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A novel way of computing the shape derivative for a class of non-smooth PDEs and its impact on deriving necessary conditions for locally optimal shapes

Livia Betz

TL;DR

The paper develops a novel approach to compute shape derivatives for a class of non-smooth PDEs using the functional variational approach (FVA) without extending PDEs to a hold-all domain. It proves directional differentiability of the state with respect to domain perturbations, identifies a boundary-driven velocity field $W$, and characterizes the state derivative $q$ as the shape derivative, both on interior subdomains and along the boundary. It then reformulates the shape optimization as a reduced control problem in function space, establishes local optimality notions, and derives primal first-order necessary conditions for locally optimal shapes under both pointwise and distributed observations. Finally, it clarifies the relationship between FVA and the speed method, showing that the FVA perturbations reproduce the standard velocity-based shape derivative framework. The results extend sensitivity analysis and optimality conditions to non-smooth PDEs, enabling rigorous locally optimal shape design in two dimensions without PDE extensions.

Abstract

We derive necessary conditions for locally optimal shapes of a design problem governed by a non-smooth PDE. The main particularity of the state system is the lack of differentiability of the nonlinearity. We work in the framework of the functional variational approach (FVA), which has the capacity to transfer geometric optimization problems into optimal control problems, the set of admissible shapes being parametrized by a large class of continuous mappings. In the FVA setting, we introduce a sensitivity analysis technique that is novel even for smooth PDEs. We emphasize that we do not resort to extensions on the hold-all domain or any kind of approximation of the original PDE. The computation of the directional derivative of the state w.r.t. functional variations results in a new way of computing the shape derivative. The presented approach allows us to handle in the objective pointwise observation and derivatives of the state on an observation set as well as distributed observation terms. In addition, we introduce the concept of locally optimal shapes and we put into evidence its connection to locally minimizers of the corresponding control problem. With directional differentiability results for the control-to-state map at our disposal, we can then state necessary conditions for locally optimal shapes in general non-smooth settings.

A novel way of computing the shape derivative for a class of non-smooth PDEs and its impact on deriving necessary conditions for locally optimal shapes

TL;DR

The paper develops a novel approach to compute shape derivatives for a class of non-smooth PDEs using the functional variational approach (FVA) without extending PDEs to a hold-all domain. It proves directional differentiability of the state with respect to domain perturbations, identifies a boundary-driven velocity field , and characterizes the state derivative as the shape derivative, both on interior subdomains and along the boundary. It then reformulates the shape optimization as a reduced control problem in function space, establishes local optimality notions, and derives primal first-order necessary conditions for locally optimal shapes under both pointwise and distributed observations. Finally, it clarifies the relationship between FVA and the speed method, showing that the FVA perturbations reproduce the standard velocity-based shape derivative framework. The results extend sensitivity analysis and optimality conditions to non-smooth PDEs, enabling rigorous locally optimal shape design in two dimensions without PDE extensions.

Abstract

We derive necessary conditions for locally optimal shapes of a design problem governed by a non-smooth PDE. The main particularity of the state system is the lack of differentiability of the nonlinearity. We work in the framework of the functional variational approach (FVA), which has the capacity to transfer geometric optimization problems into optimal control problems, the set of admissible shapes being parametrized by a large class of continuous mappings. In the FVA setting, we introduce a sensitivity analysis technique that is novel even for smooth PDEs. We emphasize that we do not resort to extensions on the hold-all domain or any kind of approximation of the original PDE. The computation of the directional derivative of the state w.r.t. functional variations results in a new way of computing the shape derivative. The presented approach allows us to handle in the objective pointwise observation and derivatives of the state on an observation set as well as distributed observation terms. In addition, we introduce the concept of locally optimal shapes and we put into evidence its connection to locally minimizers of the corresponding control problem. With directional differentiability results for the control-to-state map at our disposal, we can then state necessary conditions for locally optimal shapes in general non-smooth settings.

Paper Structure

This paper contains 11 sections, 25 theorems, 189 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Dirichlet problem
  4. Continuity properties of $\mathcal{S}$
  5. Directional differentiability of $\mathcal{S}$
  6. Equivalence of the shape optimization problem with a control problem with admissible set consisting of functions
  7. Local optimality
  8. Necessary optimality conditions for locally optimal shapes
  9. $H^2$ norms and pointwise observation on a given set
  10. Distributed observation
  11. The FVA and the speed method

Key Result

Proposition 2.1

\newlabelprop:ham For each $g \in \mathcal{F}$, the level set $\{g=0\}$ is a finite union of disjoint closed $\mathcal{C}^2$ curves, without self intersections and not intersecting $\partial D$. Each component of $\{g=0\}$ is parametrized by the solution $z :[0,\infty)\to \{g=0\}^\circ$ of the Hami when some initial point $x_0 \in \{g=0\}^\circ$ is chosen.

Figures (3)

  • Figure 2.1: The zero level set of $h$ intersects each component of $\{g=0\}$.
  • Figure 3.1: The set $\Gamma_2$ (the part of the boundary of $\partial \Omega_g$ included in $\bar{\Omega}_{g+\lambda h}$)
  • Figure 3.2: The blue curves describe $\partial \Omega_{g+\lambda h}$, while the black curves represent $\partial \Omega_g$. The doted blue lines are those parts of $\partial \Omega_{g+\lambda h}$ that are outside $\bar{\Omega}_g$

Theorems & Definitions (55)

  • Proposition 2.1: pen
  • Remark 2.2
  • Lemma 2.3: p3
  • proof
  • Proposition 2.4: p3, Uniqueness of the approximating curve
  • Proposition 2.5: to, p3
  • Proposition 2.6: oc_t
  • Definition 3.2: The control-to-state map associated to \ref{['eq']}
  • Remark 3.3
  • Lemma 3.4: gt
  • ...and 45 more