonstrated that the ILU(p) strategy is much more efficient in the current context.…”

“…In computations of two-dimensional flows, levels of fill between 2 and 4 are typically optimal. Lower values are preferred in three-dimensions [19,20]. In the present work, a value of p equal to 4 is used in all cases.…”

“…If they disagree, then Jacobian-free breakdown has occurred and consideration should be given to increasing d in Eq. (19). Forming the Jacobian matrix explicitly can be time consuming, and it may be worthwhile to examine the impact of varying d first.…”

“…In most cases, the residual was reduced by twelve orders of magnitude in fewer than 1000 equivalent residual evaluations. Closely related algorithms were applied to two-dimensional unstructured meshes by Blanco and Zingg [17,18], three-dimensional unstructured meshes by Wong and Zingg [19], three-dimensional inviscid flows on structured meshes by Nichols and Zingg [20], parallel computations of three-dimensional inviscid flows on structured meshes by Hicken and Zingg [21] and shape optimization based on steady [12] and unsteady [22] flows. Although the work of Pueyo and Zingg provided a solid foundation for further development of Newton-Krylov methods for the compressible Reynolds-averaged Navier-Stokes equations, it was restricted in several respects.…”