• Title, Summary, Keyword: reproducing kernel method

### A study on convergence and complexity of reproducing kernel collocation method

• Hu, Hsin-Yun;Lai, Chiu-Kai;Chen, Jiun-Shyan
• Interaction and multiscale mechanics
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• v.2 no.3
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• pp.295-319
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• 2009
• In this work, we discuss a reproducing kernel collocation method (RKCM) for solving $2^{nd}$ order PDE based on strong formulation, where the reproducing kernel shape functions with compact support are used as approximation functions. The method based on strong form collocation avoids the domain integration, and leads to well-conditioned discrete system of equations. We investigate the convergence and the computational complexity for this proposed method. An important result obtained from the analysis is that the degree of basis in the reproducing kernel approximation has to be greater than one for the method to converge. Some numerical experiments are provided to validate the error analysis. The complexity of RKCM is also analyzed, and the complexity comparison with the weak formulation using reproducing kernel approximation is presented.

### 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.

### Kirchhoff Plate Analysis by Using Hermite Reproducing Kernel Particle Method (HRKPM을 이용한 키르히호프 판의 해석)

• 석병호;송태한
• Transactions of the Korean Society of Machine Tool Engineers
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• v.12 no.5
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• pp.67-72
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• 2003
• For the analysis of Kirchhoff plate bending problems, a new meshless method is implemented. For the satisfaction of the $C^1$ continuity condition in which the first derivative is treated an another primary variable, Hermite interpolation is enforced on standard reproducing kernel particle method. In order to impose essential boundary conditions on solving $C^1$ continuity problems, shape function modifications are adopted. Through numerical tests, the characteristics and accuracy of the HRKPM are investigated and compared with the finite element analysis. By this implementatioa it is shown that high accuracy is achieved by using HRKPM for solving Kirchhoff plate bending problems.

### Analysis of Bulk Metal Forming Process by Reproducing Kernel Particle Method (재생커널입자법을 이용한 체적성형공정의 해석)

• Han, Kyu-Taek
• Journal of the Korean Society of Manufacturing Process Engineers
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• v.8 no.3
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• pp.21-26
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• 2009
• The finite element analysis of metal forming processes often fails because of severe mesh distortion at large deformation. As the concept of meshless methods, only nodal point data are used for modeling and solving. As the main feature of these methods, the domain of the problem is represented by a set of nodes, and a finite element mesh is unnecessary. This computational methods reduces time-consuming model generation and refinement effort. It provides a higher rate of convergence than the conventional finite element methods. The displacement shape functions are constructed by the reproducing kernel approximation that satisfies consistency conditions. In this research, A meshless method approach based on the reproducing kernel particle method (RKPM) is applied with metal forming analysis. Numerical examples are analyzed to verify the performance of meshless method for metal forming analysis.

### 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.

### SOLUTION OF THE SYSTEM OF FOURTH ORDER BOUNDARY VALUE PROBLEM USING REPRODUCING KERNEL SPACE

• Akram, Ghazala;Ur Rehman, Hamood
• Journal of applied mathematics & informatics
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• v.31 no.1_2
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• pp.55-63
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• 2013
• In this paper, a general technique is proposed for solving a system of fourth-order boundary value problems. The solution is given in the form of series and its approximate solution is obtained by truncating the series. Advantages of the method are that the representation of exact solution is obtained in a new reproducing kernel Hilbert space and accuracy of numerical computation is higher. Numerical results show that the method employed in the paper is valid. Numerical evidence is presented to show the applicability and superiority of the new method.

### The coupling of complex variable-reproducing kernel particle method and finite element method for two-dimensional potential problems

• Chen, Li;Liew, K.M.;Cheng, Yumin
• Interaction and multiscale mechanics
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• v.3 no.3
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• pp.277-298
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• 2010
• The complex variable reproducing kernel particle method (CVRKPM) and the FEM are coupled in this paper to analyze the two-dimensional potential problems. The coupled method not only conveniently imposes the essential boundary conditions, but also exploits the advantages of the individual methods while avoiding their disadvantages, resulting in improved computational efficiency. A hybrid approximation function is applied to combine the CVRKPM with the FEM. Formulations of the coupled method are presented in detail. Three numerical examples of the two-dimensional potential problems are presented to demonstrate the effectiveness of the new method.

### 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.

### SOLVING SINGULAR NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS IN THE REPRODUCING KERNEL SPACE

• Geng, Fazhan;Cui, Minggen
• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.631-644
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• 2008
• In this paper, we present a new method for solving a nonlinear two-point boundary value problem with finitely many singularities. Its exact solution is represented in the form of series in the reproducing kernel space. In the mean time, the n-term approximation $u_n(x)$ to the exact solution u(x) is obtained and is proved to converge to the exact solution. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other.

### FCM for the Multi-Scale Problems (고속 최소자승 점별계산법을 이용한 멀티 스케일 문제의 해석)

• 김도완;김용식
• Proceedings of the Computational Structural Engineering Institute Conference
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• pp.599-603
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• 2002
• We propose a new meshfree method to be called the fast moving least square reproducing kernel collocation method(FCM). This methodology is composed of the fast moving least square reproducing kernel(FMLSRK) approximation and the point collocation scheme. Using point collocation makes the meshfree method really come true. In this paper, FCM Is shown to be a good method at least to calculate the numerical solutions governed by second order elliptic partial differential equations with geometric singularity or geometric multi-scales. To treat such problems, we use the concept of variable dilation parameter.