On domain symmetry and its use in homogenization

Journal Article · Wed Mar 08 00:00:00 EST 2017 · Computer Methods in Applied Mechanics and Engineering

DOI:https://doi.org/10.1016/j.cma.2017.01.009· OSTI ID:1399739

Barbarosie, Cristian A. ^[1]; Tortorelli, Daniel A. ^[2]; Watts, Seth E. ^[3]

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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)

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.

• Research Organization: Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
• Sponsoring Organization: USDOE
• Grant/Contract Number: AC52-07NA27344
• OSTI ID: 1399739
• Alternate ID(s): OSTI ID: 1398149
• Report Number(s): LLNL-JRNL-690921; TRN: US1702850
• Journal Information: Computer Methods in Applied Mechanics and Engineering, Vol. 320, Issue C; ISSN 0045-7825
• Publisher: Elsevier
• Country of Publication: United States
• Language: English

Related Subjects

42 ENGINEERING
97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE
36 MATERIALS SCIENCE
Symmetric domain
Finite element method
Inverse homogenization