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

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.



@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},