Algebraic multigrid methods for the solution of the Navier-Stokes equations in complicated geometries

Griebel, Michael; Neunhoeffer, T.; Regler, Hans

doi:10.1002/(sici)1097-0363(19980215)26:3<281::aid-fld632>3.0.co;2-2

Cited by 28 publications

(22 citation statements)

References 34 publications

(30 reference statements)

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“…Other methods are also possible like multilevel methods, whether used as stand alone methods to substitute projection methods, as proposed in [24], coupled with projection methods as in [25], or coupled with Krylov methods as preconditioners as proposed in [26]. These methods have been successfully used in a parallel context in compressible flows in [27,28].…”

Section: Projection Methodsmentioning

confidence: 99%

A massively parallel fractional step solver for incompressible flows

Houzeaux

Vázquez

Aubry

et al. 2009

Journal of Computational Physics

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Section: Projection Methodsmentioning

confidence: 99%

A massively parallel fractional step solver for incompressible flows

Houzeaux

Vázquez

Aubry

et al. 2009

Journal of Computational Physics

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“…With these definitions we can introduce the first step of the coarse grid selection [13,17], the setup phase one.…”

Section: Algebraic Multigrid Methods For Scalar Pdesmentioning

confidence: 99%

“…This approach is based only on information available from the linear system to be solved. These methods can be applied successfully to many linear systems which come from a discretization of a scalar (elliptic) partial differential equation (PDE) [12,13,20]. But in the case of systems of PDEs most of the AMG methods fail.…”

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confidence: 99%

An Algebraic Multigrid Method for Linear Elasticity

Griebel¹,

Oeltz²,

Schweitzer³

2003

SIAM J. Sci. Comput.

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Abstract. We present an algebraic multigrid (AMG) method for the efficient solution of linear (block-)systems stemming from a discretization of a system of partial differential equations (PDEs). It generalizes the classical AMG approach for scalar problems to systems of PDEs in a natural blockwise fashion. We apply this approach to linear elasticity and show that the block-interpolation, described in this paper, reproduces the rigid body modes, i.e., the kernel elements of the discrete linear elasticity operator. It is well-known from geometric multigrid methods that this reproduction of the kernel elements is an essential property to obtain convergence rates which are independent of the problem size. We furthermore present results of various numerical experiments in two and three dimensions. They confirm that the method is robust with respect to variations of the Poisson ratio ν. We obtain rates ρ < 0.4 for ν < 0.4. These measured rates clearly show that the method provides fast convergence for a large variety of discretized elasticity problems.

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“…Furthermore, if the GCL is fulfilled, then we can easily employ a standard projection method, because the equation for conservation of mass (1) reduces to the wellknown divergence-free constraint for the velocity field. This can be seen immediately by using (25) in (1):…”

Section: Discretization Of the Geometric Conservation Lawmentioning

confidence: 99%

Flow simulation on moving boundary-fitted grids and application to fluid-structure interaction problems

Engel

Griebel

2005

Int. J. Numer. Meth. Fluids

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SUMMARYWe present a method for the parallel numerical simulation of transient three-dimensional fluidstructure interaction problems. Here, we consider the interaction of incompressible flow in the fluid domain and linear elastic deformation in the solid domain. The coupled problem is tackled by an approach based on the classical alternating Schwarz method with non-overlapping subdomains, the subproblems are solved alternatingly and the coupling conditions are realized via the exchange of boundary conditions. The elasticity problem is solved by a standard linear finite element method. A main issue is that the flow solver has to be able to handle time-dependent domains. To this end, we present a technique to solve the incompressible Navier-Stokes equation in three-dimensional domains with moving boundaries. This numerical method is a generalization of a finite volume discretization using curvilinear coordinates to time-dependent coordinate transformations. It corresponds to a discretization of the Arbitrary Lagrangian-Eulerian formulation of the Navier-Stokes equations. Here the grid velocity is treated in such a way that the so-called Geometric Conservation Law is implicitly satisfied. Altogether, our approach results in a scheme which is an extension of the well-known MACmethod to a staggered mesh in moving boundary-fitted coordinates which uses grid-dependent velocity components as the primary variables.To validate our method, we present some numerical results which show that second order convergence in space is obtained on moving grids. Finally, we give the results of a fully coupled fluidstructure interaction problem. It turns out that already a simple explicit coupling with one iteration of the Schwarz method, i.e. one solution of the fluid problem and one solution of the elasticity problem per time step, yields a convergent, simple, yet efficient overall method for fluid-structure interaction problems.

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Algebraic multigrid methods for the solution of the Navier-Stokes equations in complicated geometries

Cited by 28 publications

References 34 publications

A massively parallel fractional step solver for incompressible flows

A massively parallel fractional step solver for incompressible flows

An Algebraic Multigrid Method for Linear Elasticity

Flow simulation on moving boundary-fitted grids and application to fluid-structure interaction problems

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