Smooth, second order, non-negative meshfree approximants selected by maximum entropy

We present a family of approximation schemes, which we refer to as second‐order maximum‐entropy (max‐ent) approximation schemes, that extends the first‐order local max‐ent approximation schemes to second‐order consistency. This method retains the fundamental properties of first‐order max‐ent schemes...

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Bibliographic Details
Published in:International journal for numerical methods in engineering 2009-09, Vol.79 (13), p.1605-1632
Main Authors: Cyron, C. J., Arroyo, M., Ortiz, M.
Format: Article
Language:English
Subjects:
Anàlisi numèrica
Approximants
Approximation
B-Splines
Computational techniques
Convex approximants
Entropia
Exact sciences and technology
Fundamental areas of phenomenology (including applications)
Matemàtiques i estadística
Mathematical analysis
Mathematical methods in physics
Mathematical models
Maximum entropy
Maximum entropy method
Meshfree methods
Meshless methods
Numerical analysis
Physics
Shape functions
Solid mechanics
Structural and continuum mechanics
Structural vibration
Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...)
Àrees temàtiques de la UPC
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Summary:We present a family of approximation schemes, which we refer to as second‐order maximum‐entropy (max‐ent) approximation schemes, that extends the first‐order local max‐ent approximation schemes to second‐order consistency. This method retains the fundamental properties of first‐order max‐ent schemes, namely the shape functions are smooth, non‐negative, and satisfy a weak Kronecker‐delta property at the boundary. This last property makes the imposition of essential boundary conditions in the numerical solution of partial differential equations trivial. The evaluation of the shape functions is not explicit, but it is very efficient and robust. To our knowledge, the proposed method is the first higher‐order scheme for function approximation from unstructured data in arbitrary dimensions with non‐negative shape functions. As a consequence, the approximants exhibit variation diminishing properties, as well as an excellent behavior in structural vibrations problems as compared with the Lagrange finite elements, MLS‐based meshfree methods and even B‐Spline approximations, as shown through numerical experiments. When compared with usual MLS‐based second‐order meshfree methods, the shape functions presented here are much easier to integrate in a Galerkin approach, as illustrated by the standard benchmark problems. Copyright © 2009 John Wiley & Sons, Ltd.
ISSN:0029-5981
1097-0207
DOI:10.1002/nme.2597