• Structural Engineering and Mechanics
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• v.22 no.1
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• pp.1-16
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• 2006

This paper presents a virtual boundary element-equivalent collocation method (VBEM) for the plane magnetoelectroelastic solids, which is based on the fundamental solutions of the plane magnetoelectroelastic solids and the basic idea of the virtual boundary element method for elasticity. Besides all the advantages of the conventional boundary element method (BEM) over domain discretization methods, this method avoids the computation of singular integral on the boundary by introducing the virtual boundary. In the end, several numerical examples are performed to demonstrate the performance of this method, and the results show that they agree well with the exact solutions. So the method is one of the efficient numerical methods used to analyze megnatoelectroelastic solids.

• Bulletin of the Korean Mathematical Society
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• v.38 no.1
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• pp.17-27
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• 2001

We discuss a finite difference preconditioner for the$C^1$ Lagrance quadratic spline collocation method for a uniformly elliptic operator with homogeneous Dirichlet boundary conditions. Using the generalized field of values argument, we analyzed eigenvalues of the matrix preconditioned by the matrix corresponding to a finite difference operator with zero boundary condition.

• Journal of the Chungcheong Mathematical Society
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• v.9 no.1
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• pp.153-164
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• 1996

An $O(h^4)$ cubic spline collocation method for biharmonic equation with a special boundary conditions is formulated and a fast direct method is proposed for the linear system arising when the cubic spline collocation method is employed. This method requires $O(N^2\;{\log}\;N)$ arithmatic operations over an $N{\times}N$ uniform partition.

• Interaction and multiscale mechanics
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• v.2 no.4
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• pp.333-352
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• 2009

We introduce a radial basis collocation method to solve axially moving beam problems which involve $2^{nd}$ order differentiation in time and $4^{th}$ order differentiation in space. The discrete equation is constructed based on the strong form of the governing equation. The employment of multiquadrics radial basis function allows approximation of higher order derivatives in the strong form. Unlike the other approximation functions used in the meshfree methods, such as the moving least-squares approximation, $4^{th}$ order derivative of multiquadrics radial basis function is straightforward. We also show that the standard weighted boundary collocation approach for imposition of boundary conditions in static problems yields significant errors in the transient problems. This inaccuracy in dynamic problems can be corrected by a statically condensed semi-discrete equation resulting from an exact imposition of boundary conditions. The effectiveness of this approach is examined in the numerical examples.

• Journal of the korean Society of Automotive Engineers
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• v.4 no.1
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• pp.31-39
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• 1982

In this paper, the stress intensity factors at the crack tips in a finite plate having a straight crack with arbitrary lengths and directions are analyzed by means of Boundary Collocation Method in Cartesian coordinates under the following two cases. A) Case of only considering stress components at the boundary collocation points B) Case of considering stress resultant at the boundary, in addition to case of )A) and analyzed by means of F.E.M. To certify the Boundary Collocation Method the solutions of B.C.M. are compared with the solutions of F.E.M. demonstrates the simplicity of input data preparation and the reduction of cpu time against F.E.M.

• Structural Engineering and Mechanics
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• v.17 no.5
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• pp.671-690
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• 2004

A meshless method with novel variation of point collocation by finite mixture approximation is developed in this paper, termed the meshless finite mixture (MFM) method. It is based on the finite mixture theorem and consists of two or more existing meshless techniques for exploitation of their respective merits for the numerical solution of partial differential boundary value (PDBV) problems. In this representation, the classical reproducing kernel particle and differential quadrature techniques are mixed in a point collocation framework. The least-square method is used to optimize the value of the weight coefficient to construct the final finite mixture approximation with higher accuracy and numerical stability. In order to validate the developed MFM method, several one- and two-dimensional PDBV problems are studied with different mixed boundary conditions. From the numerical results, it is observed that the optimized MFM weight coefficient can improve significantly the numerical stability and accuracy of the newly developed MFM method for the various PDBV problems.

• Journal of applied mathematics & informatics
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• v.8 no.2
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• pp.371-386
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• 2001

In this work we investigate the finite element preconditioning method for the $C^1$-cubic spline collocation discretizations for an elliptic operator A defined by $Au := -{\Delta}u + a_1u_x+a_2u_y+a_0u$ in the unit square with some boundary conditions. We compute the condition number and the numerical solution of the preconditioning system for the several example problems. Finally, we compare the this preconditioning system with the another preconditioning system.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 1995.04a
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• pp.573-577
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• 1995

An integration method for the hypersingular kernels, in the boundary integralequations used for the solution of crack-like problems in elasticity, has been developed. To isolate the stronger singularities, the actual boundaries are replaced by the smoothly curved auxiliary boundaries which provide the detoured, non-singular integration paths. The auxiliary boundary can be interpreted as a contracted form of the actual boundaries except for the singular element where the collocating point is located. For an optimal integration path for every singular collocation point, the auxiliary boundary may have different shape depending on the position of the collocation point on the singular element.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.27 no.1
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• pp.66-76
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• 2003

The arbitrary V-notched crack problem is considered. The general expressions for the stress components on this problem are obtained as explicit series forms composed of independent unknown coefficients which are denoted by coefficients of eigenvector. For this results eigenvalue equation is performed first through introducing complex stress functions and applying the traction free boundary conditions. Next solving this equation, eigenvalues and corresponding eigenvectors are obtained respectively, and finally inserting these results into stress components, the general equations are obtained. These results are also shown to be applicable to the symmetric V-notched crack or straight crack. It can be shown that this solutions are composed of the linear combination of Mode I and Mode II solutions which are obtained from different characteristic equations, respectively. Through performing asymptotic analysis for stresses, the stress intensity factor is given as a closed form equipped with the unknown coefficients of eigenvector. In order to calculate the unknown coefficients. based on these general explicit equations, numerical programming using the overdetermined boundary collocation method which is algorithmed originally by Carpenter is also worked out. As this programming requires the input data, the commercial FE analysis for stresses is performed. From this study, for some V-notched problems, unknown coefficients can be calculated numerically and also fracture parameters are determined.

• Solar Energy
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• v.8 no.1
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• pp.82-94
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• 1988

The hydrodynamic stability equations for natural convection flows adjacent to a vertical isothermal surface in cold or warm water (Boussinesq or non-Boussinesq situation for density relation), constitute a two-point-boundary-value (eigenvalue) problem, which was solved numerically using the simple shooting and the orthogonal collocation method. This is the first instance in which these stability equations have been solved using a computer code COLSYS, that is based on the orthogonal collocation method, designed to solve accurately two-point-boundary-value problem. Use of the orthogonal collocation method significantly reduces the error propagation which occurs in solving the initial value problem and avoids the inaccuracy of superposition of asymptotic solutions using the conventional technique of simple shooting.