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A study on convergence and complexity of reproducing kernel collocation method
Hu, Hsin-Yun,Lai, Chiu-Kai,Chen, Jiun-Shyan Techno-Press 2009 Interaction and multiscale mechanics Vol.2 No.3
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.
Reproducing kernel based evaluation of incompatibility tensor in field theory of plasticity
Aoyagi, Y.,Hasebe, T.,Guan, P.C.,Chen, J.S. Techno-Press 2008 Interaction and multiscale mechanics Vol.1 No.4
This paper employs the reproducing kernel (RK) approximation for evaluation of field theory-based incompatibility tensor in a polycrystalline plasticity simulation. The modulation patterns, which is interpreted as mimicking geometrical-type dislocation substructures, are obtained based on the proposed method. Comparisons are made using FEM and RK based approximation methods among different support sizes and other evaluation conditions of the strain gradients. It is demonstrated that the evolution of the modulation patterns needs to be accurately calculated at each time step to yield a correct physical interpretation. The effect of the higher order strain derivative processing zone on the predicted modulation patterns is also discussed.
Chen, Li,Liew, K.M.,Cheng, Yumin Techno-Press 2010 Interaction and multiscale mechanics Vol.3 No.3
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.
SOLUTION OF THE SYSTEM OF FOURTH ORDER BOUNDARY VALUE PROBLEM USING REPRODUCING KERNEL SPACE
Akram, Ghazala,Ur Rehman, Hamood The Korean Society for Computational and Applied M 2013 Journal of applied mathematics & informatics Vol.31 No.1
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.
SOLUTION OF THE SYSTEM OF FOURTH ORDER BOUNDARY VALUE PROBLEM USING REPRODUCING KERNEL SPACE
Ghazala Akram,Hamood Ur Rehman 한국전산응용수학회 2013 Journal of applied mathematics & informatics Vol.31 No.1
In this paper, a general technique is proposed for solving asystem of fourth-order boundary value problems. The solution is given inthe form of series and its approximate solution is obtained by truncatingthe series. Advantages of the method are that the representation of exactsolution is obtained in a new reproducing kernel Hilbert space and accuracyof numerical computation is higher. Numerical results show that themethod employed in the paper is valid. Numerical evidence is presented toshow the applicability and superiority of the new method..