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Computation of shape Taylor expansions

Gang Bao, Jun Lai, Haoran Ma

TL;DR

This work develops a unified, recurrence-based framework to compute shape Taylor expansions of arbitrary order $N$ for 2D acoustic scattering under sound-soft, sound-hard, impedance, and transmission boundaries. Leveraging boundary integral equations and potential theory, it provides explicit recurrence formulas for boundary data and a method to evaluate higher-order normal derivatives, enabling high-fidelity approximations of the scattered field under boundary perturbations. The approach is extended to uncertainty quantification by deriving moment expansions for random boundary perturbations, with error terms scaling as $\mathcal{O}(\epsilon^{N+2})$ for even $N$, and demonstrated through numerical examples on circles and ellipses. The results indicate that higher-order shape Taylor expansions improve accuracy for moderately large perturbations and enable reliable moment estimations, with potential extensions to 3D acoustic and electromagnetic scattering.

Abstract

Shape derivative is an important analytical tool for studying scattering problems involving perturbations in scatterers. Many applications, including inverse scattering, optimal design, and uncertainty quantification, are based on shape derivatives. However, computing high order shape derivatives is challenging due to the complexity of shape calculus. This work introduces a comprehensive method for computing shape Taylor expansions in two dimensions using recurrence formulas. The approach is developed under sound-soft, sound-hard, impedance, and transmission boundary conditions. Additionally, we apply the shape Taylor expansion to uncertainty quantification in wave scattering, enabling high order moment estimation for the scattered field under random boundary perturbations. Numerical examples are provided to illustrate the effectiveness of the shape Taylor expansion in achieving high order approximations.

Computation of shape Taylor expansions

TL;DR

This work develops a unified, recurrence-based framework to compute shape Taylor expansions of arbitrary order for 2D acoustic scattering under sound-soft, sound-hard, impedance, and transmission boundaries. Leveraging boundary integral equations and potential theory, it provides explicit recurrence formulas for boundary data and a method to evaluate higher-order normal derivatives, enabling high-fidelity approximations of the scattered field under boundary perturbations. The approach is extended to uncertainty quantification by deriving moment expansions for random boundary perturbations, with error terms scaling as for even , and demonstrated through numerical examples on circles and ellipses. The results indicate that higher-order shape Taylor expansions improve accuracy for moderately large perturbations and enable reliable moment estimations, with potential extensions to 3D acoustic and electromagnetic scattering.

Abstract

Shape derivative is an important analytical tool for studying scattering problems involving perturbations in scatterers. Many applications, including inverse scattering, optimal design, and uncertainty quantification, are based on shape derivatives. However, computing high order shape derivatives is challenging due to the complexity of shape calculus. This work introduces a comprehensive method for computing shape Taylor expansions in two dimensions using recurrence formulas. The approach is developed under sound-soft, sound-hard, impedance, and transmission boundary conditions. Additionally, we apply the shape Taylor expansion to uncertainty quantification in wave scattering, enabling high order moment estimation for the scattered field under random boundary perturbations. Numerical examples are provided to illustrate the effectiveness of the shape Taylor expansion in achieving high order approximations.

Paper Structure

This paper contains 15 sections, 8 theorems, 93 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Recurrences of shape derivatives
  3. Computation of the shape Taylor expansion
  4. Sound-soft boundary
  5. Sound-hard boundary
  6. Impedance and transmission boundaries
  7. Evaluating normal derivatives
  8. Random boundary scattering problem
  9. Numerical examples
  10. Numerical implementation
  11. Sound-hard boundary
  12. Sound-soft boundary
  13. Penetrable scattering
  14. Moment estimations of a random field
  15. Conclusion

Key Result

Theorem 2.1

Let $u_{\rm tot}$ be the solution of scattering_eqn2 and $u$ be the corresponding scattered field. Suppose the $N$th order shape derivative of $u$ w.r.t. $\mathbf{v}_{[N]}$ satisfies with boundary conditions on $\boldsymbol{\Gamma}$ given by one of the following: where $\delta_{\mathbf{v}_{[N]}}u = u$ for $N=0$. Then the $N+1$th order shape derivative $\delta_{\mathbf{v}_{[N+1]}}u$ also satisfie

Figures (7)

  • Figure 1: Results for the scattering of a perturbed circle with sound-hard boundary condition.
  • Figure 2: Results for the scattering of a perturbed ellipse with sound-soft boundary condition.
  • Figure 3: Illustration for the scattering of a perturbed circle with transmission boundary condition.
  • Figure 4: Approximation errors of the shape Taylor expansion for the transmission scattering problem at observation points with different radii.
  • Figure 5: Scattering with a random boundary perturbation under the impedance boundary condition
  • ...and 2 more figures

Theorems & Definitions (13)

  • Theorem 2.1: Impenetrable cases
  • Theorem 2.2: Penetrable case
  • Theorem 3.1
  • Remark 3.2
  • proof
  • Corollary 3.3
  • Corollary 3.4
  • Theorem 3.5
  • proof
  • Remark 3.6
  • ...and 3 more