• Title/Summary/Keyword: reproducing kernel method

### REPRODUCING KERNEL METHOD FOR SOLVING TENTH-ORDER BOUNDARY VALUE PROBLEMS

• Geng, Fazhan;Cui, Minggen
• Journal of applied mathematics & informatics
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• v.28 no.3_4
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• pp.813-821
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• 2010
• In this paper, the tenth-order linear boundary value problems are solved using reproducing kernel method. The algorithm developed approximates the solutions, and their higher-order derivatives, of differential equations and it avoids the complexity provided by other numerical approaches. First a new reproducing kernel space is constructed to solve this class of tenth-order linear boundary value problems; then the approximate solutions of such problems are given in the form of series using the present method. Three examples compared with those considered by Siddiqi, Twizell and Akram [S.S. Siddiqi, E.H. Twizell, Spline solutions of linear tenth order boundary value problems, Int. J. Comput. Math. 68 (1998) 345-362; S.S.Siddiqi, G.Akram, Solutions of tenth-order boundary value problems using eleventh degree spline, Applied Mathematics and Computation 185 (1)(2007) 115-127] show that the method developed in this paper is more efficient.

### Analysis of a Gas Circuit Breaker Using the Fast Moving Least Square Reproducing Kernel Method

• Lee, Chany;Kim, Do-Wan;Park, Sang-Hun;Kim, Hong-Kyu;Jung, Hyun-Kyo
• Journal of Electrical Engineering and Technology
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• v.4 no.2
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• pp.272-276
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• 2009
• In this paper, the arc region of a gas circuit breaker (GCB) is analyzed using the fast moving least square reproducing kernel method (FMLSRKM) which can simultaneously calculate an approximated solution and its derivatives. For this problem, an axisymmetric and inhomogeneous formulation of the FMLSRKM is used and applied. The field distribution obtained by the FMLSRKM is compared to that of the finite element method. Then, a whole breaking period of a GCB is simulated, including analysis of the arc gas flow by finite volume fluid in the cell, and the electric field of the arc region using the FMLSRKM.

### Analysis on a Simple Waveguide Using Meshfree Method (무요소법을 이용한 waveguide 내의 필드 분포 해석)

• Lee, Chany;Woo, Dong-Kyun;Jung, Hyun-Kyo
• 한국정보통신설비학회:학술대회논문집
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• pp.190-192
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• 2008
• This paper shows the formulation of fast moving least square reproducing kernel method (FMLSRKM) which is a kind of meshfree methods. FMLSRKM has some advantages compared to conventional numerical techniques such as finite element method. For simple analysis on a rectangular waveguide, point collocation scheme is introduced and applied.

### An Euler Solution Using Moving Least Square Reproducing Kernel Method

• Ahn Mu-Young;Kim Yongsik;Chang Keun-Shik;Choe Hi Jun;Chung Dong-Kyu
• 한국전산유체공학회:학술대회논문집
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• pp.130-130
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• 2003

### A NUMERICAL ALGORITHM FOR SINGULAR MULTI-POINT BVPS USING THE REPRODUCING KERNEL METHOD

• Jia, Yuntao;Lin, Yingzhen
• The Pure and Applied Mathematics
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• v.21 no.1
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• pp.51-60
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• 2014
• In this paper, we construct a complex reproducing kernel space for singular multi-point BVPs, and skillfully obtain reproducing kernel expressions. Then, we transform the problem into an equivalent operator equation, and give a numerical algorithm to provide the approximate solution. The uniform convergence of this algorithm is proved, and complexity analysis is done. Lastly, we show the validity and feasibility of the numerical algorithm by two numerical examples.

### Shape Design Optimization of Crack Propagation Problems Using Meshfree Methods (무요소법을 이용한 균열진전 문제의 형상 최적설계)

• Kim, Jae-Hyun;Ha, Seung-Hyun;Cho, Seonho
• Journal of the Computational Structural Engineering Institute of Korea
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• v.27 no.5
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• pp.337-343
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• 2014
• This paper presents a continuum-based shape design sensitivity analysis(DSA) method for crack propagation problems using a reproducing kernel method(RKM), which facilitates the remeshing problem required for finite element analysis(FEA) and provides the higher order shape functions by increasing the continuity of the kernel functions. A linear elasticity is considered to obtain the required stress field around the crack tip for the evaluation of J-integral. The sensitivity of displacement field and stress intensity factor(SIF) with respect to shape design variables are derived using a material derivative approach. For efficient computation of design sensitivity, an adjoint variable method is employed tather than the direct differentiation method. Through numerical examples, The mesh-free and the DSA methods show excellent agreement with finite difference results. The DSA results are further extended to a shape optimization of crack propagation problems to control the propagation path.

### Level Set Based Topological Shape Optimization Combined with Meshfree Method (레벨셋과 무요소법을 결합한 위상 및 형상 최적설계)

• Ahn, Seung-Ho;Ha, Seung-Hyun;Cho, Seonho
• Journal of the Computational Structural Engineering Institute of Korea
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• v.27 no.1
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• pp.1-8
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• 2014
• Using the level set and the meshfree methods, we develop a topological shape optimization method applied to linear elasticity problems. Design gradients are computed using an efficient adjoint design sensitivity analysis(DSA) method. The boundaries are represented by an implicit moving boundary(IMB) embedded in the level set function obtainable from the "Hamilton-Jacobi type" equation with the "Up-wind scheme". Then, using the implicit function, explicit boundaries are generated to obtain the response and sensitivity of the structures. Global nodal shape function derived on a basis of the reproducing kernel(RK) method is employed to discretize the displacement field in the governing continuum equation. Thus, the material points can be located everywhere in the continuum domain, which enables to generate the explicit boundaries and leads to a precise design result. The developed method defines a Lagrangian functional for the constrained optimization. It minimizes the compliance, satisfying the constraint of allowable volume through the variations of boundary. During the optimization, the velocity to integrate the Hamilton-Jacobi equation is obtained from the optimality condition for the Lagrangian functional. Compared with the conventional shape optimization method, the developed one can easily represent the topological shape variations.