A time-adaptive finite volume method for the Cahn–Hilliard and Kuramoto–Sivashinsky equations

Cueto‐Felgueroso, Luis; Peraire, J.

doi:10.1016/j.jcp.2008.07.024

Cited by 68 publications

(65 citation statements)

References 53 publications

(127 reference statements)

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“…As a consequence, different procedures have been investigated that aim at keeping the computational cost as low as possible. Relevant examples are the continuous/discontinuous Galerkin method (Wells et al, 2006), finite volume methods (Cueto-Felgueroso and Peraire, 2008) or meshless methods (Rajagopal et al, 2010;Zhou and Li, 2006). Here, we will focus on a novel technology, namely isogeometric analysis, which permits simple discretizations of higher-order operators on non-trivial geometries.…”

Section: Discretization Of Higher-order Partial-differential Operatorsmentioning

confidence: 99%

“…Their difference can then be employed as an estimate for the temporal error and as a guide to decrease or increase the time-step size. For phase-field models such techniques have been employed in, e.g., (Ceniceros and Roma, 2007;Cueto-Felgueroso and Peraire, 2008;Gomez et al, 2008;Wodo and Ganapathysubramanian, 2011).…”

Section: Adaptive Mesh and Time-step Refinementmentioning

confidence: 99%

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Computational Phase‐Field Modeling

Gómez

Zee

2017

Encyclopedia of Computational Mechanics Second Edition

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Phase-field modeling is emerging as a promising tool for the treatment of problems with interfaces. The classical description of interface problems requires the numerical solution of partial differential equations on moving domains in which the domain motions are also unknowns. The computational treatment of these problems requires moving meshes and is very difficult when the moving domains undergo topological changes. Phase-field modeling may be understood as a methodology to reformulate interface problems as equations posed on fixed domains. In some cases, the phase-field model may be shown to converge to the moving-boundary problem as a regularization parameter tends to zero, which shows the mathematical soundness of the approach. However, this is only part of the story because phase-field models do not need to have a moving-boundary problem associated and can be rigorously derived from classical thermomechanics. In this context, the distinguishing feature is that constitutive models depend on the variational derivative of the free energy. In all, phase-field models open the opportunity for the efficient treatment of outstanding problems in computational mechanics, such as, the interaction of a large number of cracks in three dimensions, cavitation, film and nucleate boiling, tumor growth or fully three-dimensional air-water flows with surface tension. In addition, phase-field models bring a new set of challenges for numerical discretization that will excite the computational mechanics community.

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Section: Discretization Of Higher-order Partial-differential Operatorsmentioning

confidence: 99%

Section: Adaptive Mesh and Time-step Refinementmentioning

confidence: 99%

Computational Phase‐Field Modeling

Gómez

Zee

2017

Encyclopedia of Computational Mechanics Second Edition

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“…The basic idea behind the Finite Volume Method (FVM) ( [22], [23], [24]) is an application of the Ostrogradski-Gauss divergence theorem to replace the volumetric integrals inherent to the governing equation with the surface integrals rewritten for all the finite volumes completely composing the entire computational domain. A contribution of each finite volume to the global equilibrium equation is represented here as the contribution of its center as well as its outer faces, which differs from the Finite Element Method discretization, where a contribution of each element traditionally results from its nodal points contributions.…”

Section: Cfd Analysismentioning

confidence: 99%

Coupled Finite Volume and Finite Element Method Analysis of a Complex Large-Span Roof Structure

Szafran

Juszczyk

Kamiński

2017

International Journal of Applied Mechanics and Engineering

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The main goal of this paper is to present coupled Computational Fluid Dynamics and structural analysis for the precise determination of wind impact on internal forces and deformations of structural elements of a longspan roof structure. The Finite Volume Method (FVM) serves for a solution of the fluid flow problem to model the air flow around the structure, whose results are applied in turn as the boundary tractions in the Finite Element Method problem structural solution for the linear elastostatics with small deformations. The first part is carried out with the use of ANSYS 15.0 computer system, whereas the FEM system Robot supports stress analysis in particular roof members. A comparison of the wind pressure distribution throughout the roof surface shows some differences with respect to that available in the engineering designing codes like Eurocode, which deserves separate further numerical studies. Coupling of these two separate numerical techniques appears to be promising in view of future computational models of stochastic nature in large scale structural systems due to the stochastic perturbation method.

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“…Most of these theoretical results were validated computationally by several authors (see De Mello and Silveira Filho (2005); Eyre (1998); Lee et al (2011) among others). The numerical solution of entire cCH or CH equation have been investigated by various researchers using finite difference (Christlieb et al, 2012;Cueto-Felgueroso and Peraire, 2008;Du and Nicolaides, 1991;Eyre, 1998), finite elements French, 1987, 1989;Wells et al, 2006), boundary integral (Dehghan and Mirzaei, 2009), Fourier spectral (Zhu et al, 1999) methods -to name a few. In this work we consider finite difference schemes for both the cCH and CH equations.…”

Section: Introductionmentioning

confidence: 99%

“…This same method was employed later by De Mello and Silveira Filho (2005) to solve the CH equation (1.2) in one, two and three dimensions based on a free boundary condition. Cueto-Felgueroso and Peraire (2008) employed a time adaptive procedure to solve a two dimensional CH problem. Elliott and French (1987) observed computationally that the solution will blow up in a finite time if the initial data is large and α < 0.…”

Section: Introductionmentioning

confidence: 99%

On a fractional step-splitting scheme for the Cahn-Hilliard equation

Aderogba

Chapwanya

Djoko

2014

Engineering Computations

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For a partial differential equation with a fourth order derivative such as the CahnHilliard equation, it is always a challenge to design numerical schemes that can handle the restrictive time step introduced by this higher order term. In this work, a fractional splitting method is employed to isolate the convective, the nonlinear second order and the fourth order differential terms. The full equation is then solved by consistent schemes for each differential term independently. In addition to validating the second order accuracy, we will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzburg-Landau energy and the coarsening properties of the solution.

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A time-adaptive finite volume method for the Cahn–Hilliard and Kuramoto–Sivashinsky equations

Cited by 68 publications

References 53 publications

Computational Phase‐Field Modeling

Computational Phase‐Field Modeling

Coupled Finite Volume and Finite Element Method Analysis of a Complex Large-Span Roof Structure

On a fractional step-splitting scheme for the Cahn-Hilliard equation

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