This work is concerned with developing the hierarchical basis for meshless methods. A reproducingkernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discret...
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This work is concerned with developing the hierarchical basis for meshless methods. A reproducingkernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart. To form such hierarchical partition, a class of basic wavelet functions are introduced. Based upon the built-in consistency conditions, the differential consistency conditions for the hierarchical kernel functions are derived. It serves as an indispensable instrument in establishing the interpolation error estimate, which is theoretically proven and numerically validated. For a special interpolant with different combinations of the hierarchical kernels, a synchronized convergence effect may be observed. Being different from the conventional Legendre function based p-type hierarchical basis, the new hierarchical basis is an intrinsic pseudo-spectral basis, which can remain as a partition of unity in a focal region, because the discrete wavelet kernels form a 'partition of nullity'. These newly developed kernels can be used as the multiscale basis to solve partial differential equations in numerical computation as a p-type refinement. Copyright (C) 1999 John Wiley & Sons, Ltd.
Conventional finite element analysis of metal forming processes often breaks down due to severe mesh distortion. Since 1993, meshless methods have been considerably developed for structural applications. The main feat...
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Conventional finite element analysis of metal forming processes often breaks down due to severe mesh distortion. Since 1993, meshless methods have been considerably developed for structural applications. The main feature of these methods is that the domain of the problem is represented by a set of nodes, and a finite element mesh is unnecessary. This new generation of computational methods reduces time-consuming model generation and refinement effort, and it provides a higher rate of convergence than that of the conventional finite element methods. A meshless method based on the reproducing kernel particle method (RKPM) is applied to metal forming analysis. The displacement shape functions are developed from a reproducingkernel approximation that satisfies consistency conditions. The use of smooth shape functions with large support size are particularly effective in dealing with large material distortion in metal forming analysis. In this work, a collocation formulation is used in the boundary integral of the contact constraint equations formulated by a penalty method. Metal forming examples, such as ring compression test and upsetting, are analyzed to demonstrate the performance of the method. (C) 1998 Elsevier Science S.A. All rights reserved.
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