• Structural Engineering and Mechanics
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• v.14 no.3
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• pp.263-285
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• 2002

This paper presents a numerical procedure for solving initial-value problems using the special functions which belong to a class of Rvachev's basis functions $R_{bf}$ based on algebraic and trigonometric polynomials. Because of infinite derivability of these functions, derivatives of all orders, required by differential equation of the problem and initial conditions, are used directly in the numerical procedure. The accuracy and stability of the proposed numerical procedure are proved on an example of a single degree of freedom system. Critical time step was also determined. An algorithm for solving multiple degree of freedom systems by the collocation method was developed. Numerical results obtained by $R_{bf}$ functions are compared with exact solutions and results obtained by the most commonly used numerical procedures for solving initial-value problems.

• Proceedings of the Korea Concrete Institute Conference
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• 1998.10a
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• pp.412-417
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• 1998

This paper provides accurate flexural vibration solutions for thick (Mindlin) sectorial plates. A Ritz method is employed which incorporates a complete set of admissible algebraic-trigonometric polynomials in conjunction with an admissible set of Mindlin “corner functions". These corner functions model the singular vibratory moments and shear forces, which simultaneously exist at the vertex of corner angle exceeding 180$^{\circ}$. The first set guarantees convergence to the exact frequencies as sufficient terms are taken. The second set represents the corner singularities, and accelerates convergence substantially. Numerical results are obtained for completely free sectorial plates. Accurate frequencies are presented for a wide spectrum of vertex angles (90$^{\circ}$, 180$^{\circ}$, 270$^{\circ}$, 300$^{\circ}$, 330$^{\circ}$, 350$^{\circ}$, 35 5$^{\circ}$,and 359$^{\circ}$)and thickness ratios.tios.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.14 no.4
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• pp.336-343
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• 2004

This paper reports the first known free vibration data for thin rectangular plates with V-notches. The classical Ritz method is employed with two sets of admissible functions assumed for the transverse vibratory displacements. These sets include (1) mathematically complete algebraic-trigonometric polynomials which guarantee convergence to exact frequencies as sufficient terms are retained, and (2) corner functions which account for the bending moment singularities at the sharp reentrant corner of the Y-notch. Extensive convergence studies summarized herein confirm that the corner functions substantially enhance the convergence and accuracy of nondirectional frequencies for rectangular plates having the V-notch. In this paper, accurate frequencies and normalized contours of vibratory transverse displacement are presented for various notched plates, so that the effect of corner stress singularities may be understood.

• Nuclear Engineering and Technology
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• v.31 no.2
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• pp.214-225
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• 1999

An accurate method is presented for flexural vibrations of sectorial plates having simply supported-free and free-free radial edges, when the circular edge is either clamped or simply supported. The classical Ritz method is employed with two sets of admissible functions assumed for the transverse vibratory displacements. These sets consist of : (1) mathematically complete algebraic-trigonometric polynomials which gurantee convergence to exact frequencies as sufficient terms are retained, and (2) comer functions which account for the bending moment singularities at re-entrant comer of the radial edges having arbitrary edge conditions. Accurate (at least four significant figures) frequencies and normalized contours of the transverse vibratory displacement are presented for the spectra of corner angles [90$^{\circ}$, 180$^{\circ}$(semi-circular), 270$^{\circ}$, 300$^{\circ}$, 330$^{\circ}$, 350$^{\circ}$, 355$^{\circ}$, 360$^{\circ}$ (complete circular)] causing a re-entrant comer of the radial edges. Future solutions drawn from alternative numerical procedures and finite element techniques may be compared with these accurate results.

• Smart Structures and Systems
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• v.13 no.1
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• pp.111-135
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• 2014

Based on the Reissner mixed variational theorem (RMVT), finite rectangular layer methods (FRLMs) are developed for the three-dimensional (3D) linear buckling analysis of simply-supported, fiber-reinforced composite material (FRCM) and functionally graded material (FGM) sandwich plates subjected to bi-axial compressive loads. In this work, the material properties of the FGM layers are assumed to obey the power-law distributions of the volume fractions of the constituents through the thickness, and the plate is divided into a number of finite rectangular layers, in which the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-plane variations of the field variables of each individual layer, respectively, and an h-refinement process is adopted to yield the convergent solutions. The accuracy and convergence of the RMVT-based FRLMs with various orders used for expansions of each field variables through the thickness are assessed by comparing their solutions with the exact 3D and accurate two-dimensional ones available in the literature.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2003.04a
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• pp.35-42
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• 2003

This paper provides the first known flexural vibration data for thick (Mindlin) rectangular plates having V-notches. The V-notch has bending moment and shear force singularities at its sharp corner due to the transverse vibratory bending motion. Based upon Mindlin plate theory, in which transverse shear deformation and rotary inertia effects are considered, the Ritz procedure is employed with a hybrid set of admissible functions assumed for the rotational and transverse vibratory displacements. This set includes: (1) a mathematically complete set of admissible algebraic-trigonometric polynomials which guarantee convergence to exact frequencies as sufficient terms are retained; and (2) an admissible set of Mindlin corner functions which account for the bending moment and shear force singularities at the sharp corner of the V-notch. Extensive convergence studies demonstrate the necessity of adding the Mindlin corner functions to achieve accurate frequencies for rectangular plates having sharp V-notches.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.12 no.1
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• pp.41-53
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• 2008

Polynomials are one of most important and widely used numerical tools in dealing with a smooth function on a bounded domain and trigonometric functions work for smooth periodic functions. However, they are not the best choice if a function has a bounded support in space and in frequency domain. The Prolate Spheroidal wave function (PSWF) of order zero has been known as a best candidate as a basis for band-limited functions. In this paper, we review some basic properties of PSWFs defined as eigenfunctions of bounded Fourier transformation. We also propose numerical inversion schemes based on PSWF and present some numerical examples to show their feasibilities as signal processing tools.

• Steel and Composite Structures
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• v.39 no.3
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• pp.291-306
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• 2021

Based on Reissner's mixed variational theorem (RMVT), the authors develop a semi-analytical finite element (FE) method for a three-dimensional (3D) bending analysis of nonhomogeneous orthotropic, complete and incomplete toroidal shells subjected to uniformly-distributed loads. In this formulation, the toroidal shell is divided into several finite annular prisms (FAPs) with quadrilateral cross-sections, where trigonometric functions and serendipity polynomials are used to interpolate the circumferential direction and meridian-radial surface variations in the primary field variables of each individual prism, respectively. The material properties of the toroidal shell are considered to be nonhomogeneous orthotropic over the meridianradial surface, such that homogeneous isotropic toroidal shells, laminated cross-ply toroidal shells, and single- and bi-directional functionally graded toroidal shells can be included as special cases in this work. Implementation of the current FAP methods shows that their solutions converge rapidly, and the convergent FAP solutions closely agree with the 3D elasticity solutions available in the literature.

• Smart Structures and Systems
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• v.16 no.5
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• pp.781-806
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• 2015

A unified formulation of finite layer methods (FLMs), based on the Reissner mixed variational theorem (RMVT), is developed for the three-dimensional (3D) coupled electro-elastic analysis of simply-supported, functionally graded piezoelectric material (FGPM) plates with open- and closed-circuit surface conditions and under electro-mechanical loads. In this formulation, the material properties of the plate are assumed to obey an exponent-law varying exponentially through the thickness coordinate, and the plate is divided into a number of finite rectangular layers, in which the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-plane variations of the primary field variables of each individual layer, respectively, such as the elastic displacement, transverse shear and normal stress, electric potential, and normal electric displacement components. The relevant orders used for expanding these variables in the thickness coordinate can be freely chosen as the linear, quadratic and cubic orders. Four different mechanical/electrical loading conditions applied on the top and bottom surfaces of the plate are considered, and the corresponding coupled electro-elastic analysis of the loaded FGPM plates is undertaken. The accuracy and convergence rate of the RMVT-based FLMs are assessed by comparing their solutions with the exact 3D piezoelectricity ones available in the literature.

• Journal of Korean Society of Steel Construction
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• v.21 no.4
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• pp.433-442
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• 2009

In this paper, new three-dimensional vibration data for the solid cylinders of the N-sided polygonal cross-section with V-notches or sharp cracks are presented, and a Ritz procedure is employed, which incorporates a mathematically complete set of algebraic-trigonometric polynomials in conjunction with an admissible set of edge functions that explicitly model the tri-axial stress singularities that exist along a terminus edge of the V-notch. Convergence studies demonstrate the necessity of adding the edge functions to achieve the accurate frequencies and mode shapes of N-sided polygonal cylindrical solids with stress singularities.