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Numerical solution by shape optimization method to an inverse shape problem in multi-dimensional advection-diffusion problem with space dependent coefficients

Elmehdi Cherrat, Lekbir Afraites, Julius Fergy Tiongson Rabago

Abstract

This work focuses on numerically solving a shape identification problem related to advection-diffusion processes with space-dependent coefficients using shape optimization techniques. Two boundary-type cost functionals are considered, and their corresponding variations with respect to shapes are derived using the adjoint method, employing the chain rule approach. This involves firstly utilizing the material derivative of the state system and secondly using its shape derivative. Subsequently, an alternating direction method of multipliers (ADMM) combined with the Sobolev-gradient-descent algorithm is applied to stably solve the shape reconstruction problem. Numerical experiments in two and three dimensions are conducted to demonstrate the feasibility of the methods.

Numerical solution by shape optimization method to an inverse shape problem in multi-dimensional advection-diffusion problem with space dependent coefficients

Abstract

This work focuses on numerically solving a shape identification problem related to advection-diffusion processes with space-dependent coefficients using shape optimization techniques. Two boundary-type cost functionals are considered, and their corresponding variations with respect to shapes are derived using the adjoint method, employing the chain rule approach. This involves firstly utilizing the material derivative of the state system and secondly using its shape derivative. Subsequently, an alternating direction method of multipliers (ADMM) combined with the Sobolev-gradient-descent algorithm is applied to stably solve the shape reconstruction problem. Numerical experiments in two and three dimensions are conducted to demonstrate the feasibility of the methods.

Paper Structure

This paper contains 25 sections, 8 theorems, 118 equations, 14 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. The problem setting
  3. The main problem
  4. Shape optimization reformulations
  5. Weak forms of the state systems
  6. Shape Derivatives
  7. Some elements of shape calculus
  8. Material and shape derivatives of the state variables
  9. Shape derivatives of the shape functionals
  10. Proof of Proposition \ref{['prop:shape_gradients']} via material derivative
  11. Proof of Proposition \ref{['prop:shape_gradients']} via shape derivative of state
  12. Numerical Experiments
  13. Conventional numerical algorithm
  14. Numerical examples in 2D
  15. Alternating direction method of multipliers
  16. ...and 10 more sections

Key Result

Theorem 3.1

The Neumann solution $u_{N} \in {\color{black}{V_{\Gamma}(\Omega)}}$ of eq:state_un has a derivative $\dot{u}_{N} \in {\color{black}{V_{\Gamma}(\Omega)}}$ that satisfies where

Figures (14)

  • Figure 1: Reconstruction of $\Gamma^{\ast}_{1}$ in the absence and presence of noise ($\delta = 0\%, 10\%, 30\%$). The plot on the right shows the cost value histories for each case.
  • Figure 2: Reconstruction of $\Gamma^{\ast}_{2}$ in the absence and presence of noise ($\delta = 0\%, 10\%, 30\%$). The plot on the right shows the cost value histories for each case.
  • Figure 3: Reconstruction of $\Gamma^{\ast}_{2}$ with noisy data ($\delta = 30\%$) with perimeter regularization and with adaptive mesh refinement. The plot on the right shows the cost value histories for each case.
  • Figure 4: Reconstruction of $\Gamma^{\ast}_{1}$ with $\delta = 0\%, 10\%, 30\%$ using \ref{['eq:Dirichlet_cost_function']} with perimeter regularization but without adaptive mesh refinement. The right plot shows the histories of cost values for each noise levels.
  • Figure 5: Reconstruction of $\Gamma^{\ast}_{2}$ with $\delta = 0\%, 10\%, 30\%$ using \ref{['eq:Dirichlet_cost_function']} with perimeter regularization but without adaptive mesh refinement. The right plot shows the histories of cost values for each noise levels.
  • ...and 9 more figures

Theorems & Definitions (23)

  • Remark 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.3
  • proof
  • proof : Proof of Theorem \ref{['theo:form_un']}
  • Remark 3.4
  • Remark 3.5
  • Proposition 1: Shape gradient of $J$
  • Remark 3.6
  • ...and 13 more