• Korean Journal of Mathematics
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• v.13 no.2
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• pp.139-148
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• 2005

In this paper we extend the mixed finite element method and its $L_2$-error estimate for postprocessed solutions by using Crank-Nicolson time-discretization method. Global $O(h^2+({\Delta}t)^2)$-superconvergence for the lowest order Raviart-Thomas element ($Q_0-Q_{1,0}{\times}Q_{0,1}$) are obtained. Numerical examples are presented to confirm superconvergence phenomena.

• The Pure and Applied Mathematics
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• v.13 no.4 s.34
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• pp.281-294
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• 2006

Many partial differential equations defined on a rectangular domain can be solved numerically by using a domain decomposition method. The most commonly used decompositions are the domain being decomposed in stripwise and rectangular way. Theories for non-overlapping domain decomposition(in which two adjacent subdomains share an interface) were often focused on the stripwise decomposition and claimed that extensions could be made to the rectangular decomposition without further discussions. In this paper we focus on the comparisons of the two ways of decompositions. We consider the unconditionally stable scheme, the MIP algorithm, for solving parabolic partial differential equations. The SOR iterative method is used in the MIP algorithm. Even though the theories are the same but the performances are different. We found out that the stripwise decomposition has better performance.

• Communications of the Korean Mathematical Society
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• v.35 no.4
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• pp.1143-1158
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• 2020

In this paper, we are concerned with the blow-up phenomena for a class of pseudo-parabolic equations with weighted source u_t - △u - △u_t = a(x)f(u) subject to Dirichlet (or Neumann) boundary conditions in any smooth bounded domain Ω ⊂ ℝⁿ (n ≥ 1). Firstly, we obtain the upper and lower bounds for blow-up time of solutions to these problems. Moreover, we also give the estimates of blow-up rate of solutions under some suitable conditions. Finally, three models are presented to illustrate our main results. In some special cases, we can even get some exact values of blow-up time and blow-up rate.

• Architectural research
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• v.8 no.1
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• pp.47-56
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• 2006

In this research, the analysis of post-buckling behavior of tapered columns has been performed under a combined load of uniformly distributed axial load along the length and concentric axial load at free end by solving the nonlinear differential equation with the differential transformation technique. The buckling load at various slopes at free end of column is calculated and the results of the analysis using the differential transformation technique is verified with those of previous studies. It is also shown through the results that the buckling load of sinusoidal tapered columns is largest, the linear is second largest, and the parabolic is small in the all ranges of slopes at free end and the deflection of parabolic tapered columns in the x coordinates is largest, the sinusoidal is second largest, and the linear is smallest in the range of slope 0 to 140 degrees at free end. However, when the range of the slope is 160 to 176 degrees at the free end, the deflection of sinusoidal tapered columns in the x coordinates is largest, the linear is second largest, and the parabolic is smallest. In addition, for the linear tapered column, the buckling load increases along with the flexural stiffness ratio. Also, for the parabolic and the sinusoidal tapered column, the buckling loads increase and decrease as the flexural ratios increase in the range of flexural stiffness ratio n = 1.0 to n = 2.0. Through this research, it is verified that the differential transformation technique can be applied to solve the nonlinear differential equation problems, such as analysis of post-buckling behavior of tapered columns. It is also expected that the differential transformation technique apply to various more complicated problems in future.

• Magazine of the Korean Society of Agricultural Engineers
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• v.44 no.3
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• pp.92-100
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• 2002

The advent or high-strength steel has enabled the arch structures to be relatively light, durable and long-spanned by reducing the cross sectional area. On the other hand, the possibility of collapse may be increased due to the slender members which may cause the stability problems. The limit analysis to estimate the ultimate load is based on the concept of collapse mechanism that forms the plastic zone through the full transverse sections. So, it is not appropriate to apply it directly to the instability analysis of arch structures that are composed with compressive members. The objective of this study is to evaluate the ultimate load carrying capacity of the parabolic arch by using the elasto-plastic finite element model. As the rise to span ratio (h/L) varies from 0.0 to 0.5 with the increment of 0.05, the ultimate load has been calculated fur arch structures subjected to uniformly distributed vertical loads. Also, the disco-elasto-plastic analysis has been carried out to find the duration time until the behavior of arch begins to show the stable state when the estimated ultimate load is applied. It may be noted that the maximum ultimate lead of the parabolic arch occurs at h/L=0.2, and the appropriate ratio can be recommended between 0.2 and 0.3. Moreover, it is shown that the circular arch may be more suitable when the h/L ratio is less than 0.2, however, the parabolic arch can be suggested when the h/L ratio is greater than 0.3. The ultimate load carrying capacity of parabolic arch can be estimated by the well-known formula of kEI/L$^3$where the values of k have been reported in this study. In addition, there is no general tendency to obtain the duration time of arch structures subjected to the ultimate load in order to reach the steady state. Merely, it is observed that the duration time is the shortest when the h/L ratio is 0.1, and the longest when the h/L ratio is 0.2.

• Journal of the Korean Mathematical Society
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• v.33 no.3
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• pp.495-505
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• 1996

Let $\Omega$ be a bounded domain in $R^d, d \geq 1$, with smooth boundary $\partial\Omega$, and let $0 < T < \infty$ be given.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2005.05a
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• pp.703-706
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• 2005

Since soil structure interactions are one of the most important subjects in the structural/foundation engineering, much study concerning the soil structure interactions had been carried out. One of typical structures related to the soil structure interactions is the strip foundation which is basically defined as the beam or strip rested on or supported by the soils. At the present time, lack of studies on dynamic problems related to the strip foundations is still found in the literature. From these viewpoint this paper aims to theoretically investigate dynamics of the parabolic strip foundations and also to present the practical engineering data for the design purpose. Differential equations governing the free, out o plane vibrations of such strip foundations are derived, in which effects of the rotatory and torsional inertias and also shear deformation are included although the warping of the cross-section is excluded. Governing differential equations subjected to the boundary conditions of free-free end constraints are numerically solved for obtaining the natural frequencies and mode shapes by using the numerical integration technique and the numerical method of nonlinear equation.

• Steel and Composite Structures
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• v.50 no.1
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• pp.25-43
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• 2024

In this article, a mathematical model is developed to predict the dynamic behavior of bio-inspired composite beam with helicoidal orientation scheme under variable axial load using a unified higher order shear deformation beam theory. The geometrical kinematic relations of displacements are portrayed with higher parabolic shear deformation beam theory. Constitutive equation of composite beam is proposed based on plane stress problem. The variable axial load is distributed through the axial direction by constant, linear, and parabolic functions. The equations of motion and associated boundary conditions are derived in detail by Hamilton's principle. Using the differential quadrature method (DQM), the governing equations, which are integro-differential equations are discretized in spatial direction, then they are transformed into linear eigenvalue problems. The proposed model is verified with previous works available in literatures. Parametric analyses are developed to present the influence of axial load type, orthotropic ratio, slenderness ratio, lamination scheme, and boundary conditions on the natural frequencies of composite beam structures. The present enhanced model can be used especially in designing spacecrafts, naval, automotive, helicopter, the wind turbine, musical instruments, and civil structures subjected to the variable axial loads.

• Geomechanics and Engineering
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• v.7 no.6
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• pp.665-678
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• 2014

In discrete element modeling, 2D software has been widely used in order to gain further insights into the fundamental mechanisms with less computational time. The porosities used in 2D DEM studies should be determined with appropriate approaches based on 3D laboratory porosities. This paper summarizes the main approaches for converting porosities from 3D to 2D for DEM studies and theoretical evaluations show that none of the current approaches can be widely used in dealing with soil mechanical problems. Therefore, a parabolic equation and a criterion have been suggested for the determination of 2D porosities in this paper. Moreover, a case study has been used to validate that the 2D porosity obtained from the above suggestion to be rational with both the realistic contact force distribution in the specimen and the good agreement of the DEM simulation results of direct shear tests with the corresponding experimental data. Therefore, the parabolic equation and the criterion are suggested for the determination of 2D porosities in a wide range of polydisperse particle systems, especially in dealing with soil mechanical problems.

• Structural Engineering and Mechanics
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• v.39 no.6
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• pp.751-765
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• 2011

The present paper deals with the identification of a concentrated damage in an elastic parabolic arch through the minimization of an objective function which measures the differences between numerical and experimental values of static displacements. The damage consists in a notch that reduces the height of the cross section at a given abscissa and therefore causes a variation in the flexural stiffness of the structure. The analytical values of static displacements due to applied loads are calculated by means of the principle of virtual work for both the undamaged and damaged arch. First, pseudo-experimental data are used to study the inverse problem and investigate whether a unique solution can occur or not. Various damage intensities are considered to assess the reliability of the identification procedure. Then, the identification procedure is applied to an experimental case, where displacements are measured on a prototype arch. The identified values of damage parameters, i.e., location and intensity, are compared to those obtained by means of a dynamic identification technique performed on the same structure.