Fast Multipole Boundary Element Method Research Articles

A low-frequency fast multipole boundary element method for acoustic problems in a subsonic uniform flow

The fast multipole method (FMM) based on the plane wave expansion is known to suffer from numerical instability in the low-frequency regime. This paper presents a low-frequency fast multipole boundary element method (LF-FMBEM) for acoustic problems in a subsonic uniform flow. First, a hybrid convected boundary integral formula based on the Burton-Miller method is derived to overcome the non-uniqueness difficulty at fictitious eigenfrequencies. The explicit evaluation of hypersingular integrals in the convected boundary integral formulae is also introduced. Then, the formulae of FMM based on the series expansion for convected BEM are derived to improve the calculation efficiency. The recursive calculation method is derived in the expansion of the derivative of the integrands. Besides, the rotation-coaxial translation-rotation back (RCR) technique is employed to accelerate the multipole translation. The numerical implementation process of the developed algorithm is presented in detail. Several numerical experiments are performed to validate the computational efficiency and accuracy of the developed LF-FMBEM. Results show that the proposed algorithm can achieve large-scale computation of one million degrees of freedom (DOF) on a personal computer, and the computational accuracy is still high when the Mach number reaches 0.95. The non-uniqueness problem for convected acoustics problems is also effectively overcome.

Sensitivity Analysis of Underwater Structural-Acoustic Problems Based on Coupled Finite Element Method/Fast Multipole Boundary Element Method with Non-Uniform Rational B-Splines

For the direct differentiation technique-based modeling of acoustic fluid–structure interaction and structural-acoustic sensitivity analysis, a coupling algorithm based on the finite element method (FEM) and the fast multipole boundary element method (FMBEM) is suggested. By bypassing the challenging volume parameterization process in isogeometric finite element techniques and the laborious meshing process in traditional FEM/BEM, non-uniform rational B-splines (NURBS) isogeometric analysis (IGA) is utilized to immediately perform numerical analysis on CAD models. The matrix-vector products in the boundary element analysis are accelerated using the fast multipole method (FMM). To hasten the solution of the linear system of equations, the iterative solver GMRES is used. The numerical prediction of the effects of arbitrarily shaped vibrating structures on the sound field is made feasible by the FEM/FMBEM technique. A number of numerical examples are provided to show the applicability and effectiveness of the suggested approach.

FEM/Wideband FMBEM coupling based on subdivision isogeometry for structural-acoustic design sensitivity analysis

A computer simulation approach known as the isogeometric (IGA) method may directly use the surface information of geometric model. In 3D computer graphics, Loop subdivision surfaces are a common method for creating complicated shapes. In this study, we propose a coupling algorithm that utilizes Loop subdivision surfaces and a direct differentiation method for the computation of acoustic-fluid-structure interaction and the performance of structural-acoustic sensitivity analysis. This algorithm combines the finite element method (FEM) and wideband fast multipole boundary element method (FMBEM). Because of that the proposed method is of a great ability of integrating the numerical calculation and computer-aided modeling, the current technique can deliver results quickly and accurately. The numerical prediction of the effects of vibrating structures with arbitrary shape within sound field is made feasible by the FEM/Wideband FMBEM technique. Calculation examples are provided to show the applicability and effectiveness of the suggested method.

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.

Effect of embankment slope on road traffic noise propagation: numerical investigation and construction of correction formula for difference by slope angle

The propagation characteristics of road traffic noise from an embankment road are investigated through three-dimensional wave-based numerical analysis. Since this is a large-scale problem, we use the fast multipole boundary element method (FMBEM) as a highly efficient wave-based numerical method. We analyze sound fields around embankments with various slope angles and clarify that the sound pressure gradient changes with the slope angle, and in particular, the sound pressure level (SPL) distribution changes significantly near the line where the embankment slope meets the ground surface. When the prediction plane is an orthogonal cross-section to the direction of the lane, the distribution of the difference in A-weighted SPL between the case where the embankment is sloped and the case where it has a right angle is generally similar regardless of the position of the prediction plane. Based on this result, we propose an empirical formula to express the difference between the case with a slope and the case with a right angle. We confirm that the proposed formula provides a certain level of accuracy regardless of the location of the prediction plane when we appropriately determine the coefficients in the formula for each case from the viewpoint of error minimization.

FMBEM simulation of the laboratory measurement of sound transmission loss in rectangular test rooms

In order to predict sound transmission loss of building elements, the fast multipole boundary element method is applied to simulate the laboratory measurement method in rectangular test rooms specified by ISO 10140-2 (JIS A 1416) for all the frequency bands from 100 to 4 kHz. Firstly, fine and simplified models are validated in comparison with theoretical and measured results for reference samples of membrane materials. Secondly, in addition to the sound pressure method, the sound intensity method is numerically simulated. As a result, it was shown that both models are applicable at mid and high frequency bands, but a simplified model overestimates the transmission loss more than a fine model at low frequency bands. Regarding the sound intensity method, the simulated values are lower than those with the sound pressure method, however approaching the theoretical values.

Wideband fast multipole boundary element method for flow-induced noise analysis based on Lighthill's equation

In order to reduce the computational load for the analysis of three-dimensional flow-induced noise based on the Lighthill equation using the boundary element method (BEM), an efficient method is developed on the basis of the fast multipole BEM (FMBEM). The common hierarchical cell structure for grouping boundary element nodes, sound-receiving points and flow-induced noise sources is used; this enables coefficients of the FMBEM to be reused during the calculation of sound pressure at boundary element nodes and receiving points, thereby further accelerating the analysis. In addition, to deal with the increasing hierarchical level for encompassing flow-induced noise sources, which are often widely distributed around the object to be analyzed, a wideband version of the FMBEM is applied. Following the computational procedure, two application examples are presented for discussion. First, noise from a cylinder located in a flow is analyzed to discuss the accuracy of the method. Then, a problem with large number of degrees-of-freedom and flow-induced noise sources is analyzed to demonstrate its applicability to large-scale problems.

Determination of sound-field diffusion indices based on FMBEM incidence directivity analysis

Regarding the diffuseness of sound field, several indices and their measurement methods have been proposed in terms of isotropy of traveling waves. In this paper, we attempt to calculate two kinds of diffusion indices based on the incidence directivity analysis employing the fast multipole boundary element method (FMBEM). The directivity analysis is performed for steady-state sound fields in a rectangular room with changing the absorption condition of the walls. As for diffusion indices, the directional diffusion coefficient is determined from the deviation of directivity of sound intensity, while alternative isotropy indicator is calculated through the spherical expansion of the directivity. The numerical results show general tendencies that the former index is almost constant, whereas the latter index decreases with increasing the frequency. As a common tendency, it is confirmed that the indices become higher in a reflective room condition, and remarkable dips occur around the natural frequencies when absorbing walls are unevenly distributed. Moreover, concerning the receiving cell, the size of it has little effect on the indices. On the other hand, it is verified that the indices become higher with the receiving cell near the point source.

A fast multipole boundary element method for acoustics in viscothermal fluids

Standard numerical models in acoustics rely on the isentropic Helmholtz equation. Its derivation assumes adiabatic and reversible, i.e., dissipation-free, wave propagation. Sound waves in fluids are, however, subject to viscous and thermal losses. These losses originate from viscous friction and heat conduction, leading to the formation of acoustic boundary layers. Considering these effects becomes significant in setups with acoustic cavities of similar dimension as the boundary layers. Recently, boundary element methods (BEM) accounting for the viscothermal dissipation have been proposed. These methods limit the discretization to the surface of the fluid domain and require significantly fewer degrees of freedom than comparable finite element models. However, the BEM coefficient matrices are fully populated, resulting in high computational costs and storage requirements. This study develops a new BEM formulation for acoustics in viscothermal fluids that uses the fast multipole method - a low-rank approximation technique based on a hierarchical subdivision of the computational domain - to alleviate this shortcoming. It is shown that the fast viscothermal BEM improves the algorithmic complexity over the conventional formulation, reducing both the memory requirements and solution time. The results indicate good scalability of the method making it feasible for larger applications such as acoustic metamaterials.

Evaluation of photogrammetry-based near-field head related transfer function estimation for immersive sound reproduction

Binaural reproduction algorithms are commonly applied in virtual and augmented reality applications to create immersive audio environments. Nonetheless, flexible near-field binaural rendering remains challenging. In this work, we explore the acoustic accuracy of photogrammetry-based, near-field head-related transfer function estimation. Photogrammetry measurements of the Head and Torso Simulator (HATS) are taken using a photogrammetry rig consisting of 44 synchronized Canon 1300D DSLR cameras mounted on ten aluminium poles arranged in a circle of 0.8m radius. An Apple iPhone 14 Pro mobile phone is used to acquire the ear data. Commercial photogrammetry software is then used to create a 3D mesh from the still images. Finally, numerical acoustic simulations using the fast multipole boundary element method (FM-BEM) technique are applied to the mesh to generate the near-field HRTFs. We present comparisons of acoustically-measured and photogrammetry-based near-field HRTFs.

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.

Installed Fan Noise Simulation of a Supersonic Business Aircraft

Overcoming the problem of excessive engine noise at low altitudes is a formidable task on the way to developing a supersonic passenger aircraft. The focus of this paper is on the fan noise shielding during take-off, investigated as part of the DLR project ELTON SST (estimation of landing and take-off noise of supersonic transport) for an in-house aircraft design. The supersonic inlet is required to provide the proper quantity and uniformity of air to the engine over a wider range of flight conditions than the subsonic inlet. For passenger aircraft, the noise problem influences engine integration and placement, and the new generation of supersonic transport would require innovative engineering solutions in order to come up with an efficient low-noise design. Potential solutions are evaluated using DLR tools capable of accurate source generation and noise propagation to the far-field. For low-speed aircraft operation, the method of choice is a strongly coupled volume-resolving discontinuous Galerkin (DG) and fast multipole boundary element method (FM-BEM) which is applied due to a large disparity between the Mach numbers on the interior and exterior of the inlet. The method is used for obtaining the acoustic signature of the full-scale model at realistic flight points, including the application of the programmed lapse rate (PLR), which involves simulations at higher pitch angles than for the reference flight path. The results show that the proposed method is highly suitable for obtaining accurate noise footprints during the low-speed phase and could be used to assist with certification procedures of future supersonic aircraft.

Data-driven analysis of parametrized acoustic systems in the frequency domain

A data-driven method combined with the formulations of boundary integral equations is developed for the frequency-domain analysis of parametrized acoustic systems, arising from the spatial discretization of the linear Helmholtz equation. The method derives surrogate models for the approximation of frequency response functions at selected field points via the construction of neural networks with radial basis function interpolation. This requires the offline collection of a training database consisting of discrete frequency-parameter samples and their associated response values, whereas the information about system matrices and interference with boundary element kernels are no longer necessary, leading to a matrix-free and non-intrusive nature. To ensure accurate fitting results, a tensor decomposition technique is employed to decouple the naturally formed tensor grid between the frequency and parameter inputs in the database. After establishing two sets of radial basis networks separately, the final reduced-order solutions as a continuous function of the frequency and parameters are represented by the linear combinations of tensor products. In the online phase, the original transfer function at any testing frequency-parameter location can be quickly predicted as direct output from the compact surrogate model. With the analytical solution, the traditional boundary element method and the fast multipole boundary element method as solvers for the generation of training data, two benchmark problems and one complex system are investigated. In all cases, the simplicity, versatility and efficiency of the proposed method are demonstrated through examples of multi-frequency solutions of acoustic systems with parametrized material and geometric properties.

The Accuracy and Efficiency of a Single-Level Fast Multipole Boundary Element Model for Analyzing Cathodic Protection of Large Pipeline Networks

A single-level fast multipole boundary element method was developed for analyzing cathodic protection (CP) systems of large pipeline networks. This method was obtained by embedding far-field approximation within the traditional single-level fast multipole method. The far-field approximation was used for computing the coefficients for far elements within adjacent cells and determining the moments of the elements within far cells. This approximation reduced the difficulty of the procedures and programming leading to a significant decrease in computational time. The Newton-Raphson method and generalized minimal residual method were combined based on the proposed method to solve the nonlinear boundary conditions due to the polarization curve. Several CP problems were considered to verify and evaluate the method. The calculated potentials of this method were in good agreement with the conventional boundary element method, which was achieved by using pipe elements and quadrilateral elements to mesh the surfaces. Finally, the impressed current CP systems of a large network (more than 100,000 elements) and a complex urban gas network were investigated. The results indicated the capability, efficiency, and precision of the present method for solving large and complicated problems on a common desktop computer.

Study on the structure-borne noise of U-shaped girder bridges with fully-enclosed sound barriers

To investigate the structure-borne noise characteristics of U-shaped girder bridges (USGBs) with fully-enclosed sound barriers (FESBs), an analysis model of the USGB-FESB coupling system was established using the fast multipole boundary element method (FMBEM). The structure-borne noise analysis model was validated using a noise field test of a 27-m-long simply-supported concrete USGB under the loading of moving trains in an urban rail transit. The structure-borne noise radiation characteristics of the USGB-FESB coupling system were studied, and the influences of sound barrier type (vertical sound barrier and fully-enclosed sound barrier) and bridge type (U-shaped girder and box girder) on these characteristics were analyzed. The results show that the noise radiated from the USGB with FESBs was higher than the noise radiated from the USGB without FESBs by 0.28 to 2.53 dB (L) in the frequency band of 20 to 100 Hz. In comparison to vertical sound barriers, the structure-borne noise generated by the FESBs contributes more to the structure-borne noise. The structure-borne noise of a box-girder bridge with FESBs is more significant than that of the USGB-FESB.

A new periodic FM-BEM for solving the acoustic transmission problems in periodic media

This paper presents a new fast multipole boundary element method (FM-BEM) for solving the acoustic transmission problems in 2-D periodic media. We divide the periodic media into fundamental blocks, and then construct the boundary integral equations in the fundamental block. The fast multipole algorithm is proposed for the square periodic and hexagonal periodic structures, the convergence and complexity of the algorithm are analyzed. Our approach is applied to solve the acoustic transmission problems for liquid phononic crystals, the acoustic band gaps of the phononic crystals are derived.

Analysis of 2-D elastic solid with multiple V-notches by a fast multipole BEM with a novel singular element with multi-order asymptotic terms

A novel fast multipole boundary element method (BEM) is developed to analyze the stress fields of plane structures with multiple V-notches. To accurately evaluate the singular stresses in the vicinity of the V-notch tip, a novel singular element with multi-order asymptotic terms (SEMAT) is first proposed. The tractions and displacements on the SEMAT are expressed as the asymptotic expansions, where both the first singular term and other higher-order terms are considered. The SEMAT is then introduced into the fast multipole BEM, and several singularity problems of boundary integrals on the SEMAT are treated with caution. Finally, the fast multipole BEM with the SEMAT is used to obtain the whole stress and displacement fields for the plane V-notch/crack structures. Numerical examples show that the present method is accurate for analyzing the V-notch with opening angle of arbitrary size in both single and bonded materials. It is not necessary to use a dense mesh in modeling, and the computational efficiency of the present method is high for large-scale problems.

D Fast Multipole Boundary Element Method Analysis of Heat Exchange Performance of Buried Pipe Groups

D Fast Multipole Boundary Element Method Analysis of Heat Exchange Performance of Buried Pipe Groups

The Application of Fast Multipole Boundary Element Method to the Calculation of Slamming Load of Three-Dimensional Elastic Sphere

An efficient fluid-structure interaction (FSI) method is established to quickly and accurately calculate the fluid load of three-dimensional (3D) elastic structure in the process of water slamming. In this method, the boundary element method (BEM) and computational structural dynamics (CSD) are coupled by thin plate spline (TPS) interpolation method and flow field reconstruction. In order to improve the calculation efficiency, the fast multipole algorithm is introduced into the BEM, the finite element mesh of the wetted part of the structure is directly used as the flow field boundary, and the structural dynamic response is analyzed by the direct integration method of state equation based on modal superposition. The numerical results are compared with the results of Nisewanger’s sphere water entry test and the existing literature, which proves the accuracy of the numerical calculation method established in this paper. The effects of time-domain iteration step size, mesh size, slamming velocity, fluid density and structural elastic effect on water slamming load are studied. The results show that the slamming velocity and fluid density have a great influence on the simulation results; the iteration step size has little influence on the accuracy of simulation calculation, and the mesh size has a certain influence on the accuracy of simulation calculation; elastic effect reduces the peak value of slamming load, considering elastic effect is necessary for accurate calculation of slamming load.

Generalized isogeometric boundary element method for uncertainty analysis of time-harmonic wave propagation in infinite domains

This paper proposes a novel generalized nth order perturbation isogeometric fast multipole boundary element method for time harmonic wave propagation in infinite domains. The non-uniform rational B-splines are employed to construct structural geometries and discretize boundary integral equations. The randomness of wave number for incident plane wave is considered as the source of system uncertainty. The generalized nth order perturbation method is employed to model the physical field depending on the input random variable. The nth order derivatives of field functions and kernel functions in the boundary integral equations are derived by generalized nth order Taylor series expansions with a small perturbation parameter. The subtraction of singularity technique is used to evaluate the singular integrals and the fast multipole method is applied to accelerate the solution. The Monte Carlo simulations are conducted in numerical examples to demonstrate the validity and correctness of the proposed algorithm.