Optimal Design of Magnetoelastic Rods
It is well known from the mechanics of materials and structures that softness and slenderness are two key features that permit an elastic body to attain a large range of motion under moderate external forcing. The first feature, a material (i.e., local) property, is related to the small growth of the strain-energy density at a point as a function of the deformation tensor at the same point. Typical examples of soft materials are polymers, such as rubbers and gels, in contrast to hard materials such as wood or steel. The second feature, a geometric property, is related to the degeneracy, for thin domains, of the best constants in Korn-type inequalities which bound the displacement field in terms of the deformation field. Thus, softness allows strain to be large with moderate involvement of energy, whereas slenderness allows displacement to be large with moderate involvement of strain.
\[\begin{cases} -\vartheta^{\prime \prime}-\vec{h} \cdot D \vec{m}(\alpha+\vartheta)=0 & \text{in } (0,1) \\ \vartheta(0)=0 \\ \vartheta^{\prime}(1)=0 \end{cases}\]where
\[D \vec{m}(v)=(-\sin v, \cos v), \quad \text{for all } v \in \mathbb{R}\]is the derivative of the function $\vec{m}: \mathbb{R} \rightarrow \mathbb{R}^{2}$ defined by $\vec{m}(v)=(\cos v, \sin v)$. Moreover, such a minimizer is unique if:
\[|\vec{h}| < c_{p}^{-2}\]where $c_{p}=2/\pi$ is the best constant in the Poincaré-type inequality:
\[\int_{0}^{1} v^{2} \leq c_{p}^{2} \int_{0}^{1}\left(v^{\prime}\right)^{2} \quad \text{for all } v \in C^{1}([0,1]) \text{ such that } v(0)=0\]| Thus, if $ | \vec{h} | < c_{p}^{-2}$, then the state equation defines a solution operator: |
which maps the control $(\vec{h}, \alpha)$ into the state $\vartheta$.
Toy Problem
Given a target shape $\bar{\vartheta} \in C^{2}((0,1)) \cap C^{1}([0,1])$ such that:
\[\bar{\vartheta}(0)=0, \quad \bar{\vartheta}^{\prime}(1)=0\]find a control $(\vec{h}, \alpha)$ such that the equilibrium equation has a unique solution $\vartheta$, and this solution is as close as possible to $\bar{\vartheta}$.
It is not difficult to check that a solution exists regardless of the particular norm selected if there is a pair $(\vec{h}, \alpha)$ which nullifies the residual:
\[r := -\bar{\vartheta}^{\prime \prime} - \vec{h} \cdot \vec{m}(\alpha+\bar{\vartheta})\]with $\vec{h}$ satisfying the condition. Precisely, assume that:
\[M = \max_{s \in [0,1]} |\bar{\vartheta}^{\prime \prime}(s)| < c_{p}^{-2}\]For $\vec{h} = (M, 0)$ and $\alpha(s) = \arcsin\left(\frac{\bar{\vartheta}^{\prime \prime}}{M}\right) - \bar{\vartheta}(s)$, we have $r=0$. Thus $\bar{\vartheta}$ is a unique solution.