Abstract2018-05-24T12:52:58+00:00

Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods

Author (s): Marino Arroyo and Michael Ortiz
Journal: International Journal for Numerical Methods in Engineering

Volume: 65, Issue 13
Pages: 2167 – 2202
Date: 2006

Abstract:
We present a one-parameter family of approximation schemes, which we refer to as local maximum-entropy (max-ent) approximation schemes, that bridges continuously two important limits: Delaunay triangulation and maximum-entropy statistical inference.
Local max-ent approximation schemes represent a compromise—in the sense of Pareto optimality—between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width. Local max-ent approximation schemes are entirely defined by the node set and the domain of analysis, and the shape functions are positive, interpolate affine functions
exactly, and have a weak Kronecker-delta property at the boundary. Local max-ent approximation may be regarded as a regularization, or thermalization, of Delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation. Local max-ent approximation schemes can be taken as a convenient basis for the numerical solution of PDEs in the style of meshfree Galerkin methods. In test cases characterized by smooth solutions we find that the accuracy of local max-ent approximation schemes is vastly superior to that of finite elements.

  
  

Bibtex:

@article {NME:NME1534,
author = {Arroyo, M. and Ortiz, M.},
title = {Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods},
journal = {International Journal for Numerical Methods in Engineering},
volume = {65},
number = {13},
publisher = {John Wiley & Sons, Ltd.},
issn = {1097-0207},
url = {http://dx.doi.org/10.1002/nme.1534},
doi = {10.1002/nme.1534},
pages = {2167--2202},
year = {2006},
}