TY - JOUR

T1 - Quantifying force networks in particulate systems

AU - Kramár, Miroslav

AU - Goullet, Arnaud

AU - Kondic, Lou

AU - Mischaikow, Konstantin

N1 - Funding Information:
This work was partially supported by NSF - DMS-0835621 , 0915019 , 1125174 , AFOSR Grant Nos. FA9550-09-1-0148 , FA9550-10-1-0436 and DARPA (A.G., M.K., and K.M) and NSF Grant No. DMS-0835611 , and DTRA Grant No. 1-10-1-0021 (A.G. and L.K.).

PY - 2014/8/15

Y1 - 2014/8/15

N2 - We present mathematical models based on persistent homology for analyzing force distributions in particulate systems. We define three distinct chain complexes of these distributions: digital, position, and interaction, motivated by different types of data that may be available from experiments and simulations, e.g. digital images, location of the particles, and the forces between the particles, respectively. We describe how algebraic topology, in particular, homology allows one to obtain algebraic representations of the geometry captured by these complexes. For each complex we define an associated force network from which persistent homology is computed. Using numerical data obtained from discrete element simulations of a system of particles undergoing slow compression, we demonstrate how persistent homology can be used to compare the force distributions in different systems, and discuss the differences between the properties of digital, position, and interaction force networks. To conclude, we formulate well-defined measures quantifying differences between force networks corresponding to the different states of a system, and therefore allow to analyze in precise terms dynamical properties of force networks.

AB - We present mathematical models based on persistent homology for analyzing force distributions in particulate systems. We define three distinct chain complexes of these distributions: digital, position, and interaction, motivated by different types of data that may be available from experiments and simulations, e.g. digital images, location of the particles, and the forces between the particles, respectively. We describe how algebraic topology, in particular, homology allows one to obtain algebraic representations of the geometry captured by these complexes. For each complex we define an associated force network from which persistent homology is computed. Using numerical data obtained from discrete element simulations of a system of particles undergoing slow compression, we demonstrate how persistent homology can be used to compare the force distributions in different systems, and discuss the differences between the properties of digital, position, and interaction force networks. To conclude, we formulate well-defined measures quantifying differences between force networks corresponding to the different states of a system, and therefore allow to analyze in precise terms dynamical properties of force networks.

KW - Force networks

KW - Particulate systems

KW - Persistence diagram

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U2 - 10.1016/j.physd.2014.05.009

DO - 10.1016/j.physd.2014.05.009

M3 - Article

AN - SCOPUS:84903627180

SN - 0167-2789

VL - 283

SP - 37

EP - 55

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

ER -