M2P 2023

Shape optimization method for strength problems considering thermal expansion in a multiscale structure

  • Torisaki, Mihiro (Toyota Technological Institute)
  • Shimoda, Masatoshi (Toyota Technological Institute)

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In this study, we propose a solution to a shape optimization problem for the strength design of periodic microstructures with thermal loading in a multiscale structure. The homogenization method is used to bridge the macrostructure and the microstructure and to calculate the homogenized elastic tensor and thermoelastic tensor of the microstructures. By replacing the maximum stress value with a Kreisselmeier-Steinhauser function, the difficulty of non-differentiability of maximum stress is avoided. This problem is formulated as a distributed parameter optimization problem subject to an area constraint including the whole microstructure. The shape gradient functions for this design problems are derived using Lagrange's undetermined multiplier method, the material derivative method, and the adjoint variable method. The H1 gradient method is used to determine the unit cell shapes of the microstructure, while reducing the objective function and maintaining smooth design boundaries. Numerical examples confirm the effectiveness of the microstructure shape optimization method of multiscale structures.