Dual Boundary Integral Equation Research Articles

On the BEM for acoustic wave problems

The progress of the boundary element method (BEM) for solving acoustic wave problems is reviewed in this paper. The BEM is in a unique position among all the numerical methods available for solving acoustic wave problems. During the last few decades, research on the acoustic BEM has overcome many of the difficulties, and it is now an accurate and efficient numerical method in modeling many large-scale acoustic problems. This paper focuses on reviewing the dual boundary integral equation (BIE) formulation pioneered by Burton and Miller, treatment of the singular integrals involved in the BIEs, discretization considerations, and fast solution methods for solving the acoustic BEM equations. New directions in the research on the acoustic BEM are also discussed, with a few examples to show the potentials of the BEM in modeling aeroacoustics, acoustic metamaterials, bioacoustics, and sound rendering in computer animations.

Application of the dual reciprocity boundary integral equation approach to solve fourth-order time-fractional partial differential equations

ABSTRACTThis paper proposes a numerical approach to approximate the unknown solution of some high-order fractional partial differential equations. The main idea of this approach is to transform the original problem into an equivalent integral equation that depends only on the boundary values. The linear radial basis functions are used as the main tool for approximating the non-homogeneous terms and time derivative. Also the Caputo's sense is applied to approximate time derivatives. Numerical results demonstrate the order of time steps is and when and , respectively. Finally to overcome the nonlinear terms, predictor–corrector scheme is employed. The efficiency and usefulness of proposed method are demonstrated by some numerical examples.

Modeling of multiple crack propagation in 2-D elastic solids by the fast multipole boundary element method

In this paper, a fast multipole boundary element method (BEM) is presented for modeling crack propagation in two-dimensional (2-D) linear elastic solids. A dual boundary integral equation (BIE) formulation using a linear combination of the displacement and traction BIEs is applied to model cracks in this BEM. Constant boundary elements are used to discretize the BIEs and the fast multipole method (FMM) is applied to accelerate the solution of the BEM system of equations. Numerical examples of multiple crack propagation in 2-D elastic domains and under cyclic loading, including perforated plates with multiple holes and cracks, are presented to show the effectiveness and efficiency of the developed fast multipole BEM.

Investigation on the dynamics of air-gun array bubbles based on the dual fast multipole boundary element method

The air-gun array has important applications in ocean engineering. In this work, new dynamic behaviors of the air-gun array bubbles were investigated. The new adaptive fast multipole boundary element method (FMBEM) was implemented to accelerate the calculation of large scale moving boundary problems. The dual boundary integral equation (BIE) with an optimal linear combination of the regular BIE and the gradient BIE was used to facilitate a faster convergent FMBEM. It has been verified that the accuracy of the present numerical model was satisfactory for the dynamic simulation and the overall computational cost was scaled to O(N1.1) with N being the number of unknowns, which surpassed the conventional BEM largely. The behavior of air-gun array bubbles was studied for different distribution forms of the air-guns with the gravity effect considered. By increasing the dimensions of the bubble array, it can be seen that the “shield effect” of outer bubbles on inner bubbles was more evident. Pressure gradient was the dominant factor that determined the re-entrant jet's direction. The period of bubble array was longer if array scale increased. The distance between bubbles versus the intensity of interaction effect has been studied. In addition, the evolutions and the movements of the bubbles are quite different in this kind of large scale problem.

Local Eshelby matrix in eigen-variable boundary integral equations for solids with particles and cracks in full space

In order for the large scale numerical simulation of solids with particles and cracks, the concept of local Eshelby matrix has been introduced into the computational model of the eigen-variable boundary integral equation (BIE) to resolve the problem of interactions among particles and cracks by joining the two concepts of the eigenstrain and eigen COD. The local Eshelby matrix in the present paper is considered as an extension of the concept of Eshelby tensor in numerical form for both equivalent inclusions and cracks, defined on the group of near field particles and cracks in full space. Through some typical numerical examples, the computations of stress intensity factors (SIF) are carried out and compared with the results of the sub-domain method using the dual BIE. With the proposed computational model, it is verified not only the correctness and feasibility but also the high efficiency of the present solution procedure, showing the potential of the proposed method for large scale numerical simulation of solids with particles and cracks.

The dual reciprocity boundary integral equation technique to solve a class of the linear and nonlinear fractional partial differential equations

In this paper, we apply the boundary integral equation technique and the dual reciprocity boundary elements method (DRBEM) for the numerical solution of linear and nonlinear time‐fractional partial differential equations (TFPDEs). The main aim of the present paper is to examine the applicability and efficiency of DRBEM for solving TFPDEs. We employ the time‐stepping scheme to approximate the time derivative, and the method of linear radial basis functions is also used in the DRBEM technique. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity that appears in the nonlinear problems under consideration. To confirm the accuracy of the new approach, several examples are presented. The convergence of the DRBEM is studied numerically by comparing the exact solutions of the problems under investigation. Copyright © 2015 John Wiley & Sons, Ltd.

A low-frequency fast multipole boundary element method based on analytical integration of the hypersingular integral for 3D acoustic problems

A low-frequency fast multipole boundary element method (FMBEM) for 3D acoustic problems is proposed in this paper. The FMBEM adopts the explicit integration of the hypersingular integral in the dual boundary integral equation (BIE) formulation which was developed recently by Matsumoto, Zheng et al. for boundary discretization with constant element. This explicit integration formulation is analytical in nature and cancels out the divergent terms in the limit process. But two types of regular line integrals remain which are usually evaluated numerically using Gaussian quadrature. For these two types of regular line integrals, an accurate and efficient analytical method to evaluate them is developed in the present paper that does not use the Gaussian quadrature. In addition, the numerical instability of the low-frequency FMBEM using the rotation, coaxial translation and rotation back (RCR) decomposing algorithm for higher frequency acoustic problems is reported in this paper. Numerical examples are presented to validate the FMBEM based on the analytical integration of the hypersingular integral. The diagonal form moment which has analytical expression is applied in the upward pass. The improved low-frequency FMBEM delivers an algorithm with efficiency between the low-frequency FMBEM based on the RCR and the diagonal form FMBEM, and can be used for acoustic problems analysis of higher frequency.

Dual boundary integral equation formulation in plane elasticity using complex variable

This paper investigates the dual boundary integral equation formulation in plane elasticity using a complex variable. Four kinds of BIE are studied, and they are: (1) the first complex variable BIE for the interior region, (2) the second complex variable BIE for the interior region, (3) the first complex variable BIE for the exterior region, and (4) the second complex variable BIE for the exterior region. Using the Somigliana identity and letting the domain point approach a boundary point, the first complex variable BIE is obtained. Displacement versus traction operator is suggested. Using this operator and letting the domain point approach a boundary point, the second complex variable BIE is obtained. When the domain point approaches a boundary point, all limit processes are performed exactly through the generalized Sokhotski–Plemelj’s formulae. For the exterior problems, two degenerate boundary cases, the curved crack and the deformable curved line, are studied. Particularly, for the degenerate boundary case, or the shrinking curved crack case, four kinds of BIE are obtained.

Application of the dual reciprocity boundary integral equation technique to solve the nonlinear Klein–Gordon equation

This paper aims to obtain approximate solutions of the Nonlinear Klein–Gordon (NLKG) equation by employing the Boundary Integral Equation (BIE) method and the Dual Reciprocity Boundary Element Method (DRBEM). This method is improved by using a predictor–corrector scheme to the nonlinearity which appears in the problem. We employ the time stepping scheme to approximate the time derivative, and the Linear Radial Basis Functions (LRBFs), are used in the Dual Reciprocity (DR) technique. To confirm the accuracy of the new approach, the numerical results of a Double-Soliton and a problem with inhomogeneous terms are compared with analytical solutions and for the examples possessing single and periodic waves, two conserved quantities associated to the (NLKG) equation, the energy and the momentum are investigated.

Dual boundary integral equation formulation in antiplane elasticity using complex variable

This paper investigates the dual boundary integral equation formulation in antiplane elasticity using complex variable. Four kinds of boundary integral equation (BIE) are studied, and they are the first complex variable BIE for the interior region, the second complex variable BIE for the interior region, the first complex variable BIE for the exterior region, and the second complex variable BIE for the exterior region. The first BIE for the interior region is derived from the Somigliana identity, or the Betti’s reciprocal theorem in elasticity. A displacement versus traction operator is suggested. After using this operator, the second BIE for the interior region is derived. Similar derivations are performed for the first and second BIEs for the exterior region. In the case of the exterior boundary, two degenerate boundary cases are studied. One is the curved crack case, and other is the case of a deformable line. All kernels in the suggested BIEs are expressed in terms of complex variable.

Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method

In this paper, we use a numerical method based on the boundary integral equation (BIE) and an application of the dual reciprocity method (DRM) to solve the second-order one space-dimensional hyperbolic telegraph equation. Also the time stepping scheme is employed to deal with the time derivative. In this study, we have used three different types of radial basis functions (cubic, thin plate spline and linear RBFs), to approximate functions in the dual reciprocity method (DRM). To confirm the accuracy of the new approach and to show the performance of each of the RBFs, several examples are presented. The convergence of the DRBIE method is studied numerically by comparison with the exact solutions of the problems.

Stress intensity factor solutions for two-dimensional elastostatic problems by the hypersingular boundary integral equation

The boundary element method is a numerical method that can be advantageously used for a wide range of engineering problems, including the stress concentration problems encountered in fracture mechanics. In linear elastic fracture mechanics (LEFM), the stress intensity factor is an important parameter. Cracks, if present in the region experiencing the modes of deformation, increase the stress amplitude significantly and this high stress may lead to premature failure of the engineering components. If the value of the SIF is known, it is possible to predict whether the crack will propagate or not. As the conventional boundary integral equation (CBIE) degenerates when a mathematical crack is modelled, a previously developed dual boundary integral equation approach has been adopted in the current work. It utilizes the hypersingular boundary integral equation (HBIE) along with the CBIE. A weakly singular form of HBIE is utilized in the current work to eliminate the hypersingularity analytically. Stress intensity factors are evaluated using the crack opening displacement (COD), displacement extrapolation (DE), and the J-integral approaches. A stand-alone code has been developed for calculating the stress intensity factors of general two-dimensional domains with cracks. The code has been validated by evaluating the stress intensity factors for the standard components, for which the stress intensity factor values are available in the literature. Accurate and well-converged results are obtained proving the robustness of the code. A linear combination of the CBIE and HBIE was applied at the crack and a significant (87–97 per cent) reduction in the condition numbers for the system of equations was observed for the examples studied. Again, the results obtained are accurate and well converged.

A new fast multipole boundary element method for solving 2-D Stokes flow problems based on a dual BIE formulation

A fast multipole boundary element method (BEM) is presented in this paper for large-scale analysis of two-dimensional (2-D) Stokes flow problems based on a dual boundary integral equation (BIE) formulation. In this dual BIE formulation, a linear combination of the conventional BIE for velocity and the hypersingular BIE for traction is employed to achieve better conditioning for the BEM systems of equations. In both the velocity and traction BIEs, the direct formulations are used, that is, the boundary variables involved are the velocity and traction directly. The fast multipole formulations for both the velocity BIE and traction BIE for 2-D Stokes flow problems are presented in this paper based on the complex variable representations of the fundamental solutions. Several numerical examples are presented to study the accuracy and efficiency of the proposed approach. The numerical results clearly demonstrate the potentials of the developed fast multipole BEM for solving large-scale 2-D Stokes flow problems.

A dual BIE approach for large‐scale modelling of 3‐D electrostatic problems with the fast multipole boundary element method

AbstractA dual boundary integral equation (BIE) formulation is presented for the analysis of general 3‐D electrostatic problems, especially those involving thin structures. This dual BIE formulation uses a linear combination of the conventional BIE and hypersingular BIE on the entire boundary of a problem domain. Similar to crack problems in elasticity, the conventional BIE degenerates when the field outside a thin body is investigated, such as the electrostatic field around a thin conducting plate. The dual BIE formulation, however, does not degenerate in such cases. Most importantly, the dual BIE is found to have better conditioning for the equations using the boundary element method (BEM) compared with the conventional BIE, even for domains with regular shapes. Thus the dual BIE is well suited for implementation with the fast multipole BEM. The fast multipole BEM for the dual BIE formulation is developed based on an adaptive fast multiple approach for the conventional BIE. Several examples are studied with the fast multipole BEM code, including finite and infinite domain problems, bulky and thin plate structures, and simplified comb‐drive models having more than 440 thin beams with the total number of equations above 1.45 million and solved on a PC. The numerical results clearly demonstrate that the dual BIE is very effective in solving general 3‐D electrostatic problems, as well as special cases involving thin perfect conducting structures, and that the adaptive fast multipole BEM with the dual BIE formulation is very efficient and promising in solving large‐scale electrostatic problems. Copyright © 2007 John Wiley & Sons, Ltd.

A New Design Sensitivity Analysis of Acoustic Problems Based on Bem Avoiding Fictitious Eigenfrequency Issue

This paper presents a new design sensitivity analysis based on the boundary element method avoiding the fictitious eigenfrequency issue in acoustic problems. The direct differentiation method is applied to the derivation of sensitivity formulas. In solving an external acoustic problem with internal sub-domains by means of the boundary integral equation without any care, is numerical solution is violated at the so-called fictitious eigenfrequencies corresponding to the internal subdomains. The present paper proposes a new boundary element sensitivity analysis avoiding such a fictitious eigenfrequency problem. This is based on the dual boundary integral equation, proposed previously by the authors, and the integral expressions are differentiated directly with respect to the design parameters. One equation is the combined boundary integral equation proposed by Burton-Miller and the other is the normal derivative boundary integral equation multiplied by the same coupling parameter as in the Burton-Miller expression. The quadrilateral element is employed in this study. The Burton-Miller integral expression is used only at the middle nodes of element, while the normal derivative boundary integral equation multiplied by the same coupling parameter is applied to the vertex nodes of element, and vice versa. The effectiveness of the proposed approach is illustrated through some numerical examples for three-dimensional problems.

A New Boundary Element Analysis of 3-D Acoustic Fields Avoiding the Fictitious Eigenfrequency Problem

This paper is concerned with a new approach for avoiding the fictitious eigenfrequency problem in boundary element analysis of three-dimensional acoustic problems governed by Helmholtz equation. It is well known that, in solving without any care an external acoustic problem with internal sub-domains by means of the boundary integral equation, its numerical solution is violated at so-called fictitious eigenfrequencies corresponding to the internal sub-domains. The present paper proposes a new boundary element analysis to circumvent such a fictitious eigenfrequency problem by using dual boundary integral equation for nodal points on the boundary. One equation is the combined integral equation proposed by Burton-Miller. The other equation is the normal derivative boundary integral equation multiplied by the same coupling parameter as in the Burton-Miller expression. The quadrilateral element is employed in this study, and the Burton-Miller combined boundary integral equation is used at the middle nodes of element, while only the normal derivative boundary integral equation multiplied by the same coupling parameter is applied to the vertex nodes of element, and vice versa. The proposed approach is implemented, and its validity and effectiveness are demonstrated through numerical computation of the typical problems.

Dual BIE approaches for modeling electrostatic MEMS problems with thin beams and accelerated by the fast multipole method

Three boundary integral equation (BIE) formulations are investigated for the analysis of electrostatic fields exterior to thin-beam structures as found in some micro-electro-mechanical systems (MEMS). The three BIE formulations are: (1) the regular BIE using only the single-layer potential; (2) the dual BIE (a) using the regular BIE on one surface of a beam and the gradient BIE on the other surface; and (3) the dual BIE (b) using a linear combination of the regular BIE and gradient BIE on all the surfaces of the beam. Similar to crack problems in elasticity, the regular BIE degenerates when the beam thickness tends to zero, while the two dual BIE formulations do not degenerate. Most importantly, the dual BIE (b) is found to be well conditioned for all the values of the beam thickness, and thus well suited for implementation with the fast multipole BEM. The fast multipole BEM for both the regular BIE and the dual BIE (b) formulations are developed and tested on a simplified comb-drive model. The numerical results clearly show that the dual BIEs are very effective in solving MEMS problems with thin beams and the fast multipole BEM with the dual BIE (b) formulation is very efficient in solving large-scale MEMS models.

The simple boundary element method for multiple cracks in functionally graded media governed by potential theory: a three‐dimensional Galerkin approach

AbstractThe simple boundary element method consists of recycling existing codes for homogeneous media to solve problems in non‐homogeneous media while maintaining a purely boundary‐only formulation. Within this scope, this paper presents a ‘simple’ Galerkin boundary element method for multiple cracks in problems governed by potential theory in functionally graded media. Steady‐state heat conduction is investigated for thermal conductivity varying either parabolically, exponentially, or trigonometrically in one or more co‐ordinates. A three‐dimensional implementation which merges the dual boundary integral equation technique with the Galerkin approach is presented. Special emphasis is given to the treatment of crack surfaces and boundary conditions. The test examples simulated with the present method are verified with finite element results using graded finite elements. The numerical examples demonstrate the accuracy and efficiency of the present method especially when multiple interacting cracks are involved. Copyright © 2005 John Wiley & Sons, Ltd.

A New Method for Avoiding the Fictitious Eigenfrequency Problem in Boundary Element Analysis of Acoustic Problems (Consideration on 2D Problems)

This paper is concerned with a new method for avoiding the fictitious eigenfrequency problem in boundary element analysis of two-dimensional acoustic problems govened by Helmholtz equation. It is well known that in solving the external acoustic problem by means of the boundary integral equation, the solution is violated by the fictitious eigenfrequencies of the problem, which correspond to those of the internal problem. The present paper proposes a new method to circumvent the fictitious eigenfrequency problem by using the dual boundary integral equation for nodal points on the boundary. The quadratic boundary element is employed in this study. The usual boundary integral equation is applied to the end points of boundary element connected to other elements, while the derivaive boundary integral equation is applied to the other internal point of element. The proposed approach is implemented, and its effectiveness is demonstrated through comparison of the numerical results obtained by the developed computer code with other solutions.

On fictitious frequencies using circulants for radiation problems of a cylinder

Dual boundary integral equation (BIE) was developed for problems containing degenerate boundaries in 1988 by Hong and Chen [Journal of Engineering Mechanics-ASCE, 114, 6, 1988] and was termed the dual boundary element method (BEM) in 1992 by Portela et al. [International Journal for Numerical Methods in Engineering, 33, 6, 1992]. After near 30 years, the dual BIE/BEM for the problem containing a zero-thickness barrier was revisited mathematically to study the rank deficiency from the viewpoint of the updating term and the updating document of singular value decomposition (SVD) [Journal of Mechanics, 31, 5, 2015]. In this paper, we revisit the dual BEM from the physical point of view. Although there is no zero-thickness barrier in the real world, it is always required to simulate a finite-thickness degenerate boundary to be zero-thickness in comparison with sea, air or earth scale. For example, a sheet pile, a screen, a crack problem, a thin airfoil and a breakwater were modeled by the geometry of zero-thickness. The role of the dual BEM is evident since Lafe et al. [Journal of the Hydraulics Division-ASCE, 106, 6, 1980] used the conventional BEM to model the finite-thickness pile wall to geometrically approximate zero-thickness barrier but numerically yielding divergent solution. On the contrary, we physically model the finite-thickness breakwater as a zero-thickness barrier. The breakwater is employed as an illustrative case to demonstrate that the dual BEM simulated by a zero-thickness barrier can yield more acceptable results to match the experiment data in comparison with those of the finite thickness using the conventional BEM. Finally, a single horizontal plate and two dual horizontal plates in vertical direction and in horizontal direction are three illustrative cases to tell you why the dual BEM is necessary not only in mathematics but also in physics.