Two-scale topology optimization for composite plates with in-plane periodicity

Shinnosuke Nishi, Kenjiro Terada, Junji Kato, Shinji Nishiwaki, Kazuhiro Izui

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17 Citations (Scopus)


This study proposes a two-scale topology optimization method for a microstructure (an in-plane unit cell) that maximizes the macroscopic mechanical performance of composite plates. The proposed method is based on the in-plane homogenization method for a composite plate model in which the macrostructure is modeled using thick plate theory and the microstructures are three-dimensional solids. Macroscopic plate characteristics such as homogenized plate stiffnesses and generalized thermal strains are evaluated through the application of numerical plate tests applied to an in-plane unit cell. To handle large rotations of the composite plates, we employ a co-rotational formulation that facilitates working with the two-scale plate model formulated within a small strain framework. Two types of objective functions are tested in the presented optimization problems: one minimizes the macroscopic end compliance to maximize the macroscopic plate stiffness, whereas the other maximizes components of a macroscopic nodal displacement vector. Analytical sensitivities are derived based on in-plane homogenization formulae so that a gradient-based method can be employed to update the topology of in-plane unit cells. Several numerical examples are presented to demonstrate the proposed method's capability related to the design of optimal in-plane unit cells of composite plates.

Original languageEnglish
Pages (from-to)1164-1188
Number of pages25
JournalInternational Journal for Numerical Methods in Engineering
Issue number8
Publication statusPublished - 2018 Feb 24


  • analytical sensitivities
  • co-rotational formulation
  • composite plates
  • in-plane homogenization
  • multiscale topology optimization


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