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

Authors

  • M. Arroyo,

    1. Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, U.S.A.
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    • Current address: LaCàN, Universitat Politècnica de Catalunya, C/Jordi Girona 1-3, Barcelona 08034, Spain.

  • M. Ortiz

    Corresponding author
    1. Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, U.S.A.
    • Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, U.S.A.
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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. Copyright © 2005 John Wiley & Sons, Ltd.

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