A Novel Form of Reproducing Kernel Interpolation Method with Applications to Nonlinear Mechanics
Amit Shaw; and D Roy

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 19, No. 1, pp. 69-98, 2007
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Abstract A novel discretization strategy and derivative reproduction based on reproducing kernel (RK) particle approximations of functions are proposed. The proposed scheme is in the form of an RK interpolation that offers significant numerical advantages over a recent version of the strategy by Chen et al. (2003), wherein the authors added a set of primitive functions to the reproducing kernel (enrichment) functions. It was also required that the support size of the primitive function be less than the smallest distance between two successive grid points. Since the primitive function was required to vary from 0 to 1 within half of this support size, this potentially led to considerable numerical corruption of the algorithm. In contrast, the present version of the interpolating strategy, which is far less prone to such numerical ill-conditioning, linearly combines two different families of RK basis functions and determines the coefficients of the linear combination using the interpolating conditions. Apart from the interpolation scheme, a new technique for approximating derivatives of RK basis functions is also proposed. Such an approximation is based on a direct reproduction of derivatives within any given polynomial space. Detailed error estimates are provided and convergence studies are performed for a couple of test boundary value problems with known exact solutions. The proposed method is next applied to a class of one-dimensional beam equations (Elastica and Plastica) and a two-dimensional von Karman plate equation. Some of these results are compared with those obtained from a few other competing algorithms, such as the standard form of RK particle method, the interpolating RK method by Chen et al. (2003) as well as the classical finite element method. The relative numerical advantages of the new method are brought out in the process.
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