• Title/Summary/Keyword: Divergence Free Element

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HERMITE BICUBIC STREAM FUNCTION METHOD FOR INCOMPRESSIBLE FLOW COMPUTATIONS IN TWO DIMENSIONS (이차원 비압축성 유동 계산을 위한 Hermite 겹 3차 유동 함수법)

  • Kim, J.W.
    • 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].

HERMITE BICUBIC STREAM FUNCTION METHOD FOR INCOMPRESSIBLE FLOW COMPUTATIONS IN TWO DIMENSIONS (이차원 비압축성 유동 계산을 위한 Hermite 쌍 3차 유동 함수법)

  • Kim, J.W.
    • 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].

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HERMITE BICUBIC STREAM FUNCTION METHOD FOR INCOMPRESSIBLE FLOW COMPUTATIONS IN TWO DIMENSIONS (이차원 비압축성 유동 계산을 위한 Hermite 쌍 3차 유동 함수법)

  • Kim, J.W.
    • 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].

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A STUDY ON INCOMPRESSIBLE FLOW COMPUTATIONS USING A HERMITE STREAM FUNCTION (Hermite 유동함수를 이용한 비압축성 유동계산에 대한 연구)

  • Kim, J.W.
    • 한국전산유체공학회:학술대회논문집
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    • 2006.10a
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    • pp.61-65
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    • 2006
  • This paper describes a recent development on the divergence free basis function based on a hermite stream function. The well-known cavity problem has been used to compare the accuracy and the convergence of the present method with those of a modified residual method known as one of the stabilized finite element methods. The comparison showed the present method performs better in the accuracy and convergence.

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COMPUTATIONS OF NATURAL CONVECTION FLOW WITHIN A SQUARE CAVITY BY HERMITE STREAM FUNCTION METHOD (Hermite 유동함수법에 의한 정사각형 공동 내부의 자연대류 유동계산)

  • Kim, J.W.
    • Journal of computational fluids engineering
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    • v.14 no.4
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    • pp.67-77
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    • 2009
  • This paper is a continuation of a recent development on the Hermite-based divergence-free element method and deals with a non-isothermal fluid flow driven by the buoyancy force in a square cavity with temperature difference across the two sides. Two Hermite functions are considered for numerical computations in this paper. One is a cubic function and the other is a quartic function. The degrees-of-freedom of the cubic Hermite function are stream function and its first and second derivatives for the velocity field, and temperature and its first derivatives for the temperature field. The degrees-of-freedom of the quartic Hermite function include two second derivatives and one cross derivative of the stream function in addition to the degrees-of-freedom of the cubic stream function. This paper presents a brief review on the Hermite based divergence-free basis functions and its finite element formulations for the buoyancy driven flow. The present algorithm does not employ any upwinding or a stabilization term. However, numerical values and contour graphs for major flow variables showed good agreements with those by De Vahl Davis[6].

Incompressible How Computations using a Hermite Stream Function (Hermite 유동함수를 이용한 비압축성 유동계산)

  • Kim, Jin-Whan
    • Proceedings of the Korea Committee for Ocean Resources and Engineering Conference
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    • 2006.11a
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    • pp.411-414
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    • 2006
  • This paper describes a recent development on the divergence free basis function based on a hermite stream function. The well-known cavity problem has been used to compare the accuracy and the convergence of the present method with those of a modified residual method known as one of the stabilized finite e1ement methods. The comparison showed the present method performs better in the accuracy and convergence.

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INCOMPRESSIBLE FLOW COMPUTATIONS USING A HERMITE STREAM FUNCTION (Hermite 유동함수를 이용한 비압축성 유동계산)

  • Kim, J.W.
    • Journal of computational fluids engineering
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    • v.12 no.1
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    • pp.35-42
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    • 2007
  • This paper describes a recent development on the divergence free basis function based on a hermite stream function and verifies its validity by comparing results with those from a modified residual method known as one of stabilized finite element methods. It can be shown that a proper choice of degrees of freedom at a node with a proper arrangement of the hermite interpolation functions can yield solenoidal or divergent free interpolation functions for the velocities. The well-known cavity problem has been chosen for validity of the present algorithm. The comparisons from numerical results between the present and the modified residual showed the present method yields better results in both the velocity and the pressure within modest Reynolds numbers(Re = 1,000).

INCOMPRESSIBLE FLOW COMPUTATIONS BY HERMITE CUBIC, QUARTIC AND QUINTIC STREAM FUNCTIONS (Hermite 3차, 4차 및 5차 유동함수에 의한 비압축성 유동계산)

  • Kim, J.W.
    • 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.

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Computation of Incompressible Flows Using Higher Order Divergence-free Elements (고차의 무발산 요소를 이용한 비압축성 유동계산)

  • Kim, Jin-Whan
    • 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.

Study on the Stability of Elastic Material Subjected to Dry Friction Force (건성마찰력을 받는 탄성재료의 안정성에 관한 연구)

  • Ko, Jun-Bin;Jang, Tag-Soon;Ryu, Si-Ung
    • Transactions of the Korean Society of Mechanical Engineers A
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    • v.28 no.2
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    • pp.143-148
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    • 2004
  • This paper discussed on the stability of elastic material subjected to dry friction force for low boundary conditions: clamped free, clamped-simply supported, simply supported-simply supported, clamped-clamped. It is assumed in this paper that the dry frictional force between a tool stand and an elastic material can be modeled as a distributed follower force. The friction material is modeled for simplicity into a Winkler-type elastic foundation. The stability of beams on the elastic foundation subjected to distribute follower force is formulated by using finite element method to have a standard eigenvalue problem. It is found that the clamped-free beam loses its stability in the flutter type instability, the simply supported-simply supported beam loses its stability in the divergence type instability and the other two boundary conditions the beams lose their stability in the divergence-flutter type instability.