• Computers and Concrete
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• 제23권4호
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• pp.235-243
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• 2019

The discontinuous Galerkin method (DGM) has become widely used as it possesses several qualities, such as a natural ability to dealing with discontinuities. DGM has its major success related to fluid mechanics. Its major importance is the ability to deal with discontinuities and still provide high order of approximation. That is an important advantage when simulating cracking propagation. No remeshing is necessary during the propagation, since the crack path follows the interface of elements. However, DGM comes with the drawback of an increased number of degrees of freedom when compared to the classical continuous finite element method. Thus, it seems a natural approach to combine them in the same simulation obtaining the advantages of both methods. This paper proposes the application of the combined continuous-discontinuous Galerkin method (CDGM) to crack propagation. An important engineering problem is the simulation of crack propagation in concrete structures. The problem is characterized by discontinuities that evolve throughout the domain. Crack propagation is simulated using CDGM. Discontinuous elements are placed in regions with discontinuities and continuous elements elsewhere. The cohesive zone model describes the fracture process zone where softening effects are expressed by cohesive zones in the interface of elements. Two numerical examples demonstrate the capacities of CDGM. In the first example, a plain concrete beam is submitted to a three-point bending test. Numerical results are compared to experimental data from the literature. The second example deals with a full-scale ground slab, comparing the CDGM results to numerical and experimental data from the literature.

• 한국전산유체공학회:학술대회논문집
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• 한국전산유체공학회 2008년도 학술대회
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• pp.57-70
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• 2008

A high-order accurate Euler flow solver based on a discontinuous Galerkin finite-element method has been developed for the numerical simulations of blade-vortex interaction phenomena on unstructured meshes. A free vortex in freestream was investigated to assess the vortex-preserving property and the accuracy of the present flow solver. Blade-vortex interaction problems in subsonic and transonic freestreams were simulated by adopting a multi-level solution-adaptive dynamic mesh refinement/coarsening technique. The results were compared with those of other numerical and experimental methods. It was shown that the present discontinuous Galerkin flow solver can preserve the vortex structure for significantly longer vortex convection time and can accurately capture the complex unsteady blade-vortex interaction flows, including generation and propagation of acoustic waves.

• 한국전산유체공학회:학술대회논문집
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• 한국전산유체공학회 2008년 추계학술대회논문집
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• pp.57-70
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• 2008

A high-order accurate Euler flow solver based on a discontinuous Galerkin finite-element method has been developed for the numerical simulations of blade-vortex interaction phenomena on unstructured meshes. A free vortex in freestream was investigated to assess the vortex-preserving property and the accuracy of the present flow solver. Blade-vortex interaction problems in subsonic and transonic freestreams were simulated by adopting a multi-level solution-adaptive dynamic mesh refinement/coarsening technique. The results were compared with those of other numerical and experimental methods. It was shown that the present discontinuous Galerkin flow solver can preserve the vortex structure for significantly longer vortex convection time and can accurately capture the complex unsteady blade-vortex interaction flows, including generation and propagation of acoustic waves.

• 한국콘텐츠학회논문지
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• 제13권3호
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• pp.428-434
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• 2013

빈발하고 있는 대규모 홍수와 자연재해는 정확도가 높은 하천 흐름 수치해석 모델에 대한 관심의 증대로 이어지고 있다. 현재 하천에서 발생하는 일반적인 흐름은 기존에 개발된 여러 형태의 천수방정식을 지배방정식으로 하는 수치기법에 의해 해석되고 있으나, 연속적이지 않은 형태의 흐름을 해석하거나 매우 정확한 해석을 필요로 하는 경우에는 기존의 수치해석기법은 많은 한계를 보여 주고 있다. 본 연구에서는 불연속 갤러킨 기법 기반의 흐름 모델을 개발하고, 이를 이용하여 전통적으로 1차원 천이류로 분류되는, 댐 붕괴파, 둔덕위 흐름 모의에 적용하여 기존의 수치해와 대체로 잘 일치함을 확인하였다.

• Korean Journal of Mathematics
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• 제13권2호
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• pp.161-176
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• 2005

We consider a model of population dynamics whose mortality function is unbounded. We note that the regularity of the solution depends on the growth rate of the mortality near the maximum age. We propose Gauss-Legendre methods along the characteristics to approximate the solution when the solution is smooth enough. It is proven that the scheme is convergent at fourth-order rate in the maximum norm. We also propose discontinuous Galerkin finite element methods to approximate the solution which is not smooth enough. The stability of the method is discussed. Several numerical examples are presented.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제13권3호
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• pp.169-180
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• 2009

A discontinuous Galerkin method with interior penalty terms is presented for linear Sobolev equation. On appropriate finite element spaces, we apply a symmetric interior penalty Galerkin method to formulate semidiscrete approximate solutions. To deal with a damping term $\nabla{\cdot}({\nabla}u_t)$ included in Sobolev equations, which is the distinct character compared to parabolic differential equations, we choose special test functions. A priori error estimate for the semidiscrete time scheme is analyzed and an optimal $L^\infty(L^2)$ error estimation is derived.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제23권3호
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• pp.211-236
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• 2019

This paper is devoted to the study and investigation of the Runge-Kutta discontinuous Galerkin method for a system of differential equations consisting of two hyperbolic conservation laws. The numerical coupling flux which is used at a given interface (x = 0) is the upwind flux. Moreover, in the linear case, we derive optimal convergence rates in the $L_2$-norm, showing an error estimate of order ${\mathcal{O}}(h^{k+1})$ in domains where the exact solution is smooth; here h is the mesh width and k is the degree of the (orthogonal Legendre) polynomial functions spanning the finite element subspace. The underlying temporal discretization scheme in time is the third-order total variation diminishing Runge-Kutta scheme. We justify the advantages of the Runge-Kutta discontinuous Galerkin method in a series of numerical examples.

• East Asian mathematical journal
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• 제31권5호
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• pp.685-693
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• 2015

In this paper, we introduce fully discrete mixed discontinuous Galerkin approximations for parabolic problems. And we analyze the error estimates in $l^{\infty}(L^2)$ norm for the primary variable and the error estimates in the energy norm for the primary variable and the flux variable.

• Journal of applied mathematics & informatics
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• 제22권1_2호
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• pp.317-329
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• 2006

In this paper, a singularly perturbed ordinary differential equation with non-smooth data is considered. The numerical method is generated by means of a Petrov-Galerkin finite element method with the piecewise-exponential test function and the piecewise-linear trial function. At the discontinuous point of the coefficient, a special technique is used. The method is shown to be first-order accurate and singular perturbation parameter uniform convergence. Finally, numerical results are presented, which are in agreement with theoretical results.

• 한국소성가공학회:학술대회논문집
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• 한국소성가공학회 2005년도 춘계학술대회 논문집
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• pp.43-46
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• 2005

Micro molding technology is a promising mass production technology for polymer based microstructures. Mass production technologies such as the micro injection/compression molding, hot embossing, and micro reaction molding are already in use. In the present study, we have developed a numerical analysis system to simulate three-dimensional non-isothermal cavity filling for hot embossing, with a special emphasis on the free surface capturing. Precise free surface capturing has been successfully accomplished with the level set method, which is solved by means of the Runge-Kutta discontinuous Galerkin (RKDG) method. The RKDG method turns out to be excellent from the viewpoint of both numerical stability and accuracy of volume conservation. The Stokes equations are solved by the stabilized finite element method using the equal order tri-linear interpolation function. To prevent possible numerical oscillation in temperature Held we employ the streamline upwind Petrov-Galerkin (SUPG) method. With the developed code we investigated the detailed change of free surface shape in time during the mold filling. In the filling simulation of a simple rectangular cavity with repeating protruded parts, we find out that filling patterns are significantly influenced by the geometric characteristics such as the thickness of base plate and the aspect ratio and pitch of repeating microstructures. The numerical analysis system enables us to understand the basic flow and material deformation taking place during the cavity filling stage in microstructure fabrications.