Second order convex maximum entropy approximants with applications to high order PDE

Author (s): Rosolen, A.; Millan, D. and Arroyo, M.
Journal: International Journal for Numerical Methods in Engineering

Volume: 94, Issue 2
Pages: 150 – 182
Date: 2013

Abstract:
We present a new approach for second order maximum entropy (max-ent) meshfree approximants that produces positive and smooth basis functions of uniform aspect ratio even for non-uniform node sets, and prescribes robustly feasible constraints for the entropy maximization program defining the approximants. We examine the performance of the proposed approximation scheme in the numerical solution by a direct Galerkin method of a number of partial differential equations (PDEs), including structural vibrations, elliptic second order PDEs, and fourth order PDEs for Kirchhoff-Love thin shells and for a phase field model describing the mechanics of biomembranes. The examples highlight the ability of the method to deal with non-uniform node distributions, and the high accuracy of the solutions. Surprisingly, the first order meshfree max-ent approximants with large supports are competitive when compared to the proposed second order approach in all the tested examples, even in the higher order PDEs.

Journal paper (from the publisher)  
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Bibtex:

@article {NME:NME4443,
	author = {Rosolen, A. and Millán, D. and Arroyo, M.},
	title = {Second-order convex maximum entropy approximants with applications to high-order PDE},
	journal = {International Journal for Numerical Methods in Engineering},
	volume = {94},
	number = {2},
	issn = {1097-0207},
	url = {http://dx.doi.org/10.1002/nme.4443},
	doi = {10.1002/nme.4443},
	pages = {150--182},
	keywords = {meshfree methods, maximum entropy approximants, higher order PDE, structural vibrations, phase field, thin shells},
	year = {2013},
}