二维能量约束最优冷却问题的边界优化
Optimizing bounds for energy-constrained optimal cooling problems in two dimensions
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中文总结 AI 辅助
该研究针对二维能量约束最优冷却问题，通过拉格朗日对偶性构建对偶问题并结合半定规划，得到了任意区域及圆盘/环形区域冷却流的冷却效率上界，改进了现有结果。
中文摘要 AI 辅助
我们研究二维区域内不可压缩流体的最优控制问题，该区域具有冷边界和内部热源与热汇。在以无量纲佩克莱数（Péclet number）平方衡量的动能预算约束下，目标是最大化与流体温度的均方梯度成反比的冷却效率。利用拉格朗日对偶性，我们构建了一个适定的对偶问题，其解可给出最大冷却效率$\boldsymbol{\textit{E}}(\boldsymbol{\textit{Pe}})$的上界。随后，我们通过离散化得到的半定规划收敛层次结构对该对偶问题进行数值近似。我们在正方形和环形区域的最优冷却问题上展示了该计算方法，同时说明如何利用问题对称性降低计算复杂度。最后，我们为对偶问题构造容许点，以获得最优冷却效率$\boldsymbol{\textit{E}}(\boldsymbol{\textit{Pe}})$的新解析上界。具体而言，我们证明对于任意区域和热分布，$\boldsymbol{\textit{E}}(\boldsymbol{\textit{Pe}}) \boldsymbol{\text{lesssim}} \boldsymbol{\textit{Pe}}^{2}$；对于圆盘和环形区域中具有正方位平均的热源/热汇分布的冷却流，$\boldsymbol{\textit{E}}(\boldsymbol{\textit{Pe}}) \boldsymbol{\text{lesssim}} \boldsymbol{\textit{Pe}}^{2}/ \boldsymbol{\text{ln}}^{2}\boldsymbol{\textit{Pe}}$。这些结果推广并改进了已知的圆盘能量约束冷却流的效率边界。
英文摘要
We study optimal control problems for incompressible fluids in two-dimensional domains with a cold boundary and internal heat sources and sinks. Given a kinetic energy budget, measured by the square of a nondimensional Péclet number $\mathrm{Pe}$, the goal is to maximize a cooling efficiency inversely proportional to the mean square gradient of the fluid's temperature. Using Lagrange duality, we formulate a well-posed dual problem whose solution yields an upper bound on the maximum cooling efficiency $\mathcal{E}(\mathrm{Pe})$. We then numerically approximate the dual problem using a convergent hierarchy of semidefinite programs obtained via discretization. We illustrate this computational approach on optimal cooling problems in a square and in an annulus, explaining also how problem symmetries can be exploited to reduce computational complexity. Finally, we construct admissible points for the dual problem to obtain new analytical upper bounds on the optimal cooling efficiency $\mathcal{E}(\mathrm{Pe})$. Specifically, we prove that $\mathcal{E}(\mathrm{Pe}) \lesssim \mathrm{Pe}^{2}$ for arbitrary domains and heat distributions, and that $\mathcal{E}(\mathrm{Pe})\lesssim \mathrm{Pe}^{2}/ \ln^2\mathrm{Pe}$ for cooling flows in disks and annuli with heat source/sink distributions with a positive azimuthal average. These results generalize and improve known efficiency bounds for energy-constrained cooling flows in a disk.
