TY - JOUR

T1 - On domain symmetry and its use in homogenization

AU - Barbarosie, Cristian

AU - Tortorelli, Daniel A.

AU - Watts, Seth

N1 - Funding Information:
The first author acknowledges his support from the Funda??o para a Ci?ncia e a Tecnologia, UID/MAT/04561/2013. The second and third authors acknowledge their support from the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and from DARPA's Materials with Controlled Microstructural Architecture program managed by Dr. Judah Goldwasser.
Publisher Copyright:
© 2017

PY - 2017/6/15

Y1 - 2017/6/15

N2 - The present paper 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 abelianD2, 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.

AB - The present paper 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 abelianD2, 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.

KW - Finite element method

KW - Inverse homogenization

KW - Symmetric domain

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U2 - 10.1016/j.cma.2017.01.009

DO - 10.1016/j.cma.2017.01.009

M3 - Article

AN - SCOPUS:85016992040

VL - 320

SP - 1

EP - 45

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0374-2830

ER -