• Korean Journal of Air-Conditioning and Refrigeration Engineering
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• v.9 no.1
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• pp.43-54
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• 1997

This paper presents an approximate analytical solution to a two-region one-dimensional model for the charging process of stratified thermal storage tanks with variable inlet temperature in the presence of momentum-induced mixing. Based on the superposition principle, an arbitrary-varying inlet temperature is decomposed into inherent discontinuous steps and continuous intervals approximated as a finite number of piecewise linear functions. This approximation allows the temperature of the upper perfectly-mixed layer to be expressed in terms of constant, linear and exponential functions with respect to time. Applying the Laplace transform technique to the model equation for the lower thermocline layer subject to each of three representative interfacial conditions yields compact-form solutions, a linear combination of which constitutes the final temperature profile. A systematic method for deriving solutions to the plug-flow problem having polynomial-type boundary conditions is also established. The effect of adiabatic exit boundary on solution behaviors proves to be negligible under the actual working conditions, which justifies the assumption of semi-infinite domain introduced in the solution procedure. Finally, the approximate solution is validated by comparing it with an exact solution obtained for a specific variation of inlet temperature. Excellent agreements between them suffice to show the necessity and utility of this work.

• Journal of the Korean Geotechnical Society
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• v.27 no.10
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• pp.105-116
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• 2011

No analytical solution exists for evaluating in-situ hydraulic conductivity of vertical cutoff walls by analyzing slug test results. Recently, an analytical solution to interpret slug tests has been proposed for a partially penetrated well in an aquifer. However, this analytical solution cannot be directly applied to the cutoff wall because the solution has been developed exclusively for an infinite aquifer instead of a narrow cutoff wall. To consider the cutoff wall boundary conditions, the analytical solution has been modified in this study to take into account the narrow boundaries by introducing the imaginary well theory. Two boundary conditions are considered according to the existence of filter cakes: constant head boundary and no flux boundary. Generalized steady-state shape factors are presented for each geometric condition, which can be used for evaluating the in-situ hydraulic conductivity of cutoff walls. The constant head boundary condition provides higher shape factors and no flux boundary condition provides lower shape factors than the infinite aquifer, which enables to adjust the in-situ hydraulic conductivity of the cutoff wall. The hydraulic conductivities calculated from the analytical solution in this paper give about 1.2~1.7 times higher than those from the Bouwer and Rice method, one of the semi-empirical formulas. Considering the compressibility of the backfill material, the analytical solution developed in this study was proved to correspond to the case of incompressible backfill materials.

• Structural Engineering and Mechanics
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• v.59 no.2
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• pp.227-241
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• 2016

A circular plate with constant thickness, finite radius and stiff edge lying on an elastic halfspace is considered. The half-space consists of a soft functionally graded (FGM) layer with arbitrary varying elastic properties and a homogeneous elastic substrate. The plate bends under the action of arbitrary axisymmetric distributed load and response from the elastic half-space. A semi-analytical solution for the problem effective in whole range of geometric (relative layer thickness) and mechanical (elastic properties of coating and substrate, stiffness of the plate) properties is constructed using the bilateral asymptotic method (Aizikovich et al. 2009). Approximated analytical expressions for the contact stresses and deflections of the plate are provided. Numerical results showing the qualitative dependence of the solution from the initial parameters of the problem are obtained with high precision.

• Steel and Composite Structures
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• v.50 no.2
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• pp.135-148
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• 2024

This paper analytically investigates the free vibration analysis of functionally graded-carbon nanotube reinforced composite (FG-CNTRC) plates by dynamic stiffness method (DSM). The properties of CNTRC are determined with the extended rule of mixture. The governing differential equations of motion based on the first-order shear deformation theory of CNTRC plate are derived using Hamilton's principle. The FG-CNTRC plates are studied for a uniform and two different distributions of carbon nanotubes (CNTs). The accuracy and performance of the DSM are compared with the results obtained from closed closed-form and semi-analytical solution methods in previous studies. In this study, the effects of boundary condition, distribution type of CNTs, plate aspect ratio, plate length to thickness ratio, and different values of CNTs volume fraction on the natural frequencies of the FG-CNTRC plates are investigated. Finally, various natural frequencies of the plates in different conditions are provided as a benchmark for comparing the accuracy and precision of the other analytical and numerical methods.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.17 no.4
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• pp.232-242
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• 2005

In this study, we get an analytical solution for the diffraction and multi-reflection around a semi-infinite breakwater and breakwaters with a gap by using the solution of Penney and Price (1952). We find analytical solutions for single- and multi-reflections around the breakwaters by assuming that the reflected waves are regarded to be those diffracting through a breakwater gap. On the basis of these solutions, it is possible to understand the wave diffraction with different cases of incident wave direction and breakwater layout. These solutions may help harbor engineers to understand the phenomena of diffraction and multi-reflections around the breakwaters. These solutions may also be used to evaluate the applicability of wave transformation models which are used in designing coastal structures.

• Advances in Computational Design
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• v.4 no.1
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• pp.15-32
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• 2019

One of the efficient and useful tools to achieve the optimal design of structures is employing the sensitivity analysis in the finite element model. In the numerical optimization process, often the semi-analytical method is used for estimation of derivatives of the objective function with respect to design variables. Numerical methods for calculation of sensitivities are susceptible to the step size in design parameters perturbation and this is one of the great disadvantages of these methods. This article uses complex variables method to calculate the sensitivity analysis and combine it with discrete sensitivity analysis. Finally, it provides a new method to obtain the sensitivity analysis for linear structures. The use of complex variables method for sensitivity analysis has several advantages compared to other numerical methods. Implementing the finite element to calculate first derivatives of sensitivity using this method has no complexity and only requires the change in finite element meshing in the imaginary axis. This means that the real value of coordinates does not change. Second, this method has the lower dependency on the step size. In this research, the process of sensitivity analysis calculation using a finite element model based on complex variables is explained for linear problems, and some examples that have known analytical solution are solved. Results obtained by using the presented method in comparison with exact solution and also finite difference method indicate the excellent efficiency of the proposed method, and it can predict the sustainable and accurate results with the several different step sizes, despite low dependence on step size.

• Economic and Environmental Geology
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• v.53 no.1
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• pp.11-21
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• 2020

One-dimensional analytical solutions were used for forward and back diffusion of trichloroethylene (TCE) and tetrachloroethylene (PCE) in a single system with high- and low-permeability layers. Concentration profiles in a low-permeability layer, diffusive fluxes at the interface between the high- and low-permeability layers, and contaminant persistence in the high-permeability layer due to back diffusion were simulated with a comparison of semi-infinite and finite analytical solutions. In order to validate the analytical solutions used in this study, the results of one-dimensional analytical solutions developed by Yang et al. (2015) were compared with Nash-Sutcliffe model efficiency coefficient (NSE). When compared with Yang et al. (2015), the analytical solutions used in this study showed good agreements (NSE = 0.99). When compared with semi-infinite analytical solutions, TCE and PCE concentration profiles in the low-permeability layer, the diffusive fluxes, and the contaminant tailings of the high-permeability layer were underestimated. In order to determine the appropriate analytical solutions based on the effective diffusion coefficient, the thickness of the low-permeability layer, and the diffusion time in the TCE and PCE contaminated site, a term of dimensionless diffusion length (Z_d) was used. If the Z_d is less than 0.7, the semi-infinite solutions can be used to simulate accurate concentration profiles in low-permeability layers. If the Z_d is greater than 0.7, the reliability of simulations may be improved by using the finite solutions.

• Journal of computational fluids engineering
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• v.14 no.3
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• pp.86-94
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• 2009

The non-conservative form of momentum equations is often used for some two-phase flow codes instead of a conservative form because of numerical convenience. Another non-conservative form, so called, a semi-conservative form can improve the numerical solution of these codes maintaining the numerical convenience. It is close to the conservative form but still maintains the feature of the non-conservative form. A semi-conservative form of the momentum equations and a non-conservative form of the momentum equations are implemented in CUPID[1] code. The numerical results of the semi-conservative and the non-conservative forms are compared against analytical solutions and the solutions of the FLUENT code that uses the conservative form. The results clearly showed that the semi-conservative form of the momentum equations provides better solutions than the non-conservative form, especially for heterogeneous two-phase flows.

• Journal of the Korea Computer Graphics Society
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• v.2 no.1
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• pp.1-6
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• 1996

In ray tracing of 3D volume data, the color of each pixel in the image is typically obtained by accumulating the contributions of sample points on the ray cast from the pixel point. This accumulation is most naturally represented by integration. In most methods, however, it is done by numerical summation because analytical solution to the integration are hard to find. This paper shows that a semi-analytical solution can be obtained for a typical ray tracing of volume data. Tentative conclusions about the significance and usefulness of our approach are presented based on our experiments.

• Geomechanics and Engineering
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• v.29 no.1
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• pp.1-11
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• 2022

A semi-analytical solution to responses of overconsolidated (OC) unsaturated soils surrounding an expanding spherical cavity under constant suction condition is presented. To capture the elastoplastic hydro-mechanical property of OC unsaturated soils, the unified hardening (UH) model for OC unsaturated soil is adopted in corporation with a soil-water characteristic curve (SWCC) and two suction yield surfaces. Taking the specific volume, radial stress, tangential stress and degree of saturation as the four basic unknowns, the problem investigated is formulated by solving a set of first-order ordinary differential equations with the help of an auxiliary variable and an iterative algorithm. The present solution is validated by comparing with available solution based on the modified Cam Clay (MCC) model. Parametric studies reveal that the hydraulic and mechanical responses of spherical cavity expanding in unsaturated soils are not only coupled, but also affected by suction and overconsolidation ratio (OCR) significantly. More importantly, whether hydraulic yield will occur or not depends only on the initial relationship between suction yield stress and suction. The presented solution can be used for calibration of some insitu tests in OC unsaturated soil.