Building blocks for a leading edge high‐order flow solver

Huismann, Immo; Stiller, Jörg; Fröhlich, Jochen

doi:10.1002/pamm.201710037

Cited by 4 publications

(5 citation statements)

References 16 publications

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“…The locally-preconditioned solver dCG exhibits a high robustness against increases in the aspect ratio, only requiring twice as long for an aspect ratio of AR = 48. This is to be expected, as it bears similarity to so-called wirebasket solvers, even sharing their poly-logarithmic bounds regarding the polynomial degree [21]. The multigrid solvers do not fare as well.…”

Section: Solver Runtimes For Anisotropic Meshesmentioning

confidence: 96%

Scaling to the stars – a linearly scaling elliptic solver for p-multigrid

Huismann

Stiller

Fröhlich

2019

Journal of Computational Physics

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High-order methods gain increased attention in computational fluid dynamics. However, due to the time step restrictions arising from the semi-implicit time stepping for the incompressible case, the potential advantage of these methods depends critically on efficient elliptic solvers. Due to the operation counts of operators scaling with with the polynomial degree p times the number of degrees of freedom nDOF, the runtime of the best available multigrid solvers scales with O(p · nDOF). This scaling with p significantly lowers the applicability of high-order methods to high orders. While the operators for residual evaluation can be linearized when using static condensation, Schwarz-type smoothers require their inverses on fixed subdomains. No explicit inverse is known in the condensed case and matrix-matrix multiplications scale with p · nDOF. This paper derives a matrix-free explicit inverse for the static condensed operator in a cuboidal subdomain. It scales with p 3 per element, i.e. nDOF globally, and allows for a linearly scaling additive Schwarz smoother, yielding a p-multigrid cycle with an operation count of O(nDOF). The resulting solver uses fewer than four iterations for all polynomial degrees to reduce the residual by ten orders and has a runtime scaling linearly with nDOF for polynomial degrees at least up to 48. Furthermore the runtime is less than one microsecond per unknown over wide parameter ranges when using one core of a CPU, leading to time-stepping for the incompressible Navier-Stokes equations using as much time for explicitly treated convection terms as for the elliptic solvers. * Corresponding author: Immo.Huismann@tu-dresden.de arXiv:1808.03595v1 [math.NA] 10 Aug 2018 to h-multigrid. In both cases, smoothers are required and overlapping Schwarz smoothers are a good option [18,29,41]. These decompose the grid into overlapping blocks on which the exact inverse is applied, producing exceptional smoothing rates. But these methods not only require the super-linearly scaling operator for residual evaluation, but a super-linearly scaling solver as well. This super-linear scaling prevents the respective multigrid techniques from achieving their full potential and prohibits the usage of large polynomial degrees.The goal of this paper is to devise a linearly scaling multigrid cycle for three-dimensional structured Cartesian grids, where residual evaluation, smoothing, restriction, and prolongation all scale linearly with the number degrees of freedom, without a further factor of p present in other methods. To this end the residual evaluation derived in [22] for the static condensed case is combined with the p-multigrid block smoother technique proposed in [18,17], leaving only the inverse on a 2 3 element block as non-matrix-free, super-linearly scaling operator. This operator is embedded in the full system and then factorized and rearranged, attaining a matrix-free inverse that is then factorized to linear complexity. The inverse leads to a linearly scaling multigrid cycle, which is confirmed in run...

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Section: Solver Runtimes For Anisotropic Meshesmentioning

confidence: 96%

Scaling to the stars – a linearly scaling elliptic solver for p-multigrid

Huismann

Stiller

Fröhlich

2019

Journal of Computational Physics

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“…However, it is yet unknown whether the approach extends to the discontinuous case. Iterative substructuring [32], the Cascadic multigrid method [2], or multigrid conjugate gradient methods [31] are further candidates to take into account and were investigated for the continuous methed in [19] and [21], respectively. Work in this direction is under way and will be published in a following paper.…”

Section: Robustness Against Increases In the Number Of Elementsmentioning

confidence: 99%

Linearizing the hybridizable discontinuous Galerkin method: A linearly scaling operator

Huismann,

Stiller,

Fröhlich

2020

Preprint

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This paper proposes a matrix-free residual evaluation technique for the hybridizable discontinuous Galerkin method requiring a number of operations scaling only linearly with the number of degrees of freedom. The method results from application of tensor-product bases on cuboidal Cartesian elements, a specific choice for the penalty parameter, and the fast diagonalization technique. In combination with a linearly scaling, face-wise preconditioner, a linearly scaling iteration time for a conjugate gradient method is attained. This allows for solutions in 1 µs per unknown on one CPU core -a number typically associated with low-order methods.

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“…2,15 A proper designed multigrid scheme for the linear solver in CFD tools is a key component for scalability and high efficiency. 16 Furthermore, the majority of practical simulation scenarios requires adaptive unstructured grids. Other high-order numerical methods that addresses the wave propagation problem and the wave-body problem in a single solver strategy is the high-order boundary element method [17][18][19] that is particularly strong in handling the geometry using unstructured grids, but is limited in terms of numerical efficiency due to inefficient asymptotic scaling of work effort as a result of high computational complexity in the discrete solution of the resulting system of equations in the solver.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, numerical schemes that have fast convergence rates and scale linearly with problem size (measured in terms of degrees of freedom in the discretization) has significant potential to reduce the cost of CFD and enable simulations of increasing fidelity in practical times 2,15 . A proper designed multigrid scheme for the linear solver in CFD tools is a key component for scalability and high efficiency 16 . Furthermore, the majority of practical simulation scenarios requires adaptive unstructured grids.…”

Section: Introductionmentioning

confidence: 99%

An efficient p‐multigrid spectral element model for fully nonlinear water waves and fixed bodies

Engsig-Karup

Laskowski

2021

Numerical Methods in Fluids

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In marine offshore engineering, cost‐efficient simulation of unsteady water waves and their nonlinear interaction with bodies are important to address a broad range of engineering applications at increasing fidelity and scale. We consider a fully nonlinear potential flow (FNPF) model discretized using a Galerkin spectral element method to serve as a basis for handling both wave propagation and wave‐body interaction with high computational efficiency within a single modeling approach. We design and propose an efficient 𝒪(n)‐scalable computational procedure based on geometric p‐multigrid for solving the Laplace problem in the numerical scheme. The fluid volume and the geometric features of complex bodies is represented accurately using high‐order polynomial basis functions and unstructured meshes with curvilinear prism elements. The new p‐multigrid spectral element model can take advantage of the high‐order polynomial basis and thereby avoid generating a hierarchy of geometric meshes with changing number of elements as required in geometric h‐multigrid approaches. We provide numerical benchmarks for the algorithmic and numerical efficiency of the iterative geometric p‐multigrid solver. Results of numerical experiments are presented for wave propagation and for wave‐body interaction in an advanced case for focusing design waves interacting with a floating production storage and offloading. Our study shows, that the use of iterative geometric p‐multigrid methods for the Laplace problem can significantly improve run‐time efficiency of FNPF simulators.

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Building blocks for a leading edge high‐order flow solver

Cited by 4 publications

References 16 publications

Scaling to the stars – a linearly scaling elliptic solver for p-multigrid

Scaling to the stars – a linearly scaling elliptic solver for p-multigrid

Linearizing the hybridizable discontinuous Galerkin method: A linearly scaling operator

An efficient p‐multigrid spectral element model for fully nonlinear water waves and fixed bodies

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