不可伸长杆的多材料形状优化的相场模型
Phase field model for multi-material shape optimization of inextensible rods
AI总结：
本文提出一种优化非均匀弹性杆弯曲与扭转刚度的相场模型，通过 $Γ$-收敛证明其收敛至清晰界面模型，并利用最速下降法数值求解，将最优形状与植物茎形态发育相联系。
摘要：
我们推导了一个优化非均匀弹性杆的弯曲刚度和扭转刚度的模型。这是通过研究一个带有周长惩罚的清晰界面形状优化问题来实现的，该问题将两种刚度都作为目标。然后我们提出该优化问题的相场近似，并通过 $Γ$-收敛证明其收敛至上述清晰界面模型。在最后部分，我们使用最速下降法数值逼近相场问题的极小值解，并将所得最优形状与植物茎形态的发育联系起来。
英文摘要：
We derive a model for the optimization of the bending and torsional rigidities of non-homogeneous elastic rods. This is achieved by studying a sharp interface shape optimization problem with perimeter penalization, that treats both rigidities as objectives. We then formulate a phase field approximation of the optimization problem and show the convergence to the aforementioned sharp interface model via $Γ$-convergence. In the final part of this work we numerically approximate minimizers of the phase field problem by using a steepest descent approach and relate the resulting optimal shapes to the development of the morphology of plant stems.
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