Origin Intensity Factor Research Articles

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.

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.

Numerical simulation of anti-plane wave propagation in heterogeneous media

This paper formulates a semi-analytical boundary meshless method, the singular boundary method (SBM), to simulate the anti-plane wave motion in the heterogeneous media. Two different continuous inhomogeneity variations are studied, and the corresponding fundamental solutions are derived by means of a simple variable transformation. The derivation reveals that the fundamental solutions are determined by the wavenumbers of the resulted governing equations after the variable transformation. The source singularities of these fundamental solutions are overcome by the novel origin intensity factors (OIFs). The fundamental solutions and OIFs make the SBM applicable for the simulation of the anti-plane wave scattering and diffraction in nonhomogeneous media. Numerical experiments, including the finite and semi-infinite cases with different variations and boundary conditions, are presented to illustrate the convergence, accuracy and effectiveness of the proposed SBM.

A half-space singular boundary method for predicting ground-borne vibrations

In this study, the singular boundary method (SBM) is extended to simulate the dynamic response of the half-space saturated soils in the frequency domain. The SBM employs the fundamental solutions of the governing equations as basis functions. Thus, the domain of interest can be discretized into several source points merely on the boundary, which makes the SBM available to describe the wave propagation in the half-space soil medium. The source singularity caused by the coincidence of collocation points and source points is overcome by the origin intensity factors (OIFs), which is given for the unclosed boundary. Two numerical examples exhibit the feasibility and advantages of the proposed SBM.

2.5D singular boundary method for exterior acoustic radiation and scattering problems

In this paper, a numerical methodology based on a two-and-a-half-dimensional (2.5D) singular boundary method (SBM) to deal with acoustic radiation and scattering problems in the context of longitudinally invariant structures is proposed and studied. In the proposed 2.5D SBM, the desingularisation provided by the subtracting and adding-back technique is used to determine the origin intensity factors (OIFs). These OIFs are derived by means of the OIFs of the Laplace equation. The feasibility, validity and accuracy of the proposed method are demonstrated for three acoustic benchmark problems, in which detailed comparisons with analytical solutions, the 2.5D boundary element method (BEM) and the 2.5D method of fundamental solutions (MFS) are performed. As a novelty of the present study, it is found that the 2.5D SBM provides a higher numerical accuracy than the 2.5D linear-element BEM and lower than the 2.5D quadratic-element BEM. Although the results obtained depict that a nodal approximation of the boundary geometry leads to a significant reduction in the accuracy of the 2.5D SBM, the delivered errors are still acceptable. For complex geometries, the 2.5D SBM is found to be simpler and more robust than the 2.5D MFS, since no optimization procedure is required.

Singular boundary method for 2D and 3D acoustic design sensitivity analysis

In this paper, a novel Burton-Miller-type singular boundary method (BM-SBM) formulation is proposed for the acoustic design sensitivity analysis, with the help of the direct differentiation method. The Burton-Miller formulation is employed to overcome the fictitious frequency problem in the numerical solutions of exterior acoustic problems. The simple empirical formulas are used to estimate the origin intensity factors in the BM-SBM. Compared with the traditional methods, the present scheme is an accurate and semi-analytical approach with the merits of truly meshless, integration-free, mathematical simplicity, and easy-to-program. As a boundary-type method based on the fundamental solution, furthermore, the BM-SBM is straightforward for addressing the exterior Helmholtz problems encountered in the acoustic design sensitivity analysis. Numerical experiments, including 2D and 3D geometries with Neumann boundary conditions, indicate that the proposed methodology is an accurate, stable, and convergent numerical approach for the acoustic design sensitivity analysis.

A regularized fast multipole method of moments for rapid calculation of three-dimensional time-harmonic electromagnetic scattering from complex targets

The article applies the recent research results of the singular boundary method to rapidly calculate time-harmonic electromagnetic scattering from complex targets. The study focuses on how to efficiently solve the large-scale and completely dense linear equations obtained by the regularized method of moments. The core innovation point is to propose a regularized fast multipole method of moments by combining the regularized method of moments and the fast multipole method. Several key mechanical issues encountered in simulation of electromagnetic scattering are discussed. The Rao-Wilton-Glisson vector basis function is used as the basis function. A modified fundamental solution is employed in place of the fundamental solution of the Helmholtz equation to avoid non-uniqueness at interior resonances. The fast multipole method is used to expedite the solution process. A fast near-field approximation preconditioner is introduced to greatly reduce excessive iterations. The origin intensity factor is applied to calculate the singular terms of interpolation matrix. The radar cross section of the NASA almond and a scaled stealth fighter has been accurately calculated by the proposed method.

A semi-analytical and boundary-type meshless method with adjoint variable formulation for acoustic design sensitivity analysis

This paper proposes a novel semi-analytical method for the two- and three-dimensional acoustic design sensitivity analysis by combining the Burton–Miller-type singular boundary method (BM-SBM) and the adjoint variable method. The BM-SBM is an accurate, stable and boundary-type meshless approach, and can effectively avoid the fictitious frequency issue in the numerical solutions of exterior acoustic problems due to the introduction of the Burton–Miller formula. The origin intensity factors in the BM-SBM are evaluated by employing the empirical formulas and the subtraction and adding-back technique, which is straightforward, accurate and efficient. Unlike the direct differentiation method, the adjoint variable method can save computation time and computer resources since it avoids computing the gradients of state variables by introducing adjoint state space equations. Compared with the existing methods, the present scheme can be regarded as a competitive alternative of acoustic design sensitivity analysis, due to the aforementioned merits. Three numerical examples are provided to illustrate the effectiveness and accuracy of the developed method.

A simple formula for obtaining OIFs on Neumann boundary in 2D potential problems and its applications

Prior to this study, an empirical formula has been proposed to evaluate origin intensity factors on the Dirichlet boundary. In this work, a simple empirical formula for obtaining origin intensity factors on the Neumann boundary is provided. As a result, origin intensity factors both on Dirichlet boundary and Neumann boundary could be directly achieved. It makes the numerical implementation of the singular boundary method easier. Numerical results are carried out to verify the accuracy and stability of the present scheme.

High-precision calculation of electromagnetic scattering by the Burton-Miller type regularized method of moments

High-precision calculation of electromagnetic scattering plays an important role in industrial field. This study aims to apply the latest research results based on semi-analytical boundary collocation method to calculate computational electromagnetics. The article focuses on discussing some common mechanical bottlenecks encountered by semi-analytical boundary collocation method in simulating electromagnetic scattering, such as the non-uniqueness at internal resonance, the singularity and near singularity of the fundamental solution and the preconditioning problem. The article constructs a regularized method of moments based on the Burton-Miller formulation to avoid the non-uniqueness at internal resonance. The origin intensity factor and nearly singular factor are used to replace the singular and nearly singular terms of the interpolation matrix. A novel fast near-field approximation preconditioner is introduced to greatly reduce the number of iterations without significantly increasing storage and computational complexity. Numerical experiment shows that the Burton-Miller type regularized method of moments can accurately calculate radar cross section scattered by the NASA almond model and real human head.

Localized singular boundary method for solving Laplace and Helmholtz equations in arbitrary 2D domains

In this research, the localized singular boundary method (LSBM) is proposed to solve the Laplace and Helmholtz equations in 2D arbitrary domains. In the traditional SBM, the resultant matrix system is a dense matrix, and it is unsuited for solving the large-scale problems. As a localized domain-type meshless method, a local subdomain for every node can be composed by its own and several nearest nodes. To each of the subdomains, the SBM formulation is applied to derive an implicit expression of the variable at each node in conjunction with the moving least-square approximation. To satisfy the boundary conditions at every boundary node and the governing equation at every node, a sparse linear algebraic system can be obtained. Thus, the numerical solutions at all nodes can be achieved by solving it. Compared with the traditional SBM, the LSBM involves only the origin intensity factor on a circular boundary associated with Dirichlet boundary conditions. It can also effectively avoid the boundary layer effect in the conventional SBM. Furthermore, the proposed LSBM requires less memory storage and computational cost due to the sparse and banded matrix system. Several numerical examples are tested to verify the accuracy and stability of the proposed LSBM.

A regularized method of moments for three-dimensional time-harmonic electromagnetic scattering

A regularized method of moments based on the modified fundamental solution of the Helmholtz equation is proposed in this article. The regularized method of moments uses the origin intensity factor technique which is free of mesh and integration to deal with the singularity at origin of the basis function. Thus, the time-consuming singular integration can be avoided. In addition, the non-uniqueness at internal resonance is also fixed using the constructed modified fundamental solution. In comparison with the traditional method of moments, the regularized method of moments can reduce the computational time by half, while the stability and accuracy stay about the same. Numerical experiments demonstrate that the regularized method of moments can accurately and efficiently compute the radar cross section of perfect conducting scatter in all frequency ranges.

2.5D singular boundary method for acoustic wave propagation

This study proposes a two-and-a-half-dimensional (2.5D) singular boundary method (SBM) to solve three dimensional (3D) acoustic problems with a constant cross section excited by harmonic point sources. The SBM expands the solution of a problem with a linear combination of fundamental solutions of the governing equation. The singular terms in the fundamental solutions are replaced by the origin intensity factors (OIFs). Nevertheless, the SBM suffers quadratically increasing memory consumption and operation count with respect to the degree of freedoms owing to the fully-populated coefficient matrices. This makes the SBM not applicable in road and rail traffic systems, where the domain of interest always is a few hundred meters long. Taking advantage of the invariant cross section in these problems, the 2.5D method is devised to decompose the 3D problem into a summation of a set of simpler 2D problems, and thus enables the SBM to solve 3D traffic acoustic problems. To recast the results of 3D problems, a new formulation for numerical integration is proposed. The proposed 2.5D SBM is tested to two problems in both full-space and half-space to illustrate its effectiveness and feasibility.

Improved singular boundary method and dual reciprocity method for fractional derivative Rayleigh–Stokes problem

The improved singular boundary method (ISBM) and dual reciprocity method (DRM) are coupled to solve fractional derivative the Rayleigh–Stokes problem with nonhomogeneous term. This method is free of mesh and integration, mathematically simple, and easy to program. Also, origin intensity factors (OIFs) significant techniques in ISBM make the method as a strong meshless method. First, the time-fractional derivative term in mentioned equation is discretized; then, ISBM–DRM is utilized to solve consequent equation. It is proved the method is unconditionally stable and convergent with convergence order $${\mathcal {O}}(\tau ^{1+\alpha })$$. In addition, numerical results confirm the accuracy and efficiency of the presented scheme.

Singular boundary method for 2D and 3D heat source reconstruction

This paper presents the singular boundary method (SBM) in conjunction with the regularization method to recover the heat source in steady-state heat conduction problems from boundary temperature and heat flux measurements. To avoid the internal measurements, the heat source is assumed to satisfy a second-order partial differential equation on a physical basis. As a result, the source inverse problems can be transformed to a direct fourth-order governing equation, which can be tackled by the SBM. The SBM utilizes the fundamental solutions to approximate the solutions and desingularizes the source singularities of fundamental solutions with origin intensity factors (OIFs), which are derived for both 2D and 3D cases. The regularization method is introduced to deal with noisy boundary. The feasibility and accuracy of the proposed method is validated by both 2D and 3D cases.

Singular boundary method for 3D time-harmonic electromagnetic scattering problems

A frequency domain singular boundary method is presented for solving 3D time-harmonic electromagnetic scattering problem from perfect electric conductors. To avoid solving the coupled partial differential equations with fundamental solutions involving hypersingular terms, we decompose the governing equation into a system of independent Helmholtz equations with mutually coupled boundary conditions. Then the singular boundary method employs the fundamental solutions of the Helmholtz equations to approximate the scattered electric field variables. To desingularize the source singularity in the fundamental solutions, the origin intensity factors are introduced. In the novel formulation, only the origin intensity factors for fundamental solutions of 3D Helmholtz equations and its derivatives need to be considered which have been derived in the paper. Several numerical examples involving various perfectly conducting obstacles are carried out to demonstrate the validity and accuracy of the present method.

A regularized approach evaluating origin intensity factor of singular boundary method for Helmholtz equation with high wavenumbers

Evaluation of the origin intensity factor of the singular boundary method for Helmholtz equation with high wavenumbers has been a difficult task for a long time. In this study, a regularized approach is provided to bypass this limitation. The core idea of the subtraction and adding-back technique is to substitute an artificially constructed general solution of the Helmholtz equation into the boundary integral equation or the hyper boundary integral equation to evaluate the non-singular expressions of the fundamental solutions at origin. The core difficulty is to derive the appropriate artificially constructed general solution. The regularized approach avoids the unstable inverse interpolation and has strict mathematical derivation process. Therefore, it is easy-to-program and free of mesh dependency. Numerical experiments show that the proposed technique can be used successfully to avoid singularity and hyper singularity difficulties encountered in the boundary element method and the singular boundary method.

A frequency domain formulation of the singular boundary method for dynamic analysis of thin elastic plate

This paper presents a frequency domain formulation of the singular boundary method (SBM) for the analysis of thin elastic plates subjected to dynamic loads. In the SBM, the fundamental solutions for the dynamic thin plate problem in frequency domain are taken as basis functions. To isolate the singularities, we derive the origin intensity factors (OIFs) for the kernels of the SBM which allows the source points to coincide with the collocation points. For the transient analysis, the exponential window method (EWM) is applied in conjunction with the SBM to obtain the time-domain result. To demonstrate the efficiency and accuracy of the present method, several examples under harmonic and transient loads are carried out in comparison with the exact solutions and finite element solutions.