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Filters on satake graphs

This is an extension of a previous work* dealing with the homogenization of quasi-static granular mechanics based on a graph-theoretical treatment of the type pioneered by M. Satake. In the previous work, an attempt was made to derive continuum-level gradients from more or less standard polynomial representations of the kinematics of the underlying graph (or network) of grain centroids and contacts.

The present work adopts a more general approach based on the notion that homogenization of discrete or other heterogeneous systems are essentially filtering operations. Typically, these involve low-pass filters which eliminate small spatial or temporal scales (or large spatial or temporal frequencies) while ideally conserving energy (Hill-Mandel condition).

We focus attention on a class of linear filters on granular kinematics involving homogenization kernels which map simplicial complexes (points, lines, planes and volumes) on granular graphs into continuum kinematic fields endowed with conjugate stress and hyperstress. Specifically, we consider kernels derived by spectral methods from orthogonal polynomials or filtered Fourier transforms. As a prelude to the actual application to granular micro-mechanics, some simple examples are presented involving artificial particle motions on on spatially graphs, with randomness in both.
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*J. Goddard, “From Granular Matter to Generalized Continuum”, in Capriz, P. and Mariano, P. (eds.), Mathematical Models of Granular Matter. Lecture Notes in Applied Mathematics, Vol. 1937, 1-20, Springer, 2008.

Author(s):

Joe Goddard    
University of California, San Diego
United States