*2.1. Problem Formulation*

The mechanical behavior of the region *V* in *R* <sup>3</sup> with the boundary *∂V*, within the linear theory of elasticity, can be described by the following system of equations [11]:

$$\nabla \cdot \widetilde{\sigma} = 0, \forall \overrightarrow{\widetilde{\mathfrak{x}}} \in V^0 \tag{1}$$

$$\widetilde{\varepsilon} = \frac{1}{2} \left( \nabla \overline{\boldsymbol{u}}^{\flat} + \left( \nabla \overline{\boldsymbol{u}}^{\flat} \right)^{T} \right) , \forall \overline{\boldsymbol{x}}^{\flat} \in \mathcal{V}^{0} \tag{2}$$

$$\widetilde{\sigma} = \mathbb{C} : \widetilde{\mathfrak{e}} , \,\forall \widetilde{\mathfrak{x}} \in V^0 \tag{3}$$

$$
\overrightarrow{\boldsymbol{\mu}} = \mathbf{0}, \forall \overrightarrow{\boldsymbol{\pi}} \in \mathbb{S}\_{\text{kin}} \tag{4}
$$

$$
\widetilde{\boldsymbol{\sigma}} \cdot \overrightarrow{\boldsymbol{n}} = \overrightarrow{\boldsymbol{p}} , \forall \overrightarrow{\boldsymbol{x}} \in \mathbb{S}\_{sta} \tag{5}
$$

$$\mathcal{S}\_{\text{sta}} \cup \mathcal{S}\_{\text{kin}} = \partial V \tag{6}$$

where *V* ◦ = *V* ∪ *∂V*; *u* is the displacement vector; *σ* is the stress tensor; *ε* is the elastic strain tensor; and *C* is the stiffness tensor. *Ssta* is the surface on which static boundary conditions are specified, and *Skin* is the surface on which kinematic boundary conditions are specified (see Figure 1).

**Figure 1.** Scheme for problem formulation.

It is necessary to find a distribution of the stiffness tensor *C* in the volume *V* such that the stress invariant (in our case, the von Mises stress) reaches a minimum at the constant boundary conditions.

$$\mathcal{C} = \mathcal{C}(\stackrel{\rightarrow}{\mathfrak{x}})\_{\prime \underset{\mathfrak{x} \in V^{\prime}}{\max}} \|\tilde{\sigma}\| \to \min \tag{7}$$

Applying the design conditions, it is necessary to determine the region *Vcon*, in which the components of the tensor of elastic properties remain unchanged:

$$V\_{\rm con} \in V^0 \tag{8}$$

Let us call the region *Vcon* the constant region. So, *V* 0 in (7) can be determined as *V* ◦\*Vcon*. Adding Equation (7) to Equations (1)–(6) allows the formulating of the optimization problem for the structure.
