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：我们推导了一个优化非均匀弹性杆弯曲刚度与扭转刚度的模型。该模型通过研究一个以周长惩罚项处理的尖锐界面形状优化问题来实现，将两种刚度均作为优化目标。随后我们构建了该优化问题的相场近似，并通过Γ-收敛证明了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相关联。