Realizations of isostatic material frameworks

Mahdi Sadjadi; Varda F. Hagh; Mingyu Kang; Meera Sitharam; Robert Connelly; Steven J. Gortler; Louis Theran; Miranda Holmes-Cerfon; Michael F. Thorpe

doi:10.1002/pssb.202000555

Realizations of isostatic material frameworks

Mahdi Sadjadi, Varda F. Hagh, Mingyu Kang, Meera Sitharam, Robert Connelly, Steven J. Gortler, Louis Theran, Miranda Holmes-Cerfon, Michael F. Thorpe

Research output: Contribution to journal › Article › peer-review

Abstract

This article studies the set of equivalent realizations of isostatic frameworks and algorithms for finding all such realizations. It is shown that an isostatic framework has an even number of equivalent realizations that preserve edge lengths and connectivity. The complete set of equivalent realizations for a toy framework with pinned boundary in two dimensions is enumerated and the impact of boundary length on the emergence of these realizations is studied. To ameliorate the computational complexity of finding a solution to a large multivariate quadratic system corresponding to the constraints, alternative methods—based on constraint reduction and distance-based covering map or Cayley parameterization of the search space—are presented. The application of these methods is studied on atomic clusters, a model of 2D glasses and jamming.

Original language	English
Article number	2000555
Number of pages	17
Journal	Physica Status Solidi. B
Volume	258
Issue number	9
Early online date	2 Mar 2021
DOIs	https://doi.org/10.1002/pssb.202000555
Publication status	Published - 24 Sept 2021

Keywords

Glasses
Isostatic
Jamming
Networks
Tunneling

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Sadjadi_2021_PSSB_Realizations_AAM
Copyright © 2021 Wiley-VCH GmbH. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1002/pssb.202000555
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https://arxiv.org/abs/2102.06295Licence: CC BY

Cite this

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title = "Realizations of isostatic material frameworks",

abstract = "This article studies the set of equivalent realizations of isostatic frameworks and algorithms for finding all such realizations. It is shown that an isostatic framework has an even number of equivalent realizations that preserve edge lengths and connectivity. The complete set of equivalent realizations for a toy framework with pinned boundary in two dimensions is enumerated and the impact of boundary length on the emergence of these realizations is studied. To ameliorate the computational complexity of finding a solution to a large multivariate quadratic system corresponding to the constraints, alternative methods—based on constraint reduction and distance-based covering map or Cayley parameterization of the search space—are presented. The application of these methods is studied on atomic clusters, a model of 2D glasses and jamming.",

keywords = "Glasses, Isostatic, Jamming, Networks, Tunneling",

author = "Mahdi Sadjadi and Hagh, {Varda F.} and Mingyu Kang and Meera Sitharam and Robert Connelly and Gortler, {Steven J.} and Louis Theran and Miranda Holmes-Cerfon and Thorpe, {Michael F.}",

note = "The authors would like to thank financial support though NSF grant no. DMS 1564468 (Connelly, Gortler, Holmes‐Cerfon, Sitharam, Thorpe). ",

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AU - Sadjadi, Mahdi

AU - Hagh, Varda F.

AU - Kang, Mingyu

AU - Sitharam, Meera

AU - Connelly, Robert

AU - Gortler, Steven J.

AU - Theran, Louis

AU - Holmes-Cerfon, Miranda

AU - Thorpe, Michael F.

N1 - The authors would like to thank financial support though NSF grant no. DMS 1564468 (Connelly, Gortler, Holmes‐Cerfon, Sitharam, Thorpe).

PY - 2021/9/24

Y1 - 2021/9/24

N2 - This article studies the set of equivalent realizations of isostatic frameworks and algorithms for finding all such realizations. It is shown that an isostatic framework has an even number of equivalent realizations that preserve edge lengths and connectivity. The complete set of equivalent realizations for a toy framework with pinned boundary in two dimensions is enumerated and the impact of boundary length on the emergence of these realizations is studied. To ameliorate the computational complexity of finding a solution to a large multivariate quadratic system corresponding to the constraints, alternative methods—based on constraint reduction and distance-based covering map or Cayley parameterization of the search space—are presented. The application of these methods is studied on atomic clusters, a model of 2D glasses and jamming.

AB - This article studies the set of equivalent realizations of isostatic frameworks and algorithms for finding all such realizations. It is shown that an isostatic framework has an even number of equivalent realizations that preserve edge lengths and connectivity. The complete set of equivalent realizations for a toy framework with pinned boundary in two dimensions is enumerated and the impact of boundary length on the emergence of these realizations is studied. To ameliorate the computational complexity of finding a solution to a large multivariate quadratic system corresponding to the constraints, alternative methods—based on constraint reduction and distance-based covering map or Cayley parameterization of the search space—are presented. The application of these methods is studied on atomic clusters, a model of 2D glasses and jamming.

KW - Glasses

KW - Isostatic

KW - Jamming

KW - Networks

KW - Tunneling

U2 - 10.1002/pssb.202000555

DO - 10.1002/pssb.202000555

M3 - Article

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VL - 258

JO - Physica Status Solidi. B

JF - Physica Status Solidi. B

IS - 9

M1 - 2000555

ER -

Realizations of isostatic material frameworks

Abstract

Keywords

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