Singular Boundary Method Research Articles

A fast semi-analytical meshless method in two-dimensions

This paper introduces a fast direct algorithm for the singular boundary method (SBM) in two-dimensional (2D) problems, utilizing the hierarchical off-diagonal low-rank (HODLR) matrix concept as the foundation of the fast direct solver. The HODLR matrix is constructed by hierarchically partitioning the coefficient matrix into blocks using a binary tree, with all off-diagonal blocks at each level being represented as low-rank factors. Furthermore, The Sherman-Morrison-Woodbury formula can be used to efficiently compute the inverse of the HODLR matrix. The numerical experiment results demonstrate that the new fast solver can significantly improve the computational efficiency of the SBM while maintaining the same level of precision.

A novel hybrid SBM-MFS methodology for acoustic wave propagation problems

In this paper, a novel hybrid meshless approach that combines the singular boundary method (SBM) and the method of fundamental solutions (MFS) to deal with two-dimensional (2D) exterior acoustic wave propagation problems is proposed and studied. The methodology is particularly devised to solve problems with complex boundary geometries containing geometric singularities such as corners and sharp edges. It employs the SBM to model intricate segments of these geometries and the MFS for the smooth ones. The proposed hybrid SBM-MFS method is studied in a 2D context in the framework of three benchmark examples involving acoustic radiation problems of circular-, square- and L-shaped objects in a full-space acoustic medium. In addition, the applicability of the proposed hybrid SBM-MFS methodology to predict the acoustic performance of a T-shaped thin barrier is also investigated. These examples are specifically designed to assess the feasibility, validity and accuracy of the hybrid SBM-MFS approach in comparison with the available analytical solutions and alternative numerical strategies such as the MFS, the SBM and the boundary element method (BEM). Numerical simulations demonstrate that the proposed method can match the level of accuracy of the MFS while keeps the robustness of the SBM when dealing with complex geometries, overcoming one the most important drawbacks of the traditional MFS. Moreover, the proposed hybrid SBM-MFS naturally avoids the non-uniqueness problem arising at the fictitious eigenfrequencies associated with the corresponding interior problems, a feature that neither the SBM nor the BEM inherently possesses.

A 2.5D hybrid SBM-MFS methodology for elastic wave propagation problems

This paper proposes a novel hybrid methodology that combines the singular boundary method (SBM) and the method of fundamental solutions (MFS) for the computational simulation of elastic wave propagation. Particularly, the methodology aims to address radiation or scattering problems of longitudinally invariant systems in the wavenumber–frequency domain involving boundaries with intricate geometries. The approach uses the SBM to deal with the complex parts of these geometries and the MFS for the smooth ones. The method is studied in the framework of three case studies involving longitudinally infinite cavities in a homogeneous full-space with circular, square-shaped and five-cusped hypocycloid cross-sections. These three examples are selected to assess the accuracy and robustness of the hybrid SBM-MFS approach in comparison with alternative modelling strategies. The comparisons show that the proposed method inherits the accuracy of the MFS while keeping the robustness of the SBM when dealing with complex geometries. The method is found to be computationally more efficient than the SBM or the boundary element method (BEM). Moreover, the hybrid approach naturally mitigates the effect of fictitious eigenfrequencies, a feature that neither conventional versions of the SBM nor the BEM have.

Nonlinear coupled multi-physics field analysis of permanent magnet synchronous motor combining finite element analysis with optimized meshless method

Aiming at significant disadvantages of finite element analysis (FEA) performed in coupled multiphysics field calculation, including high computing cost and strong dependency on meshing quality. Taking a high-speed permanent magnet synchronous motor (P MSM) as an example, an approach combining FEA with optimized meshless method is proposed to solve the heat transfer problem. Firstly, electromagnetic losses under rated and 1.2 times overloaded operation conditions are calculated using FEA respectively. Secondly, three-dimensional FEA flow models considering the influence of viscosity varying with temperature are established. Thirdly, convective heat transfer coefficients are calculated as the boundary condition, after which the electromagnetic losses as heat sources are mapped to the threedimensional steady-state temperature field calculations using FEA and conventional meshless method respectively. Applying the above two methods, a singular boundary method (SBM) is employed to replace the temporal derivative term of the transient calculation equation. Finally, to mitigate the illconditioned matrix appearing in the conventional meshless method coupled with FEA, an optimized meshless method is proposed to recalculate the heat transfer equation. After that, the calculation results of the above methods are compared with experimental results. This research is beneficial for the multiphysical coupling analysis of electrical equipment with complex structures considering nonlinear factors.

Shape optimization of sound barriers using an isogeometric meshless method

The sound barrier is an important means to reduce noise caused by traveling vehicles on roads or railways. Structural design and optimization of the sound barrier can effectively reduce the use of materials and improve the noise reduction effect. In this paper, a new isogeometric singular boundary method is proposed and applied to the shape optimization of sound barriers. The geometric structure is accurately represented by using non-uniform rational B-splines. The acoustic shape sensitivity of the control points was calculated using the direct differentiation method and the adjoint variable method. After that, the method of moving asymptotes is adopted as an optimizer to search for the optimal layout of the design objective. In the numerical procedure, the shoelace formula is introduced to calculate the area of the closed structure, which only uses the discrete node information on the boundary. The proposed approach completely avoids the mesh division in the finite element method as well as the singular integral calculation in the boundary element method. More importantly, it can be seamlessly connected with the computer-aided design system for the subsequent treatment by engineers. Three numerical examples are provided to illustrate the accuracy and effectiveness of the proposed isogeometric method. This work provides a simple and effective way for the structural optimization design of sound barriers.

Cutoff wavenumber analysis of arbitrarily shaped waveguides using regularized method of fundamental solutions with excitation sources

AbstractThe method of fundamental solutions with excitation source (MFS‐ES) is a reliable method for determining the eigenvalues of a two‐dimensional hollow waveguide. However, the accuracy in MFS‐ES is extremely dependent on the choice of the auxiliary boundary. In this work, to avoid the problem, a regularized MFS‐ES (RMFS‐ES) is proposed. First, to overcome the shortcomings of MFS, a regularized method of fundamental Solutions (RMFS) is proposed by borrowing the ideas of singular boundary method (SBM) and single layer regularized meshless method (SRMM). Second, the expressions for the fundamental solutions at the singularities are given by utilizing the subtraction and adding‐back technique (SAB). Third, the excitation source method is introduced to overcome the problem that the RMFS may also have spurious frequencies. Finally, to improve the accuracy of RMFS‐ES in solving polygonal waveguide eigenvalue, a wedge scheme (WS) is given. Numerical results show that the meshless RMFS‐ES is a promising numerical method for waveguide eigenvalue problems.

Regularized singular boundary method for calculating wave forces on three-dimensional large offshore structure

How to quickly calculate the singularity of the special Green's function at origin has been a difficult problem that has plagued ocean dynamics for many years. This letter proposes a regularized singular boundary method (RSBM) based on the special Green's function and origin intensity factor (OIF) technique. The source points of the RSBM only need to be arranged on the boundary of the obstacle. By linearly combining the OIF of the modified Helmholtz equation and the OIF of the Helmholtz equation, an explicit OIF suitable for three-dimensional ocean dynamics is derived. The derived OIF does not contain singular integrals and is easy to use. The RSBM replaces the Green's function at origin with the OIF. Therefore, it is no longer necessary to reuse the integral form of Green's function as basis function at origin. The computational efficiency and accuracy have been significantly improved. Numerical experiments show that the RSBM can accurately calculate the wave forces on large three-dimensional offshore structure.

Performance analysis and material distribution optimization for sound barriers using a semianalytical meshless method

AbstractWith the increase in car ownership, traffic noise pollution has increased considerably and is one of the most severe types of noise pollution that affects living standards. Noise reduction by sound barriers is a common protective measure used in this country and abroad. The acoustic performance of a sound barrier is highly dependent on its shape and material. In this paper, a semianalytical meshless Burton–Miller‐type singular boundary method is proposed to analyze the acoustic performance of various shapes of sound barriers, and the distribution of sound‐absorbing materials on the surface of sound barriers is optimized by combining a solid isotropic material with a penalization method. The acoustic effect of the sound‐absorbing material is simplified as the acoustical impedance boundary condition. The objective of optimization is to minimize the sound pressure in a given reference plane. The volume of the sound‐absorbing material is used as a constraint. The density of the nodes covered with the sound‐absorbing material is used as the design variable. The method of moving asymptotes was used to update the design variables. This model completely avoids the mesh discretization process in the finite element method and requires only boundary nodes. In addition, the approach also does not require the singular integral calculation in the boundary element method. The method is illustrated and validated using numerical examples to demonstrate its accuracy and efficiency.

Acoustic simulation using singular boundary method based on loop subdivision surfaces: A seamless integration of CAD and CAE

This paper firstly combines the Burton-Miller-type singular boundary method (BMSBM) with the Loop subdivision surfaces (LSS) for the acoustic simulation of 3D complicated structures. As a modern modeling technique, the LSS provides a flexible and simple means to represent model surfaces by controlling meshes and subdivision rules. Thus, it can achieve arbitrary topology and multi-resolution structures. As a semi-analytical and boundary-type meshless technique, the BMSBM using the fundamental solutions is simple, accurate, and straightforward for solving acoustic radiation and scattering problems. The new approach constructs a geometry model using a subdivision scheme, and therefore can create a multi-resolution hierarchical model meeting various accuracy requirements without a repetitive meshing procedure. The proposed method effectively breaks through the technical bottleneck that the BMSBM makes it difficult to collate boundary nodes of complex structures and achieves a seamless integration between CAD and CAE. The numerical results of three examples demonstrated the effectiveness and accuracy of the proposed method for acoustic simulation. The present study provides an innovative idea for the subsequent optimization of materials and shapes of acoustic structures, which has the potential to enhance precision and efficiency significantly.

An efficient hybrid collocation scheme for vibro-acoustic analysis of the underwater functionally graded structures in the shallow ocean

In this paper, a novel hybrid collocation scheme based on the generalized finite difference method (GFDM) and the Burton–Miller singular boundary method (BMSBM) is developed to study the vibration and acoustic radiation characteristics of an underwater functionally graded (FG) structure immersed in the shallow ocean. By utilizing the individual strengths of both methods, the GFDM is adopted for the vibration analysis of the FG structures, while the BMSBM is applied for the acoustic wave radiation analysis. Numerical examples verify the accuracy and efficiency of the proposed hybrid collocation scheme for the natural frequency and acoustic radiation behavior of the underwater structures consisting of the functionally graded materials (FGMs). The influences of the material parameters, structural geometries and shallow ocean environment on the natural frequencies of the structural vibration and the underwater acoustic wave radiation field are investigated in detail. Numerical study shows that the variation trend of the natural frequencies of the structural vibration is consistent with that of the resonance peak frequencies of the underwater acoustic radiation field. The present results may provide valuable references for the design and application of underwater marine structures consisting of the FGMs.

A pile–soil interaction model for ground-borne vibration problems based on the singular boundary method

An efficient three-dimensional approach for solving pile–soil interaction problems is proposed. In the approach, the soil is modelled as an elastic half-space, and its response in the presence of the pile’s corresponding cavity is computed by employing the singular boundary method. The pile is modelled analytically using the classic rod and Euler–Bernoulli beam theories. For the coupling with the soil, the pile is divided in a set of rigid segments that interact along their circumference with the soil. The methodology allows the rotational motions and reaction torques at these segments to be accounted for and their contribution in the accuracy of the scheme is assessed. A criterion to define the minimum number of collocation points that offers an acceptable trade-off between accuracy and numerical performance is also proposed. The method is validated against well-established methodologies and using the reciprocity principle that relates the wave radiation from the pile to the ground field with the incident wave problem due to a load on the ground surface. Results are shown for different soil stiffnesses and different pile length to diameter ratios. The employment of the singular boundary method is shown to provide strong computational advantages to detailed modelling approaches such as the three-dimensional finite element-boundary element method, as well as overcoming the fundamental limitations of plane-strain and axisymmetric methods.

A novel semi-analytical meshless method for the thickness optimization of porous material distributed on sound barriers

This paper presents a novel semi-analytical meshless method to optimize the thickness of porous material distributed on sound barriers, by employing the Burton–Miller-type singular boundary method in conjunction with the method of moving asymptotes. Firstly, the Delany–Bazley–Miki model is utilized to characterize the acoustic property of sound barrier with a porous sound-absorbing material. Then, the sensitivity formula with respect to the thickness of the porous layer is derived based on the analytical computation and the adjoint variable formula, in which the design variable is a thickness parameter distributed between [0,1]. Finally, the optimal thickness distribution is achieved by solving the optimization model using the method of moving asymptotes. Compared to traditional algorithms, the proposed new method is simple, accurate, easy-to-program, and free of mesh and integration. Numerical experiments demonstrate the feasibility and effectiveness of the proposed algorithm.

Acoustic sensitivity analysis for 3D structure with constant cross-section using 2.5D singular boundary method

In this paper, a 2.5D singular boundary method (SBM) in conjunction with the direct differentiation method (DDM) and adjoint variable method (AVM) are formulated for the sensitivity analysis of 3D longitudinally invariant structures. The assumption of a constant cross-section in the longitude direction enables it possible to decouple the 3D system into 2D problems at every wavenumber, and thus makes it a 2.5D-problem. The 2D sensitivity problem is solved by a boundary-type method, SBM. The SBM solves a problem with a linear combination of the fundamental solutions with respect to boundary collocation points. The fundamental solution makes the SBM available for exterior problems without the artificial truncated boundary. The source singularity issue of fundamental solution is overcome by a simple analytical formula and its derivatives. Numerical experiments present the accuracy, efficiency and feasibility of the proposed methods, and indicate that the proposed methods can be considered as an effective alternative in 3D problems with a longitudinally invariant cross-section.

A Novel Coupled Meshless Model for Simulation of Acoustic Wave Propagation in Infinite Domain Containing Multiple Heterogeneous Media

This study presents a novel coupled meshless model for simulating acoustic wave propagation in heterogeneous media, based on the singular boundary method (SBM) and Kansa’s method (KS). In the proposed approach, the SBM was used to model the homogeneous part of the propagation domain, while KS was employed to model a heterogeneity. The interface compatibility conditions associated with velocities and pressures were imposed to couple the two methods. The proposed SBM–KS coupled approach combines the respective advantages of the SBM and KS. The SBM is especially suitable for solving external sound field problems, while KS is attractive for nonlinear problems in bounded non-homogeneous media. Moreover, the new methodology completely avoids grid generation and numerical integration compared with the finite element method and boundary element method. Numerical experiments verified the accuracy and effectiveness of the proposed scheme.

Modified 2.5D singular boundary methods to deal with spurious eigensolutions in exterior acoustic problems

This paper studies the problem of spurious eigensolutions in the context of the singular boundary method (SBM) formulated in the two-and-a-half-dimensional (2.5D) domain and proposes two numerical schemes to overcome this numerical difficulty. The SBM can be seen as a version of the method of fundamental solutions where the source points are located at the physical boundary. Similar to other boundary-type discretization schemes, the SBM also encounters the non-uniqueness problem at the vicinity of the eigensolutions of the corresponding interior problem. In the 2.5D domain framework, the so-called fictitious eigenfrequencies appearing in 3D problems arise in the form of spurious dispersion curves associated with propagation modes of the corresponding interior problem. The two enhanced 2.5D SBM approaches proposed in this work, based on the Burton–Miller method in one case and the dual surface method in the other, are designed to filter out the spurious eigenvalues from the simulation results and deliver accurate solutions along the wavenumber-frequency spectrum. Three benchmark examples including the radiation problems of an infinitely long cylinder under Dirichlet and Neumann boundary conditions and the radiation problem of a longitudinally infinite object with a constant star-like cross section subjected to a Dirichlet boundary condition are considered to study the proposed methods. The results demonstrate the capability of the proposed numerical schemes to successfully avoid the non-uniqueness problem when the 2.5D SBM is employed.

A 2.5D automatic FEM-SBM method for the evaluation of free-field vibrations induced by underground railway infrastructures

This paper presents an efficient method to predict underground railway-induced vibrations. The method uses the finite element method (FEM) to model the railway tunnel structure and the singular boundary method (SBM) to model the wave propagation in the surrounding soil. The FEM mesh and the distribution of SBM collocation points at the tunnel/soil interface are generated using an automatic meshing strategy. The presented method is one of the main components of VIBWAY, a user-friendly prediction tool to address railway-induced vibration problems. This paper presents three calculation examples in which the soil response due to forces applied on the tunnel structure are computed in terms of transfer functions. The results obtained for each one of the calculation examples are compared with those computed using a model based on a 2.5D FEM-BEM approach. The presented comparisons show that the proposed approach is a suitable strategy for predicting underground railway-induced vibrations, both in terms of accuracy and computational efficiency. Moreover, the use of an automatic meshing strategy and the SBM formulation not only eases the implementation of the approach but it also makes it easier to use, which is one of the key features of the VIBWAY tool.

A Modified Formulation of Singular Boundary Method for Exterior Acoustics

This paper proposes a modified formulation of the singular boundary method (SBM) by introducing the combined Helmholtz integral equation formulation (CHIEF) and the self-regularization technique to exterior acoustics. In the SBM, the concept of the origin intensity factor (OIF) is introduced to avoid the singularities of the fundamental solutions. The SBM belongs to the meshless boundary collocation methods. The additional use of the CHIEF scheme and the self-regularization technique in the SBM guarantees the unique solution of the exterior acoustics accurately and efficiently. Consequently, by using the SBM coupled with the CHIEF scheme and the self-regularization technique, the accuracy of the numerical solution can be improved, especially near the corresponding internal characteristic frequencies. Several numerical examples of two-dimensional and three-dimensional benchmark examples about exterior acoustics are used to verify the effectiveness and accuracy of the proposed method. The proposed numerical results are compared with the analytical solutions and the solutions obtained by the other numerical methods.

An ACA-BM-SBM for 2D acoustic sensitivity analysis

<abstract> <p>In this paper, we present a novel computational approach (named ACA-BM-SBM) for the calculation of 2D acoustic sensitivity by combining the Burton-Miller-type singular boundary method (BM-SBM) with the adaptive cross-approximation (ACA) algorithm. The BM-SBM circumvents the source singularities of the fundamental solutions by introducing the origin intensity factors, and it eliminates the fictitious frequency problem in external acoustic fields by introducing the Burton-Miller formula. As a semi-analysis meshless method, the BM-SBM can accurately solve the external acoustic problem governed by the Helmholtz equation. Nevertheless, the computational inefficiency introduced by the dense coefficient matrix renders this method suboptimal, particularly for large-scale simulations. As the number of nodes increases, the computation time and store memory increase dramatically. ACA is a purely algebraic method based on hierarchical matrices which can be used to partition the coefficient matrix step by step. By employing ACA, the BM-SBM can be effectively accelerated, and this results in less computation time, as well as fewer memory requirements. Numerical experiments, including Dirichlet and Neumann boundary conditions, illustrate that the proposed approach is an accurate, efficient and fast numerical method for acoustic sensitivity analysis.</p> </abstract>

Transient Dynamic Response Analysis of Two-Dimensional Saturated Soil with Singular Boundary Method

In this paper, the singular boundary method (SBM) in conjunction with the exponential window method (EWM) is firstly extended to simulate the transient dynamic response of two-dimensional saturated soil. The frequency-domain (Fourier space) governing equations of Biot theory is solved by the SBM with a linear combination of the fundamental solutions. In order to avoid the perplexing fictitious boundary in the method of fundamental solution (MFS), the SBM places the source point on the physical boundary and eliminates the source singularity of the fundamental solution via the origin intensity factors (OIFs). The EWM is carried out for the inverse Fourier transform, which transforms the frequency-domain solutions into the time-domain solutions. The accuracy and feasibility of the SBM-EWM are verified by three numerical examples. The numerical comparison between the MFS and SBM indicates that the SBM takes a quarter of the time taken by the MFS.

Singular boundary method for band structure calculations of in-plane waves in 2D phononic crystals

In this study, the singular boundary method, which is a boundary collocation method, is first employed to calculate the band structures of in-plane waves in two-dimensional phononic crystals. When a unit cell of a phononic crystal is considered, the whole boundaries, including the continuity and periodic boundaries are discretized by the singular boundary method. Then, a linear eigenvalue equation is derived. For a given frequency, an eigenvalue that involves the Block wave can be calculated. Therefore, by sweeping the frequencies, the band structure can be obtained. The accuracy, efficiency and convergence of the proposed method are tested by several numerical examples, and the results demonstrate that the singular boundary method can provide stable and efficient results for band structure calculations of in-plane waves in two-dimensional phononic crystals.