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Shape optimization for contact problem involving Signorini unilateral conditions

Aymeric Jacob de Cordemoy

TL;DR

This work addresses shape optimization for a linear elastic body under Signorini unilateral contact, without penalization. It develops a shape-sensitivity framework based on proximal operators and twice epi-differentiability, proving that the state solution admits a material derivative that solves a Signorini problem and a shape derivative that is computable via an explicit shape gradient. The main contributions are the characterization of material and shape derivatives as Signorini problems and an explicit expression for the energy’s shape gradient, enabling descent directions without adjoints. Numerical tests in 2D demonstrate the method’s feasibility and show energy reduction under volume-constrained shape updates. This penalization-free approach advances shape optimization in contact mechanics by providing exact gradient information and a practicable computational path.

Abstract

This paper investigates a shape optimization problem involving the Signorini unilateral conditions in a linear elastic model, without any penalization procedure. The shape sensitivity analysis is performed using tools from convex and variational analysis such as proximal operators and the notion of twice epi-differentiability. We prove that the solution to the Signorini problem admits a directional derivative with respect to the shape which moreover coincides with the solution to another Signorini problem. Then, the shape gradient of the corresponding energy functional is explicitly characterized which allows us to perform numerical simulations to illustrate this methodology.

Shape optimization for contact problem involving Signorini unilateral conditions

TL;DR

This work addresses shape optimization for a linear elastic body under Signorini unilateral contact, without penalization. It develops a shape-sensitivity framework based on proximal operators and twice epi-differentiability, proving that the state solution admits a material derivative that solves a Signorini problem and a shape derivative that is computable via an explicit shape gradient. The main contributions are the characterization of material and shape derivatives as Signorini problems and an explicit expression for the energy’s shape gradient, enabling descent directions without adjoints. Numerical tests in 2D demonstrate the method’s feasibility and show energy reduction under volume-constrained shape updates. This penalization-free approach advances shape optimization in contact mechanics by providing exact gradient information and a practicable computational path.

Abstract

This paper investigates a shape optimization problem involving the Signorini unilateral conditions in a linear elastic model, without any penalization procedure. The shape sensitivity analysis is performed using tools from convex and variational analysis such as proximal operators and the notion of twice epi-differentiability. We prove that the solution to the Signorini problem admits a directional derivative with respect to the shape which moreover coincides with the solution to another Signorini problem. Then, the shape gradient of the corresponding energy functional is explicitly characterized which allows us to perform numerical simulations to illustrate this methodology.

Paper Structure

This paper contains 21 sections, 18 theorems, 121 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation.
  3. Novelties compared with existing literature.
  4. Description of the shape optimization problem and methodology.
  5. Main theoretical results.
  6. Organization of the paper.
  7. Preliminaries
  8. Reminders on proximal operator and twice epi-differentiability
  9. Functional framework and some required boundary value problems
  10. A Problem with Dirichlet-Neumann Conditions
  11. A Signorini problem
  12. Main results
  13. Setting of the shape perturbation
  14. Twice epi-differentiability, material and shape derivatives
  15. Shape gradient of the Signorini energy functional
  16. ...and 6 more sections

Key Result

Lemma 2.13

Let $\mathrm{C}$ be a nonempty closed convex subset of $\mathcal{H}$ and $\iota_{\mathrm{C}}$ be the indicator function of $\mathrm{C}$. If $\mathrm{C}$ is polyhedric at $x\in\mathrm{C}$ for $y\in\mathrm{N}_{\mathrm{C}}(x)$, then $\iota_\mathrm{C}$ is twice epi-differentiable at $x$ for $y$ and

Figures (3)

  • Figure 1: Unit disk $\Omega_{\mathrm{ref}}$ and its boundary $\Gamma_{\mathrm{ref}}=\Gamma_{\mathrm{D}}\cup\Gamma_{\mathrm{S}_{\mathrm{ref}}}$.
  • Figure 2: Initial shape (left) and the shape minimizing $\mathcal{J}$ (right), under the volume constraint $\vert \Omega_{\mathrm{ref}} \vert=\pi$. On top is shown the vector values of the Signorini solution, while at bottom is shown the values of the integrand of $\mathcal{J}$, i.e., the map $x\in\Omega\to \frac{1}{2}\mathrm{A}\mathrm{e}\left(u(x)\right):\mathrm{e}\left(u(x)\right)-f(x)\cdot u(x)$.
  • Figure 3: Values of the energy functional (left) and the volume (right) with respect to the iterations.

Theorems & Definitions (52)

  • Definition 2.1: Domain and epigraph
  • Definition 2.2: Convex subdifferential operator
  • Definition 2.3: Proximal operator
  • Remark 2.4
  • Definition 2.5: Mosco-convergence
  • Definition 2.6: Mosco epi-convergence
  • Definition 2.7: Twice epi-differentiability
  • Definition 2.8: Normal cone
  • Example 2.9
  • Definition 2.10: Tangent cone
  • ...and 42 more