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

Arroyo, M. and Ortiz, M. (2006) Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods. International Journal for Numerical Methods in Engineering, 65 (13). pp. 2167-2202. ISSN 0029-5981. https://resolver.caltech.edu/CaltechAUTHORS:20171128-101329930

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Abstract

We 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.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://dx.doi.org/10.1002/nme.1534DOIArticle
http://onlinelibrary.wiley.com/doi/10.1002/nme.1534/abstractPublisherArticle
ORCID:
AuthorORCID
Ortiz, M.0000-0001-5877-4824
Additional Information: © 2005 John Wiley & Sons. Received 17 December 2004. Revised 27 April 2005. Accepted 17 August 2005. The authors gratefully acknowledge the support of the Department of Energy through Caltech’s ASCI ASAP Center for the Simulation of the Dynamic Response of Materials, and the support received from NSF through an ITR grant on Multiscale Modelling and Simulation and Caltech’s Center for Integrative Multiscale Modelling and Simulation.
Group:GALCIT
Funders:
Funding AgencyGrant Number
Department of Energy (DOE)UNSPECIFIED
NSFUNSPECIFIED
Subject Keywords:maximum entropy; information theory; approximation theory; meshfree methods; Delaunay triangulation
Issue or Number:13
Record Number:CaltechAUTHORS:20171128-101329930
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20171128-101329930
Official Citation:Arroyo, M. and Ortiz, M. (2006), Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods. Int. J. Numer. Meth. Engng., 65: 2167–2202. doi:10.1002/nme.1534
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:83494
Collection:CaltechAUTHORS
Deposited By: Lydia Suarez
Deposited On:28 Nov 2017 18:37
Last Modified:24 Feb 2020 10:30

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