• Kyungpook Mathematical Journal
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• v.59 no.4
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• pp.735-769
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• 2019

In this paper, we consider the Robin-Dirichlet problem for a nonlinear wave equation of Kirchhoff-Carrier type with a viscoelastic term. Using the Faedo-Galerkin method and the linearization method for nonlinear terms, the existence and uniqueness of a weak solution are proved. An asymptotic expansion of high order in a small parameter of a weak solution is also discussed.

• Structural Engineering and Mechanics
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• v.38 no.5
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• pp.573-592
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• 2011

Comprehensive and accurate analysis of a finite foundation beam is a challenging engineering problem and an important subject in foundation design. One of the limitation of the traditional Winkler elastic foundation model is that the model neglects the effect of the interface resistance between the beam and the underneath foundation soil. By taking the beam-soil interface resistance into account, a deformation governing differential equation for a finite beam resting on the Winkler elastic foundation is developed. The coupling effect between vertical and horizontal displacements is also considered in the presented method. Using Galerkin method, semi-analytical solutions for vertical and horizontal displacements, axial force, shear force and bending moment of the beam under symmetric loads are presented. The influences of the interface resistance on the behavior of foundation beam are also investigated.

• Structural Engineering and Mechanics
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• v.56 no.3
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• pp.461-472
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• 2015

In this paper, the nonlinear static response of curled cantilever beam actuators subjected to the one-sided electrostatic field is focused on. By assuming the deflection function of electrostatically actuated beam, analytical approximate solutions are established via using Galerkin method to solve the equilibrium equation. The Pull-In voltages which determine the stability of the curled beam actuators are also obtained. These approximate solutions show excellent agreements with numerical solutions obtained by the shooting method and the experimental data for a wide range of beam length. Expressions of these analytical approximate solutions are brief and could easily be used to derive the effects of various physical parameters on MEMS structures.

• Coupled systems mechanics
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• v.8 no.1
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• pp.71-93
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• 2019

Finite element simulations of solid mechanics problems often involve the use of Non-Confirming Meshes (NCM) to increase accuracy in capturing nonlinear behavior, including damage and plasticity, in part of a solid domain without an undue increase in computational costs. In the presence of material nonlinearity and plasticity, higher-order variables are often needed to capture nonlinear behavior and material history on non-conforming interfaces. The most popular formulations for coupling non-conforming meshes are dual methods that involve the interpolation of a traction field on the interface. These methods are subject to the Ladyzhenskaya-Babuska-Brezzi (LBB) stability condition, and are therefore limited in their implementation with the higher-order elements needed to capture nonlinear material behavior. Alternatively, the enriched discontinuous Galerkin approach (EDGA) (Haikal and Hjelmstad 2010) is a primal method that provides higher order kinematic fields on the interface, and in which interface tractions are computed from local finite element estimates, therefore facilitating its implementation with nonlinear material models. The inclusion of higher-order interface variables, however, presents the issue of preserving material history at integration points when a increase in integration order is needed. In this study, the enriched discontinuous Galerkin approach (EDGA) is extended to the case of small-deformation plasticity. An interface-driven Gauss-Kronrod integration rule is proposed to enable adaptive enrichment on the interface while preserving history-dependent material data at existing integration points. The method is implemented using classical J2 plasticity theory as well as the pressure-dependent Drucker-Prager material model. We show that an efficient treatment of interface variables can improve algorithmic performance and provide a consistent approach for coupling non-conforming meshes in inelasticity.

• Journal of the Korean Mathematical Society
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• v.61 no.4
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• pp.621-657
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• 2024

In this study, the numerical solution of a space-fractional Burgers' equation with initial and boundary conditions is considered. This equation is the simplest nonlinear model for diffusive waves in fluid dynamics. It occurs in a variety of physical phenomena, including viscous sound waves, waves in fluid-filled viscous elastic pipes, magneto-hydrodynamic waves in a medium with finite electrical conductivity, and one-dimensional turbulence. The proposed QBS/CNG technique consists of the Galerkin method with a function basis of quadratic B-splines for the spatial discretization of the space-fractional Burgers' equation. This is then followed by the Crank-Nicolson approach for time-stepping. A linearized scheme is fully constructed to reduce computational costs. Stability analysis, error estimates, and convergence rates are studied. Finally, some test problems are used to confirm the theoretical results and the proposed method's effectiveness, with the results displayed in tables, 2D, and 3D graphs.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.12 no.4
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• pp.181-189
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• 2000

A weakly nonlinear mild-slope equation has been derived directly from the continuity equation with the aid of the Galerkin's method. The equation is combined with the momentum equations defined at the mean water level. A single component model has also been obtained in terms of the surface displacement. The linearized form is completely identical with the time-dependent mild-slope equation proposed by Smith and Sprinks(1975). For the verification purposes of the present nonlinear model, the degenerate forms were compared with Airy(1845)'s non-dispersive nonlinear wave equation, classical Boussinesq equation, andsecond¬order permanent Stokes waves. In this study, the present nonlinear wave equations are discretized by the approximate factorization techniques so that a tridiagonal matrix solver is used for each direction. Through the comparison with physical experiments, nonlinear wave model capacity was examined and the overall agreement was obtained.

• Journal of applied mathematics & informatics
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• v.22 no.1_2
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• pp.223-235
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• 2006

By applying the Landau-type transformation, we transform a Stefan problem with nonlinear free boundary condition into a system consisting of a parabolic equation and the ordinary differential equations. Fully discrete finite element method is developed to approximate the solution of a system of a parabolic equation and the ordinary differential equations. We derive optimal orders of convergence of fully discrete approximations in $L_2,\;H^1$ and $H^2$ normed spaces.

• Structural Engineering and Mechanics
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• v.35 no.4
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• pp.387-407
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• 2010

In this paper, using perturbation and Galerkin method, the response of a resonant viscoelastic microbeam to an electric actuation is obtained. The microbeam is under axial load and electrical load. It is assumed that midplane is stretched, when the beam is deflected. The equation of motion is derived using the Newton's second law. The viscoelastic model is taken to be the Kelvin-Voigt model. In the first section, the static deflection is obtained using the Galerkin method. Exact linear symmetric mode shape of a straight beam and its deflection function under constant transverse load are used as admissible functions. So, an analytical expression that describes the static deflection at all points is obtained. Comparing the result with previous research show that using deflection function as admissible function decreases the computation errors and previous calculations volume. In the second section, the response of a microbeam resonator system under primary and secondary resonance excitation has been obtained by analytical multiple scale perturbation method combined with the Galerkin method. It is shown, that a small amount of viscoelastic damping has an important effect and causes to decrease the maximum amplitude of response, and to shift the resonance frequency. Also, it shown, that an increase of the DC voltage, ratio of the air gap to the microbeam thickness, tensile axial load, would increase the effect of viscoelastic damping, and an increase of the compressive axial load would decrease the effect of viscoelastic damping.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.19 no.7
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• pp.699-709
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• 2009

In this paper, the nonlinear dynamics and the stability of nanopipes conveying fluid and modelled as a thin-walled beam is investigated. Effects of boundary conditions, geometric nonlinearity, non-classical transverse shear and rotary inertia are incorporated in this study. The governing equations and the three different boundary conditions are derived through Hamilton's principle. Numerical analysis is performed by using extend Galerkin method which enables us to obtain more exact solutions compared with conventional Galerkin method. Variations of critical flow velocity for different boundary conditions of carbon nanopipes are investigated and compared with linear case.

• Steel and Composite Structures
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• v.44 no.4
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• pp.557-567
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• 2022

Nonlinear free vibration analysis of a functionally graded beam resting on the nonlinear viscoelastic foundation is studied with uniform temperature rising. The non-linear strain-displacement relationship is considered in the finite strain theory. The governing nonlinear dynamic equation is derived based on the finite strain theory with using of Hamilton's principle. The Galerkin's decomposition technique is utilized to discretize the governing nonlinear partial differential equation to nonlinear ordinary differential equation and then is solved by using of multiple time scale method. The influences of temperature rising, material distribution parameter, nonlinear viscoelastic foundation parameters on the nonlinear free response and phase trajectory are investigated. In this paper, it is aimed that a contribution to the literature for nonlinear thermal vibration solutions of a functionally graded beam resting on the nonlinear viscoelastic foundation by using of multiple time scale method.