Geometric constraints for shape and topology optimization in architectural design

Abstract

This work proposes a shape and topology optimization framework oriented towards conceptual architectural design. A particular emphasis is put on the possibility for the user to interfere on the optimization process by supplying information about his personal taste. More precisely, we formulate three novel constraints on the geometry of shapes; while the first two are mainly related to aesthetics, the third one may also be used to handle several fabrication issues that are of special interest in the device of civil structures. The common mathematical ingredient to all three models is the signed distance function to a domain, and its sensitivity analysis with respect to perturbations of this domain; in the present work, this material is extended to the case where the ambient space is equipped with an anisotropic metric tensor. Numerical examples are discussed in two and three space dimensions.

Appendix: Mathematical details

In this appendix, we present the mathematical proofs of the results stated in Sects. 4.3.1 and 5.

1.1 Proofs of the properties of the anisotropic, signed distance function

Proof of Proposition 4

The proof can be achieved in two ways: one could directly adapt the proofs of the results of Propositions 1 and 2, or observe that the anisotropic context can be derived from the isotropic one via a suitable transformation. We rely on this second point of view.

More precisely, we consider the mapping ${\mathbb {R}}^d \ni x \mapsto M^{\frac{1}{2}}x \in {\mathbb {R}}^d$, and set $x_M := M^{\frac{1}{2}}x$ for short. As a straightforward consequence of definitions, for any two points $x,y\in {\mathbb {R}}^d$, on has $|x - y |_M = |x_M - y_M |$. It follows that the signed distance functions $d_\varOmega$ and $d_\varOmega ^M$ (resp. sets of projections $\varPi _{\partial \varOmega }$ and $\varPi _{\partial \varOmega }^M$) are related as:

$$\begin{aligned} \forall x \in {\mathbb {R}}^d, \,\, d_\varOmega ^M(x)= & {} d_{\varOmega _M}(x_M),\nonumber \\ \text { and } \varPi _{\partial \varOmega }^M(x)= & {} M^{-\frac{1}{2}}\varPi _{\partial \varOmega _M}(x_M), \end{aligned}$$

(35)

where we have introduced the transformed domain $\varOmega _M := M^{\frac{1}{2}} \varOmega$.

The following lemma will be used repeatedly in the following. Its proof is postponed to the end of this section.

Lemma 1

Let $\varOmega$ be a bounded domain of class ${\mathcal {C}}^2$ and M be a symmetric, positive definite matrix. Then,

• 1. The unit normal vector $n_{\varOmega _M}$ to $\partial \varOmega _M$, pointing outward $\varOmega _M$ reads:

  $$\begin{aligned}&\forall x \in \partial \varOmega , \,\, n_{\varOmega _M}(x_M) = \frac{\text {com}(M^{\frac{1}{2}}) n_\varOmega (x)}{|\text {com}(M^{\frac{1}{2}}) n_\varOmega (x) \vert } \nonumber \\&\quad = \frac{M^{-\frac{1}{2}} n_\varOmega (x)}{|M^{-\frac{1}{2}} n_\varOmega (x)|}, \end{aligned}$$

  (36)

  where $\text {com}(B)$ stands for the cofactor matrix of a $d \times d$ matrix B.

• 2. The second fundamental form $II^M$ of the boundary $\partial \varOmega _M$ satisfies the identity:

  $$\begin{aligned} \forall x \in \partial \varOmega , \,\, M^{\frac{1}{2}} II^M_{x_M} M^{\frac{1}{2}} = \frac{1}{|M^{-\frac{1}{2}}n_\varOmega (x) \vert } II_{x}. \end{aligned}$$

It follows immediately from (35) that the skeleton $\varSigma ^M$ is obtained from that $\varSigma _M$ of $\varOmega _M$ via $\varSigma ^M = M^{-\frac{1}{2}} \varSigma _M$. Hence, for $x \notin \partial \varOmega$, $d_\varOmega ^M$ is differentiable at x if and only if $x \notin \varSigma ^M$; using again (35), one has then:

$$\begin{aligned} p_{\partial \varOmega }^M(x) = M^{-\frac{1}{2}} p_{\partial \varOmega _M}(x_M). \end{aligned}$$

(37)

Hence, for $x \notin \partial \varOmega$, $x \notin \varSigma ^M$, (16) follows and (i) is proved.

To obtain (ii), we simply apply (11) to the domain $\varOmega _M$, and rely on Lemma 1; for $x \notin \partial \varOmega$, $x \notin \varSigma ^M$, the sequence of equalities

$$\begin{aligned} \nabla d_{\varOmega }^M(x)= & {} M^{\frac{1}{2}}\nabla d_{\varOmega _M}(x_M) =M^{\frac{1}{2}} n_{\varOmega _M}(p_{\partial \varOmega _M}(x_M)) \\= & {} \frac{1}{|M^{-\frac{1}{2}}n_\varOmega (p_{\partial \varOmega }^M(x)) |} n_\varOmega (p_{\partial \varOmega }^M(x)) \end{aligned}$$

proves the first part of (17). As for the second part, one has:

$$\begin{aligned} x_M = p_{\partial \varOmega _M}(x_M) + d_{\varOmega _M}(x_M) n_{\varOmega _M}(p_{\partial \varOmega _M}(x_M)). \end{aligned}$$

Then, using (35), (37) and Lemma 1, it follows:

$$\begin{aligned} \begin{array}{ccl} x &{}=&{} M^{-\frac{1}{2}} p_{\partial \varOmega _M}(x_M) + d_{\varOmega _M}(x_M) M^{-\frac{1}{2}} n_{\varOmega _M}(p_{\partial \varOmega _M}(x_M)), \\ &{}=&{} p_{\partial \varOmega }^M(x) + \frac{d_\varOmega ^M(x)}{|\text {com}(M^{\frac{1}{2}}) n_\varOmega (p_{\partial \varOmega }^M(x))|} M^{-\frac{1}{2}} \text {com}(M^{\frac{1}{2}}) n_\varOmega (p_{\partial \varOmega }^M(x)), \\ &{}=&{} p_{\partial \varOmega }^M(x) + \frac{d_\varOmega ^M(x)}{|M^{-\frac{1}{2}} n_\varOmega (p_{\partial \varOmega }^M(x))|} M^{-1}n_\varOmega (p_{\partial \varOmega }^M(x)), \end{array} \end{aligned}$$

which is the desired result.

Point (iii) is a straightforward consequence of the first two points (and notably the fact that $\varSigma ^M = M^{-\frac{1}{2}} \varSigma _M$), and Proposition 1.

Likewise, point (iv) follows from Proposition 2, and the transformation rule of Lemma 1 for the second fundamental form of $\varOmega _M$.

Eventually, point (v) follows easily from a combination of (37), Lemma 1, and the expression (13) of the derivative of the usual projection mapping. $\square$

We now provide the missing ingredient in the proof of Proposition 4.

Proof of Lemma 1

The first point is classical (and actually holds in a much larger context than that of linear mappings), and we concentrate on that of the second one. To this end, we use the following elementary characterization: for a given tangent vector $v_M = M^{\frac{1}{2}}v$ to $\partial \varOmega _M$ (i.e. $v_M \in T_{x_M}\partial \varOmega _M$), where $v \in T_x \partial \varOmega$,

$$\begin{aligned} II^M_{x_M}(v_M,v_M) = \langle \gamma ^{\prime \prime }(0), n_{\varOmega _M}(x_M) \rangle , \end{aligned}$$

(38)

for any curve $\gamma : (-t_0,t_0) \rightarrow \partial \varOmega _M$ of class ${\mathcal {C}}^2$ such that $\gamma (0) = x_M$, $\gamma ^\prime (0) = v_M$. Let us choose $\gamma (t) = M^{\frac{1}{2}} \zeta (t)$, where $\zeta : (-t_0,t_0) \rightarrow \partial \varOmega$ is a curve of class ${\mathcal {C}}^2$ such that $\zeta (0) = x$, $\zeta ^\prime (0) = v$. Then, (38) together with (36) yield:

$$\begin{aligned} \begin{array}{ccl} II^M_{x_M}(v_M,v_M) &{}=&{} \frac{1}{|\text {com}(M^{\frac{1}{2}}) n_\varOmega (x) \vert } \langle \zeta ^{\prime \prime }(0), M^{\frac{1}{2}}\text {com}(M^{\frac{1}{2}}) n_\varOmega (x) \rangle ,\\ &{}=&{} \frac{\text {det}(M^{\frac{1}{2}})}{|\text {com}(M^{\frac{1}{2}}) n_\varOmega (x) \vert } \langle \zeta ^{\prime \prime }(0), n_\varOmega (x) \rangle ,\\ &{}=&{} \frac{1}{|M^{-\frac{1}{2}}n_\varOmega (x) \vert } \langle \zeta ^{\prime \prime }(0), n_\varOmega (x) \rangle , \end{array} \end{aligned}$$

and the desired result follows. $\square$

Proofs of Propositions 5 and 6

Again, both results may be equivalently inferred from their isotropic counterparts, by using the isomorphism $x \mapsto M^{\frac{1}{2}}x$ in the same way as in the proof of Proposition 4, or by adapting their original proofs (see [32] and [27]) to the present anisotropic context, and we omit further details about these points. $\square$

1.2 Details of the calculations of the shape derivatives of Sect. 5

Proof of Proposition 8

The proof is carried out in the work [4], and is reproduced in here for convenience.

For $\theta \in \varTheta _{ad}$, a change of variables in the boundary integral (22) yields (see e.g. [40] Prop. 5.4.3):

$$\begin{aligned} P(\varOmega _\theta ) = \int _{\varGamma }{\varphi ( n_{\varOmega _\theta }\circ (\text {Id}+\theta ))\, |\text {com}(\text {Id}+\theta ) |\,ds}. \end{aligned}$$

Using the well-known identity over matrices:

$$\begin{aligned} \text {com}(M) = \text {det}(M) M^{-T}, \end{aligned}$$

we obtain that $\theta \mapsto |\text {com}(I+\nabla \theta ) |$ is Fréchet-differentiable at $\theta = 0$, and

$$\begin{aligned}&|\text {com}(I+\nabla \theta ) |= 1+ \text {div}_{\partial \varOmega }(\theta ) \\&\quad +\, o(\theta ), \text { where } \frac{|o(\theta )|}{||\theta ||_{{\mathcal {C}}^{1,\infty }({\mathbb {R}}^d,{\mathbb {R}}^d) }} \mathop {\longrightarrow }\limits ^{\theta \rightarrow 0} 0. \end{aligned}$$

In this formula, we recall that $\text {div}_{\partial \varOmega } V := \text {div}(V) - \langle \nabla V n ,n\rangle$ stands for the tangential divergence of a (smooth enough) vector field $V : \partial \varOmega \rightarrow {\mathbb {R}}^d$. Hence, using Lebesgue dominated convergence theorem and Proposition 3, it follows that $\theta \mapsto P(\varOmega _\theta )$ is Fréchet-differentiable at 0, and that the corresponding derivative reads:

$$\begin{aligned}&\forall \theta \in \varTheta _{ad},\,\, P^\prime (\varOmega )(\theta ) = \\&\quad \int _\varGamma {\left( \nabla \varphi (n)\cdot \left( \left( \left( \nabla \theta ^T n \right) \cdot n \right) n \!- \!\nabla \theta ^T n\right) \!+\! \,\varphi (n) \text {div}_{\partial \varOmega }(\theta )\!\right) \,ds}. \end{aligned}$$

Using integration by parts on the boundary $\varGamma$ (see [40], Prop. 5.4.9), together with the identity:

$$\begin{aligned} \nabla _{\partial \varOmega }(\theta \cdot n) = \nabla \theta ^T n + \nabla n^T \theta - (\nabla \theta ^T n \cdot n) n, \end{aligned}$$

we obtain:

$$\begin{aligned} P^\prime (\varOmega )(\theta )= & {} \int _\varGamma {\kappa \varphi (n) \theta \cdot n\,ds} - \int _\varGamma { \nabla \varphi (n) \cdot \nabla _{\partial \varOmega }(\theta \cdot n) \,ds} \\&+ \int _{\varGamma }{\left( \nabla \varphi (n) - \nabla _{\partial \varOmega }(\varphi (n))\right) \cdot \theta \,ds}. \end{aligned}$$

Eventually, using the symmetry property (14), the last integral in the right-hand side of the above equality vanishes, and the expected result follows. $\square$

Proof of Corollary 1

Using $\varphi (n) = |n - n_g |^2$, $\nabla \varphi (n) = 2(n-n_g)$ in Proposition 8 yields:

$$\begin{aligned} P_p^\prime (\varOmega )(\theta )= & {} \int _{\varGamma }{\kappa |n_\varOmega - n_g |^2 \theta \cdot n \,ds} \\&- \int _{\varGamma }{(n_\varOmega - n_g) \cdot \nabla _{\partial \varOmega }(\theta \cdot n)\,ds}. \end{aligned}$$

Using again integration by parts on the boundary on the second term in the right-hand side of the above equality yields:

$$\begin{aligned}&P_p^\prime (\varOmega )(\theta ) \\&\quad =\int _{\varGamma }{\kappa |n_\varOmega \!- \!n_g |^2 \theta \cdot n \,ds} \!-\! \int _{\varGamma }{(n_\varOmega \!-\! n_g) \!\cdot \!\nabla _{\partial \varOmega }(\theta \cdot n)\,ds},\\&\quad = \int _{\varGamma }\left( \kappa |n_\varOmega - n_g |^2 - 2\kappa (n_\varOmega - n_g) \cdot n_\varOmega \right. \\&\qquad \left. +\, 2(\kappa - \text {div}_{\partial \varOmega }(n_g))\right) \,\theta \cdot n \,ds, \end{aligned}$$

which leads to the desired result. $\square$

Proof of Theorem 1

It follows from Proposition 6 that $P_a^M(\varOmega )$ is shape differentiable at an arbitrary shape $\varOmega \in {{\mathcal {U}}}_{ad}$. An easy calculation then yields:

$$\begin{aligned}&\forall \theta \in \varTheta _{ad}, \,\, {P_a^M}^\prime (\varOmega )(\theta ) \\&\quad = \int _\varOmega {G(d_\varOmega ^M(x)) \langle M n^M(p_{\partial \varOmega }^M(x)), \theta (p_{\partial \varOmega }^M(x))\rangle \,dx}. \end{aligned}$$

We now proceed to achieve an expression of the above derivative with the convenient structure (5). To this end, recalling that $\overline{\varSigma ^M}$ has zero Lebesgue measure (see Proposition 4), we apply the coarea formula to the mapping $p_{\partial \varOmega }^M : \varOmega \setminus \overline{\varSigma ^M} \rightarrow \partial \varOmega$, as a curvilinear version of the classical Fubini theorem (see [21], or [3] in the same context):

$$\begin{aligned}&{P_a^M}^\prime (\varOmega )(\theta ) =\int _{\partial \varOmega }\langle Mn^M(x), \theta (x) \rangle \\&\quad \left( \int _{\text {ray}^M_{\partial \varOmega }(x) \cap \varOmega } {\frac{G(d_\varOmega ^M(z))}{\text {Jac}(p_{\partial \varOmega }^M(z)) }\,d\ell (z)}\right) \,ds(x), \end{aligned}$$

where we recall that $\text {ray}_{\partial \varOmega }^M(x) = (p_{\partial \varOmega }^M)^{-1}(x)$, and the Jacobian $\text {Jac}(p_{\partial \varOmega }^M(x))$ is defined as:

$$\begin{aligned} \text {Jac}(p_{\partial \varOmega }^M(z)) = \sqrt{\nabla p_{\partial \varOmega }^M(z)\nabla p_{\partial \varOmega }^M(z)^T}, \,\, z \in {\mathbb {R}}^d \setminus \overline{\varSigma ^M}. \end{aligned}$$

The only remaining task is then to calculate this Jacobian, which we do by using the representation (18) of $\nabla p_{\partial \varOmega }^M$ in an orthonormal basis of ${\mathbb {R}}^d$:

$$\begin{aligned} \text {Jac}(p_{\partial \varOmega }^M(z)) \!= \!\prod _{i=1}^{d-1}{\left( 1 \!+ \! \frac{ d_\varOmega ^M(z)}{|M^{-\frac{1}{2}} n(p_{\partial \varOmega ^M}(z)) |}\lambda _i(p_{\partial \varOmega ^M}(z)) \! \right) ^{-1}}. \end{aligned}$$

This ends the proof. $\square$