Efficiently enhancing heat conduction through optimized distribution of a limited quantity of high thermal conductivity material is paramount in cooling electronic devices and numerous other applications. This paper introduces a target-driven all-at-once approach for PDE-constrained optimization and derives a splitting smoothed particle hydrodynamics (SPH) method for optimizing the distribution of thermal conductivity in heat conduction problems. In this method, the optimization iteration of the system is split into several easily addressed steps. A targeting step is employed to progressively enforce the direct target, which potentially leads to increased PDE residuals. Then, these residuals are recovered through an evolution step of the design variable. After this, a PDE solution step is carried out to further decrease the PDE residuals, and the system is ready for the next iteration. Unlike the simulation-based approaches, the present method does not rely on the adjoint state equation and converged state variable field in each iteration, and the optimization process is significantly simplified and accelerated. With the utilization of an implicit SPH splitting operator and a general numerical regularization formulation, the information propagation is further accelerated and the numerical stability is greatly enhanced. Typical examples of heat conduction optimization demonstrate that the current method yields optimal results comparable to previous methods and exhibits considerable computational efficiency. Moreover, the optimal results feature more moderate extreme values, which offers distinct advantages for the easier selection of appropriate material with high thermal conductivity.

中文翻译：

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目标驱动的导热系数分布分割SPH优化

通过有限数量的高导热材料的优化分布来有效增强热传导对于冷却电子设备和许多其他应用至关重要。本文介绍了一种用于 PDE 约束优化的目标驱动一次性方法，并推导了一种用于优化热传导问题中热导率分布的分裂平滑粒子流体动力学 (SPH) 方法。在该方法中，系统的优化迭代被分成几个容易解决的步骤。采用目标步骤来逐步强制执行直接目标，这可能会导致 PDE 残差增加。然后，通过设计变量的演化步骤恢复这些残差。此后，执行 PDE 求解步骤以进一步减少 PDE 残差，系统为下一次迭代做好准备。与基于仿真的方法不同，本方法在每次迭代中不依赖伴随状态方程和收敛状态变量场，并且优化过程显着简化和加速。通过使用隐式SPH分裂算子和通用数值正则化公式，信息传播进一步加速，数值稳定性大大增强。热传导优化的典型例子表明，当前的方法可以产生与以前的方法相当的最佳结果，并且具有相当大的计算效率。此外，最佳结果具有更温和的极值，这为更轻松地选择具有高导热率的合适材料提供了明显的优势。