zbMATH — the first resource for mathematics

Smooth, second order, non-negative meshfree approximants selected by maximum entropy. (English) Zbl 1176.74208
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.

74S30 Other numerical methods in solid mechanics (MSC2010)
74H45 Vibrations in dynamical problems in solid mechanics
Full Text: DOI
[1] Arroyo, 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– (2006) · Zbl 1146.74048
[2] Belytschko, Element-free Galerkin methods, International Journal for Numerical Methods in Engineering 37 (2) pp 229– (1994) · Zbl 0796.73077
[3] Liu, Reproducing kernel particle methods, International Journal for Numerical Methods in Engineering 20 pp 1081– (1995) · Zbl 0881.76072
[4] Huerta, Meshfree Methods pp 279– (2004)
[5] DeVore, The Approximation of Continuous Functions by Positive Linear Operators (1972) · Zbl 0276.41011 · doi:10.1007/BFb0059493
[6] Hughes, Isogeometric analysis: Cad, finite elements, nurbs, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering 194 pp 4135– (2005) · Zbl 1151.74419
[7] Cottrell, Isogeometric analysis of structural vibrations, Computer Methods in Applied Mechanics and Engineering 195 (41-43) pp 5257– (2006) · Zbl 1119.74024
[8] Shannon, A mathematical theory of communication, The Bell System Technical Journal 27 (3) pp 379– (1948) · Zbl 1154.94303 · doi:10.1002/j.1538-7305.1948.tb01338.x
[9] Jaynes, Information theory and statistical mechanics, Physical Review 106 (4) pp 620– (1957) · Zbl 0084.43701
[10] Arroyo, Meshfree Methods for Partial Differential Equations III (2006)
[11] Sukumar, Overview and construction of meshfree basis functions: from moving least squares to entropy approximants, International Journal for Numerical Methods in Engineering 70 (2) pp 181– (2007) · Zbl 1194.65149
[12] Sukumar, Construction of polygonal interpolants: a maximum entropy approach, International Journal for Numerical Methods in Engineering 61 (12) pp 2159– (2004) · Zbl 1073.65505
[13] Prautzch, Bézier and B-spline Techniques (2002) · doi:10.1007/978-3-662-04919-8
[14] Sukumar, The natural element method in solid mechanics, International Journal for Numerical Methods in Engineering 43 (5) pp 839– (1998) · Zbl 0940.74078
[15] Cirak, Subdivision surfaces: a new paradigm for thin-shell finite-element analysis, International Journal for Numerical Methods in Engineering 47 (12) pp 2039– (2000) · Zbl 0983.74063
[16] Cottin, Global smoothness preservation and variation-diminishing property, Journal of Inequalities and Applications 4 (2) pp 91– (1999) · Zbl 0970.41015
[17] Boyd, Convex Optimization (2004) · doi:10.1017/CBO9780511804441
[18] Hölling, Introduction to the Web-method and its applications, Advances in Computational Mathematics 23 (1) pp 215– (2005)
[19] Babuška, Survey of meshless and generalized finite element methods: a unified approach, Acta Numerica 12 pp 1– (2003)
[20] Born, Dynamical Theory of Crystal Lattices (1954)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.