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Detecting immersed obstacle in Stokes fluid flow using the coupled complex boundary method

Julius Fergy Tiongson Rabago, Lekbir Afraites, Hirofumi Notsu

Abstract

A non-conventional shape optimization approach is introduced to address the identification of an obstacle immersed in a fluid described by the Stokes equation within a larger bounded domain, relying on boundary measurements on the accessible surface. The approach employs tools from shape optimization, utilizing the coupled complex boundary method to transform the over-specified problem into a complex boundary value problem by incorporating a complex Robin boundary condition. This condition is derived by coupling the Dirichlet and Neumann boundary conditions along the accessible boundary. The identification of the obstacle involves optimizing a cost function constructed based on the imaginary part of the solution across the entire domain. The subsequent calculation of the shape gradient of this cost function, rigorously performed via the rearrangement method, enables the iterative solution of the optimization problem using a Sobolev gradient descent algorithm. The feasibility of the method is illustrated through numerical experiments in both two and three spatial dimensions, demonstrating its effectiveness in reconstructing obstacles with pronounced concavities under high-level noise-contaminated data, all without perimeter or volume functional penalization.

Detecting immersed obstacle in Stokes fluid flow using the coupled complex boundary method

Abstract

A non-conventional shape optimization approach is introduced to address the identification of an obstacle immersed in a fluid described by the Stokes equation within a larger bounded domain, relying on boundary measurements on the accessible surface. The approach employs tools from shape optimization, utilizing the coupled complex boundary method to transform the over-specified problem into a complex boundary value problem by incorporating a complex Robin boundary condition. This condition is derived by coupling the Dirichlet and Neumann boundary conditions along the accessible boundary. The identification of the obstacle involves optimizing a cost function constructed based on the imaginary part of the solution across the entire domain. The subsequent calculation of the shape gradient of this cost function, rigorously performed via the rearrangement method, enables the iterative solution of the optimization problem using a Sobolev gradient descent algorithm. The feasibility of the method is illustrated through numerical experiments in both two and three spatial dimensions, demonstrating its effectiveness in reconstructing obstacles with pronounced concavities under high-level noise-contaminated data, all without perimeter or volume functional penalization.
Paper Structure (33 sections, 13 theorems, 87 equations, 14 figures)

This paper contains 33 sections, 13 theorems, 87 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Shape optimization techniques for shape recovery
  4. The CCBM and contribution to the literature
  5. Preliminaries
  6. The CCBM formulation
  7. Well-posedness of the CCBM formulation
  8. The shape optimization problem
  9. First-order shape sensitivity analysis
  10. Some concepts from shape calculus
  11. The shape gradient of $J$
  12. Computation of the shape gradient: Proof of Theorem \ref{['thm:shape_derivative']}
  13. Numerical experiments
  14. Numerical algorithm
  15. Forward problem
  16. ...and 18 more sections

Key Result

Theorem 1.2

Let ${D} \subset \mathbb{R}^{d}$, $d \in \{2,3\}$, be a bounded $\mathcal{C}^{1,1}$ domain, and $\Sigma^{0}$ be a non-empty open subset of $\Sigma := \partial{D}$. Moreover, let and $\boldsymbol{\mathbf{f}} \in H^{1/2}(\Sigma)^{d}$ with $\boldsymbol{\mathbf{f}} \neq \boldsymbol{\mathbf{0}}$, satisfying the flux condition $\int_{\Sigma}{\boldsymbol{\mathbf{f}} \cdot \boldsymbol{\mathbf{n}}}{\, {d}

Figures (14)

  • Figure 1: Shape reconstructions with large convex obstacles
  • Figure 2: Shape reconstructions with complicated-shaped obstacles
  • Figure 3: Effect of the choice of initial guess
  • Figure 4: Shape reconstructions with noisy data
  • Figure 5: Influence of the size of the obstacle
  • ...and 9 more figures

Theorems & Definitions (26)

  • Theorem 1.2: Alvarezetal2005
  • Remark 1.3: Existence, uniqueness, and regularity
  • Remark 2.1
  • Lemma 2.3
  • Remark 2.5
  • Theorem 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 16 more