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

This is the pre-peer reviewed version of the following article: Arroyo, M.; Ortiz, M. Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods. "International journal for numerical methods in engineering", Març 2006, vol. 65, núm. 13, p. 2167-2202, which has been published in final form at http://www3.interscience.wiley.com/journal/112159842/abstractWe present a one-parameter family of approximation schemes, which we refer to as local maximum-entropy approximation schemes, that bridges continuously two important limits: Delaunay triangulation and maximum-entropy (max-ent) 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.Peer Reviewe