Precisely, we consider the replacement of $J(\Gamma_D, \Gamma_N)$ in (7.27) by the function:

$$J_{\varepsilon}(\Gamma_D, \Gamma_N) := \frac{1}{2\mathrm{Vol}(\Omega)} \int_{\Omega} |u_{\Gamma_D, \Gamma_N, \varepsilon}|^2 dx,$$

where $u_{\Gamma_D, \Gamma_N, \varepsilon} \in H^1(\Omega)^3$ is characterized by the following boundary value problem:

$$\left\{
\begin{array}{ll}
-\mathrm{div}(Ae(u_{\Gamma_D, \Gamma_N, \varepsilon})) = 0 & \text{in } \Omega, \\
Ae(u_{\Gamma_D, \Gamma_N, \varepsilon})n = fn & \text{on } \Gamma_N, \\
Ae(u_{\Gamma_D, \Gamma_N, \varepsilon})n = g & \text{on } \Gamma_T, \\
Ae(u_{\Gamma_D, \Gamma_N, \varepsilon})n = 0 & \text{on } \Gamma_F, \\
Ae(u_{\Gamma_D, \Gamma_N, \varepsilon})n + h_{\Gamma_D, \varepsilon}u_{\Gamma_D, \Gamma_N, \varepsilon} = 0 & \text{on } \Gamma_D \cup \Gamma.
\end{array}
\right.$$

The function $h_{\Gamma_D, \varepsilon} : \partial\Omega \to \mathbb{R}$ is defined by:

$$\forall x \in \partial\Omega, \quad h_{\Gamma_D, \varepsilon}(x) = h\left(\frac{d_{\Gamma_D}^{\partial\Omega}(x)}{\varepsilon}\right),$$

where the geodesic signed distance function $d_{\Gamma_D}^{\partial\Omega}$ to $\Gamma_D$ is defined in Section 3.2 and the transition profile $h \in C^{\infty}(\mathbb{R})$ satisfies (3.12).

The shape derivative of $J_{\varepsilon}(\Gamma_D, \Gamma_N)$ is the subject of the following proposition, whose proof is omitted for brevity.

**Proposition 7.7.** The functional $J_{\varepsilon}(\Gamma_D, \Gamma_N)$ is shape differentiable and its derivative reads, for an arbitrary tangential deformation $\theta$ (i.e. such that $\theta \cdot n = 0$):

$$J'_{\varepsilon}(\Gamma_D, \Gamma_N)(\theta) = -\frac{1}{\varepsilon^2} \int_{\partial\Omega} h'\left(\frac{d_{\Gamma_D}^{\partial\Omega}(x)}{\varepsilon}\right) \theta(\pi_{\Sigma_D}(x)) \cdot n_{\Sigma_D}(\pi_{\Sigma}(x)) u_{\Gamma_D, \Gamma_N, \varepsilon}(x) \cdot p_{\Gamma_D, \Gamma_N, \varepsilon}(x) \,ds(x) \\ - \int_{\Sigma_N} f(x)(p_{\Gamma_D, \Gamma_N, \varepsilon}(x) \cdot n(x)) (\theta \cdot n_{\Sigma_N})(x) \,d\ell(x)$$

where the adjoint state $p_{\Gamma_D, \Gamma_N, \varepsilon} \in H^1(\Omega)^3$ is the solution to the following boundary value problem:

$$\left\{
\begin{array}{ll}
-\mathrm{div}(Ae(p_{\Gamma_D, \Gamma_N, \varepsilon})) = -\frac{u_{\Gamma_D, \Gamma_N, \varepsilon}}{\mathrm{Vol}(\Omega)} & \text{in } \Omega, \\
Ae(p_{\Gamma_D, \Gamma_N, \varepsilon})n + h_{\Gamma_D, \varepsilon}p_{\Gamma_D, \Gamma_N, \varepsilon} = 0 & \text{on } \partial\Omega.
\end{array}
\right.$$

### 7.6.3. The topological derivative

The sensitivities of the shape functional $J(\Gamma_D, \Gamma_N)$ with respect to the addition of a small surface disk $\omega_{x_0, \varepsilon}$ centered at $x_0 \in \Gamma$ to $\Gamma_D$ or $\Gamma_N$ are provided in the next result; its proof is similar to those of the results of Section 6.3, and it is omitted for brevity.

**Proposition 7.8.** Let $\Gamma_D, \Gamma_N$ be disjoint regions of the smooth boundary $\partial\Omega$ as in (7.26), and let $x_0 \in \Gamma$ be given. Then,
<table><caption>TABLE 3. Values of the parameters used in the optimal design example of the clampings and locators on the boundary of a mechanical part considered in Section 7.6.</caption><thead><tr><th>Parameter</th><th>ε</th><th>l<sub>D</sub></th><th>l<sub>N</sub></th><th>f</th><th>g</th><th>h<sub>max</sub></th><th>h<sub>min</sub></th></tr></thead><tbody><tr><td>Value</td><td>0.00001</td><td>0.0001</td><td>0.0001</td><td>-1/10</td><td>(1, 0, 0)</td><td>0.2</td><td>0.02</td></tr></tbody></table>

<figure><img src="image_32.png" /><figcaption>FIGURE 27. (a) Side views of the initial mesh $T^0$ of the mechanical structure $\Omega$ considered in the optimization example of clamping-locator regions of Section 7.6; (b) Front and back views. The pink regions are the zones $\Gamma_F$ which are not subject to optimization; the blue region is the initial locator region $\Gamma_D^0$ and the orange region in the back of the structure is that $\Gamma_T$ where the tool is operating.</figcaption></figure>

(i) The perturbed criterion $J((\Gamma_D)_{x_0,\varepsilon}, \Gamma_N)$, accounting for the addition of the surface disk $\omega_{x_0,\varepsilon} \subset \Gamma$ to $\Gamma_D$, has the following asymptotic expansion:

$$J((\Gamma_D)_{x_0,\varepsilon}, \Gamma_N) = J(\Gamma_D, \Gamma_N) + \frac{1}{|\log \varepsilon|} \frac{\pi \mu}{1 - \bar{\nu}} u_{\Gamma_D, \Gamma_N}(x_0) \cdot p_{\Gamma_D, \Gamma_N}(x_0) + o\left(\frac{1}{|\log \varepsilon|}\right) \text{ if } d = 2.$$

and

$$J((\Gamma_D)_{x_0,\varepsilon}, \Gamma_N) = J(\Gamma_D, \Gamma_N) + \varepsilon M u_{\Gamma_D, \Gamma_N}(x_0) \cdot p_{\Gamma_D, \Gamma_N}(x_0) + o(\varepsilon) \text{ if } d = 3.$$
<figure><img src="image_33.png" /><figcaption>FIGURE 28. A few intermediate shapes obtained in the optimization process of the clamping and locator regions of a mechanical structure considered in Section 7.6. The clamping and locator regions $\Gamma_N$, $\Gamma_D$ are depicted in green and blue, respectively.</figcaption></figure>

(ii) The perturbed criterion $J(\Gamma_D, (\Gamma_N)_{x_0, \epsilon})$, accounting for the addition of $\omega_{x_0, \epsilon} \subset \Gamma$ to $\Gamma_N$, has the following asymptotic expansion:

$$J(\Gamma_D, (\Gamma_N)_{x_0, \epsilon}) = J(\Gamma_D, \Gamma_N) - 2\epsilon f(x_0)n(x_0) \cdot p_{\Gamma_D, \Gamma_N}(x_0) + o(\epsilon) \quad \text{if } d = 2.$$

and

$$J(\Gamma_D, (\Gamma_N)_{x_0, \epsilon}) = J(\Gamma_D, \Gamma_N) - \pi\epsilon^2 f(x_0)n(x_0) \cdot p_{\Gamma_D, \Gamma_N}(x_0) + o(\epsilon^2) \quad \text{if } d = 3.$$

In the above formulas, the polarization tensor $M$ is defined by (6.10) and the adjoint state $p_{\Gamma_D, \Gamma_N} \in H^1(\Omega)^3$ is the solution to the following boundary value problem:

$$\begin{cases} -\operatorname{div}(Ae(p_{\Gamma_D, \Gamma_N})) = -\frac{u_{\Gamma_D, \Gamma_N}}{\operatorname{Vol}(\Omega)} & \text{in } \Omega, \\ Ae(p_{\Gamma_D, \Gamma_N})n = 0 & \text{on } \Gamma \cup \Gamma_N \cup \Gamma_T \cup \Gamma_F, \\ p_{\Gamma_D, \Gamma_N} = 0 & \text{on } \Gamma_D. \end{cases}$$

### 7.6.4. Numerical results

The initial configuration is depicted on Figure 27; it is supplied by a tetrahedral mesh $T^0$ of $\Omega$ comprising 28,000 vertices and 146,000 tetrahedra. The top and bottom regions (highlighted in pink) of $\partial\Omega$ correspond to the zones $\Gamma_F$ which are free of effort and are not subject to optimization. The region in blue indicates the initial locator area, while the zone in orange is that $\Gamma_T$ where the tool applies force. The numerical values of the parameters of the computation are supplied in Table 3. Like in Section 7.3, we solve (7.27) by alternating between the individual optimizations of the clamping and locator regions. We apply Algorithm 2 with the parameter $n_{\text{top}} = 10$, i.e. every 10 iterations, the geometric update process is replaced by a topological update step, during which a small surface disk is attached to either $\Gamma_D$ or $\Gamma_N$. This process continues until 100 iterations are completed, after which only geometric optimization updates are carried out.

This problem is challenging because many configurations could potentially stabilize the system and minimize the average displacement (7.27), making it difficult to identify an optimal configuration. Further complicating the issue, our experiments reveal that adding a new locator region could destabilize the system, leading to an
<figure><img src="image_34.png" /><figcaption>FIGURE 28. (cont.) A few intermediate shapes obtained in the optimization process of the clamp-
ing and locator regions of a mechanical structure considered in Section 7.6. The clamping and
locator regions $\Gamma_N, \Gamma_D$ are depicted in green and blue, respectively.</figcaption></figure>

increase in the objective functional, as the creation of a new clamping region might push the piece in a direction
that is not yet adequately located.

Our algorithm successfully minimizes the mean displacement $J(\Gamma_D, \Gamma_N)$ of the mechanical part $\Omega$, reducing the value of this objective from 46.1951 to approximately 0.08. A few snapshots of the optimization process are shown in Figure 28; the final design and the convergence history are reported in Figure 29. The total simulation takes around 5 hours. Interestingly, the optimized design predominantly relies on clamping regions to achieve this minimization: only small locator regions are present at the front and back of the structure, positioned opposite the direction of the load applied by the tool.

8. CONCLUSION AND PERSPECTIVES

In this article, we have explored the relatively seldom addressed issue of optimizing the shape and topol-
ogy of a region G on the boundary of a given domain Ω which supports the boundary conditions attached
to a physical partial differential equation. Elaborating on the “classical” notions of shape and topological
sensitivities involved in the optimal design of a bulk domain, we have introduced suitable notions of shape
and topological derivatives, accounting for the sensitivity of a functional with respect to either “regular”
<figure><img src="image_35.png" /><figcaption>FIGURE 29. (a) Optimized design (n = 245) in the clamping-locator example of Section 7.6;
(b) Associated convergence history.</figcaption></figure>

perturbations of the boundary of G, or “singular” perturbations, by the addition to G of a new, tiny con-
nected component. We have proposed practical calculation methods for our derivatives; these were thoroughly
presented in the model mathematical setting of the conductivity equation, and adapted to more realistic phys-
ical situations, described by the Helmholtz equation and the linear elasticity system. Several 3d numerical
examples have been addressed thanks to a shape and topology optimization algorithm combining both types
of sensitivity with an efficient body-fitted mesh evolution method for tracking the motion of a region on a
surface.

This work paves the way to extensions of various natures. From the methodological viewpoint, our topological
derivatives could be exploited with a different numerical strategy than that outlined in Algorithm 1, where they
allow to occasionally add a tiny connected component to the optimized region G in the course of a workflow
driven by a boundary variation method, based on shape derivatives. For instance, variants of the fixed point
algorithm proposed in [155, 156] or of the reaction-diffusion method in [157] would rely on these topological
derivatives in a standalone fashion.
From the perspective of applications, the ambition to optimize regions bearing the boundary conditions of a physical problem arises in multiple physical situations beyond those exemplified in this article. One of these, still concerned with mechanical structures, is surface texturing [158]. This technique creates small-scale surface patterns (e.g., dimples, grooves) to improve physical properties of the structure Ω, such as the stress concentration within, or the load bearing capabilities. For instance, in sliding bearings, surface texturing influences the tribological properties of the structure, such as friction, wear and lubrication [159, 160]. In the language of the present study, the optimal design of a surface texturing pattern could be realized by optimizing the repartition of a traction-free region and a textured region (bearing specific boundary conditions, e.g. related to its behavior with respect to friction) on the boundary of the structure Ω. Also, such optimal design issues arise in fluid mechanics, whose mathematical description involves the Stokes or the Navier-Stokes equations. One could for instance optimize the placement of the inlet and outlet regions Γ<sub>in</sub> and Γ<sub>out</sub> on the boundary of a duct Ω conveying a fluid, respectively bearing inhomogeneous Dirichlet and Neumann conditions, in order to reduce energy dissipation by managing viscous forces more effectively.

ACKNOWLEDGMENTS

The work of E.B., C.B.-P. and C.D. is partially supported by the projects ANR-24-CE40-2216 STOIQUES and ANR-
22-CE46-0006 StableProxies, financed by the French Agence Nationale de la Recherche (ANR). E.B. and C.D. also
gratefully acknowledge support from the MatAmSud project AMSUD220032. Part of this work was conducted while
C.D. was visiting the Laboratoire Jacques-Louis Lions from Sorbonne Universités, whose hospitality is gratefully
acknowledged.

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