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Title: On domain symmetry and its use in homogenization

Abstract

The present study focuses on solving partial differential equations in domains exhibiting symmetries and periodic boundary conditions for the purpose of homogenization. We show in a systematic manner how the symmetry can be exploited to significantly reduce the complexity of the problem and the computational burden. This is especially relevant in inverse problems, when one needs to solve the partial differential equation (the primal problem) many times in an optimization algorithm. The main motivation of our study is inverse homogenization used to design architected composite materials with novel properties which are being fabricated at ever increasing rates thanks to recent advances in additive manufacturing. For example, one may optimize the morphology of a two-phase composite unit cell to achieve isotropic homogenized properties with maximal bulk modulus and minimal Poisson ratio. Typically, the isotropy is enforced by applying constraints to the optimization problem. However, in two dimensions, one can alternatively optimize the morphology of an equilateral triangle and then rotate and reflect the triangle to form a space filling D3 symmetric hexagonal unit cell that necessarily exhibits isotropic homogenized properties. One can further use this D3 symmetry to reduce the computational expense by performing the “unit strain” periodic boundary condition simulationsmore » on the single triangle symmetry sector rather than the six fold larger hexagon. In this paper we use group representation theory to derive the necessary periodic boundary conditions on the symmetry sectors of unit cells. The developments are done in a general setting, and specialized to the two-dimensional dihedral symmetries of the abelian D2, i.e. orthotropic, square unit cell and nonabelian D3, i.e. trigonal, hexagon unit cell. We then demonstrate how this theory can be applied by evaluating the homogenized properties of a two-phase planar composite over the triangle symmetry sector of a D3 symmetric hexagonal unit cell.« less

Authors:
 [1];  [2];  [3]
+ Show Author Affiliations
  1. Univ. of Lisbon (Portugal)
  2. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Univ. of Illinois, Urbana-Champaign, IL (United States)
  3. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1399739
Alternate Identifier(s):
OSTI ID: 1398149
Report Number(s):
LLNL-JRNL-690921
Journal ID: ISSN 0045-7825; TRN: US1702850
Grant/Contract Number:  
AC52-07NA27344
Resource Type:
Accepted Manuscript
Journal Name:
Computer Methods in Applied Mechanics and Engineering
Additional Journal Information:
Journal Volume: 320; Journal Issue: C; Journal ID: ISSN 0045-7825
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
42 ENGINEERING; 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; 36 MATERIALS SCIENCE; Symmetric domain; Finite element method; Inverse homogenization

Citation Formats

Barbarosie, Cristian A., Tortorelli, Daniel A., and Watts, Seth E. On domain symmetry and its use in homogenization. United States: N. p., 2017. Web. doi:10.1016/j.cma.2017.01.009.
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Barbarosie, Cristian A., Tortorelli, Daniel A., & Watts, Seth E. On domain symmetry and its use in homogenization. United States. https://doi.org/10.1016/j.cma.2017.01.009
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Barbarosie, Cristian A., Tortorelli, Daniel A., and Watts, Seth E. Wed . "On domain symmetry and its use in homogenization". United States. https://doi.org/10.1016/j.cma.2017.01.009. https://www.osti.gov/servlets/purl/1399739.
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@article{osti_1399739,
title = {On domain symmetry and its use in homogenization},
author = {Barbarosie, Cristian A. and Tortorelli, Daniel A. and Watts, Seth E.},
abstractNote = {The present study focuses on solving partial differential equations in domains exhibiting symmetries and periodic boundary conditions for the purpose of homogenization. We show in a systematic manner how the symmetry can be exploited to significantly reduce the complexity of the problem and the computational burden. This is especially relevant in inverse problems, when one needs to solve the partial differential equation (the primal problem) many times in an optimization algorithm. The main motivation of our study is inverse homogenization used to design architected composite materials with novel properties which are being fabricated at ever increasing rates thanks to recent advances in additive manufacturing. For example, one may optimize the morphology of a two-phase composite unit cell to achieve isotropic homogenized properties with maximal bulk modulus and minimal Poisson ratio. Typically, the isotropy is enforced by applying constraints to the optimization problem. However, in two dimensions, one can alternatively optimize the morphology of an equilateral triangle and then rotate and reflect the triangle to form a space filling D3 symmetric hexagonal unit cell that necessarily exhibits isotropic homogenized properties. One can further use this D3 symmetry to reduce the computational expense by performing the “unit strain” periodic boundary condition simulations on the single triangle symmetry sector rather than the six fold larger hexagon. In this paper we use group representation theory to derive the necessary periodic boundary conditions on the symmetry sectors of unit cells. The developments are done in a general setting, and specialized to the two-dimensional dihedral symmetries of the abelian D2, i.e. orthotropic, square unit cell and nonabelian D3, i.e. trigonal, hexagon unit cell. We then demonstrate how this theory can be applied by evaluating the homogenized properties of a two-phase planar composite over the triangle symmetry sector of a D3 symmetric hexagonal unit cell.},
doi = {10.1016/j.cma.2017.01.009},
journal = {Computer Methods in Applied Mechanics and Engineering},
number = C,
volume = 320,
place = {United States},
year = {Wed Mar 08 00:00:00 EST 2017},
month = {Wed Mar 08 00:00:00 EST 2017}
}
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    Works referencing / citing this record:

    A geometric projection method for designing three-dimensional open lattices with inverse homogenization: A geometric projection method
    journal, July 2017

    • Watts, Seth; Tortorelli, Daniel A.
    • International Journal for Numerical Methods in Engineering, Vol. 112, Issue 11
    • DOI: 10.1002/nme.5569

    A geometric projection method for designing three-dimensional open lattices with inverse homogenization: A geometric projection method
    journal, January 2018

    • Watts, Seth; Tortorelli, Daniel A.
    • International Journal for Numerical Methods in Engineering, Vol. 113, Issue 8
    • DOI: 10.1002/nme.5761
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