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kFEM: Adaptive meshfree finite‐element methods using local kernels on arbitrary subdomains
Author(s) -
Quaglino A.,
Krause R.
Publication year - 2018
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.5755
Subject(s) - polygon mesh , finite element method , mathematics , meshfree methods , polygon (computer graphics) , degrees of freedom (physics and chemistry) , quadrilateral , regular polygon , boundary (topology) , algorithm , mathematical optimization , computer science , mathematical analysis , geometry , structural engineering , telecommunications , physics , frame (networking) , quantum mechanics , engineering
Summary We propose a novel finite‐element method for polygonal meshes. The resulting scheme is h p ‐adaptive, where h and p are a measure of, respectively, the size and the number of degrees of freedom of each polygon. Moreover, it is locally meshfree, since it is possible to arbitrarily choose the locations of the degrees of freedom inside each polygon. Our construction is based on nodal kernel functions, whose support consists of all polygons that contain a given node. This ensures a significantly higher sparsity compared to standard meshfree approximations. In this work, we choose axis‐aligned quadrilaterals as polygonal primitives and maximum entropy approximants as kernels. However, any other convex approximation scheme and convex polygons can be employed. We study the optimal placement of nodes for regular elements, ie, those that are not intersected by the boundary, and propose a method to generate a suitable mesh. Finally, we show via numerical experiments that the proposed approach provides good accuracy without undermining the sparsity of the resulting matrices.