Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Acoustic shape optimization using energy stable curvilinear finite differences

Gustav Eriksson, Vidar Stiernström

TL;DR

This work develops a gradient-based PDE-constrained shape optimization framework for the acoustic wave equation using energy-stable, high-order SBP finite differences on multiblock curvilinear grids. Geometry is encoded by a coordinate mapping from a reference domain, enabling optimization of the bottom boundary while computing gradients efficiently through an adjoint method and ensuring dual consistency of the semi-discrete scheme. The contributions include a rigorous energy-stability analysis, a DO/OD equivalence at the semi-discrete level, and three numerical experiments (bathymetry reconstruction and horn-shape optimization) that demonstrate accurate gradients and practical performance in time-domain settings. The approach offers a path toward accurate, scalable PDE-constrained design in acoustics with potential extensions to three dimensions and time-adjoint formulations.

Abstract

A gradient-based method for shape optimization problems constrained by the acoustic wave equation is presented. The method makes use of high-order accurate finite differences with summation-by-parts properties on multiblock curvilinear grids to discretize in space. Representing the design domain through a coordinate mapping from a reference domain, the design shape is obtained by inverting for the discretized coordinate map. The adjoint state framework is employed to efficiently compute the gradient of the loss functional. Using the summation-by-parts properties of the finite difference discretization, we prove stability and dual consistency for the semi-discrete forward and adjoint problems. Numerical experiments verify the accuracy of the finite difference scheme and demonstrate the capabilities of the shape optimization method on two model problems with real-world relevance.

Acoustic shape optimization using energy stable curvilinear finite differences

TL;DR

This work develops a gradient-based PDE-constrained shape optimization framework for the acoustic wave equation using energy-stable, high-order SBP finite differences on multiblock curvilinear grids. Geometry is encoded by a coordinate mapping from a reference domain, enabling optimization of the bottom boundary while computing gradients efficiently through an adjoint method and ensuring dual consistency of the semi-discrete scheme. The contributions include a rigorous energy-stability analysis, a DO/OD equivalence at the semi-discrete level, and three numerical experiments (bathymetry reconstruction and horn-shape optimization) that demonstrate accurate gradients and practical performance in time-domain settings. The approach offers a path toward accurate, scalable PDE-constrained design in acoustics with potential extensions to three dimensions and time-adjoint formulations.

Abstract

A gradient-based method for shape optimization problems constrained by the acoustic wave equation is presented. The method makes use of high-order accurate finite differences with summation-by-parts properties on multiblock curvilinear grids to discretize in space. Representing the design domain through a coordinate mapping from a reference domain, the design shape is obtained by inverting for the discretized coordinate map. The adjoint state framework is employed to efficiently compute the gradient of the loss functional. Using the summation-by-parts properties of the finite difference discretization, we prove stability and dual consistency for the semi-discrete forward and adjoint problems. Numerical experiments verify the accuracy of the finite difference scheme and demonstrate the capabilities of the shape optimization method on two model problems with real-world relevance.
Paper Structure (21 sections, 5 theorems, 140 equations, 11 figures, 1 table)

This paper contains 21 sections, 5 theorems, 140 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem setup
  3. The forward problem
  4. Continuous analysis
  5. One-dimensional SBP operators
  6. Two-dimensional operators on $\Omega^+$
  7. Curvilinear operators on $\Omega^-_p$
  8. SBP-P-SAT discretization
  9. The discrete optimization problem
  10. Shape parameterization and regularization
  11. The adjoint equation method
  12. Dual consistency
  13. Temporal discretization
  14. Optimization algorithm
  15. Numerical experiments
  16. ...and 6 more sections

Key Result

Lemma 1

The SBP-P-SAT scheme eq: disc_ODE_system is stable.

Figures (11)

  • Figure 1: Boundary conditions of the bathymetry problem. The dotted line indicate the interface between $\Omega^+$ and $\Omega^-_p$.
  • Figure 2: Grid of the circular domain. The different colors of the grid indicate the block decomposition.
  • Figure 3: Discretization of bathymetry problem with parameterization of the seabed. The different colors of the grid indicate the block decomposition. The cross ($\bm{\times}$) and diamond ($\bm{\Diamond}$) indicate the location of the source and the receiver, respectively.
  • Figure 4: Relative error between gradient approximated using finite differences and gradient computed using the adjoint method for varying CFL constants $k$.
  • Figure 5: Shape of the seabed after 0, 5, 20, and 177 iterations.
  • ...and 6 more figures

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 5 more