Introduction
Home
CV
Publications
Directions

Research
NEM X-FEM MAXENT
POLYFEM
QUANTUM

Teaching
ECI212A ENG104

Programming
C++ C++ FAQ
STL Python
PS CTAN CPAN

E-Journals
Linux RPMs
Websters
Dilbert

Societies
USACM IUTAM

Sports
Running [Marathons]
Chess TT NBA

Weather
Davis

MAXIMUM-ENTROPY MESHFREE METHOD IN MECHANICS COMPUTATIONS

### Maximum-entropy computations

(Research-work supported by NSF Grants CMMI-0626481 and CMMI-0826513)

The maximum entropy principle (Shannon (1948), Jaynes (1957)) provides a means to obtain least-biased statistical inference when insufficient information is available. Stemming from my prior work (NEM, polygonal FEM), the principle of maximum entropy was used to construct basis functions. The basis functions are viewed as a discrete probability distribution, and for n distinct nodes, the linear reproducing (precision) conditions are the constraints. For n > 3, the constraints represent an under-determined linear system. The maximum entropy variational principle is invoked, which leads to a unique solution with an exponential form for the basis functions. The maximum entropy approximant is valid for any point within the convex hull of the set of nodes (Sukumar, 2004), with interior nodal basis functions vanishing on the boundary of the convex hull (Fig. 1b). The use of variational principles (finite elements, conjugate gradient methods, graphical models, dynamic programming, statistical mechanics) is also appealing in data approximation (for example, Kriging, thin-plate splines, RBFs, MLS, Laplace, etc.).

In an independent study, Arroyo and Ortiz (2006) have shown the promise of local maximum entropy (convex) approximation schemes in a meshfree Galerkin method. Furthermore, links to convex analysis, extension to higher-order approximations, and the key properties of convex approximants are established. A general prescription of locally- and globally-supported convex approximation schemes can be derived using the Kullback-Leibler distance or directed divergence (principle of minimum relative entropy), which is presented in Sukumar (2005) and further elaborated in Sukumar and Wright (2007). The above studies share common elements with the Ph.D. thesis-research of Gupta (2003). For background on probability theory and Bayesian inductive inference, the books by Jaynes and Sivia are highly recommended.

Reports and Publications

• R. Silva-Valenzuela, A. Ortiz-Bernardin, N. Sukumar, E. Artioli and N. Hitschfeld-Kahler (2020), "A Nodal Integration Scheme for Meshfree Galerkin Methods using the Virtual Element Decomposition," International Journal for Numerical Methods in Engineering, Vol. 121, Number 10, pp. 2174–2205.
• A. Ortiz-Bernardin, A. Russo and N. Sukumar (2017), "Consistent and Stable Meshfree Galerkin Methods using the Virtual Element Decomposition," International Journal for Numerical Methods in Engineering, Vol. 112, Number 7, pp. 655–684.
• A. Ortiz-Bernardin, M. A. Puso and N. Sukumar (2015), "Improved Robustness for Nearly-Incompressible Large Deformation Meshfree Simulations on Delaunay Tessellations," Comput. Meth. Appl. Mech. Engrg., Vol. 293, pp. 348–374.
• D. Millán, N. Sukumar and M. Arroyo (2015), "Cell-Based Maximum-Entropy Approximants," Comput. Meth. Appl. Mech. Engrg., Vol. 284, pp. 712–731.
• N. Sukumar (2013), "Quadratic Maximum-Entropy Serendipity Shape Functions for Arbitrary Planar Polygons," Comput. Meth. Appl. Mech. Engrg., Vol. 263, pp. 27–41.
• F. Greco and N. Sukumar (2013), "Derivatives of Maximum-Entropy Basis Functions on the Boundary: Theory and Computations," International Journal for Numerical Methods in Engineering, Vol. 94, Number 12, pp. 1123–1149.
• G. Quaranta, S. K. Kunnath and N. Sukumar (2012), "Maximum-Entropy Meshfree Method for Nonlinear Static Analysis of Planar Reinforced Concrete Structures," Engineering Structures, Vol. 42, pp. 179–189.
• A. Ortiz, M. A. Puso and N. Sukumar (2011), "Maximum-Entropy Meshfree Method for Incompressible Media Problems," Finite Element in Analysis and Design, Vol. 47, Number 6, pp. 572–585.
• A. Ortiz, M. A. Puso and N. Sukumar (2010), "Maximum-Entropy Meshfree Method for Compressible and Near-Incompressible Elasticity," Comput. Meth. Appl. Mech. Engrg., Vol. 199, Number 2528, pp. 1859–1871.
• L. L. Yaw, N. Sukumar and S. K. Kunnath (2009), "Meshfree Co-Rotational Formulation for Two-Dimensional Continua," International Journal for Numerical Methods in Engineering, Vol. 79, Number 8, pp. 979–1003.
• K. Hormann and N. Sukumar (2008), "Maximum Entropy Coordinates for Arbitrary Polytopes," Computer Graphics Forum, Vol. 27, Number 5, pp. 1513–1520. Proceedings of SGP 2008.
• N. Sukumar and R. J-B Wets (2007), "Deriving the Continuity of Maximum-Entropy Basis Functions via Variational Analysis," SIAM Journal of Optimization, Vol. 18, Number 3, pp. 914–925. or Journal
• N. Sukumar and R. W. Wright (2007), "Overview and Construction of Meshfree Basis Functions: From Moving Least Squares to Entropy Approximants," International Journal for Numerical Methods in Engineering, Vol. 70, Number 2, pp. 181–205.
• N. Sukumar (September 2006), "Where Do We Stand on Meshfree Approximation Schemes?" in Online Blog on Meshfree Methods or or
• N. Sukumar and E. A. Malsch (2006), "Recent Advances in the Construction of Polygonal Finite Element Interpolants," Archives of Computational Methods in Engineering, Vol. 13, Number 1, pp. 129–163. (proof)
• N. Sukumar (2005), "Maximum Entropy Approximation," in Proceedings of the 25th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Eds. K. H. Knuth, A. E. Abbas, R. D. Morris and J. P. Castle, AIP Conference Proceedings, Vol. 803, Number 1, pp. 337–344.
• N. Sukumar (2004), "Construction of Polygonal Interpolants: A Maximum Entropy Approach," International Journal for Numerical Methods in Engineering, Vol. 61, Number 12, pp. 2159–2181

Presentations

Software

Teaching

Disclaimer: Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation