A consistent relaxation of optimal design problems for coupling shape and topological derivatives

Abstract

In this article, we introduce and analyze a general procedure for approximating a ‘black and white’ shape and topology optimization problem with a density optimization problem, allowing for the presence of ‘grayscale’ regions. Our construction relies on a regularizing operator for smearing the characteristic functions involved in the exact optimization problem, and on an interpolation scheme, which endows the intermediate density regions with fictitious material properties. Under mild hypotheses on the smoothing operator and on the interpolation scheme, we prove that the features of the approximate density optimization problem (material properties, objective function, etc.) converge to their exact counterparts as the smoothing parameter vanishes. In particular, the gradient of the approximate objective functional with respect to the density function converges to either the shape or the topological derivative of the exact objective. These results shed new light on the connections between these two different notions of sensitivities for functions of the domain, and they give rise to different numerical algorithms which are illustrated by several experiments.

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Acknowledgements

C.D. was partially supported by the ANR OptiForm. A.F. has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 320815 (ERC Advanced Grant Project “Advanced tools for computational design of engineering materials” COMP-DES-MAT).

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Correspondence to Charles Dapogny.

Sketch of the proof of Theorem 4

Sketch of the proof of Theorem 4

The proof is a little technical, but rests on classical arguments, and notably Nirenberg’s technique of difference quotients [39], as exposed, e.g. in [22] (see also the discussion in [27], §4.8). We therefore limit ourselves to sketches of proofs.

Let us start with a useful result about the characterization of \(W^{1,p}\) spaces. Let \(1<p\le \infty \), and \( 1 \le p^\prime < \infty \) be such that \(\frac{1}{p} + \frac{1}{p^\prime } = 1\). For a function \(\varphi \in L^p(D)\), a subset \(V \Subset D\), and a vector \(h \in \mathbb {R}^d\) with \(|h |< \text { dist}(V,\partial D)\), we define the difference quotient \(D_h \varphi \in L^p(V)\) as:

$$\begin{aligned} D_h \varphi (x) = \frac{\tau _h\varphi (x) - \varphi (x)}{|h |}, \text { a.e. } x \in V, \text { where }\tau _h\varphi (x) := \varphi (x+h). \end{aligned}$$

In the above context, it holds that, if V is convex and \(\varphi \in W^{1,p}(D)\):

$$\begin{aligned} ||D_h \varphi ||_{L^p(V)} \le ||\nabla \varphi ||_{L^p(D)}. \end{aligned}$$
(63)

Then (see [22], Prop. 9.3):

Proposition 10

Let \(\varphi \in L^p(D)\); the following assertions are equivalent:

  1. 1.

    \(\varphi \) belongs to \(W^{1,p}(D)\).

  2. 2.

    There exists a constant \(C >0\) such that:

    $$\begin{aligned} \left|\int _D{\varphi \frac{\partial \psi }{\partial x_i} \,dx} \right|\le C ||\psi ||_{L^{p^\prime }(D)}, \text { for any } v \in {{\mathcal {C}}}^\infty _c(D),\,\, i=1,\ldots ,d. \end{aligned}$$
  3. 3.

    There exists a constant \(C >0\) such that, for any open subset \(V \Subset D\),

    $$\begin{aligned} \limsup _{h \rightarrow 0}{ ||D_h \varphi ||_{L^p(V)}} \le C. \end{aligned}$$

Moreover, the smallest constant satisfying points (2) and (3) is \(C = ||\nabla \varphi ||_{L^p(D)^d}\).

Let us pass to the proof of Theorem 4 properly speaking. Throughout this appendix, a shape \(\varOmega \in {{\mathcal {U}}}_{ad}\) is fixed; \(L_\varepsilon \) stands for either one of the operators \(L_\varepsilon ^{\text { conv}}\) or \(L_\varepsilon ^{\text { ell}}\) constructed in Sect. 3.2, and we rely on the shorthands:

$$\begin{aligned} \gamma _\varepsilon \equiv \gamma _{\varOmega ,\varepsilon }, \text { and } u_\varepsilon \equiv u_{\varOmega ,\varepsilon }. \end{aligned}$$

Also, C consistently denotes a positive constant, that may change from one line to the other, but is in any event independent of \(\varepsilon \) and the parameter h (to be introduced).

Proof of (i):

Without loss of generality, we assume that \(U \Subset \varOmega ^0\), and we introduce two other subsets VW of D such that \(U \Subset V \Subset W \Subset \varOmega ^0\). Let \(\chi \) be a smooth cutoff function such that \( \chi \equiv 1\) on U, and \(\chi \equiv 0\) on V. Our aim is to prove that:

$$\begin{aligned} v_\varepsilon {\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}} v_\varOmega \text { strongly in } H^m(D), \end{aligned}$$
(64)

for any \(m \ge 1\), where we have defined \(v_\varepsilon = \chi u_\varepsilon \) and \(v_\varOmega = \chi u_\varOmega \). Note that we have already proved in Proposition 5 that (64) holds for \(m=1\).

Using test functions of the form \(\chi \varphi \), for arbitrary \(\varphi \in H^1(D)\), easy manipulations lead to the following variational formulations for \(v_\varepsilon \) and \(v_\varOmega \):

$$\begin{aligned} \forall \varphi \in H^1(D), \, \, \int _D{\gamma _\varepsilon \nabla v_\varepsilon \cdot \nabla \varphi \,dx} = \int _D{f_\varepsilon \varphi \,dx} + \int _D{h_\varepsilon \cdot \nabla \varphi \,dx}, \text { and } \end{aligned}$$
(65)
$$\begin{aligned} \forall \varphi \in H^1(D), \, \, \int _D{\gamma _\varOmega \nabla v_\varOmega \cdot \nabla \varphi \,dx} = \int _D{f_\varOmega \varphi \,dx} + \int _D{h_\varOmega \cdot \nabla \varphi \,dx}, \end{aligned}$$
(66)

where \(f_\varepsilon , f_\varOmega \in L^2(D)\) and \(h_\varepsilon , h_\varOmega \in H^1(D)^d\) are defined by:

$$\begin{aligned} f_\varepsilon = -\gamma _\varepsilon \nabla u_\varepsilon \cdot \nabla \chi , \,\, f_\varOmega = -\gamma _\varOmega \nabla u_\varOmega \cdot \nabla \chi , \,\, h_\varepsilon = \gamma _\varepsilon u_\varepsilon \nabla \chi , \text { and } h_\varOmega = \gamma _\varOmega u_\varOmega \nabla \chi . \end{aligned}$$
(67)

Subtracting (66) to (65) yields the following variational formulation for \(w_\varepsilon := v_\varepsilon - v_\varOmega \): for any \( \varphi \in H^1(D)\),

$$\begin{aligned}&\int _D{\gamma _\varepsilon \nabla w_\varepsilon \cdot \nabla \varphi \,dx} = - \int _D{(\gamma _\varepsilon - \gamma _\varOmega ) \nabla v_\varOmega \cdot \nabla \varphi \,dx} \\&\quad + \int _D{(f_\varepsilon - f_\varOmega ) \varphi \,dx} + \int _D{(h_\varepsilon - h_\varOmega ) \cdot \nabla \varphi \,dx}. \end{aligned}$$

Now, for any vector \(h \in \mathbb {R}^d\) with sufficiently small length, let us insert \(\varphi = D_{-h} D_h w_\varepsilon \) in this variational formulation:

$$\begin{aligned}&\int _D{\gamma _\varepsilon |\nabla D_h w_\varepsilon |^2 \,dx} = -\int _D{D_h\gamma _\varepsilon \,\tau _h(\nabla w_\varepsilon ) \cdot \nabla D_h w_\varepsilon \,dx}\nonumber \\&\quad - \int _D{\tau _h(\gamma _\varepsilon - \gamma _\varOmega ) \nabla D_h v_\varOmega \cdot \nabla D_h w_\varepsilon \,dx} \nonumber \\&\quad - \int _D{D_h(\gamma _\varepsilon - \gamma _\varOmega ) \nabla v_\varOmega \cdot \nabla D_h w_\varepsilon \,dx} + \int _D{(f_\varepsilon - f_\varOmega )D_{-h} D_h w_\varepsilon \,dx} \nonumber \\&\quad + \int _D{D_h (h_\varepsilon - h_\varOmega ) \cdot \nabla D_h w_\varepsilon \,dx}. \end{aligned}$$
(68)

To achieve the last identity, we have used the ‘discrete integration by parts’:

$$\begin{aligned} \int _D{w(D_hz) \,dx} = -\int _D{(D_{-h}w)z \,dx}, \end{aligned}$$

for arbitrary functions \(w,z \in L^2(D)\) vanishing outside W (which just follows from a change of variables in the corresponding integrals), as well as the discrete product rule:

$$\begin{aligned} D_h(uv) = D_hu \,\tau _h v + u D_h v. \end{aligned}$$

Using Hölder’s inequality with \(\frac{1}{p} + \frac{1}{q} + \frac{1}{2} =1\) and Proposition 10, we obtain:

$$\begin{aligned}&\limsup \limits _{|h |\rightarrow 0}{||\nabla D_h w_\varepsilon ||_{L^2(W)^d}} \le ||\nabla w_\varepsilon ||_{L^q(W)^d} \, \limsup \limits _{|h |\rightarrow 0}{||\nabla D_h \gamma _\varepsilon ||_{L^p(W)^d}}\nonumber \\&\quad + \,||\gamma _\varepsilon - \gamma _\varOmega ||_{L^p(W)} \,\limsup \limits _{|h |\rightarrow 0}{||\nabla D_h v_\varOmega ||_{L^q(W)^d}}\nonumber \\&\quad +\, ||\nabla v_\varOmega ||_{L^q(W)^d} \,\limsup \limits _{|h |\rightarrow 0}{||\nabla D_h (\gamma _\varepsilon - \gamma _\varOmega ) ||_{L^p(W)^d}} + ||(f_\varepsilon - f_\varOmega ) ||_{L^2(W)}\nonumber \\&\quad + \limsup \limits _{|h |\rightarrow 0}{||D_h (h_\varepsilon - h_\varOmega ) ||_{L^2(W)^d}}, \end{aligned}$$
(69)

and we now have to prove that each term in the right-hand side of this inequality tends to 0 as \(\varepsilon \rightarrow 0\), which follows quite easily from repeated uses of Proposition 10 together with the convergences of Proposition 4 for the conductivity \(\gamma _\varepsilon \) and the convergence results for \(w_\varepsilon \) expressed in Proposition 5. We omit the details, referring to the proof of (ii), where similar ones (yet in any point more involved) are handled.

As a result, one has:

$$\begin{aligned} \limsup \limits _{|h |\rightarrow 0}{||\nabla D_h w_\varepsilon ||_{L^2(W)^d}} {\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}} 0, \end{aligned}$$

which implies, from Proposition 10,

$$\begin{aligned} ||\nabla w_\varepsilon ||_{H^1(W)^d} {\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}} 0. \end{aligned}$$

Therefore, (64) holds for \(m =2\). The case \(m>2\) is obtained by iterating the previous argument. \(\square \)

Proof of (ii):

Since \(\varGamma \) is compact, it is enough to prove that the estimates (32) hold in an open neighborhood U of any arbitrary point \(x_0 \in \varGamma \); namely, we introduce two other subset VW of D such that \(U \Subset V \Subset W \Subset D\); let \(\chi \) be a smooth cutoff function, which equals 1 on U et 0 on \(D \setminus \overline{V}\); we aim to prove that:

$$\begin{aligned}&\frac{\partial v_\varepsilon }{\partial \tau } {\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}} \frac{\partial v_\varOmega }{\partial \tau } \text { in } H^{1}(W), \text { for any tangential vector field } \tau :\varGamma \rightarrow \mathbb {R}^d, \text { and }\nonumber \\ \end{aligned}$$
(70)
$$\begin{aligned}&\gamma _{\varOmega ,\varepsilon }\frac{\partial v_\varepsilon }{\partial n} {\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}} \gamma _\varOmega \frac{\partial v_\varOmega }{\partial n} \text { in } H^{1}(W) , \end{aligned}$$
(71)

where, again \(v_\varepsilon = \chi u_\varepsilon \) and \(v_\varOmega = \chi u_\varOmega \).

Without loss of generality, we assume that \(\varGamma \) is flat in U, that is: \(\varOmega \cap U = \left\{ x \in U, \,\, x_d < 0 \right\} \) is a piece of the lower half-space. The general case is recovered from this one by a standard argument of local charts for the smooth boundary \(\varGamma \). To prove (70), it is therefore enough to consider the case \(\tau = e_i\), \(i=1,\ldots ,d-1\), the \(i^{\text { th}}\)-vector of the canonical basis \((e_1,\ldots ,e_d)\) of \(\mathbb {R}^d\), which we now do.

Introducing \(f_\varepsilon , f_\varOmega \in L^2(D)\), \(h_\varepsilon , h_\varOmega \in H^1(D)^d\) as in (67), and \(w_\varepsilon = v_\varepsilon - v_\varOmega \), we now follow the exact same trail as that leading to (68) in the proof of (i), using only vectors h of the form \(h=h e_i\), with small enough \(h>0\):

$$\begin{aligned}&\int _D{\gamma _\varepsilon |\nabla D_h w_\varepsilon |^2 \,dx} = -\int _D{D_h\gamma _\varepsilon \, \tau _h(\nabla w_\varepsilon ) \cdot \nabla D_h w_\varepsilon \,dx}\nonumber \\&\quad - \int _D{\tau _h(\gamma _\varepsilon - \gamma _\varOmega ) \nabla D_h v_\varOmega \cdot \nabla D_h w_\varepsilon \,dx}\nonumber \\&\quad - \int _D{D_h(\gamma _\varepsilon - \gamma _\varOmega ) \nabla v_\varOmega \cdot \nabla D_h w_\varepsilon \,dx} + \int _D{(f_\varepsilon - f_\varOmega )D_{-h} D_h w_\varepsilon \,dx}\nonumber \\&\quad + \int _D{D_h (h_\varepsilon - h_\varOmega ) \cdot \nabla D_h w_\varepsilon \,dx} \end{aligned}$$
(72)

We now estimate each of the five terms in the right-hand side of (72):

  • Using Hölder’s inequality with \(\frac{1}{p} + \frac{1}{q} + \frac{1}{2} =1\), the first term is controlled as:

    $$\begin{aligned} \left|\int _D{D_h\gamma _\varepsilon \nabla w_\varepsilon \cdot \nabla D_h w_\varepsilon dx} \right|\le C ||\nabla v_\varepsilon -\nabla v_\varOmega ||_{L^q(W)^d} \left|\left|\frac{\partial \gamma _\varepsilon }{\partial x_i} \right|\right|_{L^p(W)} ||\nabla D_h w_\varepsilon ||_{L^2(W)^d}, \end{aligned}$$

    where we have used Proposition 10. Owing to Proposition 5, we know that \( ||\nabla v_\varepsilon -\nabla v_\varOmega ||_{L^q(D)^d} \rightarrow 0\) as \(\varepsilon \rightarrow 0\), while, from Proposition 4, \(\left|\left|\frac{\partial \gamma _\varepsilon }{\partial x_i} \right|\right|_{L^p(D)}\) is bounded since \(e_i\) is a tangential direction to \(\varGamma \).

  • The second term is controlled as:

    $$\begin{aligned}&\left|\int _D{\tau _h(\gamma _\varepsilon - \gamma _\varOmega ) \nabla D_h v_\varOmega \cdot \nabla D_h w_\varepsilon dx} \right|\le C ||\gamma _\varepsilon \\&\quad -\gamma _\varOmega ||_{L^p(D)} \left|\left|\frac{\partial (\nabla v_\varOmega )}{\partial x_i} \right|\right|_{L^q(W)^d} ||\nabla D_h w_\varepsilon ||_{L^2(W)^d}, \end{aligned}$$

    where we have used the regularity theory for \(v_\varOmega \) and the fact that \(e_i\) is tangential to \(\varGamma \), and where \(||\gamma _\varepsilon -\gamma _\varOmega ||_{L^p(D)} {\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}} 0\) (see again Proposition 4).

  • The third term in the right-hand side of (72) is estimated as:

    in which \( \left|\left|\frac{\partial \gamma _\varepsilon }{\partial x_i} - \frac{\partial \gamma _\varOmega }{\partial x_i} \right|\right|_{L^2(W)} {\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}} 0\) because of Proposition 4.

  • One has:

    where we have used (63), and it is easily seen, using Proposition 5, that \(||f_\varepsilon - f_\varOmega ||_{L^2(W)} {\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}} 0\).

  • Likewise,

    $$\begin{aligned} \left|\int _D{D_h(h_\varepsilon - h_\varOmega ) \cdot \nabla D_h w_\varepsilon \,dx} \right|\le \left|\left|\frac{\partial h_\varepsilon }{\partial x_i} - \frac{\partial h_\varOmega }{\partial x_i} \right|\right|_{L^2(W)} ||\nabla D_h w_\varepsilon ||_{L^2(W)^d} \end{aligned}$$

    and using the fact that \(e_i\) is a tangential direction to \(\varGamma \), it comes that \( \left|\left|\frac{\partial h_\varepsilon }{\partial x_i} - \frac{\partial h_\varOmega }{\partial x_i} \right|\right|_{L^2(W)} {\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}} 0\).

Putting things together, using Proposition 10, we obtain that:

$$\begin{aligned} \frac{\partial v_\varepsilon }{\partial \tau } {\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}} \frac{\partial v_\varOmega }{\partial \tau } \text { in } H^{1}(W) . \end{aligned}$$

Remark that, using Meyer’s Theorem 3 from the identity (72) together with the previous estimates for its right-hand side shows that (70) actually holds in \(W^{1,p}(W)\) for some \(p>2\).

The only thing left to prove is then (71), where we recall the simplifying hypothesis \(n=e_d\). Actually, (70) combined with Proposition 4 and the above remark already prove that, for \(i=1,\ldots ,d-1\),

$$\begin{aligned} \frac{\partial }{\partial x_i} \left( \gamma _{\varOmega ,\varepsilon }\frac{\partial v_\varepsilon }{\partial x_d}\right) = \gamma _{\varOmega ,\varepsilon } \frac{\partial ^2 v_\varepsilon }{\partial x_i\partial x_d} = \gamma _{\varOmega ,\varepsilon } \frac{\partial }{\partial x_d}\left( \frac{\partial v_\varepsilon }{\partial x_i}\right) {\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}} \gamma _\varOmega \frac{\partial ^2 v_\varOmega }{\partial x_i\partial x_d} \qquad \text{ in } L^2(W). \end{aligned}$$

Hence, the only thing left to prove is that:

$$\begin{aligned} \frac{\partial }{\partial x_d}\left( \gamma _{\varOmega ,\varepsilon }\frac{\partial v_\varepsilon }{\partial x_d} \right) {\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}} \frac{\partial }{\partial x_d}\left( \gamma _{\varOmega ,\varepsilon }\frac{\partial v_\varOmega }{\partial x_d} \right) ; \end{aligned}$$

this last convergence is obtained by using the original Eqs. (3) and (13), and notably the facts that:

$$\begin{aligned} -\text { div}(\gamma _{\varOmega ,\varepsilon } \nabla v_\varepsilon ) = -\text { div}(\gamma _{\varOmega } \nabla v_\varOmega ) =0 \text { on } U, \end{aligned}$$

together with the previous convergences in (70) and Proposition 4:

$$\begin{aligned} \frac{\partial }{\partial x_d}\left( \gamma _{\varOmega ,\varepsilon }\frac{\partial v_\varepsilon }{\partial x_d} \right)= & {} -\sum \limits _{i=1}^{d-1}{\frac{\partial }{\partial x_i}\left( \gamma _{\varOmega ,\varepsilon }\frac{\partial v_\varepsilon }{\partial x_i} \right) } {\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}} -\sum \limits _{i=1}^{d-1}{\frac{\partial }{\partial x_i}\left( \gamma _{\varOmega }\frac{\partial v_\varOmega }{\partial x_i} \right) } \\= & {} \frac{\partial }{\partial x_d}\left( \gamma _{\varOmega }\frac{\partial v_\varOmega }{\partial x_d} \right) . \end{aligned}$$

This completes the proof. \(\square \)

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Amstutz, S., Dapogny, C. & Ferrer, À. A consistent relaxation of optimal design problems for coupling shape and topological derivatives. Numer. Math. 140, 35–94 (2018). https://doi.org/10.1007/s00211-018-0964-4

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Mathematics Subject Classification

  • 74P05
  • 35Q93
  • 49Q10
  • 65N85
  • 49M20