Coupling Navier-stokes and Cahn-hilliard Equations in a Two-dimensional Annular flow Configuration

Vignal, Philippe; Sarmiento, A. F.; Cortês, Adriano M. A.; Dalcín, Lisandro; Calo, Victor M.

doi:10.1016/j.procs.2015.05.228

Cited by 21 publications

(15 citation statements)

References 33 publications

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“…The temporal integration schemes are applied to the variational formulation presented in equations (55)- (56). The fully discrete formulations for the provably-stable methods to solve the phase-field crystal equation can be defined as: given φ…”

Section: Phase-field Crystal Equationmentioning

confidence: 99%

“…The method has successfully been applied in its Galerkin version to solve the Cahn-Hilliard equation [21,23], the advective [39] and Navier-Stokes-CahnHilliard equations [19,56], the Swift-Hohenberg equation [24] and the phasefield crystal equation [25,54,57]. IGA possesses some advantages over standard finite element methods, which include being able to easily generate high-order, globally continuous basis functions as well as exact geometrical representations as the finite element space is refined.…”

Section: Introductionmentioning

confidence: 99%

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An energy-stable time-integrator for phase-field models

Vignal

Collier

Dalcín

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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We introduce a provably energy-stable time-integration method for general classes of phase-field models with polynomial potentials. We demonstrate how Taylor series expansions of the nonlinear terms present in the partial differential equations of these models can lead to expressions that guarantee energy-stability implicitly, which are second-order accurate in time. The spatial discretization relies on a mixed finite element formulation and isogeometric analysis. We also propose an adaptive time-stepping discretization that relies on a first-order backward approximation to give an error-estimator. This error estimator is accurate, robust, and does not require the computation of extra solutions to estimate the error. This methodology can be applied to any second-order accurate time-integration scheme. We present numerical examples in two and three spatial dimensions, which confirm the stability and robustness of the method. The implementation of the numerical schemes is done in PetIGA, a high-performance isogeometric analysis framework.

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Section: Phase-field Crystal Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An energy-stable time-integrator for phase-field models

Vignal

Collier

Dalcín

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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“…The strategy was implemented in PetIGA-MF [18], a toolbox built on top of PetIGA [19] extending it to support flexible and parallel multi-field discretizations based on structure-preserving B-spline spaces, in the sense of [2]. For examples of the Navier-Stokes-Cahn-Hilliard model using PetIGA-MF see [44,45] . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 As benchmark problems, we use the flows on a square and a cubic cavity, the flow in an eccentric annulus, and the flow in a hollow torus of eccentric annular cross-section.…”

Section: Performance Resultsmentioning

confidence: 99%

A scalable block-preconditioning strategy for divergence-conforming B-spline discretizations of the Stokes problem

Côrtes

Dalcín

Sarmiento

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractThe recently introduced divergence-conforming B-spline discretizations allow the construction of smooth discrete velocity-pressure pairs for viscous incompressible flows that are at the same time inf − sup stable and pointwise divergence-free. When applied to the discretized Stokes problem, these spaces generate a symmetric and indefinite saddle-point linear system. The iterative method of choice to solve such system is the Generalized Minimum Residual Method. This method lacks robustness, and one remedy is to use preconditioners. For linear systems of saddle-point type, a large family of preconditioners can be obtained by using a block factorization of the system. In this paper, we show how the nesting of "black-box" solvers and preconditioners can be put together in a block triangular strategy to build a scalable block preconditioner for the Stokes system discretized by divergence-conforming B-splines. Besides the known cavity flow problem, we used for benchmark flows defined on complex geometries: an eccentric annulus and hollow torus of an eccentric annular cross-section.

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“…On the middle pair of elements, the two basis functions that are discontinuous at x M are shown in black. The positions of the nodes are determined by d 1 and d 2 and, together with the asymptotic weights ω A 1 and ω A 2 , are computed from the (4 × 4) asymptotic system (29). The middle weight is computed from (30), which corresponds to the exactness of the rule when applied on the middle discontinuous basis function D M .…”

Section: Numerical Examples Of Derived Gaussian Rulesmentioning

confidence: 99%

“…Numerical integration is a fundamental ingredient of isogeometric analysis (IGA) and finite elements (FE), and its computational efficiency is essential. When simulating physical processes, e.g., [9,10,15,17,29], with Galerkin isogeometric discretizations, specific spline spaces appear when building mass and stiffness matrices. By construction, these spline spaces are of even degrees and the quadrature rules used to numerically integrate functions from these spaces are sub-optimal [2,5,16,26].…”

Section: Introductionmentioning

confidence: 99%

Gauss–Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis

Bartoň

Calo

2017

Computer-Aided Design

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We introduce Gaussian quadrature rules for spline spaces that are frequently used in Galerkin discretizations to build mass and stiffness matrices. By definition, these spaces are of even degrees. The optimal quadrature rules we recently derived [5] act on spaces of the smallest odd degrees and, therefore, are still slightly sub-optimal. In this work, we derive optimal rules directly for even-degree spaces and therefore further improve our recent result. We use optimal quadrature rules for spaces over two elements as elementary building blocks and use recursively the homotopy continuation concept described in [6] to derive optimal rules for arbitrary admissible number of elements. We demonstrate the proposed methodology on relevant examples, where we derive optimal rules for various even-degree spline spaces. We also discuss convergence of our rules to their asymptotic counterparts, these are the analogues of the midpoint rule of Hughes et al. [16], that are exact and optimal for infinite domains.

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Coupling Navier-stokes and Cahn-hilliard Equations in a Two-dimensional Annular flow Configuration

Cited by 21 publications

References 33 publications

An energy-stable time-integrator for phase-field models

An energy-stable time-integrator for phase-field models

A scalable block-preconditioning strategy for divergence-conforming B-spline discretizations of the Stokes problem

Gauss–Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis

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