Cell-based max-entropy

Cell-based max-entropy (CME) basis functions are used in a Galerkin method for the solution of partial differential equations. The motivation behind this work is the construction of smooth approximants with controllable support on unstructured meshes.

To build CME basis functions we work in the well known variational scheme to obtain max-ent basis functions. The particular feature of CME is due to the nodal prior weight function which is constructed from an approximate distance function to a polygonal curve in R². More precisely, we take powers of the composition of R-functions via Boolean operations. The basis functions so constructed are nonnegative, smooth, linearly complete, and compactly-supported in a neighborring of segments that enclose each node. The smoothness is controlled by two positive integer parameters: the normalization order of the approximation of the distance function and the power to which it is raised.

In the cell-max-ent.zip file you can find matlab's scripts where the properties and mathematical foundations of the new compactly-supported approximants are shown and tested, also their use to solve two-dimensional elliptic boundary-value problems is exemplified in the Poisson's equation and in linear elasticity. This has been tested in Debian 9 on March 2018. There are some mex-files that you will need to rebuild in other OSs.

Matlab/Octave codes: cell-max-ent.zip

Also, you could download a naive implementation of the Cell-based Maximum Entropy (CME) approximants in 2D and 3D for meshless weak-form based formulations (e.g. Element-free Galerkin). https://github.com/KMountris/cme

• • K. Mountris, G. Bourantas, D. Millán, G. Joldes, K. Miller, E. Pueyo, A. Wittek. "Cell‐based Maximum Entropy Approximants for Three‐Dimensional Domains: Application in Large Strain Elastodynamics using the Meshless Total Lagrangian Explicit Dynamics Method", International Journal for Numerical Methods in Engineering, Vol. 121, Nr. 3, pp. 477–491, 2020. [abstract from publisher, arXiv preprint].

  • D. Millán, N. Sukumar and M. Arroyo. "Cell-based maximum entropy approximants". Computer Methods in Applied Mechanics and Engineering, Vol. 284, pp. 712–731, 2015. [abstract from publisher, preprint]. Isogeometric Analysis Special Issue.

  • D. Millán, A. Rosolen and M. Arroyo. "Thin shell analysis from scattered points with maximum-entropy approximants", International Journal for Numerical Methods in Engineering, Vol. 85, Nr. 6, pp. 723–751, 2011. [abstract from publisher, preprint].

  • A. Rosolen, D. Millán and M. Arroyo. "On the optimum support size in meshfree methods: a variational adaptivity approach with maximum entropy approximants", International Journal for Numerical Methods in Engineering, Vol. 82, Nr. 7, pp. 868–895, 2010. [abstract from publisher, preprint].

  • Marino Arroyo and Michael Ortiz. "Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods", International Journal for Numerical Methods in Engineering, 65:2167-2202, 2006. [abstract from publisher]