• Structural Engineering and Mechanics
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• v.39 no.1
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• pp.21-46
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• 2011

The natural frequencies of continuous systems depend on the governing partial differential equation and can be numerically estimated using the finite element method. The accuracy and convergence of the finite element method depends on the choice of basis functions. A basis function will generally perform better if it is closely linked to the problem physics. The stiffness matrix is the same for either static or dynamic loading, hence the basis function can be chosen such that it satisfies the static part of the governing differential equation. However, in the case of a rotating beam, an exact closed form solution for the static part of the governing differential equation is not known. In this paper, we try to find an approximate solution for the static part of the governing differential equation for an uniform rotating beam. The error resulting from the approximation is minimized to generate relations between the constants assumed in the solution. This new function is used as a basis function which gives rise to shape functions which depend on position of the element in the beam, material, geometric properties and rotational speed of the beam. The results of finite element analysis with the new basis functions are verified with published literature for uniform and tapered rotating beams under different boundary conditions. Numerical results clearly show the advantage of the current approach at high rotation speeds with a reduction of 10 to 33% in the degrees of freedom required for convergence of the first five modes to four decimal places for an uniform rotating cantilever beam.

• Structural Engineering and Mechanics
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• v.73 no.1
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• pp.97-108
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• 2020

This study presents a 4 node, 11 DOF/node plate element based on higher order shear deformation theory for lamina composite plates. The theory accounts for parabolic distribution of the transverse shear strain through the thickness of the plate. Differential field equations of composite plates are obtained from energy methods using virtual work principle. Differential field equations of composite plates are obtained from energy methods using virtual work principle. These equations were transformed into the operator form and then transformed into functions with geometric and dynamic boundary conditions with the help of the Gâteaux differential method, after determining that they provide the potential condition. Boundary conditions were determined by performing variational operations. By using the mixed finite element method, plate element named HOPLT44 was developed. After coding in FORTRAN computer program, finite element matrices were transformed into system matrices and various analyzes were performed. The current results are verified with those results obtained in the previous work and the new results are presented in tables and graphs.

• Journal of applied mathematics & informatics
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• v.27 no.5_6
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• pp.1279-1292
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• 2009

In this paper, a numerical method that uses standard finite difference scheme defined on Shishkin mesh for a weakly coupled system of two singularly perturbed convection-diffusion second order ordinary differential equations with a discontinuous source term is presented. An error estimate is derived to show that the method is uniformly convergent with respect to the singular perturbation parameter. Numerical results are presented to illustrate the theoretical results.

• Advances in aircraft and spacecraft science
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• v.1 no.2
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• pp.125-143
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• 2014

This paper provides a new technique for solving the static analysis of arbitrarily shaped composite plates by using Strong Formulation Finite Element Method (SFEM). Several papers in literature by the authors have presented the proposed technique as an extension of the classic Generalized Differential Quadrature (GDQ) procedure. The present methodology joins the high accuracy of the strong formulation with the versatility of the well-known Finite Element Method (FEM). The continuity conditions among the elements is carried out by the compatibility or continuity conditions. The mapping technique is used to transform both the governing differential equations and the compatibility conditions between two adjacent sub-domains into the regular master element in the computational space. The numerical implementation of the global algebraic system obtained by the technique at issue is easy and straightforward. The main novelty of this paper is the application of the stress and strain recovery once the displacement parameters are evaluated. Computer investigations concerning a large number of composite plates have been carried out. SFEM results are compared with those presented in literature and a perfect agreement is observed.

• Journal of applied mathematics & informatics
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• v.9 no.2
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• pp.561-577
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• 2002

In this paper, a two-dimensional finite volume unstructured mesh method (FVUM) based on a triangular background interpolation mesh is developed for analysing the evolution of the saltwater intrusion into single and multiple coastal aquifer systems. The model formulation consists of a ground-water flow equation and a salt transport equation. These coupled and non-linear partial differential equations are transformed by FVUM into a system of differential/algebraic equations, which is solved using backward differentiation formulas of order one through five. Simulation results are compared with previously published solutions where good agreement is observed.

• Proceedings of the KSME Conference
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• 2000.11a
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• pp.529-534
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• 2000

A finite element analysis for a rotating cantilever beam is presented in this study. Based on a dynamic modelling method using the stretch deformation instead of the conventional axial deformation, three linear partial differential equations are derived from Hamilton's principle. Two of the linear differential equations show the coupling effect between stretch and chordwise deformations. The other equation is an uncoupled one for the flapwise deformation. From these partial differential equations and the associated boundary conditions, are derived two weak forms: one is for the chordwise motion and the other is for the flapwise motion. The weak forms are spatially discretized with newly defined two-node beam elements. With the discretized equations or the matrix-vector equations, the behaviours of the natural frequencies are investigated for the variation of the rotating speed.

• Journal of applied mathematics & informatics
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• v.15 no.1_2
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• pp.1-28
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• 2004

We first apply a first order splitting to a semilinear reaction-diffusion equation and then discretize the resulting system by an $H^1$-Galerkin mixed finite element method in space. This semidiscrete method yields a system of differential algebraic equations (DAEs) of index one. A priori error estimates for semidiscrete scheme are derived for both differ-ential as well as algebraic components. For fully discretization, an implicit Runge-Kutta (IRK) methods is applied to the temporal direction and the error estimates are discussed for both components. Finally, we conclude the paper with a numerical example.

• Structural Engineering and Mechanics
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• v.14 no.1
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• pp.71-84
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• 2002

A finite element formulation for the time-domain analysis of linear transient elastodynamic problems is presented based on the weak form obtained by applying the Galerkin's method to the integro-differential equations which contain the initial conditions implicitly and does not include the inertia terms. The weak form is extended temporally under the assumptions of the constant and linear time variations of field variables, since the time-stepping algorithms such as the Newmark method and the Wilson ${\theta}$-method are not necessary, obtaining two kinds of implicit finite element equations which are tested for numerical accuracy and convergency. Three classical examples having finite and infinite domains are solved and numerical results are compared with the other analytical and numerical solutions to show the versatility and accuracy of the presented formulation.

• Structural Engineering and Mechanics
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• v.72 no.4
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• pp.491-502
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• 2019

This paper explores the analytical buckling solution of rectangular thin plates by the finite integral transform method. Although several analytical and numerical developments have been made, a benchmark analytical solution is still very few due to the mathematical complexity of solving high order partial differential equations. In solution procedure, the governing high order partial differential equation with specified boundary conditions is converted into a system of linear algebraic equations and the analytical solution is obtained classically. The primary advantage of the present method is its simplicity and generality and does not need to pre-determine the deflection function which makes the solving procedure much reasonable. Another advantage of the method is that the analytical solutions obtained converge rapidly due to utilization of the sum functions. The application of the method is extensive and can also handle moderately thick and thick elastic plates as well as bending and vibration problems. The present results are validated by extensive numerical comparison with the FEA using (ABAQUS) software and the existing analytical solutions which show satisfactory agreement.

• Journal of Environmental Impact Assessment
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• v.10 no.4
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• pp.319-326
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• 2001

This study is aimed at the development of a comprehensive web-based partial differential equation solver (WPDES) using multidimensional finite element method, which can be operated on the basis of world wide web. Overall issues of engineering and environmental information management and facility control could be implemented using this solver. This paper describes the development technique of the model, which is first part on development of partial differential equation solver. Conventional commercial general solver of computational fluid dynamics problems were investigated. All the relevant environmental models were analyzed to develop integrated environmental management system using WPDES. The governing equations and the parameters of investigated models were analyzed and integrated. Several numerical modules were invented for each partial differential term in partial differential equation of many related modeling problems. Each module was coded in the fashion of object oriented method, and was combined independently for the overall governing equation. WPDES has unique characteristic, which can analyze the problem through the suitable combination of modules without development of additional models for each environment problem with different governing equation, main variables, and parameters.