Evaluation of 2‐D Green's boundary formula and its normal derivative using Legendre polynomials,with an application to acoustic scattering problems

This paper presents the non‐singular forms, in a global sense, of two‐dimensional Green's boundary formula and its normal derivative. The main advantage of the modified formulations is that they are amenable to solution by directly applying standard quadrature formulas over the entire integration domain; that is, the proposed element‐free method requires only nodal data. The approach includes expressing the unknown function as a truncated Fourier–Legendre series, together with transforming the integration interval [a, b] to [‐1,1] ; the series coefficients are thus to be determined. The hypersingular integral, interpreted in the Hadamard finite‐part sense, and some weakly singular integrals can be evaluated analytically; the remaining integrals are regular with the limiting values of the integrands defined explicitly when a source point coincides with a field point. The effectiveness of the modified formulations is examined by an elliptic cylinder subject to prescribed boundary conditions. The regularization is further applied to acoustic scattering problems. The well‐known Burton–Miller method, using a linear combination of the surface Helmholtz integral equation and its normal derivative, is adopted to overcome the non‐uniqueness problem. A general non‐singular form of the composite equation is derived. Comparisons with analytical solutions for acoustically soft and hard circular cylinders are made. Copyright © 2001 John Wiley & Sons, Ltd.     

The development of the pFFT accelerated BEM for 3-D acoustic scattering problems based on the Burton and Miller's integral formulation

The precorrected-FFT acceleration technique is successfully applied in the boundary element method for the simulation of 3-D acoustic scattering problems. The composite Helmholtz integral equation presented by Burton and Miller is employed to overcome the nonuniqueness problem occurring in the simulation of exterior acoustic problems by the boundary element method. Since the triangular constant element is employed, the hypersingular boundary integral equation is reduced into a weakly singular boundary integral equation with the application of a modified Burton and Miller's formulation. The computational cost, the consumed memory and the convergence of the current method are demonstrated and analyzed through the simulation of a plane acoustic wave scattering from a rigid sphere and from an axisymmetrical rigid structure.     

A boundary spectral method for solving exterior acoustical problems with hypersingular integrals

A boundary spectral method is developed to solve acoustical problems with arbitrary boundary conditions. A formulation, originally derived by Burton and Miller, is used to overcome the non‐uniqueness problem in the high wave number range. This formulation is further modified into a globally non‐singular form to simplify the procedure of numerical quadrature when spectral methods are applied. In the present approach, generalized Fourier coefficients are determined instead of local variables at nodes as in conventional methods. The convergence of solutions is estimated through the decay of magnitude of the generalized Fourier coefficients. Several scattering and radiation problems from a sphere are demonstrated with high wave numbers in the present paper. Copyright © 1999 John Wiey & Sons, Ltd.     

An improved form of the hypersingular boundary integral equation for exterior acoustic problems

An improved form of the hypersingular boundary integral equation (BIE) for acoustic problems is developed in this paper. One popular method for overcoming non-unique problems that occur at characteristic frequencies is the well-known Burton and Miller (1971) method [7], which consists of a linear combination of the Helmholtz equation and its normal derivative equation. The crucial part in implementing this formulation is dealing with the hypersingular integrals. This paper proposes an improved reformulation of the Burton–Miller method and is used to regularize the hypersingular integrals using a new singularity subtraction technique and properties from the associated Laplace equations. It contains only weakly singular integrals and is directly valid for acoustic problems with arbitrary boundary conditions. This work is expected to lead to considerable progress in subsequent developments of the fast multipole boundary element method (FMBEM) for acoustic problems. Numerical examples of both radiation and scattering problems clearly demonstrate that the improved BIE can provide efficient, accurate, and reliable results for 3-D acoustics.     

Preconditioned Krylov subspace methods for boundary element solution of the Helmholtz equation

Discretization of boundary integral equations leads, in general, to fully populated complex valued non-Hermitian systems of equations. In this paper we consider the efficient solution of these boundary element systems by preconditioned iterative methods of Krylov subspace type. We devise preconditioners based on the splitting of the boundary integral operators into smooth and non-smooth parts and show these to be extremely efficient. The methods are applied to the boundary element solution of the Burton and Miller formulation of the exterior Helmholtz problem which includes the derivative of the double layer Helmholtz potential—a hypersingular operator. © 1998 John Wiley & Sons, Ltd.     

An adaptive fast multipole boundary element method for three-dimensional acoustic wave problems based on the Burton–Miller formulation

The high solution costs and non-uniqueness difficulties in the boundary element method (BEM) based on the conventional boundary integral equation (CBIE) formulation are two main weaknesses in the BEM for solving exterior acoustic wave problems. To tackle these two weaknesses, an adaptive fast multipole boundary element method (FMBEM) based on the Burton–Miller formulation for 3-D acoustics is presented in this paper. In this adaptive FMBEM, the Burton–Miller formulation using a linear combination of the CBIE and hypersingular BIE (HBIE) is applied to overcome the non-uniqueness difficulties. The iterative solver generalized minimal residual (GMRES) and fast multipole method (FMM) are adopted to improve the overall computational efficiency. This adaptive FMBEM for acoustics is an extension of the adaptive FMBEM for 3-D potential problems developed by the authors recently. Several examples on large-scale acoustic radiation and scattering problems are presented in this paper which show that the developed adaptive FMBEM can be several times faster than the non-adaptive FMBEM while maintaining the accuracies of the BEM.     

A simple boundary element method for problems of potential in non‐homogeneous media

A simple boundary element method for solving potential problems in non‐homogeneous media is presented. A physical parameter (e.g. heat conductivity, permeability, permittivity, resistivity, magnetic permeability) has a spatial distribution that varies with one or more co‐ordinates. For certain classes of material variations the non‐homogeneous problem can be transformed to known homogeneous problems such as those governed by the Laplace, Helmholtz and modified Helmholtz equations. A three‐dimensional Galerkin boundary element method implementation is presented for these cases. However, the present development is not restricted to Galerkin schemes and can be readily extended to other boundary integral methods such as standard collocation. A few test examples are given to verify the proposed formulation. The paper is supplemented by an Appendix, which presents an ABAQUS user‐subroutine for graded finite elements. The results from the finite element simulations are used for comparison with the present boundary element solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

On physical and numerical resonances for water wave problems using the dual boundary element method

Scattering problems of water waves impinging on bottom-mounted vertical cylinders are solved by using the dual boundary element method (DBEM). Both resonances due to near-trapped mode (physics) and fictitious frequency (mathematics) are examined. It is found that the near-trapped mode is a physical phenomenon and the fictitious frequency stems from the numerical instability. A trapped mode is associated with a singularity that lies on the real axis of complex wave number. A near-trapped mode means a localized behavior that energy is trapped in a truncated periodical structure. Critical wave number for the near-trapped mode and fictitious frequency of numerical instability are detected in this work. Numerical oscillation of the resultant force near the fictitious frequency is also observed by using the DBEM. Fictitious frequencies depend on the formulation instead of the specified boundary condition. Both the Burton and Miller approach and the CHIEF method are employed to alleviate the problem of irregular frequencies. Highly rank-deficiency matrices for four identical cylinders are numerically examined and the rank is promoted by adding valid CHIEF constraints. Parameter study of spacing and radius of cylinders on the near-trapped mode and fictitious frequency is also addressed. Several examples of water wave interaction by circular and square cylinders are demonstrated to see the validity of the present formulation.     

hp Fast multipole boundary element method for 3D acoustics

A fast multipole boundary element method (FMBEM) extended by an adaptive mesh refinement algorithm for solving acoustic problems in three‐dimensional space is presented in this paper. The Collocation method is used, and the Burton–Miller formulation is employed to overcome the fictitious eigenfrequencies arising for exterior domain problems. Because of the application of the combined integral equation, the developed FMBEM is feasible for all positive wave numbers even up to high frequencies. In order to evaluate the hypersingular integral resulting from the Burton–Miller formulation of the boundary integral equation, an integration technique for arbitrary element order is applied. The fast multipole method combined with an arbitrary order h‐p mesh refinement strategy enables accurate computation of large‐scale systems. Numerical examples substantiate the high accuracy attainable by the developed FMBEM, while requiring only moderate computational effort at the same time. Copyright © 2016 John Wiley & Sons, Ltd.     

Coupled FE–BE method for eigenvalue analysis of elastic structures submerged in an infinite fluid domain

For thin elastic structures submerged in heavy fluid, e.g., water, a strong interaction between the structural domain and the fluid domain occurs and significantly alters the eigenfrequencies. Therefore, the eigenanalysis of the fluid–structure interaction system is necessary. In this paper, a coupled finite element and boundary element (FE–BE) method is developed for the numerical eigenanalysis of the fluid–structure interaction problems. The structure is modeled by the finite element method. The compressibility of the fluid is taken into consideration, and hence the Helmholtz equation is employed as the governing equation and solved by the boundary element method (BEM). The resulting nonlinear eigenvalue problem is converted into a small linear one by applying a contour integral method. Adequate modifications are suggested to improve the efficiency of the contour integral method and avoid missing the eigenvalues of interest. The Burton–Miller formulation is applied to tackle the fictitious eigenfrequency problem of the BEM, and the optimal choice of its coupling parameter is investigated for the coupled FE–BE method. Numerical examples are given and discussed to demonstrate the effectiveness and accuracy of the developed FE–BE method. Copyright © 2016 John Wiley & Sons, Ltd.     

A wideband FMBEM for 2D acoustic design sensitivity analysis based on direct differentiation method

This paper presents a wideband fast multipole boundary element method (FMBEM) for two dimensional acoustic design sensitivity analysis based on the direct differentiation method. The wideband fast multipole method (FMM) formed by combining the original FMM and the diagonal form FMM is used to accelerate the matrix-vector products in the boundary element analysis. The Burton–Miller formulation is used to overcome the fictitious frequency problem when using a single Helmholtz boundary integral equation for exterior boundary-value problems. The strongly singular and hypersingular integrals in the sensitivity equations can be evaluated explicitly and directly by using the piecewise constant discretization. The iterative solver GMRES is applied to accelerate the solution of the linear system of equations. A set of optimal parameters for the wideband FMBEM design sensitivity analysis are obtained by observing the performances of the wideband FMM algorithm in terms of computing time and memory usage. Numerical examples are presented to demonstrate the efficiency and validity of the proposed algorithm.     

A wideband fast multipole accelerated boundary integral equation method for time‐harmonic elastodynamics in two dimensions

This article presents a wideband fast multipole method (FMM) to accelerate the boundary integral equation method for two‐dimensional elastodynamics in frequency domain. The present wideband FMM is established by coupling the low‐frequency FMM and the high‐frequency FMM that are formulated on the ingenious decomposition of the elastodynamic fundamental solution developed by Nishimura's group. For each of the two FMMs, we estimated the approximation parameters, that is, the expansion order for the low‐frequency FMM and the quadrature order for the high‐frequency FMM according to the requested accuracy, considering the coexistence of the derivatives of the Helmholtz kernels for the longitudinal and transcendental waves in the Burton–Muller type boundary integral equation of interest. In the numerical tests, the error resulting from the fast multipole approximation was monotonically decreased as the requested accuracy level was raised. Also, the computational complexity of the present fast boundary integral equation method agreed with the theory, that is, Nlog N, where N is the number of boundary elements in a series of scattering problems. The present fast boundary integral equation method is promising for simulations of the elastic systems with subwavelength structures. As an example, the wave propagation along a waveguide fabricated in a finite‐size phononic crystal was demonstrated. Copyright © 2012 John Wiley & Sons, Ltd.     

Mathematical analysis and numerical study to free vibrations of annular plates using BIEM and BEM

In this paper, the spurious eigenequations for annular plate eigenproblems by using BIEM and BEM are studied in the continuous and discrete systems. Since any two boundary integral equations in the plate formulation (4 equations) can be chosen, 6 (C) options can be considered instead of only two approaches (single‐layer and double‐layer methods, or singular and hypersingular equations) which are adopted for the eigenproblems of the membrane and acoustic problems. The occurring mechanism of the spurious eigenequation for annular plates in the complex‐valued formulations is studied analytically. For the continuous system, degenerate kernels for the fundamental solution and the Fourier series expansion for the circular boundary density are employed to derive the true and spurious eigenequations analytically. For the discrete system, the degenerate kernels for the fundamental solution and circulants resulting from the circular boundary are employed to determine the true and spurious eigenequations. True eigenequation depends on the specified boundary condition while spurious eigenequation is embedded in each formulation. It is found that the spurious eigenvalue for the annular plate is the true eigenvalue of the associated interior problem with an inner radius of the annular domain. Also, we provide three methods (SVD updating technique, Burton and Miller method and CHIEF method) to suppress the occurrence of the spurious eigenvalues. Several examples were demonstrated to check the validity of the formulations. Copyright © 2005 John Wiley & Sons, Ltd.     

A comparison of FE–BE coupling schemes for large‐scale problems with fluid–structure interaction

To predict the sound radiation of structures, both a structural problem and an acoustic problem have to be solved. In case of thin structures and dense fluids, a strong coupling scheme between the two problems is essential, since the feedback of the acoustic pressure onto the structure is not negligible. In this paper, the structural part is modeled with the finite element (FE) method. An interface to a commercial FE package is set up to import the structural matrices. The exterior acoustic problem is efficiently modeled with the Galerkin boundary element (BE) method. To overcome the well‐known drawback of fully populated system matrices, the fast multipole method is applied. Different coupling formulations are investigated. They are either based on the Burton–Miller approach or use a mortar coupling scheme. For all cases, iterative solvers with different preconditioners are used. The efficiency with respect to their memory consumption and computation time is compared for a simple model problem. At the end of the paper, a more complex structure is simulated. Copyright © 2008 John Wiley & Sons, Ltd.     

Modelling the transient interaction of a thin elastic shell with an exterior acoustic field

Coupled finite and boundary element methods for solving transient fluid–structure interaction problems are developed. The finite element method is used to model the radiating structure, and the boundary element method (BEM) is used to determine the resulting acoustic field. The well‐known stability problems of time domain BEMs are avoided by using a Burton–Miller‐type integral equation. The stability, accuracy and efficiency of two alternative solution methods are compared using an exact solution for the case of a thin spherical elastic shell. The convergence properties of the preferred solution method are then investigated more thoroughly. Copyright © 2007 John Wiley & Sons, Ltd.     

Analytical study and numerical experiments for degenerate scale problems in the boundary element method for two‐dimensional elasticity

For a plane elasticity problem, the boundary integral equation approach has been shown to yield a non‐unique solution when geometry size is equal to a degenerate scale. In this paper, the degenerate scale problem in the boundary element method (BEM) is analytically studied using the method of stress function. For the elliptic domain problem, the numerical difficulty of the degenerate scale can be solved by using the hypersingular formulation instead of using the singular formulation in the dual BEM. A simple example is shown to demonstrate the failure using the singular integral equations of dual BEM. It is found that the degenerate scale also depends on the Poisson's ratio. By employing the hypersingular formulation in the dual BEM, no degenerate scale occurs since a zero eigenvalue is not embedded in the influence matrix for any case. Copyright © 2002 John Wiley & Sons, Ltd.     

Cell‐based volume integration for boundary integral analysis

The evaluation of volume integrals that arise in boundary integral formulations for non‐homogeneous problems was considered. Using the “Galerkin vector” to represent the Green's function, the volume integral was decomposed into a boundary integral, together with a volume integral wherein the source function was everywhere zero on the boundary. This new volume integral can be evaluated using a regular grid of cells covering the domain, with all cell integrals, including partial cells at the boundary, evaluated by simple linear interpolation of vertex values. For grid vertices that lie close to the boundary, the near‐singular integrals were handled by partial analytic integration. The method employed a Galerkin approximation and was presented in terms of the three‐dimensional Poisson problem. An axisymmetric formulation was also presented, and in this setting, the solution of a nonlinear problem was considered. Copyright © 2012 John Wiley & Sons, Ltd.     

Non‐local boundary integral formulation for softening damage

A strongly non‐local boundary element method (BEM) for structures with strain‐softening damage treated by an integral‐type operator is developed. A plasticity model with yield limit degradation is implemented in a boundary element program using the initial‐stress boundary element method with iterations in each load increment. Regularized integral representations and boundary integral equations are used to avoid the difficulties associated with numerical computation of singular integrals. A numerical example is solved to verify the physical correctness and efficiency of the proposed formulation. The example consists of a softening strip perforated by a circular hole, subjected to tension. The strain‐softening damage is described by a plasticity model with a negative hardening parameter. The local formulation is shown to exhibit spurious sensitivity to cell mesh refinements, localization of softening damage into a band of single‐cell width, and excessive dependence of energy dissipation on the cell size. By contrast, the results for the non‐local theory are shown to be free of these physically incorrect features. Compared to the classical non‐local finite element approach, an additional advantage is that the internal cells need to be introduced only within the small zone (or band) in which the strain‐softening damage tends to localize within the structure. Copyright © 2003 John Wiley & Sons, Ltd.     

Trefftz‐based dual reciprocity method for hyperbolic boundary value problems

A new dual reciprocity‐type approach to approximating the solution of non‐homogeneous hyperbolic boundary value problems is presented in this paper. Typical variants of the dual reciprocity method obtain approximate particular solutions of boundary value problems in two steps. In the first step, the source function is approximated, typically using radial basis, trigonometric or polynomial functions. In the second step, the particular solution is obtained by analytically solving the non‐homogeneous equation having the approximation of the source function as the non‐homogeneous term. However, the particular solution trial functions obtained in this way typically have complicated expressions and, in the case of hyperbolic problems, points of singularity. Conversely, the method presented here uses the same trial functions for both source function and particular solution approximations. These functions have simple expressions and need not be singular, unless a singular particular solution is physically justified. The approximation is shown to be highly convergent and robust to mesh distortion. Any boundary method can be used to approximate the complementary solution of the boundary value problem, once its particular solution is known. The option here is to use hybrid‐Trefftz finite elements for this purpose. This option secures a domain integral‐free formulation and endorses the use of super‐sized finite elements as the (hierarchical) Trefftz bases contain relevant physical information on the modeled problem. Copyright © 2015 John Wiley & Sons, Ltd.     

Two‐dimensional unsteady heat conduction analysis with heat generation by triple‐reciprocity BEM

If the initial temperature is assumed to be constant, a domain integral is not needed to solve unsteady heat conduction problems without heat generation using the boundary element method (BEM).However, with heat generation or a non‐uniform initial temperature distribution, the domain integral is necessary. This paper demonstrates that two‐dimensional problems of unsteady heat conduction with heat generation and a non‐uniform initial temperature distribution can be solved approximately without the domain integral by the triple‐reciprocity boundary element method. In this method, heat generation and the initial temperature distribution are interpolated using the boundary integral equation. Copyright © 2001 John Wiley & Sons, Ltd.