Abstract
We present a new approach to construct approximation schemes from scattered data on a node set, i.e. in the spirit of meshfree methods. The rational procedure behind these methods is to harmonize the locality of the shape functions and the information-theoretical optimality (entropy maximization) of the scheme, in a sense to be made precise in the paper. As a result, a one-parameter family of methods is defined, which smoothly and seamlessly bridges meshfree-style approximants and Delaunay approximants. Besides an appealing theoretical foundation, the method presents a number of practical advantages when it comes to solving partial differential equations. The non-negativity introduces the well-known monotonicity and variation-diminishing properties of the approximation scheme. Also, these methods satisfy ab initio a weak version of the Kronecker-delta property, which makes essential boundary conditions straightforward. The calculation of the shape functions is both efficient and robust in any spacial dimension. The implementation of a Galerkin method based on local maximum entropy approximants is illustrated by examples.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Arroyo and M. Ortiz, Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods, International Journal for Numerical Methods in Engineering in press (2005).
M. Arroyo and M. Ortiz, Maximum-entropy approximation schemes: higher order methods, International Journal for Numerical Methods in Engineering in preparation (2005).
T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, Meshless methods: An overview and recent developments, Computer Methods in Applied Mechanics and Engineering 139 (1996), no. 1–4, 3–47.
T. Belytschko, Y.Y. Lu, and L. Gu, Element-free Galerkin methods, International Journal for Numerical Methods in Engineering 37 (1994), no. 2, 229–256.
S. Boyd and L. Vandenberghe, Convex optimization, Cambridge University Press, Cambridge, UK, 2004.
P. Breitkopf, A. Rassineux, J.M. Savignat, and P. Villon, Integration constraint in diffuse element method, Computer Methods in Applied Mechanics and Engineering 193 (2004), no. 12–14, 1203–1220.
J.S. Chen, C.T. Wu, S. Yoon, and Y. You, Stabilized conforming nodal integration for Galerkin meshfree methods, International Journal for Numerical Methods in Engineering 50 (2001), 435466.
F. Cirak, M. Ortiz, and P. Schröder, Subdivision surfaces: a new paradigm for thin-shell finite-element analysis, International Journal for Numerical Methods in Engineering 47 (2000), no. 12, 2039–2072.
C. Cottin, I. Gavrea, H.H. Gonska, D.P. Kacsó, and D.X. Zhou, Global smoothness preservation and variation-diminishing property, Journal of Inequalities and Applications 4 (1999), no. 2, 91–114.
R.A. DeVore, The approximation of continuous functions by positive linear operators, Springer-Verlag, Berlin, 1972.
A. Huerta, T. Belytscko, S. Fernández-Méndez, and T. Rabczuk, Encyclopedia of computational mechanics, vol. 1, ch. Meshfree methods, pp. 279–309, Wiley, Chichester, 2004.
P. Krysl and T. Belytschko, Element-free galerkin method: Convergence of the continuous and discontinuous shape functions, Computer Methods in Applied Mechanics and Engineering 148 (1997), no. 3–4, 257–277.
W.K. Liu, S. Li, and T. Belytschko, Moving least square reproducing kernel methods Part I: Methodology and convergence, Computer Methods in Applied Mechanics and Engineering 143 (1997), no. 1–2, 113–154.
B. Nayroles, G. Touzot, and P. Villon, Generalizing the finite element method: diffuse approximation and diffuse elements, Computational Mechanics 10 (1992), no. 5, 307–318.
D. Organ, M. Fleming, T. Terry, and T. Belytschko, Continuous meshless approximations for nonconvex bodies by diffraction and transparency, Computational Mechanics 18 (1996), no. 3, 225–235.
H. Prautzch, W. Boehm, and M. Paluszny, Bézier and B-spline techniques, Springer-Verlag, Berlin, 2002.
V.T. Rajan, Optimality of the Delaunay triangulation in Rd, Discrete and Computational Geometry 12 (1994), no. 2, 189–202.
N. Sukumar, Construction of polygonal interpolants: A maximum entropy approach, International Journal for Numerical Methods in Engineering 61 (2004), no. 12, 2159–2181.
N. Sukumar, B. Moran, and T. Belytschko, The natural element method in solid mechanics, International Journal for Numerical Methods in Engineering 43 (1998), no. 5, 839–887.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer
About this chapter
Cite this chapter
Arroyo, M., Ortiz, M. (2007). Local Maximum-Entropy Approximation Schemes. In: Griebel, M., Schweitzer, M.A. (eds) Meshfree Methods for Partial Differential Equations III. Lecture Notes in Computational Science and Engineering, vol 57. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-46222-4_1
Download citation
DOI: https://doi.org/10.1007/978-3-540-46222-4_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-46214-9
Online ISBN: 978-3-540-46222-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)