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 $\Gamma$-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.

翻译：我们推导了一种用于优化非均匀弹性杆弯曲和扭转刚度的模型。该模型通过研究一个带周长惩罚的尖锐界面形状优化问题来实现，其中将两种刚度同时作为优化目标。随后，我们构建了该优化问题的相场近似，并通过Γ-收敛证明其收敛至前述尖锐界面模型。在本文最后部分，我们采用最速下降法对相场问题的极小值进行数值逼近，并将所得最优形状与植物茎秆形态发育联系起来。