• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.67-76
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• 2014

Let X be a divergence-free vector field defined on a closed, connected Riemannian manifold. In this paper, we show the equivalence between the following conditions: ${\bullet}$ X is a divergence-free vector field satisfying the shadowing property. ${\bullet}$ X is a divergence-free vector field satisfying the Lipschitz shadowing property. ${\bullet}$ X is an expansive divergence-free vector field. ${\bullet}$ X has no singularities and is Anosov.

• Journal of computational fluids engineering
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• v.13 no.4
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• pp.13-23
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• 2008

This paper is an extension of previous study[1] on a development of a divergence-free element method using a hermite interpolated stream function. Divergence-free velocity bases defined on rectangles derived herein produce pointwise divergence-free flow fields. Hence the explicit imposition of continuity constraint is not necessary and the Galerkin finite element formulation for velocities does not involve the pressure. The divergence-free element of the previous study employed hermite (serendipity) cubic for interpolation of stream function, and it has been noted a possible discontinuity in variables along element interfaces. This deficiency can be removed by use of a hermite bicubic interpolated stream function, which requires four degrees-of-freedom at each element corners. Those degrees-of-freedom are the unknown variable, its x- and y-derivatives and its cross derivative. Detailed derivations are presented for both solenoidal and irrotational basis functions from the hermite bicubic interpolated stream function. Numerical tests are performed on the lid-driven cavity flow, and results are compared with those from hermite serendipity cubics and a stabilized finite element method by Illinca et al[2].

• 한국전산유체공학회:학술대회논문집
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• 2008.03a
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• pp.33-41
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• 2008

This paper is an extension of previous study[9] on a development of a divergence-free element method using a hermite interpolated stream function. Divergence-free velocity bases defined on rectangles derived herein produce pointwise divergence-free flow fields. Hence the explicit imposition of continuity constraint is not necessary and the Galerkin finite element formulation for velocities does not involve the pressure. The divergence-free element of the previous study employed hermite serendipity cubic for interpolation of stream function, and it has been noted a possible discontinuity in variables along element interfaces. This deficiency can be removed by use of a hermite bicubic interpolated stream function, which requires at each element corners four degrees-of-freedom such as the unknown variable, its x- and y-derivatives and its cross derivative. Detailed derivations are presented for both solenoidal and irrotational bases from the hermite bicubic interpolated stream function. Numerical tests are performed on the lid-driven cavity flow, and results are compared with those from hermite serendipity cubics and a stabilized finite element method by Illinca et al[7].

• 한국전산유체공학회:학술대회논문집
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• 2008.10a
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• pp.33-41
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• 2008

This paper is an extension of previous study[9] on a development of a divergence-free element method using a hermite interpolated stream function. Divergence-free velocity bases defined on rectangles derived herein produce pointwise divergence-free flow fields. Hence the explicit imposition of continuity constraint is not necessary and the Galerkin finite element formulation for velocities does not involve the pressure. The divergence-free element of the previous study employed hermite serendipity cubic for interpolation of stream function, and it has been noted a possible discontinuity in variables along element interfaces. This deficiency can be removed by use of a hermite bicubic interpolated stream function, which requires at each element corners four degrees-of-freedom such as the unknown variable, its x- and y-derivatives and its cross derivative. Detailed derivations are presented for both solenoidal and irrotational bases from the hermite bicubic interpolated stream function. Numerical tests are performed on the lid-driven cavity flow, and results are compared with those from hermite serendipity cubics and a stabilized finite element method by Illinca et al[7].

• Communications of the Korean Mathematical Society
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• v.35 no.4
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• pp.1245-1254
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• 2020

The object of the present paper is to characterize Bach flat (k, 𝜇)-almost co-Kähler manifolds. It is proved that a Bach flat (k, 𝜇)-almost co-Kähler manifold is K-almost co-Kähler manifold under certain restriction on 𝜇 and k. We also characterize (k, 𝜇)-almost co-Kähler manifolds with divergence free Cotton tensor.

• Communications of the Korean Mathematical Society
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• v.39 no.1
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• pp.201-210
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• 2024

This paper is concerned with the study of spacetimes satisfying div 𝓜 = 0, where "div" denotes the divergence and 𝓜 is the m-projective curvature tensor. We establish that a perfect fluid spacetime with div 𝓜 = 0 is a generalized Robertson-Walker spacetime and vorticity free; whereas a four-dimensional perfect fluid spacetime becomes a Robertson-Walker spacetime. Moreover, we establish that a Ricci recurrent spacetime with div 𝓜 = 0 represents a generalized Robertson-Walker spacetime.

• 한국전산유체공학회:학술대회논문집
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• 2009.11a
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• pp.49-55
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• 2009

This paper evaluates performances of a recently developed divergence-free finite element method based on Hermite interpolated stream functions. Velocity bases are derived from Hermite interpolated stream functions to form divergence-free basis functions. These velocity basis functions constitute a solenoidal function space, and the simple gradient of the Hermite functions constitute an irrotational function space. The incompressible Navier-Stokes equation is orthogonally decomposed into a solenoidal and an irrotational parts, and the decoupled Navier-Stokes equations are projected onto their corresponding spaces to form proper variational formulations. To access accuracy and convergence of the present algorithm, three test problems are selected. They are lid-driven cavity flow, flow over a backward-facing step and buoyancy-driven flow within a square enclosure. Hermite interpolation functions from cubic to quintic are chosen to run the test problems. Numerical results are shown. In all cases it has shown that the present method has performed well in accuracies and convergences. Moreover, the present method does not require an upwinding or a stabilized term.

• Journal of Ocean Engineering and Technology
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• v.25 no.5
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• pp.9-14
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• 2011

The divergence-free finite elements introduced in this paper are derived from Hermite functions, which interpolate stream functions. Velocity bases are derived from the curl of the Hermite functions. These velocity basis functions constitute a solenoidal function space, and the gradient of the Hermite functions constitute an irrotational function space. The incompressible Navier-Stokes equation is orthogonally decomposed into its solenoidal and irrotational parts, and the decoupled Navier-Stokes equations are then projected onto their corresponding spaces to form appropriate variational formulations. The degrees of the Hermite functions we introduce in this paper are bi-cubis, quartic, and quintic. To verify the accuracy and convergence of the present method, three well-known benchmark problems are chosen. These are lid-driven cavity flow, flow over a backward facing step, and buoyancy-driven flow within a square enclosure. The numerical results show good agreement with the previously published results in all cases.

• Journal of Korea Water Resources Association
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• v.50 no.5
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• pp.289-302
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• 2017

DFB (Divergence Form for Bed slope source term) was rigorously derived and the error of mDFB using mean water depth at the cell face in DFB was clearly demonstrated. In addition, DFB technique turned out to be an exact method to the bed slope source term. The existing volume/free-surface relationship to the PSC (Partially Submerged Cell) has been corrected. It was discussed that treatment for the partially submerged edge is required to satisfy the C-property in PSC. It is expected that this study will provides a more accurate means in analyzing the shallow water equations with the approximate Riemann solver.

• Bulletin of the Korean Mathematical Society
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• v.47 no.2
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• pp.231-249
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• 2010

Asymptotic behavior of three-dimensional mixed, periodic and rotating magnetohydrodynamic system is investigated as the Rossby number goes to zero. The system presents the difficulty to be singular and mixed, that is hyperbolic in the vertical direction and parabolic in the horizontal one. The divergence free condition and the spectral properties of the penalization operator are crucial in the proofs. The main tools are the energy method, the Schochet's method and products laws in anisotropic Sobolev spaces.