Agglomeration multigrid methods with implicit Runge–Kutta smoothers applied to aerodynamic simulations on unstructured grids

Langer, Stefan

doi:10.1016/j.jcp.2014.07.050

Cited by 40 publications

(20 citation statements)

References 20 publications

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“…The presented results with SGS use the standard implementation as described here. Other implementation choices may produce slightly different figures, however the presented SGS results by Swanson et al [41], as well as Langer et al [15,30] all demonstrate a similar CFL number limitation as observed with our implementation.…”

Section: Sgs As Linear Solversupporting

confidence: 77%

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Stabilisation of discrete steady adjoint solvers

Radford

Meyer

et al. 2015

Journal of Computational Physics

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A new implicit time-stepping scheme which uses Runge-Kutta time-stepping and Krylov methods as a smoother inside FAS-cycle multigrid acceleration is proposed to stabilise the flow solver and its discrete adjoint counterpart. The algorithm can fully converge the discrete adjoint solver in a wide range of cases where conventional point-implicit methods fail due to either physical or numerical instability. This enables the discrete adjoint to be applied to a much wider range of flow regimes. In addition, the new algorithm offers improved efficiency when applied to stable cases for which the conventional Block-Jacobi solver can fully converge. Both stable and unstable cases are presented to demonstrate the improved robustness and performance of the new scheme. Eigen-analysis is presented to outline the mechanism of the adjoint stabilisation effect.

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Section: Sgs As Linear Solversupporting

confidence: 77%

“…At each iteration, the flow update is computed by inverting P directly or approximately. A hierarchy of time-marching methods can be derived by using for the preconditioning matrix P different approximations of the flow Jacobian matrix ∂R/∂U [15], as illustrated in Fig. 1.…”

Section: Convergence Acceleration Techniques For Nonlinear Flow Solversmentioning

confidence: 99%

Stabilisation of discrete steady adjoint solvers

Radford

Meyer

et al. 2015

Journal of Computational Physics

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“…In a general formulation, the solution method may be derived by a diagonally implicit Runge-Kutta method [28] and can be interpreted as a Rosenbrock method [30]. Its final formulation is…”

Section: Solution Methodsmentioning

confidence: 99%

“…Typical choices are ξ ∈ 1∕128; 1∕32. For a better understanding of the cell stretching coefficient, we refer to Langer [28]. In this paper, a similar coefficient has been investigated and the analysis carries over to the one formulated here one by one.…”

Section: A Discretizationmentioning

confidence: 99%

Investigation and Comparison of Implicit Smoothers Applied in Agglomeration Multigrid

2015

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A straightforward implicit smoothing method implemented in several codes solving the Reynolds-averaged Navier-Stokes equations is the lower-upper symmetric Gauss-Seidel method. It was proposed several years ago and is attractive to implement because of its low memory requirements and low operation count. Since, for many examples, often a severe restriction of the Courant-Friedrichs-Lewy number and loss of robustness are observed, it is the goal of this paper to revisit the lower-upper symmetric Gauss-Seidel implementation and to discuss several alternative implicit smoothing strategies used within an agglomeration multigrid for unstructured meshes. The starting point is a full implicit multistage Runge-Kutta method. Based on this method, several additional features and simplifications are developed and suggested, such that the implicit method is applicable to high-Reynolds-number viscous flows; that is, the required matrices fit into the fast memory of the cluster hardware and the arising linear systems can be approximately solved efficiently. To this end, the focus is on simplifications of the Jacobian, as well as efficient iterative approximate solution methods. To significantly improve the approximate linear solution methods, grid anisotropy is taken care of for approximately solving both the linear systems and the agglomeration strategy. The procedure creating coarse-grid meshes is extended by strategies identifying structured parts of the mesh. This seems to improve the quality of coarse-grid meshes in the way that an overall better reliability of the multigrid can be observed. Furthermore, grid information is exploited within the iterative solution methods for the linear systems. Numerical examples demonstrate the gain with respect to reliability and efficiency.

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“…Compared to the original SA model [21], this avoids the clipping of the turbulent variable to a nonnegative value which potentially prevents the full convergence of the nonlinear solver. The turbulence equation is discretized using the first-order accurate upwind scheme [22].…”

Section: A Governing Equationsmentioning

confidence: 99%

Flow Instability Prediction Via Eigenanalysis and its Application to Rotating Stall

Xu¹,

He²,

Sun³

et al. 2019

Proceedings of Global Power &Amp; Propulsion Society

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Global linear stability analysis is an effective way to predict the exact condition at which flow goes unstable. Compared to the time-domain simulation approach, eigenanalysis method can equivalently predict the destabilization condition, but at a much lower cost, since unsteady simulations are no longer required. In this work, a Newton-Krylov nonlinear flow solver is used to first solve for the steady state flow solution and then eigenanalysis is performed by applying the implicit-restart Arnoldi method to the exact Jacobian matrix. By tracking a subset of the eigenspectrum that is close to the imaginary axis, the least stable eigenmodes can be found. By perturbing the bifurcation parameter, e.g., the Reynolds number, the Hopf bifurcation point can be identified. This method is applied to find the critical Reynolds number for a laminar flow around a circular cylinder above which laminar vortex shedding appears. Time-accurate unsteady simulation confirms the correctness of the critical eigenvalue and eigenvector found. It is also applied to a quasi-3D compressor rotor annular cascade case, for which eigenanalysis is performed and flow physics is analyzed based on the unstable modes identified. Interesting correlation between the rotating perturbation pattern and cell rotating speed is found, which resembles what is observed in experiments. This work is a first step towards the study of rotating flow instabilities in turbomachines, such as rotating stall and rotating instability, and the preliminery results proved promising for future application to three-dimensional practical problems.

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Agglomeration multigrid methods with implicit Runge–Kutta smoothers applied to aerodynamic simulations on unstructured grids

Cited by 40 publications

References 20 publications

Stabilisation of discrete steady adjoint solvers

Stabilisation of discrete steady adjoint solvers

Investigation and Comparison of Implicit Smoothers Applied in Agglomeration Multigrid

Flow Instability Prediction Via Eigenanalysis and its Application to Rotating Stall

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