• Transactions of the Korean Society of Mechanical Engineers
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• v.17 no.12
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• pp.3007-3019
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• 1993

For the same configuration, the stiffness of 6-node two dimensional isoparametric is stiffer than that of 8-node two dimensional isoparametric element. This phenomenon may be called 'Relative Stiffness Stiffening Phenomenon.' In this paper, the relative stiffness stiffening phenomenon was studied, and could be corrected by modifying the position of Gauss integral points used in the numerical integration of the stiffness matrix. For the same deformation (bending) energy of 6-node and 8-node two dimensional isoparametric elements, Gauss integral points of 6-node element have to move closer, in comparison with those of 8-node element, in the case of numerical integration along the thickness direction.

• Ocean Systems Engineering
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• v.7 no.4
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• pp.399-412
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• 2017

The Gauss-Legendre integral method is applied to numerically evaluate the Green function and its derivatives in finite water depth. In this method, the singular point of the function in the traditional integral equation can be avoided. Moreover, based on the improved Gauss-Laguerre integral method proposed in the previous research, a new methodology is developed through the Gauss-Legendre integral. Using this new methodology, the Green function with the field and source points near the water surface can be obtained, which is less mentioned in the previous research. The accuracy and efficiency of this new method is investigated. The numerical results using a Gauss-Legendre integral method show good agreements with other numerical results of direct calculations and series form in the far field. Furthermore, the cases with the field and source points near the water surface are also considered. Considering the computational efficiency, the method using the Gauss-Legendre integral proposed in this paper could obtain the accurate numerical results of the Green function and its derivatives in finite water depth and can be adopted in the near field.

• Advances in concrete construction
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• v.1 no.4
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• pp.319-340
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• 2013

This paper presents a comparative study on the double-K fracture parameters of concrete obtained using four existing analytical methods such as Gauss-Chebyshev integral method, simplified Green's function method, weight function method and simplified equivalent cohesive force method. Two specimen geometries: three point bend test and compact tension specimen for sizes 100-500 mm at initial notch length to depth ratios 0.25 and 0.4 are used for the comparative study. The required input parameters for determining the double-K fracture parameters are derived from the developed fictitious crack model. It is found that the cohesive toughness and initial cracking toughness determined using weight function method and simplified equivalent cohesive force method agree well with those obtained using Gauss-Chebyshev integral method whereas these fracture parameters determined using simplified Green's function method deviates more than by 11% and 20% respectively as compared with those obtained using Gauss-Chebyshev integral method. It is also shown that all the fracture parameters related with double-K model are size dependent.

• Nonlinear Functional Analysis and Applications
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• v.26 no.5
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• pp.935-947
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• 2021

An efficient adaptive scheme based on a triple mixed quadrature rule of precision nine for approximate evaluation of line integral of analytic functions has been constructed. At first, a mixed quadrature rule SM₁(f) has been formed using Gauss-Legendre three point transformed rule and five point Booles transformed rule. A suitable linear combination of the resulting rule and Clenshaw-Curtis seven point rule gives a new mixed quadrature rule SM₁₀(f). This mixed rule is termed as triple mixed quadrature rule. An adaptive quadrature scheme is designed. Some test integrals having analytic function integrands have been evaluated using the triple mixed rule and its constituent rules in non-adaptive mode. The same set of test integrals have been evaluated using those rules as base rules in the adaptive scheme. The triple mixed rule based adaptive scheme is found to be the most effective.

• Journal of the Society of Naval Architects of Korea
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• v.32 no.2
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• pp.32-42
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• 1995

When the thickness becomes so small as in the case of the trailing edge of the propeller blade or when the curvature of the surface varies rapidly as in ship stem, the existing panel method employing a flat-surface panel, obtained by collapsing the original non-planar surface into its mean location, suffers the leakage problem and also gives inaccurate induction upon the field point very close to the panel. The hyperboloidal panel deals with the induction from the dipole distributed on the non-planar surface without approximation, overcoming the defects of the flat-surface panel. This paper introduces two distinct derivations of the formulae to compute the integral for the potential induced by a dipole of uniform density distributed on a non-planar hyperboloidal surface element. One method is based on the Gauss-Bonnet theorem and the other is based on the transformation of the surface integral into a line integral.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.27 no.3
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• pp.194-206
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• 2023

We present families of nonlinear transformations useful for numerical evaluation of weakly singular integrals. First, for end-point singular integrals, we define a prototype function with some appropriate features and then suggest a family of transformations. In addition, for interior-point singular integrals, we develop a family of nonlinear transformations based on the aforementioned prototype function. We take some examples to explore the efficiency of the proposed nonlinear transformations in using the Gauss-Legendre quadrature rule. From the numerical results, we can find the superiority of the proposed transformations compared to some existing transformations, especially for the integrals with high singularity strength.

• Transactions of the Korean Society of Mechanical Engineers
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• v.16 no.8
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• pp.1429-1437
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• 1992

A new finite element formulation based on the relaxation type hereditary integral is presented for a time-domain analysis of isotropic, linear viscoelastic problems. The semi-discrete variational approximation and elastic-viscoelastic correspondence principle are used in the theoretical development of the proposed method. In a time-stepping procedure of final, linear algebraic system equations, only a small additional computation for past history is required since the equivalent stiffness matrix is constant. The viscoelasticity matrices are derived and the stress computation algorithm is given in matrix form. The effect of time increment and Gauss point numbers on the numerical accuracy is examined. Two dimensional numerical examples of plane strain and plane stress are solved and compared with the analytical solutions to demonstrate the versatility and accuracy of the present method.

• Structural Engineering and Mechanics
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• v.18 no.6
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• pp.731-744
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• 2004

Modified virtual crack closure integral (MVCCI) technique has become very popular for computation of strain energy release rate (SERR) and stress intensity factor (SIF) for 2-D crack problems. The objective of this paper is to propose a numerical integration procedure for MVCCI so as to generalize the technique and make its application much wider. This new procedure called as numerically integrated MVCCI (NI-MVCCI) will remove the dependence of MVCCI equations on the type of finite element employed in the basic stress analysis. Numerical studies on fracture analysis of 2-D crack (mode I and II) problems have been conducted by employing 4-noded, 8-noded (regular & quarter-point), 9-noded and 12-noded finite elements. For non-singular (regular) elements at crack tip, NI-MVCCI technique generates the same results as MVCCI, but the advantage for higher order regular and singular elements is that complex equations for MVCCI need not be derived. Gauss numerical integration rule to be employed for 8-noded singular (quarter-point) element for accurate computation of SERR and SIF has been recommended based on the numerical studies.

• Geophysics and Geophysical Exploration
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• v.21 no.1
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• pp.1-7
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• 2018

In the two-dimensional (2D) modeling of electrical method, the potential in the space domain is reconstructed with the calculated potentials in the wavenumber domain using inverse Fourier transform. The inverse Fourier integral is numerically evaluated using the transformed potential at different wavenumbers. In order to improve the precision of the integration, either the logarithmic or exponential approximation has been used depending on the size of wavenumber. Two numerical methods have been generally used to evaluate the integral; interval integration and Gaussian quadrature. However, both methods do not consider the distance from the current source. Thus the resulting potential in the space domain shows some error. Especially when the distance from the current source is very small or large, the error increases abruptly and the evaluated potential becomes extremely unstable. In this study, we developed a new method to calculate the integral accurately by introducing the distance from the current source to the rescaled Gauss abscissa and weight. The numerical tests for homogeneous half-space model show that the developed method can yield the error level lower than 0.4 percent over the various distances from the current source.

• Transactions of the Korean Society of Mechanical Engineers
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• v.19 no.4
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• pp.990-1004
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• 1995

A modified 16-node element for composite plate has been proposed and compared with the 20-node element to check the validity of it. The fields of numerical inspection include mode analysis and specific damping analysis. By symetrizing the conventional unsymmetric damping matrix in the analysis of specific damping capacity, we could compute the specific damping capacity and make a program, effectively. In addition, we could predict the errors caused by reduction of integration order in thickness direction depending upon the number of layers.