326 research outputs found
Incomplete Augmented Lagrangian Preconditioner for Steady Incompressible Navier-Stokes Equations
An incomplete augmented Lagrangian preconditioner, for the steady incompressible Navier-Stokes equations discretized by stable finite elements, is proposed. The eigenvalues of the preconditioned matrix are analyzed. Numerical experiments show that the incomplete augmented Lagrangian-based preconditioner proposed is very robust and performs quite well by the Picard linearization or the Newton linearization over a wide range of values of the viscosity on both uniform and stretched grids
An assessment of solvers for saddle point problems emerging from the incompressible Navier--Stokes equations
Efficient incompressible flow simulations, using inf-sup stable pairs of finite element spaces, require the application of efficient solvers for the arising linear saddle point problems. This paper presents an assessment of different solvers: the sparse direct solver UMFPACK, the flexible GMRES (FGMRES) method with different coupled multigrid preconditioners, and FGMRES with Least Squares Commutator (LSC) preconditioners. The assessment is performed for steady-state and time-dependent flows around cylinders in 2d and 3d. Several pairs of inf-sup stable finite element spaces with second order velocity and first order pressure are used. It turns out that for the steady-state problems often FGMRES with an appropriate multigrid preconditioner was the most efficient method on finer grids. For the time-dependent problems, FGMRES with LSC preconditioners that use an inexact iterative solution of the velocity subproblem worked best for smaller time steps
Alternative Solution Algorithms for Primal and Adjoint Incompressible Navier-Stokes
Regardless of the specific discretisation framework, the discrete incompressible Navier-Stokes equations present themselves in the form of a non-linear, saddle-point Oseentype system. Traditional CFD codes typically solve the system via the well-known SIMPLE-like algorithms, which are essentially block preconditioners based on Schur complement theory. Due to their “segregated” nature, which reduces to iteratively solving a sequence of linear systems smaller than the full Oseen and better conditioned, traditional SIMPLE-like algorithms have long been considered as the only viable strategy. However, recent progress in computational power and linear solver capabilities has led researchers to develop, for Oseen-type systems (and discrete Navier-Stokes in particular), a
number of alternative preconditioners and solution schemes, found to be more efficient than SIMPLE-like strategies but previously deemed practically unfeasible in industrial contexts.
The improved efficiency of novel preconditioners entails a) faster, more stable convergence and b) the possibility of driving residuals below more strict tolerances, which is sometimes difficult with SIMPLE due to stagnating behaviour. The second aspect in particular is extremely relevant in the context of adjoint-based optimisation, as evidence suggests that an adjoint system
may be affected by convergence issues when the primal flow solution is not well converged.
In this work, we present some solution schemes (both traditional and novel) implemented for the Mixed Hybrid Finite Volumes Navier-Stokes solver we introduced in our previous work.
Performance, in terms of robustness and convergence properties, is assessed on a series of benchmark test cases. We also turn our attention to the discrete adjoint Navier-Stokes problem itself, which in essence requires solving a linear system similar to the original Oseen and therefore may benefit from the same preconditioning techniques. We show how the primal algorithms
are adapted to the adjoint system, and we run a series of adjoint test cases to compare performance of various solution scheme
Augmented Lagrangian acceleration of global-in-time Pressure Schur complement solvers for incompressible Oseen equations
This work is focused on an accelerated global-in-time solution strategy for the Oseen
equations, which highly exploits the augmented Lagrangian methodology to improve the
convergence behavior of the Schur complement iteration. The main idea of the solution
strategy is to block the individual linear systems of equations at each time step into a
single all-at-once saddle point problem. By elimination of all velocity unknowns, the
resulting implicitly defined equation can then be solved using a global-in-time pressure
Schur complement (PSC) iteration. To accelerate the convergence behavior of this
iterative scheme, the augmented Lagrangian approach is exploited by modifying the
momentum equation for all time steps in a strongly consistent manner. While the
introduced discrete grad-div stabilization does not modify the solution of the discretized
Oseen equations, the quality of customized PSC preconditioners drastically improves
and, hence, guarantees a rapid convergence. This strategy comes at the cost that the
involved auxiliary problem for the velocity field becomes ill conditioned so that standard
iterative solution strategies are no longer efficient. Therefore, a highly specialized
multigrid solver based on modified intergrid transfer operators and an additive block
preconditioner is extended to solution of the all-at-once problem. The potential of
the proposed overall solution strategy is discussed in several numerical studies as they
occur in commonly used linearization techniques for the incompressible Navier-Stokes
equations
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