Multigrid Solver Algorithms for DG Methods and Applications to Aerodynamic Flows

Wallraff, Marcel; Hartmann, Ralf; Leicht, Tobias

doi:10.1007/978-3-319-12886-3_9

Cited by 13 publications

(14 citation statements)

References 23 publications

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“…The issue of developing optimal solvers for Composite discontinuous Galerkin Methods, first developed and analyzed Antonietti et al [10] was considered by Antonietti et al [11,12]. More recently Antonietti et al [13] analysed multigrid strategies for Interior Penalty dG discretizations over agglomerated elements meshes, while Wallraff and Leicht [14] and Wallraff et al [15] applied an agglomeration based h-multigrid solver to dG discretizations of the compressible Reynolds Averaged Navier-Stokes (RANS) equations.…”

Section: Introductionmentioning

confidence: 99%

h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems

Botti

Colombo

Bassi

2017

Journal of Computational Physics

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In this work we exploit agglomeration based h-multigrid preconditioners to speed-up the iterative solution of discontinuous Galerkin discretizations of the Stokes and Navier-Stokes equations. As a distinctive feature h-coarsened mesh sequences are generated by recursive agglomeration of a fine grid, admitting arbitrarily unstructured grids of complex domains, and agglomeration based discontinuous Galerkin discretizations are employed to deal with agglomerated elements of coarse levels. Both the expense of building coarse grid operators and the performance of the resulting multigrid iteration are investigated. For the sake of efficiency coarse grid operators are inherited through element-by-element L 2 projections, avoiding the cost of numerical integration over agglomerated elements. Specific care is devoted to the projection of viscous terms discretized by means of the BR2 dG method. We demonstrate that enforcing the correct amount of stabilization on coarse grids levels is mandatory for achieving uniform convergence with respect to the number of levels. The numerical solution of steady and unsteady, linear and non-linear problems is considered tackling challenging 2D test cases and 3D real life computations on parallel architectures.Significant execution time gains are documented.

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Section: Introductionmentioning

confidence: 99%

h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems

Botti

Colombo

Bassi

2017

Journal of Computational Physics

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“…To this end, pseudo‐transient continuation strategies, as those proposed in Bassi et al and Botti, can be considered to enhance the solver performance. As recently shown by Leicht and Wallraff and Wallraff et al, the steady‐state convergence can be greatly accelerated by using multigrid techniques. In this paper, we apply h ‐multigrid to the nonlinear system directly by means of the so‐called Full Approximation Scheme (FAS).…”

Section: Basic Concepts For Dg On Agglomerated Meshesmentioning

confidence: 98%

“…The nonlinear problems of the fine‐to‐coarse levels are discretized by means of a LBE scheme where linear systems are solved with a Generalized Minimal RESidual method (GMRES) preconditioned with linear h ‐multigrid. An iterative line‐Jacobi scheme is used as linear smoother and coarse level solver within the multigrid …”

Section: Basic Concepts For Dg On Agglomerated Meshesmentioning

confidence: 99%

“…In this work, for the solution of the linear system of Equation , we rely on the same solver as used for the linearized systems arising in the nonlinear solver, ie, the system is solved by means of a GMRES Krylov subspace method, which is preconditioned by a linear h ‐multigrid method, which in turn uses an iterative line‐Jacobi approach as smoother and coarse level solver . In the numerical examples, the adjoint problem is considered converged when the linear residual norm has been reduced by 8 orders of magnitude.…”

Section: Adjoint‐based Error Estimationmentioning

confidence: 99%

“…In this work, for the solution of the linear system of Equation 17, we rely on the same solver as used for the linearized systems arising in the nonlinear solver, ie, the system is solved by means of a GMRES Krylov subspace method, which is preconditioned by a linear h-multigrid method, which in turn uses an iterative line-Jacobi approach as smoother and coarse level solver. 34,35 In the numerical examples, the adjoint problem is considered converged when the linear residual norm has been reduced by 8 orders of magnitude. Enhancing the functional of interest with the adjoint problem As a first test to assess the effectiveness of the adjoint problem, we consider the solution of the inviscid flow around the NACA0012 airfoil with M ∞ = 0.5 and = 1 • on a very coarse grid made of 512 quadrangular elements with curved edges (quadratic).…”

Section: Formulation Of the Adjoint Problemmentioning

confidence: 99%

See 2 more Smart Citations

An agglomeration‐based adaptive discontinuous Galerkin method for compressible flows

Zenoni

Leicht

Colombo

et al. 2017

Numerical Methods in Fluids

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Summary In this work, we exploit the possibility to devise discontinuous Galerkin discretizations over polytopic grids to perform grid adaptation strategies on the basis of agglomeration coarsening of a fine grid obtained via standard unstructured mesh generators. The adaptive agglomeration process is here driven by an adjoint‐based error estimator. We investigate several strategies for converting the error field estimated solving the adjoint problem into an agglomeration factor field that is an indication of the number of elements of the fine grid that should be clustered together to form an agglomerated element. As a result the size of agglomerated elements is optimized for the achievement of the best accuracy for given grid size with respect to the target quantities. To demonstrate the potential of this strategy we consider problem‐specific outputs of interest typical of aerodynamics, eg, the lift and drag coefficients in the context of inviscid and viscous flows test cases.

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Accelerating CFD solver computation time with reduced‐order modeling in a multigrid environment

Karcher

Wallraff

2020

Numerical Methods in Fluids

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SummaryReduced‐order models (ROMs) are more and more considered for use in aerodynamic applications. Benefits of these methods can be expected for optimization problems or predicting aerodynamic loads for the entire flight envelope. For these applications it is often possible to perform computations for various parameter combinations before any ROM evaluations are needed. The order reduction of the CFD solutions in this article is done using proper orthogonal decomposition. Coupled with an interpolation method predictions for unknown parameter combinations can be made. The CFD solutions are computed using a discontinuous Galerkin finite element method combined with a nonlinear multigrid scheme. The nonlinear multigrid solver algorithms depend on a good initial guess of the flow field to be computed. Typically, the initial guess on a fine mesh is obtained by solving the problem on a agglomerated mesh. Alternatively, an initial guess could be obtained from a ROM prediction, if available. The main objective is to identify the benefits of using a ROM in a higher‐order multigrid environment. Integral as well as distributed surface quantities of the ROM predictions originating from several multigrid levels will be compared with fully converged flow solutions on the top level of the multigrid algorithm. Furthermore, initializing the flow solver with predictions from several multigrid levels will be analyzed in comparison to full multigrid computations as well as to a scenario where already converged solutions for similar parameter combinations are used as initial flow solutions on the top multigrid level.

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Multigrid Solver Algorithms for DG Methods and Applications to Aerodynamic Flows

Cited by 13 publications

References 23 publications

h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems

h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems

An agglomeration‐based adaptive discontinuous Galerkin method for compressible flows

Accelerating CFD solver computation time with reduced‐order modeling in a multigrid environment

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