• Nuclear Engineering and Technology
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• v.54 no.9
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• pp.3181-3204
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• 2022

The variational nodal method for solving the neutron transport equation has evolved over 40 years. Based on a functional form of the Boltzmann neutron transport equation, the method now comprises a complete set of variants that can be employed for different problems. This paper presents an extensive review of the development of the variational nodal method. The emphasis is on summarizing the whole theoretical system rather than validating the methodologies. The paper covers the variational nodal formulation of the Boltzmann neutron transport equation, the Ritz procedure for various application purposes, the derivation of boundary conditions, the extension for adjoint and perturbation calculations, and treatments for anisotropic scattering sources. Acceleration approaches for constructing response matrices and solving the resulting system of algebraic equations are also presented.

• Nuclear Engineering and Technology
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• v.55 no.6
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• pp.2172-2194
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• 2023

A variational nodal method (VNM) with unstructured-mesh is presented for solving steady-state and dynamic neutron diffusion equations. Orthogonal polynomials are employed for spatial discretization, and the stiffness confinement method (SCM) is implemented for temporal discretization. Coordinate transformation relations are derived to map unstructured triangular nodes to a standard node. Methods for constructing triangular prism space trial functions and identifying unique nodes are elaborated. Additionally, the partitioned matrix (PM) and generalized partitioned matrix (GPM) methods are proposed to accelerate the within-group and power iterations. Neutron diffusion problems with different fuel assembly geometries validate the method. With less than 5 pcm eigenvalue (k_eff) error and 1% relative power error, the accuracy is comparable to reference methods. In addition, a test case based on the kilowatt heat pipe reactor, KRUSTY, is created, simulated, and evaluated to illustrate the method's precision and geometrical flexibility. The Dodds problem with a step transient perturbation proves that the SCM allows for sufficiently accurate power predictions even with a large time-step of approximately 0.1 s. In addition, combining the PM and GPM results in a speedup ratio of 2-3.

• Proceedings of the Korean Nuclear Society Conference
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• 1996.05a
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• pp.157-162
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• 1996

A novel nodal method is developed for the two-dimensional multi-group diffusion equations based on the Spectral-Galerkin approach. In this study, the nodal diffusion equations with Robin boundary condition are reformulated in a weak (variational) form, which is then approximated spatially by choosing appropriate basis functions. For the nodal coupling relations between the neighbouring nodes, the continuity conditions of partial currents are utilized. The resulting discrete systems with sparse structured matrices are solved by the Preconditioned Conjugate Gradient Method (PCG) and sweeping technique. The method is validated on two test problems.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.25 no.9
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• pp.1376-1383
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• 2001

Meshless methods, developed in various ways over the past decade, have been attractive as new computational methods in that they do not need mesh generation in analyzing procedure. But most of these methods were not truly meshless methods because background meshes were required for the spatial integration of a weak form. Accordingly, in this paper, nodal integration for truly meshless methods has been studied, and an improvement scheme is proposed. To improve stabilization and accuracy, which are the weak points in previous nodal integration methods, the integration area is transformed to circle and then numerically integrated. This method does not need any adding term for stabilization in the variational formulation and then simplifies the integration procedure. Numerical test results show that the proposed method is more accurate, stable, and reasonable than the existed nodal integration methods.

• Interaction and multiscale mechanics
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• v.1 no.3
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• pp.339-355
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• 2008

A Galerkin meshfree method is presented for analyzing shear deformable cylindrical panels. Based upon the analogy between the cylindrical panel and the curved beam a pure bending mode for cylindrical panel is rationally constructed. The meshfree approximation employed herein is characterized by an enhanced moving least square or reproducing kernel basis function that can exactly represent the pure bending mode and thus meets the requirement of Kirchhoff mode reproducing condition. The variational form is discretized using the efficient stabilized conforming nodal integration with a smoothed nodal gradient based curvature. The resulting meshfree formulation satisfies the integration constraint for bending exactness. Moreover, it is shown here that the smoothed gradient preserves several desired properties which are valid for the standard gradient obtained by direct differentiation, such as partition of nullity and reproduction of a constant strain field. The efficacy of the proposed approach is demonstrated by two benchmark cylindrical panel examples.

• Proceedings of the KSME Conference
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• 2004.11a
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• pp.649-654
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• 2004

A physically simple but mathematically cumbrous problem of rotating heavy chain with one fixed top point is studied. Nonlinear equation of its two-dimensional shapes of relative equilibrium is obtained and solved numerically. A linear case of small displacements is analyzed in terms of Bessel functions. The qualitative and quantitative behavior of the problem is discussed with the help of bifurcation diagram. Dynamics of the two-dimensional model near the equilibrium positions is studied with the help of simulation using the absolute nodal coordinate formulation (ANCF). The equilibriums are found instable, and the reason of instability is explained using a variational principle.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 1995.04b
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• pp.314-319
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• 1995

The effect of liquid level on the natural frequencies and mode shapes of a partially liquid-filled circular cylindrical shell with various boundary conditions is investigated by means of a theoretical analysis based upon Fourier series expansion method and a finite element analysis using ANSYS computer program. Two dimensional mode shapes of the liquid-coupled shell structure are obtained by the ANSYS finite element analysis and show that the liquid level affect the nodal point movement. It is found that the variation of normalized naturalfrequencies (natural frequencies of liquid-filled shell/antural frequencies ofempty shell) to the liquid level is depend on the axial mode numbers and circumferential wave numbers. Additionally, it is found that the number of variational steps of normalized natural frequencies is identicial to that of axial nodal points of the mode shape.

• Structural Engineering and Mechanics
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• v.20 no.4
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• pp.405-420
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• 2005

In this paper, a hybrid/mixed nonlinear shell element is developed in polar coordinate system based on Hellinger/Reissner variational principle and the large-deflection theory of plate. A numerical solution scheme is formulated using the hybrid/mixed finite element method (HMFEM), in which the nodal values of bending moments and the deflection are the unknown discrete parameters. Stability of the present element is studied. The large-deflection analyses are performed for simple supported and clamped circular plates under uniformly distributed and concentrated loads using HMFEM and the traditional displacement finite element method. A parametric study is also conducted in the research. The accuracy of the shell element is investigated using numerical computations. Comparisons of numerical solutions are made with theoretical results, finite element analysis and the available numerical results. Excellent agreements are shown.

• Journal of the Korean Society of Fisheries and Ocean Technology
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• v.18 no.1
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• pp.47-53
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• 1982

An analysis of convection, in a fully developed laminar steady flow through the vertical square duct under the condition of variational symmetric heat flux, is considered. Finite element solution algorithm by Galerkin's method with triangular elements and linear interpolation polynominals for the temperature and velocity profiles are derived for the vertical square duct. The comparison of temperature distribution due to variational symmetric heat flux in the duct were made with available the other data when the condition of peripheral heat flux were uniform and zero. Numerical values for the dimensionless temperatures and Nusselt numbers at selected Rayleigh numbers and pressure gradient parameters were obtained at a few nodal points for the vertical square ducts and effects of corner in the duct were investigated.

• Proceedings of the Korean Society For Composite Materials Conference
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• 2000.04a
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• pp.89-92
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• 2000

In this paper, an effective procedure is presented for the local recovery of displacements and stresses in multilayered composite panels, which incorporate the local refinement using mesh superposition. The mesh superposition method is used to refine the global coarse mesh by superimposing refined mesh to the localized zone of interest without transition zones. The finite element model used is a solid element based on the Hellinger-Reissner variational principle. The a posteriori computation of the through-the-thickness distributions of displacements and stresses is achieved using a predictor-corrector procedure. The procedure utilizes the superconvergent stresses and nodal displacements of the finite element patch. The element patch is generated by locally superimposing a refined local mesh to the coarse global mesh.