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p-adaptive $$\text C^{k}$$ generalized finite element method for arbitrary polygonal clouds. (English) Zbl 1162.74455
Summary: A p-Adaptive Generalized Finite Element Method (GFEM) based on a Partition of Unity (POU) of arbitrary smoothness degree is presented. The shape functions are built from the product of a Shepard POU and enrichment functions. Shepard functions have a smoothness degree directly related to the weighting functions adopted in their definition. Here the weighting functions are obtained from boolean R-functions which allow the construction of $$\text C^{k}$$ approximations, with k arbitrarily large, defined over a polygonal patch of elements, named cloud. The Element Residual Method is used to obtain error indicators by taking into account the typical nodal enrichment scheme of the method. This procedure is enhanced by using approximations with a high degree of smoothness as it eliminates the discontinuity of the stress field in the interior of each cloud. Adaptive analysis of plane elasticity problems are presented, and the performance of the technique is investigated.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74B05 Classical linear elasticity 74K20 Plates
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