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It is known that mechanical behavior of a composite structure is strongly influenced by not only the shape and properties of the macroscopic region but also the shape and properties of the micro region. In this study, we propose a multi-scale shape optimization method to design the shape of a periodic microstructure. The method consists of two-step optimization. At first, the material properties of a macrostructure are optimized using a size-optimization technique, where the components of the liner elastic tensor are set as the design variables. Objective function and the constraints are set according to design purpose. Next, the shapes of periodic-microstructures are determined to identity the material properties of the macrostructures obtained in the first step, where the inverse homogenization method and the H 1 gradient method are used. The shape optimization problem of a microstructure is formulated as a distributed-parameter optimization problem, in which the mass is set as an objective functional and the shape variation field is set as a design variable, as the constraints are the target macroscopic material properties and the homogenization equation. Through numerical calculation examples, we confirmed the effectiveness of the proposed method based on the inverse homogenization method and the H 1 gradient method.

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Multi-scale shape optimization method for designing a macrostructure with periodic-microstructures using inverse homogenization method and H&lt;sup&gt;1&lt;/sup&gt; gradient method