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A fast solver for many-particle elastic scattering in layered media
Jinrui Zhang, Yixiao He, Jun lai
TL;DR
Many-particle elastic scattering in layered media is difficult because strong particle interactions and interface effects make repeated accurate solves expensive. The paper combines Sommerfeld representations, boundary-integral scattering matrices, and fast multipole acceleration to build a solver for general-shaped particles in two and three dimensions. Numerical experiments demonstrate accuracy, efficiency, and flexibility in direct and inverse scattering, including three-dimensional and time-reversal applications.
Problem
Accurate repeated simulation is challenging when dense, wavelength-scale particles interact strongly with a layered interface, limiting effective-medium approximations and making direct discretization expensive.
Method
The solver couples Sommerfeld integral representations of layered fields with high-order boundary-integral scattering matrices for arbitrary-shaped particles and fast multipole multiple-scattering acceleration.
Results
Numerical experiments demonstrate accuracy, efficiency, and flexibility for direct and inverse scattering, including noncircular particles, large configurations, three-dimensional media, and layered time-reversal focusing.
Takeaways & Limitations
The framework supports fast elastic-scattering simulations and inverse applications such as selective focusing of buried particles in layered media.
Takeaways & Limitations
The method requires modest particle separation, and bookkeeping becomes more involved when many distinct prototype shapes are present.
Abstract
from arXiv · showhide
This paper proposes a fast solver for time-harmonic elastic scattering by multiple particles embedded in layered media, with either Dirichlet or Neumann boundary conditions imposed on the particle surfaces. Such problems arise in many important applications, including composite material optimization, nondestructive testing, and subsurface imaging. They are computationally challenging because of strong multiple scattering interactions among the particles and the layered interface. The proposed method represents the layered medium contribution through Sommerfeld integrals and couples this representation with a well-posed boundary integral formulation for the particle scattering problem. High-order integral equation discretization and scattering matrix are used to handle particles of general shape, while multiple scattering theory provides an efficient description of particle interactions. To reduce the cost for large particle systems, the resulting multiple scattering computation is further accelerated by the fast multipole method. The main formulation is developed in both two and three dimensions. Numerical experiments for rigid and traction-free particles validate the accuracy of the formulation, and demonstrate its flexibility in both direct scattering simulations and inverse scattering applications.
1 Introduction
Dense particle distributions make layered elastic scattering expensive because strong multiple scattering and long-range interface-mediated interactions must be resolved accurately. The paper develops a unified fast framework combining layered Green’s functions, general-particle scattering matrices, and fast multipole acceleration.
- Motivation: Dense particles comparable to the wavelength produce strong multiple scattering that effective-medium approximations may not capture.The full elastic scattering problem may need repeated solution during optimization, uncertainty quantification, or inversion.
- Motivation: Direct domain discretization is prohibitively expensive because it must resolve local particle geometry and long-range interactions mediated by the layered background.Fast solvers must retain boundary-integral accuracy while scaling efficiently with particle count.
- Method: The solver represents layered-background responses with Sommerfeld integrals that account for compressional- and shear-wave reflection and transmission.This representation explicitly handles the planar interface contribution.
- Method: Scattering matrices and high-order boundary-integral discretization resolve particles of non-circular shape while separating local particle scattering from global interactions.The framework is designed for complex particle geometries and large-scale configurations.
- Method: Fast multipole acceleration extends multiple-particle scattering techniques from homogeneous elastic media to layered elastic media.The construction is compatible with extensions to more general stratified backgrounds through response matrices.
2 Formulation of layered elastic scattering
The formulation models well-separated impenetrable particles in the lower layer, with fields governed by the Navier equation and continuity conditions across the interface. It supports rigid or traction-free particle boundaries and imposes radiation behavior through compressional and shear wave components.
- Geometry and fields: The two-dimensional problem contains M well-separated impenetrable particles in the lower half-space of a two-layer medium.The upper and lower regions are separated by the flat interface Γ0.
- Geometry and fields: The total displacement is the sum of the layered background field and the particle-generated scattered field.The background includes reflected and transmitted fields in the absence of particles.
- Governing equations: The displacement satisfies the Navier equation outside the particles, with piecewise constant Lamé parameters in the two layers.The formulation assumes µ > 0 and λ + µ > 0, with density normalized to one.
- Boundary conditions: Displacement and traction are continuous across Γ0, while particle boundaries impose either Dirichlet or Neumann conditions.The scattered field boundary condition is the negative incident/background contribution on the particle surfaces.
- Wave decomposition: The scattered field decomposes into compressional and shear components with wavenumbers κp and κs, and satisfies the Kupradze–Sommerfeld radiation condition.The compressional and shear wavenumbers are determined by the angular frequency and Lamé parameters.
- Assumptions: Particles must be separated by at least 10% of the particle size, and only finitely many distinct shapes are assumed.These assumptions support accurate, stable multiple-scattering representations and reusable single-particle scattering matrices.
3 The Sommerfeld integral for elastic layered media
The Sommerfeld formulation expresses layered elastic fields as spectral integrals, enabling planar-interface conditions to be enforced mode by mode. Unknown upward and downward spectral densities represent the interface-coupled fields generated by sources and particles.
- Spectral representation: The Sommerfeld representation expresses layered elastic fields as spectral integrals in the transverse variable.For planar interfaces, transmission conditions can be enforced independently in the Fourier domain.
- Free-space Green’s function: The two-dimensional free-space elastic Green’s function is converted to Sommerfeld form using Fourier transformation and contour integration.The resulting Green’s tensor is expressed through a Sommerfeld integral.
- Free-space Green’s function: The Sommerfeld integral is conditionally convergent and is valid away from the source and target having equal vertical coordinates.Its spectral representation is used for the layered-medium construction.
- Interface coupling: Interface displacement and traction continuity determines the spectral densities, with conditions imposed independently for each Fourier mode ξ.The resulting linear system uses the traction operator on the interface.
- Interface coupling: The Sommerfeld fields are coupled to multiple-particle scattering in the lower half-space to form the layered multiple-scattering system.This coupling combines interface spectral representations with particle scattering formalism.
4 Wave scattering for multiple elastic particles
The section develops a multiple-scattering framework that represents each particle through scattering matrices and couples particle interactions using multipole translations. It extends this framework to arbitrarily shaped particles and accelerates the resulting systems with FMM.
- 4.1 Scattering of a single disk: The homogeneous-medium solver uses incoming and outgoing elastic-wave expansions, with a scattering matrix mapping incoming coefficients to outgoing coefficients.For disks, the matrix is obtained by enforcing rigid or traction-free boundary conditions mode by mode, with expansions truncated to modes from −p through p.
- 4.2 Scattering of multiple disks: For multiple particles, each particle’s incoming field combines the incident wave with fields scattered by all other particles through multipole-to-local translation operators.Eliminating incoming coefficients yields a linear system involving only outgoing coefficients.
- 4.2 Scattering of multiple disks: The fast multipole method reduces the direct multiple-scattering matrix-vector cost from O(M^2p^2) to O(Mp^2) per iteration.The iterative system uses GMRES, and a block-diagonal preconditioner based on the single-particle scattering matrix improves conditioning.
- 4.3 Scattering of arbitrarily shaped particles: Arbitrarily shaped particles are handled by enclosing each particle in a nonoverlapping disk and constructing a scattering matrix with a high-precision boundary integral solver.This permits the disk-based multiple-scattering formalism to be applied while including rigid and traction-free cases.
- 4.3 Scattering of arbitrarily shaped particles: Precomputed single-particle scattering matrices reduce degrees of freedom and produce a better-conditioned global system for the multipole unknowns.Particles must be sufficiently separated, and many distinct prototype shapes make bookkeeping more involved.
5 Multi-particle scattering in a layered medium
The layered-medium formulation couples Sommerfeld representations of the interface response with multipole expansions for particle scattering. A block linear system and its Schur-complement reduction organize this coupling, while FMM and related transforms accelerate its application.
- 5.1 Coupling layered and particle scattering: The formulation combines the Sommerfeld representation of the layered background with multipole representations of fields scattered by well-separated particles.The particles are embedded in the lower half-space, and the interface is represented through spectral density variables.
- 5.2 The full linear system: The diagonal block represents independent spectral interface equations, while off-diagonal blocks map particle multipoles to the interface and Sommerfeld fields back to particles.The coupling blocks are the particle–interface communication operators in the global system.
- 5.5 Schur-complement solver: Applying GMRES to the Schur complement removes the interface-density unknowns and yields a better-conditioned system involving only particle multipole coefficients.The Schur complement has a physical interpretation in terms of the layered Green’s function.
- 5.3 Sommerfeld-to-local conversion: Direct Sommerfeld-to-local conversion costs O(MNS(4p+2)), where M is particle count, NS is quadrature-point count, and p is expansion order.A NUFFT-based scheme is noted for high-frequency problems with many inclusions.
- 5.4 Multipole-to-Sommerfeld conversion: The multipole-to-Sommerfeld conversion costs O((4p+2)NSM) directly and can also be accelerated using the NUFFT.Mode-by-mode spectral formulas provide the interface field and traction contributions from each multipole coefficient.
6 Extension to three-dimensional layered multiple elastic scattering
The 3D extension combines Sommerfeld representations, spherical-wave translations, interface conditions, and multiple-scattering coefficients into a block-structured formulation. Numerical evaluation uses polar-coordinate quadrature with radial contour deformation.
- 3D formulation: The 3D implementation uses the elastic Green’s tensor in Sommerfeld form together with multipole-to-Sommerfeld and Sommerfeld-to-multipole translation operators.These ingredients extend the layered multiple-scattering framework to three dimensions.
- Numerical evaluation: The 3D Sommerfeld integral is evaluated in polar coordinates using radial contour deformation, Nr Gauss–Legendre radial points, and Nθ equally spaced angular points.The radial variable is parameterized by kρ = λ(t), with λ(t) = t − iα tanh(βt).
- 3D formulation: Continuity of displacement and traction across the interface is enforced alongside Dirichlet or Neumann particle conditions in a block-structured 2×2 linear system.The system couples interface and particle unknowns.
- Block system: The interface unknown σt denotes spectral density, while τ collects outgoing coefficients on the particles; A and D represent interface and particle self-interactions.A is built from Sommerfeld expressions and interface continuity, whereas D follows the homogeneous-medium formulation.
- Coupling operators: Spectral representations interpret fields as integrals of plane-wave factors, whose compressional and shear components are translated into spherical waves to construct coupling matrix C.The corresponding spherical-wave components also provide the entries of coupling block B.
7 Numerical experiments
Numerical experiments test accuracy, convergence, scalability, boundary-condition behavior, interface continuity, and inverse-scattering focusing for the proposed solver in two and three dimensions.
- Experimental setup: The experiments use randomly distributed, well-separated copies of parameterized particles, with two-dimensional examples using one shape family and three-dimensional examples using spherical particles.More complicated particle boundaries are supported, although their scattering-matrix precomputation is more involved.
- Accuracy and convergence: Nearly 200 GMRES iterations are required by the direct method, while the scattering-matrix method converges much faster for the same accuracy.Excluding scattering-matrix precomputation, the proposed solver is nearly 50 times faster than the direct method at matched accuracy.
- Accuracy and convergence: The scattering-matrix method is nearly insensitive to particle shape and maintains high accuracy more quickly than the direct method.At equal boundary discretization, starfish-shaped particles are less accurate than pear-shaped particles in the direct method.
- Frequency and boundary conditions: For fixed particle count, GMRES iterations grow roughly linearly with angular frequency, while high-frequency Neumann cases require larger Nterm and tighter GMRES tolerances.The reported accuracy is lower for high-frequency problems with Neumann boundary conditions.
- Large systems and physical behavior: With 400 closely spaced particles, the scattered field behaves as if a large effective obstacle were present, particularly for rigid particles.Plane-wave experiments likewise show that rigid clusters barely transmit the incident field, whereas traction-free clusters permit deeper penetration.
- Validation and three-dimensional tests: The computed total field is continuous across the interface and has magnitude of order 10^-5 on particle surfaces for the tested Dirichlet case.A three-dimensional test with 100 buried particles again gives an almost-zero field on the particles.
- Inverse scattering: Time-reversal eigenfunctions selectively focus incident waves on unknown particles, with six significant eigenvalues for three rigid particles and reduced resolution for deeper targets.Traction-free particles show similar focusing behavior, while limited depth aperture makes y-direction focusing less sharp.
8 Conclusion
The paper develops a fast layered-medium solver that combines Sommerfeld representations, particle scattering matrices, and multipole acceleration. Numerical experiments support its accuracy, efficiency, flexibility, and use in direct, inverse, three-dimensional, and time-reversal scattering.
- 8 Conclusion: The solver combines Sommerfeld integral representations, scattering matrices for arbitrary-shaped particles, and multipole acceleration for layered multiple scattering.The formulation couples interface spectral densities and particle multipole coefficients through translation operators and solves the resulting structured system with GMRES.
- 8 Conclusion: The method simulates scattering by many rigid and traction-free particles under point-source and plane-wave incidence.Its particle and layered-medium representations are coupled within a unified multiple-scattering framework.
- 8 Conclusion: Numerical experiments demonstrate accuracy, efficiency, and flexibility for direct and inverse scattering problems.The reported applications include layered time-reversal focusing.
- 8 Conclusion: The solver achieves high accuracy for noncircular particles, scales efficiently to large configurations, and extends naturally to three-dimensional layered elastic scattering.These results indicate that the framework supports varied particle geometries and dimensional settings.