Comparison of approximate shape gradients

Abstract

Shape gradients of PDE constrained shape functionals can be stated in two equivalent ways. Both rely on the solutions of two boundary value problems (BVPs), but one involves integrating their traces on the boundary of the domain, while the other evaluates integrals in the volume. Usually, the two BVPs can only be solved approximately, for instance, by finite element methods. However, when used with finite element solutions, the equivalence of the two formulas breaks down. By means of a comprehensive convergence analysis, we establish that the volume based expression for the shape gradient generally offers better accuracy in a finite element setting. The results are confirmed by several numerical experiments.

Appendix

Closely following [10, Ch. 10, Sect. 6], we give a detailed derivation of Formulas (2.9)–(2.13). Let $u$ be the weak solution in $H^{1}({\varOmega })$ of the following state problem:

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta u + u = f &{} \text {in } \varOmega \,,\\ u = g &{} \text {on } \partial \varOmega \,. \end{array}\right. \end{aligned}$$

(5.1)

It is assumed that the Dirichlet problem (5.1) is $H^2$-regular, so that its solution $u$ is at least in $H^2(\varOmega )$ for $f\in L^{2}({\varOmega })$. We consider the shape functional

$$\begin{aligned} \mathcal{J}(\varOmega ) = \int _{\varOmega } j(u) \, d{\mathbf {x}}\,, \end{aligned}$$

and we introduce the Lagrangian

$$\begin{aligned} \fancyscript{L} (\varOmega ,v,q,\lambda ) :\,=\int _\varOmega j(v) +(\Delta v - v + f)q\, d{\mathbf {x}}+ \int _{\partial \varOmega } \lambda (g - v)\, dS\,, \end{aligned}$$

(5.2)

where the functions $v$, $q$ and $\lambda$ are in $H^2(\mathbb {R}^d)$. Performing integration by parts, the Lagrangian can be rewritten as

$$\begin{aligned}&\fancyscript{L} (\varOmega ,v,q,\lambda ) = \int _\varOmega j(v) - \nabla v \cdot \nabla q - v\,q + f\,q\, d{\mathbf {x}}+ \int _{\partial \varOmega } \frac{\partial {v}}{\partial {\mathbf {n}}}\,q + \lambda (g - v)\, dS\,,\\&\quad = \int _\varOmega j(v) +(\Delta q - q)v + f\,q\, d{\mathbf {x}}+ \int _{\partial \varOmega } \frac{\partial {v}}{\partial {\mathbf {n}}}\,q - \frac{\partial {q}}{\partial {\mathbf {n}}}\,v + \lambda (g - v)\, dS\,. \end{aligned}$$

The saddle point of $\fancyscript{L}(\varOmega ,\cdot \,,\cdot \,,\cdot )$ is characterized by

$$\begin{aligned} \left\langle \frac{\partial \fancyscript{L}(\varOmega ,v,q,\lambda )}{\partial v}, \phi \right\rangle _{\varOmega } \!\!= \left\langle \frac{\partial \fancyscript{L}(\varOmega ,v,q,\lambda )}{\partial q}, \phi \right\rangle _{\varOmega } \!\!= \left\langle \frac{\partial \fancyscript{L}(\varOmega ,v,q,\lambda )}{\partial \lambda }, \phi \right\rangle _{\partial \varOmega } \!\!= 0 \end{aligned}$$

for all $\phi \in H^2(\mathbb {R}^d)$, which, by density, leads to

$$\begin{aligned}&\left\{ \begin{array}{l@{\quad }l} -\Delta v + v = f&{} \text {in } \varOmega \,,\\ v = g&{} \text {on } \partial \varOmega \,, \end{array}\right. \end{aligned}$$

(5.3a)

$$\begin{aligned}&\left\{ \begin{array}{l@{\quad }l} -\Delta q + q = j'(v) &{}\text {in } \varOmega \,,\\ q = 0 &{}\text {on } \partial \varOmega \,, \end{array}\right. \end{aligned}$$

(5.3b)

$$\begin{aligned}&\quad \lambda = -\frac{\partial {q}}{\partial {\mathbf {n}}}\quad \text {on } \partial \varOmega \,, \end{aligned}$$

(5.3c)

weakly in $H^1(\mathbb {R}^d)$. Thus, for $\varOmega$ fixed,

$$\begin{aligned} \mathcal{J}(\varOmega ) = \inf _{v \in H^2(\mathbb {R}^d)} \sup _{q,\lambda \in H^2(\mathbb {R}^d)} \fancyscript{L}(\varOmega ,v,q,\lambda )\,, \end{aligned}$$

(5.4)

because

$$\begin{aligned} \mathcal{J}(\varOmega ) = \fancyscript{L}(\varOmega ,u,q,\lambda ) \quad \forall \, q,\lambda \in H^2(\mathbb {R}^d)\,. \end{aligned}$$

Recall that the material derivative of a generic function $f$ with respect to the deformation $T_\mathcal{V}$ is defined as

$$\begin{aligned} \dot{f} :\,=\lim _{s\searrow 0} \frac{f\circ T_{s\cdot \mathcal{V}} - f}{s}\,. \end{aligned}$$

Note that, if $f$ is independent of $\varOmega$, $\dot{f}\in H^1(\mathbb {R}^d)$ for $f\in H^2(\mathbb {R}^d)$.

To compute the Eulerian derivative of $\mathcal{J}(\varOmega )$, the Correa–Seeger theorem can be applied on the right-hand side of (5.4) [10, Ch. 10, Sect. 6.3], so that a formula for $d\mathcal{J}(\varOmega )$ can be obtained by evaluating the Eulerian derivative of the Lagrangian (5.2) in its saddle point. For $T_\mathcal{V}({\mathbf {x}}):\,={\mathbf {x}}+ \mathcal{V}({\mathbf {x}})$, the Eulerian derivative of (5.2) reads

$$\begin{aligned}&\lim _{s\searrow 0} \frac{\fancyscript{L}(T_{s\cdot \mathcal{V}}(\varOmega ),v,q,\lambda ) -\fancyscript{L}(\varOmega ,v,q,\lambda )}{s} \\&\quad = \int _\varOmega \left( j'(v) \dot{v} - \nabla \dot{v}\cdot \nabla q - \nabla v\cdot \nabla \dot{q} + \nabla v \cdot ({\mathbf {D}}\mathcal{V}+{\mathbf {D}}\mathcal{V}^T) \nabla q \right. \\&\left. \qquad - \dot{v}\,q - v\,\dot{q} + \dot{f}q + f\dot{q} + \mathrm{div }(\mathcal{V}) \left( j(v) -\nabla v \cdot \nabla q - v\, q + fq \right) \right) \, d{\mathbf {x}}\\&\qquad +\int _{\partial \varOmega } \dot{\frac{\partial {v}}{\partial {\mathbf {n}}}}q + \frac{\partial {v}}{\partial {\mathbf {n}}}\dot{q} +\lambda (\dot{g}-\dot{v}) + \dot{\lambda }(g-v) + \mathrm{div }_\Gamma (\mathcal{V}) \left( \frac{\partial {v}}{\partial {\mathbf {n}}}q + \lambda (g-v) \right) \,dS \\&\quad = \int _\varOmega j'(v) \dot{v} + \Delta q \, \dot{v} - q\, \dot{v} \, d{\mathbf {x}}+ \int _\varOmega \Delta v \, \dot{q} -v \, \dot{q} + f \, \dot{q} \, d{\mathbf {x}}\\&\qquad + \int _{\partial \varOmega } \dot{\frac{\partial {v}}{\partial {\mathbf {n}}}} q + \dot{\lambda }(g-v) + \mathrm{div }_\Gamma (\mathcal{V})\left( \frac{\partial {v}}{\partial {\mathbf {n}}} q + \lambda (g-v) \right) \, dS \\&\qquad + \int _\varOmega \nabla v \cdot ({\mathbf {D}}\mathcal{V}+{\mathbf {D}}\mathcal{V}^T) \nabla q + \dot{f}q + \mathrm{div }(\mathcal{V}) \left( j(v) -\nabla v \cdot \nabla q - v\, q + fq \right) \, d{\mathbf {x}}\\&\qquad + \int _{\partial \varOmega } \lambda (\dot{g}-\dot{v})-\frac{\partial {q}}{\partial {\mathbf {n}}}\dot{v} \, dS\,. \end{aligned}$$

So, in the saddle point defined by (5.3), we have

$$\begin{aligned}&\lim _{s\searrow 0} \frac{\fancyscript{L}(T_{s\cdot \mathcal{V}}(\varOmega ),v,q,\lambda ) -\fancyscript{L}(\varOmega ,v,q,\lambda )}{s}= \\&\quad =\int _\varOmega \nabla v \cdot ({\mathbf {D}}\mathcal{V}+{\mathbf {D}}\mathcal{V}^T) \nabla q + \dot{f}q + \mathrm{div }(\mathcal{V}) \left( j(v) -\nabla v \cdot \nabla q - v\, q + fq \right) \, d{\mathbf {x}}\\&\qquad + \int _{\partial \varOmega } -\frac{\partial {q}}{\partial {\mathbf {n}}}\dot{g} \, dS \\&\quad =\int _\varOmega \left( \nabla v \cdot ({\mathbf {D}}\mathcal{V}+{\mathbf {D}}\mathcal{V}^T) \nabla q + \dot{f}q + (j'(v) - q) \dot{g} - \nabla q \cdot \nabla \dot{g}\right. \\&\qquad \left. + \mathrm{div }(\mathcal{V}) \left( j(v) -\nabla v \cdot \nabla q - v\, q + fq \right) \right) \, d{\mathbf {x}}\,, \end{aligned}$$

which, after an additional integration by parts on the term $\dot{f}q = \nabla f\cdot \mathcal{V}q$, corresponds to Formula (2.9). Formula (2.10) is obtained performing additional integrations by parts and exploiting the vector calculus identity

$$\begin{aligned} \mathcal{V}\cdot \nabla \left( \nabla v\cdot \nabla q\right) + \nabla v \cdot \left( {\mathbf {D}}\mathcal{V}+{\mathbf {D}}\mathcal{V}^T\right) \nabla q = \nabla \left( \mathcal{V}\cdot \nabla v\right) \cdot \nabla q +\nabla v\cdot \nabla \left( \mathcal{V}\cdot \nabla v \right) . \end{aligned}$$

We refer to [5, Sect. 6] for a detailed derivation. Alternatively, (2.10) can be derived with the so-called “fast derivation” method of Céa, which, formally, does not rely on the concept of material derivative, cf. [9] and [2, Ch. 6.4.3].

Similarly, Formula (2.11) can be derived considering the Lagrangian

$$\begin{aligned} \nonumber \mathcal{L}(\varOmega ,v,q)&:= \int _\varOmega j(v) +(\Delta v - v + f)q \,d{\mathbf {x}}+\int _{\partial \varOmega } gq - \frac{\partial {v}}{\partial {\mathbf {n}}}q\,dS\,,\nonumber \\&=\int _\varOmega j(v) - \nabla v\cdot \nabla q - vq + fq\,d{\mathbf {x}}+ \int _{\partial \varOmega } gq\,dS\,,\nonumber \\&=\int _\varOmega j(v) + (\Delta q - q)v + fq \,d{\mathbf {x}}+\int _{\partial \varOmega } gq-\frac{\partial {q}}{\partial {\mathbf {n}}}v\,dS\,. \end{aligned}$$

(5.5)

Its saddle point is characterized by

$$\begin{aligned}&\left\{ \begin{array}{l@{\quad }l} -\Delta v + v = f &{} \text {in } \varOmega \,,\\ \frac{\partial {v}}{\partial {\mathbf {n}}}=g &{} \text {on } \partial \varOmega \,, \end{array}\right. \end{aligned}$$

(5.6a)

$$\begin{aligned}&\left\{ \begin{array}{l@{\quad }l} -\Delta q + q = j'(v) &{} \text {in } \varOmega \,,\\ \frac{\partial {q}}{\partial {\mathbf {n}}} = 0 &{} \text {on } \partial \varOmega \,. \end{array}\right. \end{aligned}$$

(5.6b)

Thus, the Eulerian derivative of (5.5) in (5.6) reads

$$\begin{aligned}&\lim _{s\searrow 0} \frac{\fancyscript{L}(T_{s\cdot \mathcal{V}}(\varOmega ),v,q,\lambda ) -\fancyscript{L}(\varOmega ,v,q,\lambda )}{s}= \\&\quad = \int _\varOmega \nabla v \cdot ({\mathbf {D}}\mathcal{V}+{\mathbf {D}}\mathcal{V}^T) \nabla q + \dot{f}q + \mathrm{div }(\mathcal{V}) \left( j(v) -\nabla v \cdot \nabla q - v\, q + fq \right) \, d{\mathbf {x}}\\&\qquad + \int _{\partial \varOmega } \dot{g} q + \mathrm{div }_\Gamma (\mathcal{V})\left( gq \right) \, dS\,. \end{aligned}$$

In this case, the term $\mathrm{div }_\Gamma (\mathcal{V})$ does not vanish, and to recover Formula (2.12) it is necessary to perform an integration by parts on the boundary, from which stems the mean curvature term. For piecewise smooth boundaries, this step has to be performed carefully, because, as in Remark 2.3, additional contributions of corner points appear.