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23
An Overlapping Schwarz Method for Spectral Element Solution of the Incompressible NavierStokes Equations
 J. Comp. Phys
, 1997
"... Efficient solution of the NavierStokes equations in complex domains is dependent upon the availability of fast solvers for sparse linear systems. For unsteady incompressible flows, the pressure operator is the leading contributor to stiffness, as the characteristic propagation speed is infinite. ..."
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Cited by 89 (31 self)
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Efficient solution of the NavierStokes equations in complex domains is dependent upon the availability of fast solvers for sparse linear systems. For unsteady incompressible flows, the pressure operator is the leading contributor to stiffness, as the characteristic propagation speed is infinite. In the context of operator splitting formulations, it is the pressure solve which is the most computationally challenging, despite its elliptic origins. We examine several preconditioners for the consistent L 2 Poisson operator arising in the lP N \Gamma lP N \Gamma2 spectral element formulation of the incompressible NavierStokes equations. We develop a finite element based additive Schwarz preconditioner using overlapping subdomains plus a coarse grid projection operator which is applied directly to the pressure on the interior Gauss points. For large twodimensional problems this approach can yield as much as a fivefold reduction in simulation time over previously employed methods based upon deflation. To appear in J. of Comp. Phys., 1997. Present address: Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. Email: pff@cfm.brown.edu 1 1
A Survey of Parallelization Techniques for Multigrid Solvers
, 2004
"... This paper surveys the techniques that are necessary for constructing computationally ecient parallel multigrid solvers. Both geometric and algebraic methods are considered. We rst cover the sources of parallelism, including traditional spatial partitioning and more novel additive multilevel method ..."
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Cited by 30 (0 self)
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This paper surveys the techniques that are necessary for constructing computationally ecient parallel multigrid solvers. Both geometric and algebraic methods are considered. We rst cover the sources of parallelism, including traditional spatial partitioning and more novel additive multilevel methods. We then cover the parallelism issues that must be addressed: parallel smoothing and coarsening, operator complexity, and parallelization of the coarsest grid solve. 1.1
On the miscible Rayleigh–Taylor instability: two and three dimensions
, 1999
"... We investigate the miscible Rayleigh–Taylor (RT) instability in both two and three dimensions using direct numerical simulations, where the working fluid is assumed incompressible under the Boussinesq approximation. We first consider the case of randomly perturbed interfaces. With a variety of diagn ..."
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Cited by 14 (1 self)
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We investigate the miscible Rayleigh–Taylor (RT) instability in both two and three dimensions using direct numerical simulations, where the working fluid is assumed incompressible under the Boussinesq approximation. We first consider the case of randomly perturbed interfaces. With a variety of diagnostics, we develop a physical picture for the detailed temporal development of the mixed layer: we identify three distinct evolutionary phases in this development, which can be related to detailed variations in the growth of the mixing zone. Our analysis provides an explanation for the observed differences between two and threedimensional RT instability; the analysis also leads us to concentrate on the RT models which (i) work equally well for both laminar and turbulent flows, and (ii) do not depend on turbulent scaling within the mixing layer between fluids. These candidate RT models are based on point sources within bubbles (or plumes) and their interaction with each other (or the background flow). With this motivation, we examine the evolution of single plumes, and relate our numerical results (for single plumes) to a simple analytical model for plume evolution.
Optimized multiplicative, additive and restricted additive schwarz preconditioning. in preparation
, 2005
"... Abstract. We demonstrate that a small modification of the multiplicative, additive and restricted additive Schwarz preconditioner at the algebraic level, motivated by optimized Schwarz methods defined at the continuous level, leads to a significant reduction in the iteration count of the iterative s ..."
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Abstract. We demonstrate that a small modification of the multiplicative, additive and restricted additive Schwarz preconditioner at the algebraic level, motivated by optimized Schwarz methods defined at the continuous level, leads to a significant reduction in the iteration count of the iterative solver. Numerical experiments using finite difference and spectral element discretizations of the positive definite Helmholtz problem and an idealized climate simulation illustrate the effectiveness of the new approach. Key words. Domain decomposition, restricted additive Schwarz method, optimized Schwarz methods, multiplicative Schwarz, spectral elements, highorder. AMS subject classifications. 65F19, 65N22, 65N35 1. Introduction. The
Knepley Optimal, scalable forward models for computing gravity anomalies Geophysical Journal International 187 (2011
 in midmantle Phys. Earth Planet. Inter. 180 (2010) 271282, DOI 10.1016/j.pepi.2010.04.001
"... We describe three approaches for computing a gravity signal from a density anomaly. The first approach consists of the classical “summation ” technique, whilst the remaining two methods solve the Poisson problem for the gravitational potential using either a Finite Element (FE) discretization emp ..."
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We describe three approaches for computing a gravity signal from a density anomaly. The first approach consists of the classical “summation ” technique, whilst the remaining two methods solve the Poisson problem for the gravitational potential using either a Finite Element (FE) discretization employing a multilevel preconditioner, or a Green’s function evaluated with the Fast Multipole Method (FMM). The methods utilizing the Poisson formulation described here differ from previously published approaches used in gravity modeling in that they are optimal, implying that both the memory and computational time required scale linearly with respect to the number of unknowns in the potential field. Additionally, all of the implementations presented here are developed such that the computations can be performed in a massively parallel, distributed memory computing environment. Through numerical experiments, we compare the methods on the basis of their discretization error, CPU time and parallel scalability. We demonstrate the parallel scalability of all these techniques by running forward models with up to 108 voxels on 1000’s of cores. 1 ar
Hairpin vortex formation, a case study for unsteady visualization
 41st CUG Conference
, 1999
"... To better understand the vortex dynamics of coherent structures in turbulent and transitional boundary layers, we consider direct numerical simulation of the interaction between a atplateboundarylayer ow and an isolated hemispherical roughness element. Of principal interest is the evolution of h ..."
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Cited by 3 (1 self)
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To better understand the vortex dynamics of coherent structures in turbulent and transitional boundary layers, we consider direct numerical simulation of the interaction between a atplateboundarylayer ow and an isolated hemispherical roughness element. Of principal interest is the evolution of hairpin vortices that form an interlacing pattern in the wake of the hemisphere, lift away from the wall, and are stretched by the shearing action of the boundary layer. Using animations of unsteady threedimensional representations of this ow, produced by the vtk toolkit and enhanced to operate in a CAVE virtual environment, we identify and study several key features in the evolution of this complex vortex topology not previously observed in other visualization formats. 1
Topics in Ultrascale Scientific Computing with Application in Biomedial Modeling
, 2009
"... In this Thesis we focus on simulations of blood flow in threedimensional patientspecific arterial networks. We employ highorder spectral/hpelement spatial discretization and concentrate on computational efficiency in solving multimillion degrees of freedom (DOF) flow problems on petaflop compu ..."
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In this Thesis we focus on simulations of blood flow in threedimensional patientspecific arterial networks. We employ highorder spectral/hpelement spatial discretization and concentrate on computational efficiency in solving multimillion degrees of freedom (DOF) flow problems on petaflop computers. We develop new twolevel domain decomposition method and multilevel communicating interface for ultraparallel flow simulations. Specifically, at the coarse level the computational domain is subdivided into several big patches. Within each patch a spectral element discretization (fine level) is employed. New interface conditions for the NavierStokes equations are developed. The proposed numerical approach has been tested in arterial flow simulations with up to 147 arteries. Solution of 2.87B DOF problem was computed on 18,576 processors in less than one second at each time step. A scalable and fast parallel lowenergy bases preconditioner (LEBP) in conjunction with coarsespace linear vertex solver is developed. We provide details on optimization, parallel performance and implementation of the coarsespace solver and show scalability of LEBP on thousands processors of the IBM BlueGene/L and the Cray XT3. An embarrassingly parallel but extremely efficient accelerator for iterative solver has been proposed. The new
Large Eddy Simulation of Strati¯ed Mixing in TwoDimensional DamBreak Problem in a Rectangular Enclosed Domain by
, 2006
"... Mixing in both coastal and deep ocean emerges as one of the important processes that determines the transport of pollutants, sediments and biological species, as well as the details of the global thermohaline circulation. Both the observations, due to their lack in space and time resolution, and mos ..."
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Cited by 1 (0 self)
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Mixing in both coastal and deep ocean emerges as one of the important processes that determines the transport of pollutants, sediments and biological species, as well as the details of the global thermohaline circulation. Both the observations, due to their lack in space and time resolution, and most coastal and general circulation models due to inadequate physics, can only provide partial information about oceanic mixing processes. A new class of nonhydrostatic models supplemented with physicallybased subgridscale (SGS) closures, or socalled large eddy simulation (LES), is put forth as another tool of investigation to complement observational and largescale modeling e®orts. However, SGS models have been developed primarily for homogeneous, isotropic °ows. Here, four SGS models based on Smagorinsky eddy viscosity and di®usivity are tested for strati¯ed °ows in the context of 2D dambreak problem in a rectangular enclosed domain. This idealized testbed leads to a number of simpli¯cations about the initial conditions, boundary conditions and geometry, while exhibiting the dynamically complex characteristics of strati¯ed °ows involving the interaction of shearinduced mixing and internal waves. Direct