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

Botti, Lorenzo Alessio; Colombo, Alessandro; Bassi, Francesco

doi:10.1016/j.jcp.2017.07.002

Cited by 28 publications

(35 citation statements)

References 47 publications

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“…Compared to subspace non-inheritance, which requires the re-evaluation of the Jacobians in proper coarser-space discretizations of the problem, inheritance is cheaper in processing and memory. Although previous work has shown lower convergence rates when using such cheaper operators [32,11,12], especially in the context of elliptic problems and incompressible flows, we found these operators sufficiently efficient for our target problems involving the compressible NS equations, as will be demonstrated in the results section.…”

Section: Multigrid Preconditioningmentioning

confidence: 68%

Efficient discontinuous Galerkin implementations and preconditioners for implicit unsteady compressible flow simulations

2020

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This work presents and compares efficient implementations of high-order discontinuous Galerkin methods: a modal matrix-free discontinuous Galerkin (DG) method, a hybridizable discontinuous Galerkin (HDG) method, and a primal formulation of HDG, applied to the implicit solution of unsteady compressible flows. The matrix-free implementation allows for a reduction of the memory footprint of the solver when dealing with implicit time-accurate discretizations. HDG reduces the number of globally-coupled degrees of freedom relative to DG, at high order, by statically condensing element-interior degrees of freedom from the system in favor of face unknowns. The primal formulation further reduces the element-interior degrees of freedom by eliminating the gradient as a separate unknown. This paper introduces a p-multigrid preconditioner implementation for these discretizations and presents results for various flow problems. Benefits of the p-multigrid strategy relative to simpler, less expensive, preconditioners are observed for stiff systems, such as those arising from low-Mach number flows at high-order approximation. The p-multigrid preconditioner also shows excellent scalability for parallel computations. Additional savings in both speed and memory occur with a matrix-free/reduced version of the preconditioner.

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Section: Multigrid Preconditioningmentioning

confidence: 68%

Efficient discontinuous Galerkin implementations and preconditioners for implicit unsteady compressible flow simulations

2020

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“…Remember that P = (P 1 , P 2 , P 3 ) can be different in each element and direction. Using equation (20), equation (21) can be rewritten as…”

Section: Nonlinear P-multigridmentioning

confidence: 99%

“…This is advantageous for accelerating the computations through adaptation strategies that adjust the element size (h) or the polynomial order (p) locally. Multigrid solvers have also been used to accelerate high-order DG time marching computations for a fixed number of degrees of freedom [11][12][13][14][15][16][17][18][19][20]. The Discontinuous Galerkin Spectral Element Method (DGSEM) [21,22] is a high-order nodal variant of the DG technique on hexahedral meshes that is especially suited for mesh adaptation strategies because, in addition to the mentioned properties, it handles p-anisotropic representations efficiently [1,9,23].…”

Section: Introductionmentioning

confidence: 99%

“…Craig and Zienkiewicz [47], and Rønquist and Patera [48] were the first authors working on high-order methods that proposed the use of the polynomial order, p, to define the levels of a multigrid scheme, the former for p-finite elements and the latter for nodal spectral elements. After these initial works, the use of multilevel methods spread in the high-order community; initially as p-multigrid methods [11][12][13][14][15] and more recently as hp-multigrid methods [16][17][18][19][20]. Most of these implementations use modal hierarchical shape functions [11,14,16,17], and only a small number of publications focus on nodal-based shape functions [11,15].…”

Section: Introductionmentioning

confidence: 99%

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A p-multigrid strategy with anisotropic p-adaptation based on truncation errors for high-order discontinuous Galerkin methods

Rueda-Ramírez

Manzanero

Ferrer

et al. 2019

Journal of Computational Physics

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High-order discontinuous Galerkin methods have become a popular technique in computational fluid dynamics because their accuracy increases spectrally in smooth solutions with the order of the approximation. However, their main drawback is that increasing the order also increases the computational cost. Several techniques have been introduced in the past to reduce this cost. On the one hand, local mesh adaptation strategies based on error estimation have been proposed to reduce the number of degrees of freedom while keeping a similar accuracy. On the other hand, multigrid solvers may accelerate time marching computations for a fixed number of degrees of freedom.In this paper, we combine both methods and present a novel anisotropic p-adaptation multigrid algorithm for steady-state problems that uses the multigrid scheme both as a solver and as an anisotropic error estimator. To achieve this, we show that a recently developed anisotropic truncation error estimator [1, A. M. Rueda-Ramírez, G. Rubio, E. Ferrer, E. Valero, Truncation Error Estimation in the p-Anisotropic Discontinuous Galerkin Spectral Element Method, Journal of Scientific Computing] is perfectly suited to be performed inside the multigrid cycle with negligible extra cost. Furthermore, we introduce a multi-stage p-adaptation procedure which reduces the computational time when very accurate results are required.The proposed methods are tested for the compressible Navier-Stokes equations, where we investigate two cases. First, the 2D boundary layer flow on a flat plate is studied to assess accuracy and computational cost of the algorithm, where a speed-up of 816 is achieved compared to the traditional explicit method. Second, the 3D flow around a sphere is simulated and used to test the anisotropic properties of the proposed method, where a speed-up of 152 is achieved compared to the explicit method. The proposed multi-stage procedure achieved a speed-up of 2.6 in comparison to the single-stage method in highly accurate simulations.

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“…Recently, alongside ubiquitous FV‐based CFD formulations, discontinuous Galerkin FE methods (dGFEM) are gaining momentum for their efficacy and reliability in handling complex CFD applications. Several research efforts have proposed and investigated efficient solution strategies for dGFEM discretizations of incompressible flow problems, see, eg, previous studies …”

Section: Introductionmentioning

confidence: 99%

Modeling hemodynamics in intracranial aneurysms: Comparing accuracy of CFD solvers based on finite element and finite volume schemes

Botti

Paliwal

Conti

et al. 2018

Numer Methods Biomed Eng

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Image-based computational fluid dynamics (CFD) has shown potential to aid in the clinical management of intracranial aneurysms, but its adoption in the clinical practice has been missing, partially because of lack of accuracy assessment and sensitivity analysis. To numerically solve the flow-governing equations, CFD solvers generally rely on 2 spatial discretization schemes: finite volume (FV) and finite element (FE). Since increasingly accurate numerical solutions are obtained by different means, accuracies and computational costs of FV and FE formulations cannot be compared directly. To this end, in this study, we benchmark 2 representative CFD solvers in simulating flow in a patient-specific intracranial aneurysm model: (1) ANSYS Fluent, a commercial FV-based solver, and (2) VMTKLab multidGetto, a discontinuous Galerkin (dG) FE-based solver. The FV solver's accuracy is improved by increasing the spatial mesh resolution (134k, 1.1m, 8.6m, and 68.5m tetrahedral element meshes). The dGFE solver accuracy is increased by increasing the degree of polynomials (first, second, third, and fourth degree) on the base 134k tetrahedral element mesh. Solutions from best FV and dGFE approximations are used as baseline for error quantification. On average, velocity errors for second-best approximations are approximately 1 cm/s for a [0,125] cm/s velocity magnitude field. Results show that high-order dGFE provides better accuracy per degree of freedom but worse accuracy per Jacobian nonzero entry as compared with FV. Cross-comparison of velocity errors demonstrates asymptotic convergence of both solvers to the same numerical solution. Nevertheless, the discrepancy between underresolved velocity fields suggests that mesh independence is reached following different paths.

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

Cited by 28 publications

References 47 publications

Efficient discontinuous Galerkin implementations and preconditioners for implicit unsteady compressible flow simulations

Efficient discontinuous Galerkin implementations and preconditioners for implicit unsteady compressible flow simulations

A p-multigrid strategy with anisotropic p-adaptation based on truncation errors for high-order discontinuous Galerkin methods

Modeling hemodynamics in intracranial aneurysms: Comparing accuracy of CFD solvers based on finite element and finite volume schemes

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