Parallel Multigrid Computation of the Unsteady Incompressible Navier–Stokes Equations

Degani, A. T.; Fox, Geoffrey

doi:10.1006/jcph.1996.0205

Cited by 17 publications

(14 citation statements)

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“…11]. For Stokes and Navier-Stokes, an efficient parallel implementation of multigrid box-relaxation is presented in [10]. The extension of this parallel scheme to the gel system will be pursued in a future study.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…As with Stokes (and linearized Navier-Stokes) equations, this gives rise to a large, sparse linear system of saddle point type. To treat this system we propose an extension of a multigrid method initially proposed for Stokes and Navier-Stokes equations by Vanka [37], and which has seen considerable development in the past several years (see, for example, [10,24,32,34,35,40]). The method is characterized by the smoother used and is referred to in literature as box [35, pp.…”

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confidence: 99%

“…This is mathematically and computationally useful since it means the transition between the network and solvent will be smooth even in areas where the two phases have separated. This additional assumption changes (2.1) and (2.2) to 10) where κ ≥ 0 is the diffusion constant. We use relatively small diffusion coefficients so that this modification does not change the qualitative features of the model, but ensures the solution is continuous.…”

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confidence: 99%

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An Efficient and Robust Method for Simulating Two-Phase Gel Dynamics

Wright¹,

Guy²,

Fogelson³

2008

SIAM J. Sci. Comput.

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Abstract. We develop a computational method for simulating models of gel dynamics where the gel is described by two phases, a networked polymer and a fluid solvent. The models consist of transport equations for the two phases, two coupled momentum equations, and a volume-averaged incompressibility constraint, which we discretize with finite differences/volumes. The momentum and incompressibility equations present the greatest numerical challenges since i) they involve partial derivatives with variable coefficients that can vary quite significantly throughout the domain (when the phases separate), and ii) their approximate solution requires the "inversion" of a large linear system of equations. For solving this system, we propose a box-type multigrid method to be used as a preconditioner for the generalized minimum residual (GMRES) method. Through numerical experiments of a model problem, which exhibits phase separation, we show that the computational cost of the method scales nearly linearly with the number of unknowns, and performs consistently well over a wide range of parameters. For solving the transport equation, we use a conservative finite volume method for which we derive stability bounds.Key words. multiphase flow, mixture theory, multigrid, Vanka relaxation, coupled relaxation, box relaxation, Krylov, GMRES, preconditioning AMS subject classifications. 65M55, 65F10, 65M06, 92E20, 76T991. Introduction. Polymer gels are used in a variety of industrial applications and appear naturally in many biological systems. Gels are made of two materials: a polymer network and a solvent. By weight and volume gels are mostly solvent, but the rheology of the gel can range from a very viscous fluid to an elastic solid. In addition to viscoelastic stresses, gels exhibit chemical stresses which result in swelling and deswelling behavior.A commonly used model for the mechanics of gels is the two-phase flow (or twofluid) model [3,9,11,12,13,18,33]. Each phase, network and solvent, is treated as a continuum and moves according to its own velocity field. Each region in space is composed of a mixture of the two phases that is described by the volume fractions of the phases. The equations of motion for the gel consist of two coupled momentum equations and two continuity equations. These models of gels are based on mixture theory [14,15], which has been used for many applications and is becoming increasingly popular in models of biological materials such as tissue [21], tumors [7, 23], cytoplasm [3, 11, 12, 18], and biofilms [2,8,9]. While much work has gone in to advancing these models, the same is not true for the development of efficient numerical methods for simulating them. Since repeated simulation of models is crucial for validating their usefulness, the need for fast and robust computational methods is essential.In this study, an efficient and robust computational methodology for simulating gel dynamics is proposed for a model problem that shares several characteristics of many other models in the literature [3,9,11,12...

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Section: Numerical Resultsmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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An Efficient and Robust Method for Simulating Two-Phase Gel Dynamics

Wright¹,

Guy²,

Fogelson³

2008

SIAM J. Sci. Comput.

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“…Note that vector values located at cell edges are relaxed twice while scalar values located at cell centers are relaxed once within each iteration. Parallelization similar to [18] is developed using the message passing interface (MPI) for communication on distributed memory computers.…”

Section: Smoothermentioning

confidence: 99%

A parallel computational method for simulating two‐phase gel dynamics on a staggered grid

Du¹,

Fogelson²,

Wright³

2008

Numerical Methods in Fluids

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SUMMARYWe develop a parallel computational algorithm for simulating models of gel dynamics where the gel is described by two phases, a networked polymer and a fluid solvent. The models consist of transport equations for the two phases, two coupled momentum equations, and a volume-averaged incompressibility constraint. Multigrid with Vanka-type box-relaxation scheme is used as preconditioner for the Krylov subspace solver (GMRES) to solve the momentum and incompressibility equations. Through numerical experiments of a model problem, the efficiency, robustness and scalability of the algorithm are illustrated.

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“…In [11], a number of schemes were compared, including upwind-explicit, MacCormack-Lax-Wendroff, the van Leer -TVD explicit and upwind-downwind implicit [16]. The last of these was selected to be combined with the pure propagation scheme described earlier, as it has the same accuracy and its finite volume formulation remains conveniently space-centred.…”

Section: Modelling Con6ectionmentioning

confidence: 99%