arXiv subjects

Elwin van 't Wout

Publications and source records attributed to Elwin van 't Wout.

14 recordsLinked to original sources

arXiv2026

Robust Hierarchical Matrix Compression of Acoustic Volume and Boundary Integral Operators

Discretizing integral formulations of the Helmholtz equation yields dense linear systems. Hence, simulating acoustic models at larger scales or higher frequencies is typically constrained by memory capacity. Fast algorithms, such as hierarchical matrix compression, reduce the memory footprint substantially while controlling the approximation error in matrix-vector multiplications. However, the commonly used Adaptive Cross Approximation suffers from early-convergence problems, where the iterative construction of low-rank decompositions stops before reaching the targeted error tolerance. This failure arises when the error estimator does not capture significant components of the matrix structure under partial pivoting. This manuscript proposes a new diagonal convergence criterion, additional matrix elements for the pivoting strategy, an extended admissibility condition, and a sustained convergence check to improve the robustness of hierarchical matrix compression. These modifications improve compression reliability without increasing memory. We tested our compression strategy on various discretized volume and boundary integral operators. The computational results show that our approach successfully compresses all benchmark matrices within predefined tolerances, thereby resolving the early-convergence issues encountered in standard algorithms. This robust matrix compression was achieved at the same memory footprint as alternative compression strategies. Furthermore, a complexity analysis shows log-linear memory scaling with mesh refinement at constant frequency. Finally, we successfully applied our robust matrix compression algorithm to a coupled system of volume and boundary integral operators that models transcranial ultrasound propagation. This confirms the feasibility of our robust algorithm to accelerate large-scale simulations with high-resolution meshes in a biomedical application.

math.NA↗

arXiv2026

Nested Volume-Surface Integral Equations for Acoustics

The simulation of high-frequency acoustic wave propagation in unbounded domains with local heterogeneous materials and high-contrast interfaces poses significant challenges to numerical methods. The volume-surface integral equation (VSIE) method is an attractive approach as it automatically satisfies the radiation condition at infinity via Green's functions, handles heterogeneous materials via Newton potentials, and models scattering at high-contrast interfaces via surface integral operators. However, its effectiveness in practical simulations has been limited by high computational costs, sensitivity to sharp interfaces, and insufficient computational verification. This study extends the applicability of VSIE by deriving integral formulations for nested heterogeneous materials with parameter jumps at interfaces. We also develop extensive benchmarks against coupled finite-element and boundary-element methods to verify the VSIE's accuracy and mesh convergence. The various benchmarks using open-source software demonstrate the effectiveness of VSIE for large-scale acoustic simulations.

math.NA↗

arXiv2026

Full-Wave Modeling of Transcranial Ultrasound using Volume-Surface Integral Equations and CT-Derived Heterogeneous Skull Data

Transcranial ultrasound therapy uses focused acoustic energy to induce therapeutic bioeffects in the brain. Ultrasound must be transmitted through the skull, which is highly attenuating and heterogeneous, causing beam distortion, reducing focal pressure, and shifting the target location. Computational models are frequently used to predict beam aberration, assess cranial heating, and correct the phase of ultrasound transducers. These models often rely on computed tomography (CT) images to build patient-specific geometries and estimate skull acoustic properties. However, the coarse voxel resolution of CT limits accuracy for differential equation solvers at ultrasound frequencies. This paper presents an efficient numerical method based on volume-surface integral equations to model full-wave acoustic propagation through heterogeneous skull bone. We show that our approach effectively simulates transcranial ultrasound, even when using the original CT voxels as the computational mesh, where the 0.5 mm voxel length is relatively coarse compared to the shortest wavelength of 3 mm. The method is validated against a high-resolution boundary element model using an averaged skull representation. Simulations using a CT-based skull model and a bowl transducer reveal significant beam distortion of 7.8 mm attributed to the skull's heterogeneous acoustical properties.

physics.med-ph↗

arXiv2025

Classifying acoustic cavitation with machine learning trained on multiple physical models

Acoustic cavitation threshold charts are used to map between acoustic parameters (mainly intensity and frequency) and different regimes of acoustic cavitation. The two main regimes are transient cavitation, where a bubble collapses, and stable cavitation, where a bubble undergoes periodic oscillations without collapse. The cavitation charts strongly depend on the physical model used to compute the bubble dynamics and the algorithm for classifying the cavitation threshold. The differences between modeling approaches become especially noticeable for resonant bubbles and when sonication parameters result in large-amplitude oscillations. This paper proposes a machine learning approach that integrates three physical models, i.e., the Rayleigh-Plesset, Keller-Miksis and Gilmore equations, and multiple cavitation classification techniques. Specifically, we classify the cavitation regimes based on the maximum radius, the acoustic Mach number, the kurtosis factor of acoustic emissions, and the Flynn criterion on the inertial and pressure functions. Four machine learning strategies were developed to predict the likelihood of the transient and stable cavitation, using equally weighted contributions from classification techniques. By solving the differential equations for bubble dynamics across a range of sonication and material parameters and applying cross-validation on held-out test data, our framework demonstrates high predictive accuracy for cavitation regimes. This physics-informed machine learning approach offers probabilistic insights into cavitation likelihood, combining diverse physical models and classification strategies, each contributing different levels of physical rigor and interpretability.

physics.comp-ph↗

arXiv2024

Modeling Global Surface Dust Deposition Using Physics-Informed Neural Networks

Paleoclimatic measurements serve to understand Earth System processes and evaluate climate model performances. However, their spatial coverage is generally sparse and unevenly distributed across the globe. Statistical interpolation methods are the prevalent techniques to grid such data, but these purely data-driven approaches sometimes produce results that are incoherent with our knowledge of the physical world. Physics-Informed Neural Networks (PINNs) follow an innovative approach to data analysis and physical modeling through machine learning, as they incorporate physical principles into the data-driven learning process. Here, we develop PINNs to reconstruct global maps of atmospheric dust surface deposition fluxes from measurement data in paleoclimatic archives for the Holocene and Last Glacial Maximum periods. We design an advection-diffusion equation to consider dominant wind directions at various latitudes, which prevents dust particles from flowing upwind. Our PINN improves on standard kriging interpolation by allowing variable asymmetry around data points. The reconstructions display realistic dust plumes from continental sources towards ocean basins following prevailing winds.

physics.geo-ph↗

arXiv2024

The Boundary Element Method for Acoustic Transmission with Nonconforming Grids

Acoustic wave propagation through a homogeneous material embedded in an unbounded medium can be formulated as a boundary integral equation and accurately solved with the boundary element method. The computational efficiency deteriorates at high frequencies due to the increase in mesh size with a fixed number of elements per wavelength and also at high material contrasts due to the ill-conditioning of the linear system. This study presents the design of boundary element methods feasible for nonconforming surface meshes at the material interface. The nonconforming algorithm allows for independent grid generation, improves flexibility, and reduces the degrees of freedom. It works for different boundary integral formulations for Helmholtz transmission problems, operator preconditioning, and coupling with finite element solvers. The extensive numerical benchmarks at canonical configurations and an acoustic foam model confirm the significant improvements in computational efficiency when employing the nonconforming grid coupling in the boundary element method.

math.NA↗

arXiv2023

Modeling frequency shifts of collective bubble resonances with the boundary element method

Increasing the number of closely-packed air bubbles immersed in water changes the frequency of the Minnaert resonance. The collective interactions between bubbles in a small ensemble are primarily in the same phase, causing them to radiate a spherically-symmetric field that peaks at a frequency lower than the Minnaert resonance for a single bubble. In contrast, large periodic arrays include bubbles that are further apart than half the wavelength, so that collective resonances have bubbles oscillating in opposite phases, ultimately creating a fundamental resonance at a frequency higher than the single-bubble Minnaert resonance. This work investigates the transition in resonance behavior using a modal analysis of a mass-spring system and a boundary element method. We significantly reduce the computational complexity of the full-wave solver to a linear dependence on the number of bubbles in a rectangular array. The simulated acoustic fields confirm the initial downshift in resonance frequency and the strong influence of collective resonances when the array has hundreds of bubbles covering more than half the wavelength. These results are essential in understanding the low-frequency resonance characteristics of bubble ensembles, which have important applications in diverse fields such as underwater acoustics, quantum physics, and metamaterial design.

physics.comp-ph↗

arXiv2022

Benchmarking preconditioned boundary integral formulations for acoustics

The boundary element method (BEM) is an efficient numerical method for simulating harmonic wave propagation. It uses boundary integral formulations of the Helmholtz equation at the interfaces of piecewise homogeneous domains. The discretisation of its weak formulation leads to a dense system of linear equations, which is typically solved with an iterative linear method such as GMRES. The application of BEM to simulating wave propagation through large-scale geometries is only feasible when compression and preconditioning techniques reduce the computational footprint. Furthermore, many different boundary integral equations exist that solve the same boundary value problem. The choice of preconditioner and boundary integral formulation is often optimised for a specific configuration, depending on the geometry, material characteristics, and driving frequency. On the one hand, the design flexibility for the BEM can lead to fast and accurate schemes. On the other hand, efficient and robust algorithms are difficult to achieve without expert knowledge of the BEM intricacies. This study surveys the design of boundary integral formulations for acoustics and their acceleration with operator preconditioners. Extensive benchmarks provide valuable information on the computational characteristics of several hundred different models for multiple reflection and transmission of acoustic waves.

math.NA↗

arXiv2022

Frequency-robust preconditioning of boundary integral equations for acoustic transmission

The scattering and transmission of harmonic acoustic waves at a penetrable material are commonly modelled by a set of Helmholtz equations. This system of partial differential equations can be rewritten into boundary integral equations defined at the surface of the objects and solved with the boundary element method (BEM). High frequencies or geometrical details require a fine surface mesh, which increases the number of degrees of freedom in the weak formulation. Then, matrix compression techniques need to be combined with iterative linear solvers to limit the computational footprint. Moreover, the convergence of the iterative linear solvers often depends on the frequency of the wave field and the objects' characteristic size. Here, the robust PMCHWT formulation is used to solve the acoustic transmission problem. An operator preconditioner based on on-surface radiation conditions (OSRC) is designed that yields frequency-robust convergence characteristics. Computational benchmarks compare the performance of this novel preconditioned formulation with other preconditioners and boundary integral formulations. The OSRC preconditioned PMCHWT formulation effectively simulates large-scale problems of engineering interest, such as focused ultrasound treatment of osteoid osteoma.

math.NA↗

arXiv2022

Benchmark problems for transcranial ultrasound simulation: Intercomparison of compressional wave models

Computational models of acoustic wave propagation are frequently used in transcranial ultrasound therapy, for example, to calculate the intracranial pressure field or to calculate phase delays to correct for skull distortions. To allow intercomparison between the different modeling tools and techniques used by the community, an international working group was convened to formulate a set of numerical benchmarks. Here, these benchmarks are presented, along with intercomparison results. Nine different benchmarks of increasing geometric complexity are defined. These include a single-layer planar bone immersed in water, a multi-layer bone, and a whole skull. Two transducer configurations are considered (a focused bowl and a plane piston), giving a total of 18 permutations of the benchmarks. Eleven different modeling tools are used to compute the benchmark results. The models span a wide range of numerical techniques, including the finite-difference time-domain method, angular-spectrum method, pseudospectral method, boundary-element method, and spectral-element method. Good agreement is found between the models, particularly for the position, size, and magnitude of the acoustic focus within the skull. When comparing results for each model with every other model in a cross comparison, the median values for each benchmark for the difference in focal pressure and position are less than 10\% and 1 mm, respectively. The benchmark definitions, model results, and intercomparison codes are freely available to facilitate further comparisons.

physics.comp-ph↗

arXiv2021

Boundary integral formulations for acoustic modelling of high-contrast media

The boundary element method is an efficient algorithm for simulating acoustic propagation through homogeneous objects embedded in free space. The conditioning of the system matrix strongly depends on physical parameters such as density, wavespeed and frequency. In particular, high contrast in density and wavespeed across a material interface leads to an ill-conditioned discretisation matrix. Therefore, the convergence of Krylov methods to solve the linear system is slow. Here, specialised boundary integral formulations are designed for the case of acoustic scattering at high-contrast media. The eigenvalues of the resulting system matrix accumulate at two points in the complex plane that depend on the density ratio and stay away from zero. The spectral analysis of the Calderón preconditioned PMCHWT formulation yields a single accumulation point. Benchmark simulations demonstrate the computational efficiency of the high-contrast Neumann formulation for scattering at high-contrast media.

math.NA↗

arXiv2021

Stable and efficient FEM-BEM coupling with OSRC regularisation for acoustic wave transmission

The finite element method (FEM) and the boundary element method (BEM) can numerically solve the Helmholtz system for acoustic wave propagation. When an object with heterogeneous wave speed or density is embedded in an unbounded exterior medium, the coupled FEM-BEM algorithm promises to combine the strengths of each technique. The FEM handles the heterogeneous regions while the BEM models the homogeneous exterior. Even though standard FEM-BEM algorithms are effective, they do require stabilisation at resonance frequencies. One such approach is to add a regularisation term to the system of equations. This algorithm is stable at all frequencies but also brings higher computational costs. This study proposes a regulariser based on the on-surface radiation conditions (OSRC). The OSRC operators are also used to precondition the boundary integral operators and combined with incomplete LU factorisations for the volumetric weak formulation. The proposed preconditioning strategy improves the convergence of iterative linear solvers significantly, especially at higher frequencies.

math.NA↗

arXiv2021

Accelerating frequency-domain numerical methods for weakly nonlinear focused ultrasound using nested meshes

The numerical simulation of weakly nonlinear ultrasound is important in treatment planning for focused ultrasound (FUS) therapies. However, the large domain sizes and generation of higher harmonics at the focus make these problems extremely computationally demanding. Numerical methods typically employ a uniform mesh fine enough to resolve the highest harmonic present in the problem, leading to a very large number of degrees of freedom. This paper proposes a more efficient strategy in which each harmonic is approximated on a separate mesh, the size of which is proportional to the wavelength of the harmonic. The increase in resolution required to resolve a smaller wavelength is balanced by a reduction in the domain size. This nested meshing is feasible owing to the increasingly localised nature of higher harmonics near the focus. Numerical experiments are performed for FUS transducers in homogeneous media in order to determine the size of the meshes required to accurately represent the harmonics. In particular, a fast \emph{volume potential} approach is proposed and employed to perform convergence experiments as the computation domain size is modified. This approach allows each harmonic to be computed via the evaluation of an integral over the domain. Discretising this integral using the midpoint rule allows the computations to be performed rapidly with the FFT. It is shown that at least an order of magnitude reduction in memory consumption and computation time can be achieved with nested meshing. Finally, it is demonstrated how to generalise this approach to inhomogeneous propagation domains.

math.NA↗

arXiv2017

Software frameworks for integral equations in electromagnetic scattering based on Calderón identities

In recent years there have been tremendous advances in the theoretical understanding of boundary integral equations for Maxwell problems. In particular, stable dual pairing of discretisation spaces have been developed that allow robust formulations of the preconditioned electric field, magnetic field and combined field integral equations. Within the BEM++ boundary element library we have developed implementations of these frameworks that allow an intuitive formulation of the typical Maxwell boundary integral formulations within a few lines of code. The basis of these developments is an efficient and robust implementations of Calderón identities together with a product algebra that hides and automates most technicalities involved in assembling Galerkin boundary integral equations. In this paper we demonstrate this framework and use it to derive very simple and robust software formulations of the standard preconditioned electric field, magnetic field and regularised combined field integral equations for Maxwell.

math.NA↗