• Journal of applied mathematics & informatics
• /
• v.24 no.1_2
• /
• pp.65-79
• /
• 2007

An optimal 3-point quadrature formula of closed type is derived. It is shown that the optimal quadrature formula has a better error bound than the well-known Simpson's rule. A corrected formula is also considered. Various error inequalities for these formulas are established. Applications in numerical integration are given.

• Aerospace Engineering and Technology
• /
• v.13 no.2
• /
• pp.60-65
• /
• 2014

In this paper, numerical integration method using operational matrix of integration is studied. Using the operational matrix of integration, modified fixed point iteration method is introduced in order to solve rapidly an initial value problem for non-linear equation of motion. As an example, an initial value problem for orbital motion is considered. Through the numerical example, it is shown that the algorithm is efficient from the computational time point of view.

• Structural Engineering and Mechanics
• /
• v.34 no.1
• /
• pp.85-95
• /
• 2010

In this paper, numerical solution of the singular integral equation for the multiple curved branch-cracks is investigated. If some quadrature rule is used, one difficult point in the problem is to balance the number of unknowns and equations in the solution. This difficult point was overcome by taking the following steps: (a) to place a point dislocation at the intersecting point of branches, (b) to use the curve length method to covert the integral on the curve to an integral on the real axis, (c) to use the semi-open quadrature rule in the integration. After taking these steps, the number of the unknowns is equal to the number of the resulting algebraic equations. This is a particular advantage of the suggested method. In addition, accurate results for the stress intensity factors (SIFs) at crack tips have been found in a numerical example. Finally, several numerical examples are given to illustrate the efficiency of the method presented.

• Journal of the Computational Structural Engineering Institute of Korea
• /
• v.37 no.3
• /
• pp.217-224
• /
• 2024

This paper proposes an efficient dimension reduction method (DRM) that considers the nonlinearity of the performance functions in reliability-based design optimization (RBDO). The dimension reduction method evaluates the reliability more accurately than the first-order reliability method (FORM) using integration quadrature points and weights. However, its efficiency is hindered as the number of quadrature points increases owing to the need for an additional evaluation of the performance function. In this study, we assessed the nonlinearity of the performance function in RBDO and proposed criteria for determining the number of quadrature points based on the degree of nonlinearity. This approach suggests adjusting the number of quadrature points during each iteration of the RBDO process while maintaining the accuracy of theDRM while improving the computational efficiency. The nonlinearity of the performance function was evaluated using the angle between the vectors used in the maximum probable target point (MPTP) search. Numerical tests were conducted to determine the appropriate number of quadrature points according to the degree of nonlinearity. Through a 2D numerical example, it is confirmed that the proposed method improves the efficiency while maintaining the accuracy of the dimension reduction method or Monte Carlo Simulation (MCS).

• Progress in Superconductivity and Cryogenics
• /
• v.16 no.4
• /
• pp.53-56
• /
• 2014

A numerical method of magnetic field calculation for the air-core solenoid is presented in this paper. In application of the Biot-Savart law, the magnetic field induced from the source current can be obtained by a double integration ormula. The numerical method named composite Simpson's rule for the integration is applied to the program and the adaptive quadrature method is used to adjust the step size in the calculation according to the precision we need. When the target point is in the solenoid and the intergrand's denominator may be zeroin the process of calculation, the method sill can provide an appropriate result. We have developed a program which calculates the magnetic field with at least 1ppm precision and named it as rzBI() to implement this method. The method has been used in the design of an MRI magnet, and the result show it is very flexible and convenient.

• The Mathematical Education
• /
• v.37 no.1
• /
• pp.101-114
• /
• 1998

In this thesis, We tried to introduce definite integration to the curriculum of high school mathematics with numerical integration, which had been introduced with quadrature method. For this purpose, We used new experimental mathematics approaches, so-called investigation and examination. In chapter II, We examined how much computers had been used in teaching mathematics. In chapter III, We presented the theoretical background of approximation integration within numerical integration. In chapter IV, We studied and compared various methods of numerical integration, and examined the relation between curvature of a curved line and numerical integration. In order to study more easily, We used some of computer programs. We hope that this thesis will be a turning point in developing new teaching methods and improving curriculum of mathematics in high school.

• Transactions of the Korean Society of Mechanical Engineers
• /
• v.18 no.11
• /
• pp.2922-2931
• /
• 1994

We propose a modified 6-node element, where the sampling point of Gauss quadrature moved in the thickness direction. The modified 6-node element has been applied to static problems and forced motion analyses. In this study, this method is extended to the finite element analysis of the natural frequencies of two dimensional problems. We also propose a modified 16-node element for three dimensional problems, which behaves much like a 20-node element with smaller degree of freedom. The modified 6-node and 16-node elements have been applied to the modal analyses of beams and plates, respectively. The results agree well with the results of the 8-node or 20-node element models.

• Computational Structural Engineering
• /
• v.10 no.2
• /
• pp.139-148
• /
• 1997

A finite element method is programmed to analyse the nonlinear behavior of axisymmetric structures. The lst order Mindlin shell theory which takes into account the transversal shear deformation is used to formulate a conical two node element with six degrees of freedom. To evade the shear locking phenomenon which arises in Mindlin type element when the effect of shear deformation tends to zero, the reduced integration of one point Gauss Quadrature at the center of element is employed. This method is the Updated Lagrangian formulation which refers the variables to the state of the most recent iteration. The solution is searched by Newton-Raphson iteration method. The tangent matrix of this method is obtained by a finite difference method by perturbating the degrees of freedom with small values. For the moment this program is limited to the analyses of non-linear elastic problems. For structures which could have elastic stability problem, the calculation is controled by displacement.

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

• Journal of the Computational Structural Engineering Institute of Korea
• /
• v.22 no.1
• /
• pp.117-124
• /
• 2009

The p-convergent boundary element method has been proposed to analyze two-dimensional potential problem on the basis of high order Legendre shape functions that have different property comparing with the shape functions in conventional boundary element method. The location of nodes corresponding to high order shape function are not defined along the boundary, called by nodeless node, similar to the p-convergent finite element method. As the order of shape function increases, the collocation point method is used to solve linear simultaneous equations. The collocation patterns of p-convergent boundary element method consist of non-symmetric hierarchial or symmetric non-hierarchical. As the order of shape function increases, the number of collocation point increases. The singular integral that appears in p-convergent boundary element has been calculated by special numeric quadrature technique and semi-analytical integration technique. The L-shape domain problem including singularity in the vicinity of reentrant comer is analyzed and the numerical results show that the relative error is smaller than $10^{-2}%$ range as compared with other results in literatures. In case of same condition, the symmetric p-collocation point pattern shows high accuracy of solution.