NOVEL FORMULATIONS OF MICROSCOPIC BOUNDARY-VALUE PROBLEMS
IN CONTINUOUS MULTISCALE FINITE ELEMENT METHODS
Z. Chen, P. Papadopoulos, J. Li, and L. Zhang
Fin. Elem. Anal. Des., 114, pp. 78-84, (2016)
Abstract
The maximum entropy principle has proved to be a versatile tool for solving
problems in many fields. In
this paper, we extend entropy and its use in statistical physics to evaluate
contact forces in the continuum mechanics of elastic structures.
Each potential contact node of an
elastic structure discretized by
finite elements along with the normalized contact force on the node is
considered as a system and all
potential contact nodes together with their normalized contact forces are
considered as a canonical
ensemble, with the normalized contact force of each node representing the
microstate of the node. The
product of non-penetration conditions for potential contact nodes and the
normalized nodal contact
forces then act as an expectation that its value will be zero, and maximizing
the entropy under the
constraints of the expectation and the minimum potential energy principle
results in an explicit probability distribution for the normalized contact
forces that shows the relation
between contact forces and
displacements in a formulation similar to the formulation for particles
occupying microstates in statistical physics. Moreover, an iterative procedure
that solves a series of
isolated systems to find the contact
forces is presented, with a novel termination condition. Finally, some
examples are examined to verify
the correctness and efficiency of the procedure.
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