• Title/Summary/Keyword: numerical solutions

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DEVELOPMENT OF A HIGH-ORDER IMPLICIT DISCONTINUOUS GALERKIN METHOD FOR SOLVING COMPRESSIBLE NAVIER-STOKES EQUATIONS (압축성 Navier-Stokes 방정식 해를 위한 고차 정확도 내재적 불연속 갤러킨 기법의 개발)

  • Choi, J.H.;Lee, H.D.;Kwon, O.J.
    • Journal of computational fluids engineering
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    • v.16 no.4
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    • pp.72-83
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    • 2011
  • A high-order discontinuous Galerkin method for the two-dimensional compressible Navier-Stokes equations was developed on unstructured triangular meshes. For this purpose, the BR2 methd(the second Bassi and Rebay discretization) was adopted for space discretization and an implicit Euler backward method was used for time integration. Numerical tests were conducted to estimate the convergence order of the numerical solutions of the Poiseuille flow for which analytic solutions are available for comparison. Also, the flows around a flat plate, a 2-D circular cylinder, and an NACA0012 airfoil were numerically simulated. The numerical results showed that the present implicit discontinuous Galerkin method is an efficient method to obtain very accurate numerical solutions of the compressible Navier-Stokes equations on unstructured meshes.

DEVELOPMENT OF IMPLICIT DISCONTINUOUS GALERKIN METHOD ON UNSTRUCTURED MESHES (비정렬 격자계에서 내재적 불연속 갤러킨 기법의 개발)

  • Lee, H.D.;Kwon, O.J.
    • 한국전산유체공학회:학술대회논문집
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    • 2007.04a
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    • pp.30-40
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    • 2007
  • The implicit discontinuous Galerkin method for the two-dimensional Euler equations was developed on unstructured triangular meshes, which can achieve higher-order accuracy by wing hierachical basis functions based on Legendre polynomials. Numerical tests were conducted to estimate the convergence order of numerical solutions to the Ringleb flow and the supersonic vortex flow for which analytic solutions are available. And, the flows around a circle and a NACA0012 airfoil was also numerically simulated. Numerical results show that the implicit discontinuous Galerkin methods with higher-order representation of curved solid boundaries can be an efficient higher-order method to obtain very accurate numerical solutions on unstructured meshes.

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Linearized instability analysis of frame structures under nonconservative loads: Static and dynamic approach

  • Hajdo, Emina;Mejia-Nava, Rosa Adela;Imamovic, Ismar;Ibrahimbegovic, Adnan
    • Coupled systems mechanics
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    • v.10 no.1
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    • pp.79-102
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    • 2021
  • In this paper we deal with instability problems of structures under nonconservative loading. It is shown that such class of problems should be analyzed in dynamics framework. Next to analytic solutions, provided for several simple problems, we show how to obtain the numerical solutions to more complex problems in efficient manner by using the finite element method. In particular, the numerical solution is obtained by using a modified Euler-Bernoulli beam finite element that includes the von Karman (virtual) strain in order to capture linearized instabilities (or Euler buckling). We next generalize the numerical solution to instability problems that include shear deformation by using the Timoshenko beam finite element. The proposed numerical beam models are validated against the corresponding analytic solutions.

MULTI-LEVEL ADAPTIVE SOLUTIONS TO INITIAL-VALUE PROBLEMS

  • Shamardan, A.B.;Essa, Y.M. Abo
    • Journal of applied mathematics & informatics
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    • v.7 no.1
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    • pp.215-222
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    • 2000
  • A multigrid algorithm is developed for solving the one- dimensional initial boundary value problem. The numerical solutions of linear and nonlinear Burgers; equation for various initial conditions are studied. The stability conditions are derived by Von -Neumann analysis . Numerical results are presented.

TWO-DIMENSIONAL RIEMANN PROBLEM FOR BURGERS' EQUATION

  • Yoon, Dae-Ki;Hwang, Woon-Jae
    • Bulletin of the Korean Mathematical Society
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    • v.45 no.1
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    • pp.191-205
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    • 2008
  • In this paper, we construct the analytic solutions and numerical solutions for a two-dimensional Riemann problem for Burgers' equation. In order to construct the analytic solution, we use the characteristic analysis with the shock and rarefaction base points. We apply the composite scheme suggested by Liska and Wendroff to compute numerical solutions. The result is coincident with our analytic solution. This demonstrates that the composite scheme works pretty well for Burgers' equation despite of its simplicity.

AN ACCELERATING SCHEME OF CONVERGENCE TO SOLVE FUZZY NON-LINEAR EQUATIONS

  • Jun, Younbae
    • The Pure and Applied Mathematics
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    • v.24 no.1
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    • pp.45-51
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    • 2017
  • In this paper, we propose an accelerating scheme of convergence of numerical solutions of fuzzy non-linear equations. Numerical experiments show that the new method has significant acceleration of convergence of solutions of fuzzy non-linear equation. Three-dimensional graphical representation of fuzzy solutions is also provided as a reference of visual convergence of the solution sequence.

GENERALIZED INVERSES IN NUMERICAL SOLUTIONS OF CAUCHY SINGULAR INTEGRAL EQUATIONS

  • Kim, S.
    • Communications of the Korean Mathematical Society
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    • v.13 no.4
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    • pp.875-888
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    • 1998
  • The use of the zeros of Chebyshev polynomial of the first kind $T_{4n+4(x}$ ) and second kind $U_{2n+1}$ (x) for Gauss-Chebyshev quad-rature and collocation of singular integral equations of Cauchy type yields computationally accurate solutions over other combinations of $T_{n}$ /(x) and $U_{m}$(x) as in [8]. We show that the coefficient matrix of the overdetermined system has the generalized inverse. We estimate the residual error using the norm of the generalized inverse.e.

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Effects of Spatial Discretization Schemes on Numerical Solutions of Viscoelastic Fluid Flows (공간차분도식이 점탄성 유체유동의 수치해에 미치는 영향)

  • Min, Tae-Gee;Yoo, Jung-Yul;Choi, Hae-Cheon
    • Transactions of the Korean Society of Mechanical Engineers B
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    • v.24 no.9
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    • pp.1227-1238
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    • 2000
  • This study examines the effects of the discretization schemes on numerical solutions of viscoelastic fluid flows. For this purpose, a temporally evolving mixing layer, a two-dimensional vortex pair interacting with a wall, and a turbulent channel flow are selected as the test cases. We adopt a fourth-order compact scheme (COM4) for polymeric stress derivatives in the momentum equations. For convective derivatives in the constitutive equations, the first-order upwind difference scheme (UD) and artificial diffusion scheme (AD), which are commonly used in the literature, show most stable and smooth solutions even for highly extensional flows. However, the stress fields are smeared too much and the flow fields are quite different from those obtained by higher-order upwind difference schemes for the same flow parameters. Among higher-order upwind difference schemes, a third-order compact upwind difference scheme (CUD3) shows most stable and accurate solutions. Therefore, a combination of CUD3 for the convective derivatives in the constitutive equations and COM4 for the polymeric stress derivatives in the momentum equations is recommended to be used for numerical simulation of highly extensional flows.

A Numerical Model of Large Scale Grid for Two-Dimensional Wake behind Bodies (저항물체 배후의 이차원 후류에 관한 대격자 수치모형)

  • 박일흠;이종섭;이문옥
    • Journal of Korean Society of Coastal and Ocean Engineers
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    • v.10 no.2
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    • pp.83-92
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    • 1998
  • To evaluate the hydraulic resistance behind bodies in a large scale grid numerical model, a drag stress term which is formulated by the drag force is introduced in the depth-integrated Reynolds equations. And also, the applicability and problems of this model are discussed through various numerical experiments where the analytical solutions exist. In the case of a single body, the error range of velocity difference between analytical and numerical solutions is within $\pm$10% and the wake width behind the body shows a good agreement with the analytical solution. When the drag coefficient and the eddy viscosity are precisely decided, the numerical solutions behind a row of bodies will be efficiently used in real situations.

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Deducing thick plate solutions from classical thin plate solutions

  • Wang, C.M.
    • Structural Engineering and Mechanics
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    • v.11 no.1
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    • pp.89-104
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    • 2001
  • This paper reviews the author's work on the development of relationships between solutions of the Kirchhoff (classical thin) plate theory and the Mindlin (first order shear deformation) thick plate theory. The relationships for deflections, stress-resultants, buckling loads and natural frequencies enable one to obtain the Mindlin plate solutions from the well-known Kirchhoff plate solutions for the same problem without much tedious mathematics. Sample thick plate solutions, deduced from the relationships, are presented as benchmark solutions for researchers to use in checking their numerical thick plate solutions.