Student: Midan Kaplan

Supervisor: Prof. Yaniv Brik :abstrct High fidelity modeling and integral equation solvers for acoustic scattering by complex structures Numerical simulations of acoustic wave interactions are of importance to many fields of research that range from oceanography to medical imaging. The simulated systems are often acoustically large, geometrically complex, and are immersed in an unbounded domain. The latter motivates the modeling of such systems using integral equation (IE) formulations, which inherently account for the radiation into the infinite background domain using the Green’s function and have unknowns defined only on/in the anomalous region. To enable the numerical solution of IEs, which includes dense system matrices, their efficient representation is required. For complex systems, the modeling may involve different governing equations for the various system parts, leading to hybrid IEs (HIEs) with different kernels. It is, therefore, desirable to use compression schemes that are of kernel-independent implementation and of universal asymptotic performance that is geometry and kernel indifferent. This work is on the development of a framework for the fast iterative solution of HIE for acoustic scattering. In its core is an algebraic compression engine that relies on a hierarchical matrix structure, in which compressible matrix blocks are economically expressed using their butterfly (BF) approximation – a multilevel extension of the low-rank approximation that provides quasi-linear 𝑂(𝑁log2𝑁) computational costs. This engine can be used in a straightforward manner for accelerating the solution of simple operator IEs, such as the volume IE (VIEs) [1] that describes the acoustic scattering from inhomogeneous acoustical objects. For formulations that involve chain operators, the kernel independence allows for separate compression of each operator. Such, for example is thin inertial surface IE formulation of the scattering by thin elastic shells, which involves chain application of operators as part of its regularization [2]. The compression strategy and engine are tailored to and employed for the efficient representation of coupling operators in HIE formulations [3]. The compression engine is highly suitable for the modeling of large arrays of subwavelength scatterers described as (bi-anisotropic) particles [4]. The incorporation of these models into the HIE model concludes the work. In the talk, the framework architecture and development will be reviewed. For the various IE formulations, the performance analysis and examples that validate the solver and demonstrate its error controllability and superior asymptotic performance will be presented. Examples of large and complex systems and experimental setups of practical interest will be shown. [1] M. Kaplan and Y. Brick, “Fast iterative integral equation solver for acoustic scattering by inhomogeneous objects using the butterfly approximation,” IEEE Trans. Ultrason. Ferroelect. Freq. Control., vol. 69, no. 5, pp. 1794–803, Mar. 2022. [2] M. Kaplan, et al., “Fast boundary element method solver for acoustic scattering by thin elastic shells modeled as inertial surfaces,” J. Sound Vib., vol. 637, p. 119838, Sep. 2026. [3] M. Kaplan and Y. Brick, “Fast hybrid integral equation solver for acoustic scattering,” J. Sound Vib., vol. 634, p. 119773, July 2026. [4] D. Last and Y. Brick, “Fast acoustic scattering analysis of finite and irregular Willis coupling metamaterial structures,” Available at SSRN 4940036.