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.

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 25–28, 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

"Maximum-Entropy Approximation Schemes in
Computational Mechanics," Invite Seminar,
Sandia National Laboratories,
Livermore, CA, September 2018.

"Maximum-Entropy Approximation Schemes,"
Symposium on Meshfree and Extended/ Generalized Finite Element Methods,
PACAM XII,
Port of Spain, Trinidad, January 2012.

"Maximum-Entropy Meshfree Method,"
Invited Seminar,
Student-Run Applied & Math Seminar,
Department of Mathematics,
University of California, Davis, CA, April 2010.

"Construction of Meshfree Approximation Schemes: Use of
Information-Theoretic Variational Principles,"
Invited Seminar,
Technical University of Kaiserslautern,
Germany, September 2005

Basis functions in one dimension:
[MLS,
MAXENT,
MIN RELATIVE ENTROPY]
(For instructions, type `help mls1d', `help maxent1d',
and `help relent1d' within Matlab)

JAVA applets to visualize meshfree basis functions:
[1D,
2D,
3D]
(For instructions on using the applets, scroll down the web page in the 2D
and 3D applets)

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

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