Parallel preconditioners for the unsteady Navier–Stokes equations and applications to hemodynamics simulations

Deparis, Simone; Grandperrin, Gwenol; Quarteroni, Alfio

doi:10.1016/j.compfluid.2013.10.034

Cited by 35 publications

(23 citation statements)

References 27 publications

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“…More advanced block preconditioners are the pressure convection-diffusion preconditioner, [21][22][23] the least-squares commutator preconditioner, 24 Yosida method, 25,26 the relaxed dimensional factorization preconditioner, 27 and the dimensional splitting preconditioner. 28,29 Early studies of domain decomposition methods for the Stokes problem were conducted in Reference 30. Domain-decomposition-based Schwarz preconditioners for Stokes and mixed elasticity problems have already been used for the approximation of the inverse matrices of blocks in Reference 5 and as monolithic preconditioners in References 4,31-33, and 34.…”

Section: Introductionmentioning

confidence: 99%

Reduced dimension GDSW coarse spaces for monolithic Schwarz domain decomposition methods for incompressible fluid flow problems

Heinlein

Hochmuth

Klawonn

2019

Numerical Meth Engineering

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Summary Monolithic preconditioners for incompressible fluid flow problems can significantly improve the convergence speed compared with preconditioners based on incomplete block factorizations. However, the computational costs for the setup and the application of monolithic preconditioners are typically higher. In this article, several techniques are applied to monolithic two‐level generalized Dryja‐Smith‐Widlund (GDSW) preconditioners to further improve the convergence speed and the computing time. In particular, reduced dimension GDSW coarse spaces, restricted and scaled versions of the first level, hybrid, and parallel coupling of the levels, and recycling strategies are investigated. Using a combination of all these improvements, for a small time‐dependent Navier‐Stokes problem on 240 message passing interface (MPI) ranks, a reduction of 86% of the time‐to‐solution can be obtained. Even without applying recycling strategies, the time‐to‐solution can be reduced by more than 50% for a larger steady Stokes problem on 4608 MPI ranks. For the largest problems with 11 979 MPI ranks, the scalability deteriorates drastically for the monolithic GDSW coarse space. On the other hand, using the reduced dimension coarse spaces, good scalability up to 11 979 MPI ranks, which corresponds to the largest problem configuration fitting on the employed supercomputer, could be achieved.

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Section: Introductionmentioning

confidence: 99%

Reduced dimension GDSW coarse spaces for monolithic Schwarz domain decomposition methods for incompressible fluid flow problems

Heinlein

Hochmuth

Klawonn

2019

Numerical Meth Engineering

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“…The first one shows some Matlab experiments that allow us to compare how the new estimates for α compete against the Fourier analysis estimate introduced in [5]. The second one shows some numerical experiments using up to 8192 cores on the aneurysm benchmark problem introduced in [14].…”

Section: Numerical Resultsmentioning

confidence: 99%

Parameter estimates for the Relaxed Dimensional Factorization preconditioner and application to hemodynamics

Benzi

Deparis

Grandperrin

et al. 2016

Computer Methods in Applied Mechanics and Engineering

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Abstract. We present new results on the Relaxed Dimensional Factorization (RDF) preconditioner for solving saddle point problems, first introduced in [5]. This method contains a parameter α > 0, to be chosen by the user. Previous works provided an estimate of α in the 2D case using Local Fourier Analysis. Novel algebraic estimation techniques for finding a suitable value of the RDF parameter in both the 2D and the 3D case with arbitrary geometries are proposed. These techniques are tested on a variety of discrete saddle point problems arising from the approximation of the Navier-Stokes equations using a Marker-and-Cell scheme and a finite element one. We also show results for a large-scale problem relevant for hemodynamics simulation that we solve in parallel using up to 8196 cores.

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“…For a recent comparison of the performance of different preconditioners (all described above) used for haemodynamic applications, see Deparis, Grandperrin and Quarteroni (2014).…”

Section: Rotational-incremental Chorin-temam Methodsmentioning

confidence: 99%

The cardiovascular system: Mathematical modelling, numerical algorithms and clinical applications

Quarteroni

Manzoni

Vergara³

2017

Acta Numerica

184

125

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Mathematical and numerical modelling of the cardiovascular system is a research topic that has attracted remarkable interest from the mathematical community because of its intrinsic mathematical difficulty and the increasing impact of cardiovascular diseases worldwide. In this review article we will address the two principal components of the cardiovascular system: arterial circulation and heart function. We will systematically describe all aspects of the problem, ranging from data imaging acquisition, stating the basic physical principles, analysing the associated mathematical models that comprise PDE and ODE systems, proposing sound and efficient numerical methods for their approximation, and simulating both benchmark problems and clinically inspired problems. Mathematical modelling itself imposes tremendous challenges, due to the amazing complexity of the cardiocirculatory system, the multiscale nature of the physiological processes involved, and the need to devise computational methods that are stable, reliable and efficient. Critical issues involve filtering the data, identifying the parameters of mathematical models, devising optimal treatments and accounting for uncertainties. For this reason, we will devote the last part of the paper to control and inverse problems, including parameter estimation, uncertainty quantification and the development of reduced-order models that are of paramount importance when solving problems with high complexity, which would otherwise be out of reach.

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Parallel preconditioners for the unsteady Navier–Stokes equations and applications to hemodynamics simulations

Cited by 35 publications

References 27 publications

Reduced dimension GDSW coarse spaces for monolithic Schwarz domain decomposition methods for incompressible fluid flow problems

Reduced dimension GDSW coarse spaces for monolithic Schwarz domain decomposition methods for incompressible fluid flow problems

Parameter estimates for the Relaxed Dimensional Factorization preconditioner and application to hemodynamics

The cardiovascular system: Mathematical modelling, numerical algorithms and clinical applications

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