• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.26 no.4
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• pp.244-262
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• 2022

In this paper, we analyze the hybridizable discontinuous Galerkin (HDG) method for second-order elliptic equations with nonlinear coefficients, which are used in many fields. We present the HDG method that uses a mixed formulation based on numerical trace and flux. Under assumptions on the nonlinear coefficient and H²-regularity for a dual problem, we prove that the discrete systems are well-posed and the numerical solutions have the optimal order of convergence as a mesh parameter. Also, we provide a matrix formulation that can be calculated using an iterative technique for numerical experiments. Finally, we present representative numerical examples in 2D to verify the validity of the proof of Theorem 3.10.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.20 no.2
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• pp.137-150
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• 2016

In this paper we analyze an a posteriori error estimator based on flux recovery for lowest-order finite element discretizations of elliptic interface problems. The flux recovery considered here is based on averaging the discrete normal fluxes and/or tangential derivatives at midpoints of edges with weight factors adapted to discontinuous coefficients. It is shown that the error estimator based on this flux recovery is equivalent to the error estimator of Bernardi and $Verf{\ddot{u}}rth$ based on the standard edge residuals uniformly with respect to jumps of the coefficient between subdomains. Moreover, as a byproduct, we obtain slightly modified weight factors in the edge residual estimator which are expected to produce more accurate results.

• Journal of Korea Water Resources Association
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• v.36 no.4
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• pp.597-608
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• 2003

A transcritical flow occurs when the width and slope of a channel are varying abruptly. In this study, the transcritical flow in a two-dimensional open channel is analyzed by using the shallow-water equations. A weighted average flux scheme that has flux limiter with a total variation diminishing condition is introduced for a second-order accuracy in time and space, and non- spurious oscillations at discontinuous points. A HLLC method with three wane speeds is employed to calculate the Riemann problem. To overcome difficulties resulting from variation of channel sections in a two-dimensional analysis of transcritical flow, the numerical model is developed based on a generalized grid system.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.34 no.5
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• pp.1383-1393
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• 2014

Recently, with rapid improvement in computer hardware and theoretical development in the field of computational fluid dynamics, high-order accurate schemes also have been applied in the realm of computational hydraulics. In this study, numerical solutions of 1D shallow water equations are presented with TVD Runge-Kutta discontinuous Galerkin (RKDG) finite element method. The transcritical flows such as dam-break flows due to instant dam failure and transcritical flow with bottom elevation change were studied. As a formulation of approximate Riemann solver, the local Lax-Friedrichs (LLF), Roe, HLL flux schemes were employed and MUSCL slope limiter was used to eliminate unnecessary numerical oscillations. The developed model was applied to 1D dam break and transcritical flow. The results were compared to the exact solutions and experimental data.

• Nuclear Engineering and Technology
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• v.24 no.3
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• pp.319-328
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• 1992

The Galerkin formulation of the finite element method is applied to the integral law of the first-order form of the one-group neutron transport equation in one-dimensional spherical geometry. Piecewise linear or quadratic Lagrange polynomials are utilized in the integral law for the angular flux to establish a set of linear algebraic equations. Numerical analyses are performed for the scalar flux distribution in a heterogeneous sphere as well as for the criticality problem in a uniform sphere. For the criticality problems in the uniform sphere, the results of the finite element method, with the use of continuous finite elements in space and angle, are compared with the exact solutions. In the heterogeneous problem, the scalar flux distribution obtained by using discontinuous angular and spatical finite elements is in good agreement with that from the ANISN code calculation.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.32 no.6
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• pp.569-574
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• 2020

Though the discontinuous Galerkin (DG) method has been developed and applied to shallow water equations mainly in explicit schemes, they have been criticized for the limitation in treatment of bottom friction terms and severe CFL conditions. In this study, an implicit scheme is devised and applied to some representative benchmark problems. The linear triangular elements were employed and the Roe numerical fluxes were adopted for convective fluxes. To preserve TVD property, the slope limiter was employed. As the case studies, the model is applied to the flow around the cylinders and the dam-break flow. Then, the results are compared with the experimental and numerical data of previous studies and good agreements were observed.

• Nuclear Engineering and Technology
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• v.46 no.6
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• pp.783-796
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• 2014

The coarse mesh finite difference (CMFD) method is applied to the discontinuous finite element method based discrete ordinate calculation for source convergence acceleration. The three-dimensional (3-D) DFEM-Sn code FEDONA is developed for general geometry applications as a framework for the CMFD implementation. Detailed methods for applying the CMFD acceleration are established, such as the method to acquire the coarse mesh flux and current by combining unstructured tetrahedron elements to rectangular coarse mesh geometry, and the alternating calculation method to exchange the updated flux information between the CMFD and DFEM-Sn. The partial current based CMFD (p-CMFD) is also implemented for comparison of the acceleration performance. The modified p-CMFD method is proposed to correct the weakness of the original p-CMFD formulation. The performance of CMFD acceleration is examined first for simple two-dimensional multigroup problems to investigate the effect of the problem and coarse mesh sizes. It is shown that smaller coarse meshes are more effective in the CMFD acceleration and the modified p-CMFD has similar effectiveness as the standard CMFD. The effectiveness of CMFD acceleration is then assessed for three-dimensional benchmark problems such as the IAEA (International Atomic Energy Agency) and C5G7MOX problems. It is demonstrated that a sufficiently converged solution is obtained within 7 outer iterations which would require 175 iterations with the normal DFEM-Sn calculations for the IAEA problem. It is claimed that the CMFD accelerated DFEM-Sn method can be effectively used in the practical eigenvalue calculations involving general geometries.

• Nuclear Engineering and Technology
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• v.52 no.3
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• pp.485-498
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• 2020

A diffusion synthetic acceleration (DSA) technique for the S_N transport equation discretized with the linear discontinuous expansion method with subcell balance (LDEM-SCB) on unstructured tetrahedral meshes is presented. The LDEM-SCB scheme solves the transport equation with the discrete ordinates method by using the subcell balances and linear discontinuous expansion of the flux. Discretized DSA equations are derived by consistently discretizing the continuous diffusion equation with the LDEM-SCB method, however, the discretized diffusion equations are not fully consistent with the discretized transport equations. In addition, a fine mesh rebalance (FMR) method is devised to accelerate the discretized diffusion equation coupled with the preconditioned conjugate gradient (CG) method. The DSA method is applied to various test problems to show its effectiveness in speeding up the iterative convergence of the transport equation. The results show that the DSA method gives small spectral radii for the tetrahedral meshes having various minimum aspect ratios even in highly scattering dominant mediums for the homogeneous test problems. The numerical tests for the homogeneous and heterogeneous problems show that DSA with FMR (with preconditioned CG) gives significantly higher speedups and robustness than the one with the Gauss-Seidel-like iteration.

• Journal of Korea Water Resources Association
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• v.38 no.2
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• pp.155-166
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• 2005

Upwind schemes are very well adapted to the discontinuous flow and have become popular for applications Involving dam break flow, transcritical Slow, etc. However, upwind schemes have been applied mainly to the idealized problems not to the natural channels with irregular geometry so far because of the error due to source terms. In this paper, the new type of upwind discretization of source terms, which uses the normalized Jacobian to discretize the source terms, is proposed. As results of tests to flows with source terms by the upwind models, the method proposed in this paper is proved as efficient and accurate. This generalized method for differencing source terms is simple and might beapplicable to diverse type of flux upwind discretization scheme in finite difference method.

• Korean Chemical Engineering Research
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• v.62 no.2
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• pp.191-199
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• 2024

A study was conducted on a tubular heat exchanger to improve its heat transfer rate by using a novel baffle plate design with discontinuous swirling patterns. The design consisted of perforated baffle plates with rectangular air deflectors positioned at varying angles. The tubes in the heat exchanger were arranged in a consistent alignment with the airflow direction and exposed to a uniform heat flux on their surfaces. Each baffle plate included sixteen deflectors inclined at the same angle and arranged in a clockwise pattern. This arrangement induced a swirling motion of the air inside a circular duct where the heated tubes were located, leading to increased turbulence and improved heat transfer on the tube surfaces. The spacing between the baffle plates was adjusted at different pitch ratios, and the Reynolds number was controlled within a range of 16,000 to 29,000. The effects of pitch ratios and inclination angles on the heat exchanger's performance were analyzed. The results indicated that using a baffle plate with rectangular deflectors inclined at 30° and a pitch ratio of 1.2 resulted in an average increase of 1.29 in the thermal enhancement factor.