Acoustic Shape Optimization: Methods & Applications

Updated 21 November 2025

Acoustic shape optimization is a computational discipline that designs object geometries to control wave propagation using PDE-constrained optimization.
It employs techniques such as adjoint sensitivity analysis, finite differences, level-set methods, and machine-learned surrogates to optimize acoustic metrics.
Applications span aeroacoustics, medical ultrasound, and metamaterial design, driving improvements in noise reduction, sound insulation, and energy efficiency.

Acoustic shape optimization is the computational and mathematical discipline concerned with the systematic design of object boundaries and internal geometries to optimally control acoustic wave propagation, attenuation, scattering, insulation, or acoustic power transfer. The design objective is typically formulated as a functional of the acoustic field—pressure, energy, transmission, or derived metrics such as sound pressure level (SPL) or transmission loss (STL)—subject to the underlying partial differential equations (PDEs) governing wave propagation, possibly coupled to material, structural, or flow physics. This rapidly growing research area is foundational for high-performance acoustic devices in aerodynamics, architectural acoustics, medical ultrasound, vibro-acoustic control, phononic band gap materials, and microfluidics.

1. Problem Formulation and Objective Classes

Acoustic shape optimization problems are cast as PDE-constrained optimization, where the decision variables parameterize the geometry (boundary or topology), and the objective is a functional of the acoustic state.

Mathematical Statement

State Equation: Frequently, the linear or nonlinear time-harmonic Helmholtz equation is used:

$-\nabla\cdot\left(c^{-2}(x)\nabla p\right) - \omega^2 p = 0$

with $c(x)$ denoting the (possibly spatially varying) sound speed.

Design Variables:
- Explicit boundary representations, e.g., B-splines, NURBS (control points) (Takahashi et al., 2021, Muhr et al., 2017, Eriksson et al., 2023)
- Implicit functions, e.g., level sets (Noguchi et al., 2023, Berggren et al., 14 Mar 2024)
- Material distribution or phase fields (Garcke et al., 2021, Beilina et al., 2017)
- Discrete structural parameters (resonators, attachment grids, feature widths) (Poggetto et al., 2022, Vondřejc et al., 2016)
Objective Functionals:
- Wave focusing/tracking: Minimize $J=\int |u-u_d|^2$ in a focal region (Nikolic et al., 2015, Muhr et al., 2017)
- Scattering/absorption: Minimize (or maximize) total/radiated energy at receivers or far field (Takahashi et al., 2021, Poggetto et al., 2022)
- Noise/sound insulation: Minimize near- or far-field SPL, STL, or broadband energy (Hamedi et al., 2023, Hamedi et al., 2023, Hadizadeh et al., 22 Dec 2024, Bach et al., 2020)
- Band-gap/maximal attenuation: Maximize the width of frequency intervals with suppressed propagation (Vondřejc et al., 2016, Poggetto et al., 2022)
- Microfluidics/streaming suppression: Reduce secondary flow or enable desired acoustophoretic effects (Bach et al., 2020)
- Multi-physics objectives: Simultaneous aerodynamic and acoustic optimality (Hadizadeh et al., 22 Dec 2024, Luo et al., 17 Sep 2025)

Constraints may include aerodynamical performance (e.g., preserving lift/drag (Hamedi et al., 2023, Hamedi et al., 2023, Hadizadeh et al., 22 Dec 2024)), volume fraction or manufacturing restrictions, and mesh or regularity constraints to avoid nonphysical or nonmanufacturable designs.

2. Methodological Frameworks

A variety of computational frameworks have been developed for acoustic shape optimization, tailored to problem scale, physical fidelity, and design topology.

Direct and Gradient-Based Methods

Adjoint-based Shape Sensitivity: Analytical or algorithmic derivation of the shape derivative using Lagrangian or Hadamard calculus, leading to efficient gradient-based optimization (Takahashi et al., 2021, Nikolic et al., 2015, Eriksson et al., 2023, Muhr et al., 2017, Vondřejc et al., 2016, Garcke et al., 2021). Discrete adjoint approaches ensure consistency with the numerical scheme (Eriksson et al., 2023).
Finite Differences and Differentiable Programming: Automatic differentiation (AD) through solver computational graphs enables direct gradient evaluations where adjoint formulas are impractical (Borrel-Jensen et al., 14 Nov 2025). Randomized finite difference estimators reduce the number of expensive forward-solves by projecting the boundary gradient onto random directions.
Phase-Field and Level-Set Methods: Topological and sharp/diffuse interface optimization using Ginzburg–Landau penalization or variational perimeter control, naturally handling nucleation or merging of geometric features (Garcke et al., 2021, Noguchi et al., 2023, Berggren et al., 14 Mar 2024).
Isogeometric Analysis (IGA): NURBS/spline geometry representation enables exact CAD-model fidelity and seamless use of geometrical control points as optimization variables (Takahashi et al., 2021, Muhr et al., 2017).
CutFEM and Mesh Methods: Fixed-mesh strategies (e.g., Cut Finite Element Method) avoid remeshing during geometry evolution and support fully discrete shape calculus, crucial for 3D or coupled problems with boundary-layer losses (Berggren et al., 14 Mar 2024).

Gradient-Free / Surrogate-Based Optimization

Mesh Adaptive Direct Search (MADS): Black-box, deterministic, derivative-free approaches robust to noisy, non-differentiable, or chaotic objectives (e.g., from unsteady or turbulent simulations) (Hamedi et al., 2023, Hamedi et al., 2023).
Evolutionary Algorithms & Ensemble Kalman Filters: Population-based search with regularization, multi-objective scalarization, and variance-based step-size control for complex shape spaces (Luo et al., 17 Sep 2025, Sun et al., 2021).
Machine-Learned Surrogates: Deep neural networks or graph neural networks (GNNs) as trained surrogate models for rapid evaluation of acoustic metrics, enabling multi-objective, high-throughput exploration (Hadizadeh et al., 22 Dec 2024, Sun et al., 2021).

3. Geometry Representation and Parameterization

Effective optimization relies critically on shape parameterization and computational geometry.

Representation	Description	Selected Sources
Explicit Spline/NURBS	Boundary surfaces/curves via control points	(Takahashi et al., 2021, Muhr et al., 2017)
Fourier/Modal	Periodic design variables (e.g., thickness, resonance)	(Poggetto et al., 2022, Vondřejc et al., 2016)
Level Set	Implicit zero isosurface in fixed or adapted domain	(Noguchi et al., 2023, Berggren et al., 14 Mar 2024)
Phase Field	Diffuse material field for sharp/diffuse topology	(Garcke et al., 2021)
Mesh Node Coordinates	Boundary/interior mesh nodes as degrees of freedom	(Borrel-Jensen et al., 14 Nov 2025)
Surrogate Decoding	Surrogate-to-shape mapping via generative models	(Hadizadeh et al., 22 Dec 2024, Sun et al., 2021)

4. Notable Applications

Acoustic shape optimization has been applied across diverse areas:

Aeroacoustics and Flow Noise: Airfoil trailing-edge optimization for SPL reduction at fixed lift/drag using LES+gradient-free search and surrogate GNNs, achieving >10 dB SPL reduction and up to 30% drag reduction (Hamedi et al., 2023, Hamedi et al., 2023, Hadizadeh et al., 22 Dec 2024, Luo et al., 17 Sep 2025).
Channel Flow and Streaming Control: Channel shape design to suppress acoustic streaming without loss of acoustic pressure or particle radiation force. Microfluidic handling of nm-scale particles becomes viable due to two-orders-of-magnitude reduction in streaming (Bach et al., 2020).
Functional Metamaterials and Panels: Robust sound insulation panels engineered via Bragg (thickness modulation) or local resonance (mass-spring grids), optimized for broadband STL under angle/diffuse excitation (Poggetto et al., 2022).
Bandgap Structures: Maximization of phononic bandgaps in periodic composites through inclusion shape tuning, employing homogenized effective-mass models and eigenvalue sensitivity analysis (Vondřejc et al., 2016).
Ultrasound Focusing and Medical Devices: Gradient-based, IGA- or FEM-driven design of focusing acoustic lenses governed by nonlinear (Westervelt) models, targeting focal region tracking (Nikolic et al., 2015, Muhr et al., 2017).
Topological Optimization and Programmable Devices: Phase-field or level set approaches for full shape and topology optimization, enabling programmable multi-state acoustic lenses or adaptively focusing devices under deformation (Garcke et al., 2021, Noguchi et al., 2023).
Industrial and Non-Ideal Problems: 3D viscothermal design (e.g., phase plugs of compression drivers) via CutFEM and discrete shape calculus, matching prescribed idealized frequency responses in high-fidelity simulations (Berggren et al., 14 Mar 2024).
Acoustic Material Inverse Design: Direct mapping of desired STL spectrum to resonator geometry using DNN surrogates, facilitating real-time inverse design and hybridization with evolutionary optimizers (Sun et al., 2021).

5. Algorithmic and Computational Aspects

The computational cost of acoustic shape optimization is dominated by forward PDE/CFD solves, adjoint solutions, and geometry updates. Advanced workflows leverage:

Parallelization: Gradient-free (MADS, Ensemble Kalman) search steps are parallelized, with the wall-clock per iteration set by a single expensive simulation, rather than total number of design variables (Hamedi et al., 2023, Hamedi et al., 2023, Luo et al., 17 Sep 2025).
Surrogate Models: Pre-trained GNN or DNNs reduce design-evaluation time by $10^3$ compared to direct solvers, with only minor loss in accuracy (Hadizadeh et al., 22 Dec 2024, Sun et al., 2021).
Adaptive Meshing and Refined Gradients: A posteriori error indicators localize mesh refinement where gradients are large, increasing optimization convergence and solution accuracy (Beilina et al., 2017, Borrel-Jensen et al., 14 Nov 2025).
Fully Discrete Shape Calculus: Discrete differentiation matches implementation with mathematical sensitivity, crucial in CutFEM for complex boundaries (Berggren et al., 14 Mar 2024).
Regularization and Physical Constraints: TV or area/volume preserving penalties, manufacturability bounds, and physical constraints (e.g., lift/drag, attached flow, mesh quality) maintain feasibility during optimization (Hamedi et al., 2023, Eriksson et al., 2023, Luo et al., 17 Sep 2025, Borrel-Jensen et al., 14 Nov 2025).

6. Representative Advances and Case Studies

Aeroacoustic noise reductions of up to $12$ dB were achieved for deep cavity and tandem cylinder flows at low $Re$ by MADS+FR solvers, with lift/drag maintained (Hamedi et al., 2023, Hamedi et al., 2023).
GNN surrogates for airfoil Pareto-optimization reduced $OASPL$ by $13.9\%$ (15.82 dBA), with $7.2\%$ increase in lift and $10^3\times$ speedup over CFD (Hadizadeh et al., 22 Dec 2024).
Suppression of acoustic streaming by $~100\times$ (volume with $|v_2|<0.05 v_2^0$ up from $4\%$ to $96\%$ of the channel) in shape-optimized microchannels, without reducing acoustic pressure or radiation force (Bach et al., 2020).
Full 3D shape optimization in Helmholtz–BEM with isogeometric NURBS offers rapid, surface-only control point updates, with accurate gradient formulae, demonstrated for scatterers, reflectors, and resonators (Takahashi et al., 2021).
Topology-optimized phase-field and level set designs yielded multi-state programmable acoustic devices robust to structural deformation, with sharply localized or dual-state pressure distribution (Garcke et al., 2021, Noguchi et al., 2023).
Aperiodic, locally resonant panel geometries achieved subwavelength attenuation robust under diffuse sound incidence, mimicking bioinspired cloaking surfaces (Poggetto et al., 2022).
Multi-objective ensemble Kalman methods in trailing edge noise minimization discovered bevel-reduction mechanisms that weaken both mean drag and broadband noise via modification of coherent structures and turbulence (Luo et al., 17 Sep 2025).

7. Current Challenges and Research Directions

Acoustic shape optimization remains an open frontier for several interlinked domains:

High-Fidelity Multi-Physics Coupling: Full integration of CFD/CSD, nonlinear acoustics, and multi-scale physics with robust adjoint or auto-differentiable pipelines (Borrel-Jensen et al., 14 Nov 2025, Hadizadeh et al., 22 Dec 2024).
Scalability to 3D and Manufacturing Constraints: Efficient, robust algorithms handling real-world complexity, large numbers of variables, and manufacturing fidelity (Berggren et al., 14 Mar 2024, Muhr et al., 2017).
Surrogates and Machine-Learning Integration: Training and deployment of reliable, transferable ML surrogates for gradient, topology, and parameter discovery in high-dimensional design spaces with minimal data (Hadizadeh et al., 22 Dec 2024, Sun et al., 2021).
Uncertainty Quantification: Incorporating parameter and model uncertainty, stochastic boundaries, and robust optimization in the presence of noise or fabrication tolerances.
Automatic Differentiation and Discrete Sensitivities: Further development of autodiff-enabled PDE solvers for free-boundary and topology optimization, including mesh and geometry parameterization (Borrel-Jensen et al., 14 Nov 2025, Eriksson et al., 2023).
Physics-Inspired Regularization: Applying biologically and physically inspired geometric regularization, e.g., for structural as well as acoustic optimality (Poggetto et al., 2022).