In this paper, we propose a shape optimization method to minimize the maximum thermal stress induced in the microstructures of a multiscale structure by heat conduction and thermal expansion. The weak- coupling problem is solved by applying the temperature distribution, obtained by solving the heat conduction problem, to the thermoelastic problem to find the maximum thermal stress caused by thermal expansion. The homogenization method is used to bridge the macrostructure and the porous microstructures, in which the elastic tensor, the tensor of the coefficients of thermal expansion, the thermal conductivity tensor and the thermal transfer coefficient are homogenized. The local thermal stress in the porous structure is minimized by shape optimization. The difficulty posed by non-differentiability of the local maximum stress is avoided by introducing a Kreisselmeier-Steinhauser function. It is assumed that the macrostructure consists of multiple subregions, in which the homogenized coefficients can be independently defined. This problem is formulated as a distributed-parameter optimization problem subject to a volume constraint, including all the microstructures. The shape gradient function for this design problem is derived for each subregion using the Lagrange multiplier method, the material derivative method and the adjoint method. The H gradient method is used to determine the optimal shape of the porous unit cell while reducing the objective function and maintaining smooth design boundaries. The effectiveness of the proposed method for minimizing the microstructural thermal stress of porous structures is confirmed by the numerical examples presented.

中文翻译：

---

考虑热传导的微观形状优化，以最大限度地减少微观结构热应力


在本文中，我们提出了一种形状优化方法，以最小化由于热传导和热膨胀而在多尺度结构的微结构中引起的最大热应力。通过将解决热传导问题获得的温度分布应用于热弹性问题，以找到由热膨胀引起的最大热应力，来解决弱耦合问题。均质化方法用于连接宏观结构和多孔微观结构，其中弹性张量、热膨胀系数张量、导热率张量和传热系数均质化。通过形状优化使多孔结构中的局部热应力最小化。通过引入 Kreisselmeier-Steinhauser 函数，避免了局部最大应力的不可微分带来的困难。假设宏观结构由多个子区域组成，其中均质化系数可以独立定义。该问题被表述为受体积约束（包括所有微观结构）的分布式参数优化问题。使用拉格朗日乘子法、材料导数法和伴随法针对每个子区域导出该设计问题的形状梯度函数。 H梯度法用于确定多孔晶胞的最佳形状，同时减少目标函数并保持平滑的设计边界。所提出的最小化多孔结构微观结构热应力的方法的有效性已通过所提供的数值例子得到证实。