• Kyungpook Mathematical Journal
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• v.57 no.4
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• pp.651-665
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• 2017

Some computational fluid dynamics problems concerning the thin films flow of viscous fluid with a free surface and draining or coating fluid-flow problems can be delineated by third-order ordinary differential equations. In this paper, the aim is to introduce the numerical solutions of the boundary value problems of such equations by variational iteration method. In this paper, it is shown that the third-order boundary value problems can be written as a system of integral equations, which can be solved by using the variational iteration method. These solutions are gleaned in terms of convergent series. Numerical examples are given to depict the method and their convergence.

• Structural Engineering and Mechanics
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• v.82 no.2
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• pp.225-232
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• 2022

In this paper, the physical neutral surface concept is applied to study the wave propagation of functionally graded (FG) circular plate, the wave equation is derived by Hamiltonian variational principle and the first-order shear deformation plate model. Then, we convert the equations to dimensionless equations. The exact solution of wave propagation problem is obtained by Laplace integral transformation, the first order Hankel integral transformation and the zero order Hankel integral transformation. The results obtained by the current model are very close to those obtained in the existing literature, which indicates the correctness and reliability of this study. Moreover, the effects of the functionally graded index parameters and pore volume fraction on the wave propagation are also discussed in detail.

• Journal of the Korean Magnetics Society
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• v.1 no.2
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• pp.79-84
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• 1991

A variational approach employing localized functional is presented to solve alternating magnetic field problems with open boundary. The functional used in the approach consists of the domain integral of finite element region only and the boundary integral of the interfacial boundary between the finite and infinite element regions. The boundary integral is obtained by transforming the infinite domain integral for the infinite element region into the interfacial boundary integral. The proposed algorithm is then applied to a simple two-dimensional problem where the analytic solutions are available. It is shown that the algorithm makes it possible to yield good agreements between the numerical and analytic solutions. and that it requires less computer storage memory and computation time than the conventional finite element method due to the reduction of the computing region.

• Nonlinear Functional Analysis and Applications
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• v.28 no.2
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• pp.497-520
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• 2023

Numerous problems in science and engineering defined by nonlinear functional equations can be solved by reducing them to an equivalent fixed point problem. Fixed point theory provides essential tools for solving problems arising in various branches of mathematical analysis, such as split feasibility problems, variational inequality problems, nonlinear optimization problems, equilibrium problems, complementarity problems, selection and matching problems, and problems of proving the existence of solution of integral and differential equations.The theory of fixed is known to find its applications in many fields of science and technology. For instance, the whole world has been profoundly impacted by the novel Coronavirus since 2019 and it is imperative to depict the spread of the coronavirus. Panda et al. [24] applied fractional derivatives to improve the 2019-nCoV/SARS-CoV-2 models, and by means of fixed point theory, existence and uniqueness of solutions of the models were proved. For more information on applications of fixed point theory to real life problems, authors should (see [6, 13, 24] and the references contained in).

• Interaction and multiscale mechanics
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• v.1 no.3
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• pp.329-337
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• 2008

This paper presents the derivation of the generalized governing equations in theory of elasticity, their weak forms and the some applications in the numerical analysis of structural mechanics. Unlike the differential equations in classical elasticity theory, the generalized equations of the equilibrium and compatibility equations presented here take the form of integral equations, and the generalized equilibrium equations contain the classical differential equations and the boundary conditions in a single equation. By using appropriate test functions, the weak forms of these generalized governing equations can be established. It can be shown that various variational principles in structural analysis are merely the special cases of these weak forms of generalized governing equations in elasticity. The present weak forms of elasticity equations extend greatly the choices of the trial functions for approximate solutions in the numerical analysis of various engineering problems. Therefore, the weak forms of generalized governing equations in elasticity provide a powerful modeling tool in the computational structural mechanics.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2002.04a
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• pp.123-130
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• 2002

A general formulation for shape design sensitivity analysis over three dimensional beam structure is developed based on a variational formulation of the beam in linear elasticity. Sensitivity formula is derived based on variational equations in cartesian coordinates using the material derivative concept and adjoint variable method for the displacement and Von-Mises stress functionals. Shape variation is considered for the beam shape in general 3-dimensional direction as well as for the orientation angle of the beam cross section. In the sensitivity expression, the end points evaluation at each beam segment is added to the integral formula, which are summed over the entire structure. The sensitivity formula can be evaluated with generality and ease even by employing piecewise linear design velocity field despite the bending model is fourth order differential equation. For the numerical implementation, commercial software ANSYS is used as analysis tool for the primal and adjoint analysis. Once the design variable set is defined using ANSYS language, shape and orientation variation vector at each node is generated by making finite difference to the shape with respect to each design parameter, and is used for the computation of sensitivity formula. Several numerical examples are taken to show the advantage of the method, in which the accuracy of the sensitivity is evaluated. The results are found excellent even by employing a simple linear function for the design velocity evaluation. Shape optimization is carried out for the geometric design of an archgrid and tilted bridge, which is to minimize maximum stress over the structure while maintaining constant weight. In conclusion, the proposed formulation is a useful and easy tool in finding optimum shape in a variety of the spatial frame structures.

• Honam Mathematical Journal
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• v.36 no.1
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• pp.141-146
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• 2014

In this paper, we obtain a necessary and sufficient condition for the second variation of an arbitrarily given smooth variation of a geodesic on a Riemannian manifold to be 0.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2002.05a
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• pp.790-795
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• 2002

The object of this research is limited to the reduction of compression process noise only among the main sources of compressor noise such as motor noise, compression process noise, and valve port flow noise. Thus the research is focused on the wave motion rather than the particle motion of sound wave travels. A muffler is a commonly used device to reduce the compression process noise, generated by the pressure pulsations caused by the cyclic compression process. In this research, the acoustic characteristics of the muffler are analyzed by using the normal gradient integral equation proposed by Wu and Wan. Moreover, a commercial code SYSNOISE developed by indirect variational boundary integral equation is also used to validate the results. For the noise reduction, the topology optimization technique using a genetic algorithm is used. The number, size and position of the muffler holes are considered as design variables. Compared with original design, the optimized design has very improved acoustic characteristics. Both numerical and experimental analyses are used to evaluate new design.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2003.05a
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• pp.1032-1037
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• 2003

Although the holes on the shell case are very important fer the acoustic performance, it is difficult to solve the problem because the case includes thin bodies. Hence, in the past, only the method of trial and error, which depends on the engineer's intuition and experience, was available fur the design of holes. Many researchers have tried to solve the thin-body acoustic problems, since the conventional boundary element method (BEM ) using the Helmholtz integral equation fails to yield a reliable solution fer the numerical modelling of radiation anti scattering of sound from thin bodies. In the area of the analysis of thin-body acoustic problem, three approaches are generally used; the multi-domain BEM, the indirect variational BEM, and the normal derivative integral equation And there has been just a f9w study reported on the design optimization for the acoustic radiation problems by using only the conventional BEM. For the thin-body acoustics, however, no further study in the optimization fields has been reported. In this research, the normal derivative integral equation is adopted as an analysis formulation in the thin-body acoustics, and then used fur the optimization. The analytical approaches for the design of holes are proposed by using a topology optimization technique and a genetic algorithm. The proposed approaches are implemented and validated using numerical examples.

• Steel and Composite Structures
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• v.17 no.5
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• pp.753-776
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• 2014

In this paper, nonlinear vibration and post-buckling analysis of beams made of functionally graded materials (FGMs) resting on nonlinear elastic foundation subjected to thermo-mechanical loading are studied. The thermo-mechanical material properties of the beams are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and to be temperature-dependent. The assumption of a small strain, moderate deformation is used. Based on Euler-Bernoulli beam theory and von-Karman geometric nonlinearity, the integral partial differential equation of motion is derived. Then this PDE problem which has quadratic and cubic nonlinearities is simplified into an ODE problem by using the Galerkin method. Finally, the governing equation is solved analytically using the variational iteration method (VIM). Some new results for the nonlinear natural frequencies and buckling load of the FG beams such as the influences of thermal effect, the effect of vibration amplitude, elastic coefficients of foundation, axial force, end supports and material inhomogenity are presented for future references. Results show that the thermal loading has a significant effect on the vibration and post-buckling response of FG beams.