Fast Boundary Element Method Research Articles

An advanced fast multipole dual boundary element method for analyzing multiple cracks propagation

A new fast multipole dual boundary element method (FMDBEM) is developed to analyze multiple crack propagations. To evaluate accurately the stress fields around the crack tip, a variable-order asymptotic element (VAE) is first proposed to express the singular behavior. This VAE is also suitable for the V-notches with different opening angles, requiring only minor adjustments of stress exponents. Then the VAE is introduced into the FMDBEM framework, where several singularity problems of integrals are solved. Finally, the FMDBEM with VAEs, combined with an adaptive scheme, is used to determine the crack propagation paths. Numerical examples show that the present method is accurate and easy to implement making it an appealing tool for analyzing large complex structures that include randomly distributed cracks and notches.

BEM-based fast frequency sweep for acoustic scattering by periodic slab

This paper presents a boundary element method (BEM) for computing the energy transmittance of a singly-periodic grating in 2D for a wide frequency band, which is of engineering interest in various fields with possible applications to acoustic metamaterial design. The proposed method is based on the Padé approximants of the response. The high-order frequency derivatives of the sound pressure necessary to evaluate the approximants are evaluated by a novel fast BEM accelerated by the fast-multipole and hierarchical matrix methods combined with the automatic differentiation. The target frequency band is divided adaptively, and the Padé approximation is used in each subband so as to accurately estimate the transmittance for a wide frequency range. Through some numerical examples, we confirm that the proposed method can efficiently and accurately give the transmittance even when some anomalies and stopband exist in the target band.

Finite element boundary element numerical simulation study on acoustic vibration coupling problems

Excessive structural resonance response in the acoustic vibration coupling environment can cause adverse consequences such as structural damage, local instability, and component failure. Accurately predicting the vibration response of instruments under external sound fields helps to avoid structural performance degradation or damage caused by vibration, and ensures the reliability and safety of the system, which is of great significance for the effective load of instrument equipment. This paper constructs a finite element boundary element equation for the acoustic vibration coupling problem, and adopts a series of optimization measures for it. In response to the low computational efficiency of finite element boundary element equations and their inability to meet the computational requirements of complex models, this paper introduces the acoustic vibration reciprocity theorem and the fast directional boundary element method to compensate for the aforementioned shortcomings. In addition, the study used the weighted residual method to construct a data exchange algorithm to ensure energy conservation during the transfer process. The research focuses on typical structures of spacecraft and conducts simulation analysis and application research on acoustic vibration coupling problems. The simulation showcases that the introduction of the reciprocity theorem markedly reduces the calculation time and memory occupied by the equation, with a required calculation time of 1.23 h and a data file size of approximately 299.56 MB generated by the calculation; The overall error of the data exchange process is below 10−4. Scanning frequency analysis is conducted on the surface and local openings of the model structure, and it is found that the displacement of the selected excitation point is in good agreement with the analytical solution. This study effectively reduces simulation time and memory requirements through the proposed strategy, ensures energy conservation data exchange algorithms, improves the reliability and safety of structural design, and has important practical value for evaluating the structural performance of key equipment such as spacecraft.

Modeling high-frequency backscattering from an impedance surface using Kirchhoff approximation

The Kirchhoff approximation is based on planar wave impinging on a planar surface, which is approximated for high-frequency wave scattering from a curved object. The Kirchhoff approximation is used to assess backscattering from a curved, rigid surface when exposed to high-frequency incident waves. In this presentation, we expand the rigid surface Kirchhoff approximation to accommodate impedance surfaces. The derivation is performed on a planar wave’s incident onto an impedance flat surface, resulting in a reflection coefficient, which in turn determines both surface pressure and surface normal velocity. When these two surface quantities are integrated into a Helmholtz integral formula, it allows for the evaluation of backscattering acoustic field pressures. The BeTTSi model, featuring an impedance surface, serves as an illustration for comparing the current Kirchhoff approximation with the solution obtained by a Fast Multipole Expansion Boundary Element Method (FMBEM). The latter is a numerical solution of the corresponding Helmholtz surface integral equation. The numerical discretization involves nearly a million points, representing the degrees of freedom necessary for modeling the BeTTSi under high-frequency incident waves. The comparison involves these two numerical evaluations across various incident wave directions, encompassing scenarios with single reflection and multiple reflections due to surface geometry.

Three-dimensional broadband simulation of near-fault seismic ground motion considering complete source-path-site effect: Modeled by fast multipole indirect boundary element method

Efficient and accurate numerical methods are essential for analyzing seismic wave propagation and amplification in near-fault complex sites. Understanding the complete process of ground motion simulation, including fault rupture, path propagation, and near-surface complex site response, is crucial for studying earthquake damage mechanisms, seismic zoning, and designing large-scale engineering structures. In this study, we propose a fast multipole indirect boundary element method (FMIBEM) to achieve broadband and high-efficiency simulation, enabling a comprehensive analysis of the complete process involving seismic ground motion. The FMIBEM significantly reduces the computational and storage costs associated with 3D near-fault complex site seismic wave scattering problems to O(N). We validate the accuracy of the method by comparing it with the analytical solution. Compared to the conventional indirect boundary element method (IBEM), our proposed method reduces computational time by over 95 % and storage costs by nearly 90 %. FMIBEM greatly enhances the efficiency of the boundary element method for simulating seismic wave scattering in near-fault complex sites. To demonstrate the effectiveness of our approach, we apply the FMIBEM method to two typical 3D near-fault complex site examples of seismic wave scattering. The simulation results successfully capture both the local site amplification effect and the characteristic near-fault seismic features, such as the hanging wall effect, permanent displacement effect, and large velocity pulse.

Novel and efficient implementation of multi-level fast multipole indirect BEM for industrial Helmholtz problems

Fast multipole boundary element methods (FMBEM) have been successfully applied to solve ultra-large acoustic problems. In practice, there are still a large number of industrial applications where model sizes are well below a million Degrees of freedom. Thus, it is of practical importance to further improve the solver’s efficiency for such applications. This paper presents a novel formulation of the multi-level fast multipole indirect BEM, which relies on redesigning a conventional FMBEM with considerations from several aspects. A new flexible partition algorithm is proposed to generate a flexible and efficient multi-level structure for fast multipole calculations, which also allows the explicit calculation of the far field contributions. A hybrid and efficient regularization approach for hypersingular integral is investigated and implemented, which significantly reduces the iterative solving time. The use of efficient sparse approximate inverse preconditioners is key for fast convergence. Two strategies for constructing the sparsity pattern are implemented and evaluated. Numerical results show that by bringing the aforementioned techniques all together in one package, the new flexible multi-level fast multipole indirect BEM (f-FMBEM) provides significant improvements on actual simulation time over the conventional BEM and FMBEM solvers for industrial size problems (with a non-dimensional Helmholtz number up to kA∼323) on a modern workstation. To efficiently address larger problems, a more efficient multipole translation should be considered.

A Fast Multi-Frequency Iterative Acoustic Boundary Element Method Solver Based on a Preconditioned Accelerated Recycling Strategy

Common multi-frequency solution strategies for acoustic systems are either associated with high computational costs or resort to model reduction strategies, which potentially induce considerable approximation errors. In that context, this work proposes a fast multi-frequency solver for acoustic Boundary Element Method (BEM) analyses that combines conventional preconditioners with an accelerated recycling strategy, achieving in that way a fast and accurate iterative solution. Specifically, by approximating both the system and the related preconditioner by individual affine expressions, an upfront Galerkin projection on a global deflation basis is facilitated, therefore enabling the deployment of the accelerated recycling scheme. The latter acts by accelerating the construction of the related deflation projectors, therefore leading to a significantly faster recycling procedure. Finally, the efficiency of the global deflation basis is guaranteed through an Automatic Krylov subspaces Recycling for Deflation (AKR-D) [D. Panagiotopoulos, W. Desmet and E. Deckers, An accelerated subspaces recycling strategy for the deflation of parametric linear systems based on model order reduction, Comput. Methods Appl. Mech. Eng. 403 (2023) 115765] algorithm and thus, the cumulative number of iterations required by an iterative solver within a frequency sweep is drastically decreased. The proposed multi-frequency solver is tested on two industrially relevant examples of an interior and an exterior scattering problem, benchmarked against the traditional solution strategies.

Fast and accurate boundary element methods for large-scale computational acoustics

The boundary element method (BEM) is a powerful algorithm to solve the Helmholtz equation for harmonic acoustic waves. The explicit use of Green’s functions avoids domain truncation of unbounded regions and accurately models wave propagation through homogeneous materials. Furthermore, fast multipole and hierarchical compression techniques provide efficient computations for dense matrix multiplications. However, the convergence of the iterative linear solvers deteriorates significantly when frequencies are high or materials have large contrasts in density or speed of sound. This talk presents several algorithmic improvements of the BEM. First, a preconditioner based on on-surface radiation conditions drastically reduces the iteration count of linear solvers at high frequencies. Second, anovel boundary integral formulation remains well-conditioned for high-contrast transmission problems. We used our fast and accurate BEM implementation to simulate focused ultrasound propagation in the human body, which can be translated to important biomedical applications such as the non-invasive treatment of liver cancer and neuromodulation of the brain. We validated the methodology within the benchmarking exercise of the International Transcranial Ultrasonic Stimulation Safety and Standards (ITRUSST) consortium. As a second application, we simulated the collective resonances of water-entrained arrays of air bubbles. Finally, we implemented all functionality in our open-source Python library, OptimUS.

Fast Multipole Boundary Element Method for Aerodynamic Sound Field Analysis Based on Lighthill’s Equation

The primary disadvantage of the aerodynamic sound field analysis based on the Lighthill’s equation using the boundary element method (BEM) is the computational time; this is mainly because contributions from numerous aerodynamic sound sources are computed at all boundary element nodes and sound-receiving points. This study proposes a fast method for computing source contributions based on the fast multipole method (FMM). Along with the fast multipole BEM, which is already in practical use as a fast BEM, the analysis is substantially accelerated. The use of a common hierarchical cell structure for grouping boundary element nodes, sound-receiving points and aerodynamic sound sources, enables coefficients of the FMM to be reused, thereby further accelerating the analysis. To deal with the increasing hierarchical level, a wideband FMM is applied. In the sound radiation analysis of a quadrupole source located in a free field, the accuracy is validated. Sound radiation from a cylinder located in a flow is analyzed as a practical problem; the accuracy and numerical settings are discussed. Finally, the proposed method is applied to a problem with more than 0.4 million degrees-of-freedom and more than 3 million aerodynamic sound sources to demonstrate its applicability to large-scale problems.

A fast time-domain boundary element method for three-dimensional electromagnetic scattering problems

This paper proposes a fast time-domain boundary element method (TDBEM) to solve three-dimensional transient electromagnetic scattering problems regarding perfectly electric conductors in the classical marching-on-in-time manner. The algorithm of the fast TDBEM is a time-domain variant of the interpolation-based fast multipole method (IFMM), which is similar to the time-domain IFMM for acoustic scattering problems investigated in the author's previous studies. The principle of the present IFMM is to interpolate the kernel functions of the electric and magnetic field integral equations (EFIE and MFIE, respectively) so that every kernel function is expressed in a form of separation of variables in terms of both the spatial and temporal variables. Such an expression enables to construct a fast method to evaluate the scalar and vector potentials in the EFIE and MFIE with using so-called multipole-moments and local-coefficients associated with a space-time hierarchy. As opposed to $O(N_s^2 N_t)$ of the conventional TDBEM, the computational complexity of the fast TDBEM is estimated as $O(N_s^{1+\delta}N_t)$, where $N_s$ and $N_t$ stand for the spatial and temporal degrees of freedom, respectively, and $\delta$ is typically $1/2$ or $1/3$. The numerical examples presented the advantages of the proposed fast TDBEM over the conventional TDBEM when solving large-scale problems.

Broadband seismic isolation effect of three-dimensional seismic metamaterials in a semi-infinite foundation: modelled by fast multipole indirect boundary element method

In this study, a fast multipole indirect boundary element method (FM-IBEM) is developed for studying the scattering of seismic waves by 3-D seismic metamaterials (SMs), and a 3-D rock soil composite seismic metamaterial (RSCSM) is proposed. Based on the indirect boundary element method, the fast multipole expansion technique is adopted to reduce the calculation time and memory requirements, and effectively solve the broadband scattering problem of seismic waves by large-scale SMs. The bandgap characteristics of three-dimensional SM are studied in detail, and the effects of the geometric and material parameters on the isolation effect of SMs are investigated. The results show that RSCSM can attenuate the ultralow frequency of a seismic wave, the bandgap width is 0.5–19 Hz, and the isolation effect is 37%. The cell size and material parameters significantly affect the bandgap characteristics of the RSCSM. This type of seismic metamaterial provides a new technique for seismic wave isolation and could be applied in engineering structures.

A benchmark study on room acoustic simulations with various material input complexities

This work investigates and evaluates the capabilities of an explicit finite-difference time-domain (FDTD), a fast multipole indirect boundary element method (FMBEM), and a ray-tracing (RT) solver in the context of room acoustic modeling and simulations. In room acoustic simulations, the wave-based FDTD and FMEBM methods are known for generating accurate results at low frequencies, while the RT technique is in principle more valid at higher frequencies. The numerical aspects of setting up the solutions for each solver are discussed. Special attention is given to the influence of material input data of various degrees of detail. Single/multiple frequency-independent/dependent materials are considered in the model setup. The modeling capabilities of the three solvers in handling material input with various complexities are analyzed. Numerical results are evaluated in both frequency and time domains. Room acoustic parameters, including the reverberation time (T_20), early decay time (EDT), clarity (C_80), and definition (D_50), are compared. These results are also compared with available measurement data. Last but not least, the computational efficiencies of the three solvers are briefly discussed.

A fast direct boundary element method for 3D acoustic problems based on hierarchical matrices

The boundary element method (BEM) for acoustic problems is a numerical method based on solving the discretized boundary integral equation (BIE) corresponding to the Helmholtz equation. A fast direct BEM for 3D acoustic problems is proposed in this paper, which is more suitable for broadband acoustic simulation of complex structures, such as in the design and analysis of acoustic metamaterials. The main idea of the fast direct solver is based on the hierarchical off-diagonal low-rank (HODLR) matrix, randomized interpolative decomposition and fast matrix inversion formula. Several numerical examples in solving both interior and exterior acoustic problems are presented in this paper, including radiation and scattering problems with distributed and complex structures. The numerical results show that the same level of accuracy and higher computational efficiency can be achieved by using this fast direct BEM compared with the conventional direct BEM.The second and third authors (Liu and Ye) would like to dedicate this paper to Professor Subrata Mukherjee who had enormous influences on their academic lives and research work on the BEM.

Simultaneous parametric estimation of shape and impedance of a scattering surface using a multi-frequency fast indirect boundary element method

Characterization of rough surface properties, such as surface shape or surface impedance, is relevant in many applications. Previous work has been done for characterizing these two aspects separately. This paper proposes a framework for characterizing rough surfaces of finite size in terms of their roughness shape and surface impedance in a single setup. The surface properties are estimated indirectly using a superposition of spatial sinusoidal components for the surface shape and a porous material model for the surface impedance. A fast multipole indirect boundary element method is employed to model the acoustic scattering problem. With the modeling flexibility from the indirect formulation, the target geometry can be simplified as a thin shell representation, which significantly improves computational efficiency. A subtraction method is proposed to mitigate the modeling error of this representation and enable the use of low driving frequencies (e.g. acoustic wavelength is comparable to or larger than the roughness scale) in the characterization. Considering the modeling accuracy of surface shape representation, a discretization criterion concerning both acoustic and spatial aspects is defined. The inverse problem is solved by means of a general-purpose optimizer, minimizing the difference between the simulated and reference acoustic field at multiple driving frequencies. Various numerical examples show that by using reference data with sufficient quality, the proposed framework can retrieve the surface shape and impedance separately, and also simultaneously.

Accelerating boundary element methods in wideband frequency sweep analysis by matrix-free model order reduction

In frequency-domain acoustic simulations, a wideband frequency sweep analysis is often required. This becomes particularly challenging for boundary element methods (BEM), as the system matrix has a complex dependency on frequency. Previous attempts have been made to accelerate the frequency sweep analysis for the conventional BEM i.e., without fast multipole or ℋ-matrices accelerations, which restrains the applications to limited scale problems due to the dense system matrix. This paper investigates and incorporates a rational Krylov projection based model order reduction (MOR) technology i.e., matrix-free method into various BEM formulations to address the critical challenge of accelerating frequency sweep analysis. Operating on the transfer functions of the BEM system’s input and output, the matrix-free MOR avoids the direct interference with the BEM kernels, which makes it feasible and attractive to work with existing fast BEM techniques such as ℋ-matrices and fast multipole method. The combined holistic BEM solution retains the merits of the acceleration from both MOR and fast BEM techniques. Significant speed-ups have been achieved by the matrix-free MOR for wideband BEM simulations. Numerical examples are presented to demonstrate the capability of the presented holistic approach in simulating various acoustic problems, including interior–exterior models, complex geometries and frequency dependent boundary conditions.

2-D FM-IBEM Simulation of broadband ground motions on near-fault mountain-valley coupling site

In this study, a fast multipole indirect boundary element method (FM-IBEM) was used to efficiently simulate broadband ground motions of complex sites near faults. The spatial distribution of ground motion at the mountain-valley coupling site located on the hanging wall of the strike-slip fault was quantitatively investigated. The kinematic source model was employed to simulate the fault rupture, and the scattering by the mountain-valley site was solved by the FM expansion and GMRES iterative technique. The influence of mountains on the seismic response of a valley was investigated. The effects of the fault distance and depth on the seismic response of the mountain-valley site were also studied. Moreover, the method was applied to simulate the seismic response of the location of the Liuhuanggou Bridge during the Menyuan earthquake. The results show that mountains have a "shielding effect" on near-fault waves, and valleys have a significant amplification effect on near-fault waves, especially in the 0–1 Hz frequency band. The spatial distribution of ground motion is complex owing to the near-fault and local site effects. A reasonable explanation for the seismic damage to the bridge at Liuhuanggou during the Menyuan earthquake is the significant phase difference in the spatial distribution caused by the surface waves

Fast model order reduction boundary element method for large-scale acoustic systems involving surface impedance

This paper presents a newly developed fast model order reduction boundary element method (FMORBEM) for three-dimensional acoustic wave problems with the impedance boundary condition at relatively high frequencies. This method proceeds in two phases: a precomputation phase and a solution phase. In the precomputation phase, a simple yet powerful cutoff technique which benefits from the amplitude decay property of the kernel functions in the Burton–Miller formulation is proposed. Because it does not operate on the underlying boundary element full-order model (FOM), the computational cost and memory consumption in the process of offline constructing a proper subspace are nearly O(N) complexity with N being the number of degrees of freedom of the FOM. After that, the complete system matrices are computed and projected onto the spanned subspace column-by-column, yielding acceptable computational complexity of O(N2). In the solution phase, the resulting reduced-order model (ROM) allows for efficient storage and fast analyses, which merely requires O(n2) arithmetic operations by using direct solvers with n being the dimension of the ROM. Application of this method to the high-frequency scattering analyses of acoustic systems, i.e. simple sphere models and complex scatterers, shows the improved computational efficiency and reduced memory requirement as compared to the conventional boundary element method and further illustrates that it may be competitive with the fast multipole method based on iterative solvers.

Recent Advances in Acoustic Boundary Element Methods

The modern scope of boundary element methods (BEM) for acoustics is reviewed in this paper. Over the last decades the BEM has gained popularity despite suffering from shortcomings, such as fictitious eigenfrequencies and poor scalability due to its dense and frequency-dependent coefficient matrices. Recent research activities have been focused on alleviating these drawbacks to enhance BEM usability across industry and academia. This paper reviews what is commonly known as direct BEM for linear time-harmonic acoustics. After introducing the boundary integral formulation of the Helmholtz equation for interior and exterior acoustic problems, recommendations are given regarding the boundary meshing and treatment of the non-uniqueness problem. It is shown how frequency sweeps and modal analyses can be carried out with BEM. Further extensions for efficient modeling of large-scale problems, including fast BEM and solutions methods, are surveyed. Additionally, this review paper discusses new application areas for modern BEM, such as viscothermal wave propagation, surface contribution analyses, and simulation of periodically arranged structures as found in acoustic metamaterials.

Sound shielding simulation by coupled discontinuous Galerkin and fast boundary element methods

ABSTRACT A code coupling has been established for performing efficient fan tone shielding simulations of aerial vehicles with unconventional engine installations. In particular, the Fast Multipole Boundary Element Method (FM-BEM) which is formulated to solve a surface integral based on the Kirchhoff-Helmholtz wave equation for large geometries is combined with a volume resolving Discontinuous Galerkin (DG) method which is well suited for the compact region around a jet engine intake where strong mean flow gradients are present. The Möhring-Howe acoustic analogy is utilised during the backward data exchange process for derivation of acoustic velocities in presence of a mean flow. The method can help to overcome a major difficulty related to computational complexity when solving fan noise shilding and scattering problems for a complete aircraft geometry.

Two-Dimensional Fast Multipole Indirect Boundary Element Method-Based Solution to P-Wave Scattering by a Mountain with Large-Scale Random Cracks in an Elastic Half-Space

In this study, the fast multipole indirect boundary element method (FM-IBEM) was adopted for solving the scattering problem of incident P-waves using a two-dimensional (2D) mountain terrain with large-scale random cracks. By adopting the FM-IBEM, this study resolves the bottleneck of the conventional boundary element method (BEM) for computing complex multi-degree-of-freedom models, and for the first time quantitatively analyzes the seismic wave scattering problem of large-scale narrow mountain cracks. After verifying the calculation accuracy, the horizontal and vertical displacement amplitudes of the mountain surface and its adjacent surface were accurately calculated. The results suggest the following: (1) Cracks significantly amplify the displacement amplitude of the mountain surface. The amplification mostly depends on the number of cracks and on the frequency of the incident P-wave. Compared with homogeneous mountains (no cracks), under the most unfavorable circumstances, the horizontal displacement is amplified more than 9.5-fold, while the vertical displacement is amplified more than 6.5-fold. (2) When the wave field passes through the crack area, a large amount of energy is trapped. Moreover, this energy is highly concentrated, which amplifies the local ground motion more than 7-fold. (3) The ground motion generated by the P-wave scattering by a mountain with large-scale random cracks exhibits a significant spatial difference. (4) The resonance of multi-order modes can emerge owing to large-scale random cracks. Compared with homogeneous mountains, the ground motion caused by the P-wave scattering by mountains with large-scale random cracks is much more severe in many cases. This phenomenon must be considered in practical seismic engineering applications.