Indirect Boundary Integral Equations Research Articles

Design sensitivity analysis of three-dimensional acoustic problems using a fast multipole accelerated indirect BEM

In this paper, the design sensitivity formulations for three-dimensional acoustic problems are presented based on the indirect boundary element method (IBEM). The non-unique solution difficulty of the IBEM in solving exterior acoustic problems is avoided by applying a hybrid combination of single and double layer potentials. In order to accelerate the computational procedure easily, a set of non-singular indirect boundary integral equations (IBIEs) for the acoustic design sensitivity analysis are derived and then the fast multipole method (FMM) is employed to accelerate the matrix–vector products in the IBEM. The method is further extended to the design sensitivity analysis of thin-body acoustic problems above an infinite plane. Numerical examples are given to verify the effectiveness and accuracy of the method, and the simulations of thin-body acoustic problems show its potential in engineering applications.

3D displacement discontinuity analysis of in-situ stress perturbation near a weak faul

A numerical investigation utilizing the 3D displacement discontinuity method is performed to examine the stress perturbations and induced displacements near a weak fault with arbitrary orientations and dip, assuming zero shear stress and normal displacement. The in-situ stress field near the fault is taken as known and varied with depth. The modelling is constructed based on indirect boundary integral equations. In this work, the fault plane is first modelled as a rectangular plane with negligible thickness between the adjacent surfaces. The fault plane is then divided into numerous rectangular boundary elements with imposed shear singularities on the surface, which is normal to the fault plane to simulate a traction-free scenario. The numerical results of the total induced stresses and displacements are then compared to the existing solutions of a penny-shaped crack for validation purpose. With validated results, the paper moves on to the discussion of various factors that have impacts on the induced stress and displacements, including: aspect ratio which is defined by strike over dip; orientation of the strike on the horizontal ground surface; as well as dip. The boundary integration method with modification is also used to model an elliptical distribution of singularities with inner, corner, and edge elements to accommodate more complex shape of a discontinuity; small differences are observed. Cited as: Chai, Y., Yin, S. 3D displacement discontinuity analysis of in-situ stress perturbation near a weak fault. Advances in Geo-Energy Research, 2021, 5(3): 286-296, doi: 10.46690/ager.2021.03.05

Isogeometric indirect boundary element method for solving the 3D acoustic problems

This paper provides an Isogeometric Indirect Boundary Element Method (IGIBEM) based on NURBS (Non-Uniform Rational B-Splines) and PHT-splines (polynomial splines over hierarchical T-meshes) for analyzing the three-dimensional (3D) acoustic problems. In the classical procedure, the geometries are discretized by Lagrange polynomials elements, which leads to both substantial geometrical error and time-consuming meshing steps. However, these deficiencies can be eliminated by the isogeometric analysis (IGA) directly incorporating the geometry description generated from the CAD (Computer Aided Design) software into CAE (Computer Aided Engineering) analysis. Unlike the DBEM (direct boundary element method), the IBEM (indirect boundary element method) allows different types of boundary conditions on the two sides of a surface. Moreover, the hypersingular integrals in IBEM can be transformed to a weakly singular form. In addition, the PHT-based IBEM is used to investigate the influence of local refinement on the accuracy of solutions. Finally, the non-uniqueness problem is solved, which is a fatal defect in the acoustic BEM for the exterior problem. Four different methods to handle the non-uniqueness problem are discussed and compared. The results obtained by the proposed method were compared with analytical solutions and the results computed by Lagrange-based IBEM. Several benchmark examples demonstrate: (1) the present method, i.e. IGIBEM, has super accuracy over conventional IBEM for the acoustic problems; (2) local refinement has a significant influence on the convergence rate of the solutions, and the numerical accuracy is relevant to the distance to the boundary where the local refinement acted; (3) as for the non-uniqueness problem, the imposition of specific interior boundary conditions not only obtains the best calculation result over the entire range of frequencies, but also has a simple integral formulation. • We implement an IGIBEM based on NURBS and PHT-splines for 3D acoustic problems. • The NURBS-based IBEM has super accuracy over the Lagrange-based IBEM. • The PHT-based IBEM was proposed to investigate the effect of local refinement on solution domain. • We investigated four indirect boundary integral equations to overcome the non-uniqueness problem. • The imposition of specific interior boundary conditions has simple integral formulation and the best calculation result.

Boundary stress analysis using a new regularized boundary integral equation for three-dimensional elasticity problems

This paper presents new regularized boundary integral equations (BIEs) for elastic displacement gradients in three dimensions and then combines them by the generalized Hooke’s law to calculate the boundary stress. In the new regularized BIEs, two special tangential vectors are designed with the normal vector to construct a transformation system. Based on this system, the displacement gradient in any direction can be transformed into a linear combination of the normal gradient and tangential gradients along the two special vectors. Moreover, a theorem related to some integral properties of the fundamental solution is introduced. Finally, the regularized indirect BIEs are developed by using the above-mentioned technique of linear combination and theorem. The proposed method has some advantages over the direct boundary element method, such as the relaxed continuity requirement of density function, no hypersingular integral, and being available to calculate the displacement gradient in any direction. The numerical implementation of the developed integral equation is provided, and the accuracy and convergence of the approach are also illustrated through four numerical examples.

三维位势问题新的规则化边界元法

This presentation is mainly devoted to the research on the regularization of indirect boundary integral equations (IBIEs) for three-dimensional problems and establishes the new theory and method of the regularized BEM. The two special tangential vectors, which are linearly independent and associated with the normal vectors, are constructed, and then a characteristics theorem for the contour integrations of the normal and tangential gradients of some quantities, related with the fundamental solutions for 3D potential problems, is presented. A limit theorem for the transformation from domain integral equations into boundary integral equations (BIEs) is also proposed. Based on this, together with a novel decomposition technique to the fundamental solution, the regularized BIEs with indirect unknowns, which don’t involve the direct calculation of CPV and HFP integrals, are derived for 3D potential problems. Compared with the widely practiced direct regularized BEMs, the presented method has many advantages. First, the continuity requirement for density function in the direct formulation can be reduced here. Second, it is more suitable for solving the structures of thin bodies, considering the solution process for boundary or field quantities doesn’t involve the HFP integrals and nearly HFP integrals so the regularization algorithm to the considered singular or nearly singular integrals is more effective. Third, the proposed regularized BIEs can calculate the any potential gradients on the boundary, but not limited to the normal fluxes, and also independent of the potential BIEs. A systematic approach for implementing numerical solutions is proposed by adopting the ℃ continuous elements to depict the boundary surface and the discontinuous interpolation to approximate the boundary quantities. Especially, for the boundary value problems with elliptic surfaces or piecewise plane surfaces boundary, the exact elements are developed to model their boundaries with almost no error. The validity of the proposed scheme is demonstrated by several benchmark examples. Excellent agreement between the numerical results and exact solutions is obtained even with using small amounts of element.

A NONSINGULAR BOUNDARY ELEMENT METHOD FOR THE TORSION PROBLEM OF THE ANISOTROPIC UNIFORM BAR

The presentation is mainly devoted to the research on the regularized boundary integral equations (BIEs) with indirect unknowns for torsion problem of the anisotropic uniform bar. Based on a new view and idea, a novel regularization technique is pursued, in which the nonsingular indirect BIE (IBIE) excluding the CPV and HFP integrals is established. Such torsion problems can be solved directly by using the presented technique without transforming them into isotropic ones, for this reason, no inverse transform is required. Moreover, a unique feature of the shear stress BIEs expressed by density functions is that they are independent of the warp BIEs and, as such, can be collocated at the same locations as the warp BIEs. This provides additional and concurrently useable equations for various purposes. Besides, in the numerical implementation, the boundary geometric is depicted by exact elements, while the distribution of the boundary quantity on each element is approximated by a discontinuous quadratic element. Some numerical examples will be applied to validate the current scheme. It is shown that a better precision and high-computational efficiency can be achieved by the presentation.

BOUNDARY ELEMENT ANALYSIS OF THIN STRUCTURAL PROBLEMS IN 2D ELASTOSTATICS

Thin structures, such as thin films and coatings, have been widely designed and utilized in many industries recently. However, the widespread experimental research in thin structural problems underlies a general lack of modeling efforts which can accurately and efficiently predict their performance. In this paper, the boundary element method (BEM) based on elasticity theory is developed for two-dimensional (2D) thin structures with the thickness to length ratio in the micro (10^(-6)) or nano (10^(-9)) scales. The BEM-based approach proposed in this paper is constructed using a combination of the regularized indirect boundary integral equations (BIEs) and a general nonlinear transformation which can eliminate the nearly singular properties of the integral kernels. For the test problems studied, very promising results are obtained when the thickness to length ratio is in the orders between 10^(-6) and 10^(-9), which is sufficient for modeling most thin structures in the micro- or nano-scales.

A novel boundary element approach for solving the anisotropic potential problems

This presentation is mainly devoted to the research on the regularization of indirect boundary integral equations (IBIEs) for anisotropic potential problems. Based on a new idea, a novel regularization technique is pursued, in which the regularized IBIEs excluding the CPV and HFP integrals are established. The proposed method has many advantages. First, it does not need to calculate multiple integral as the Galerkin method, so it is simple and easy for programming. Second, it can compute boundary quantities ∂u/ ∂x i ( i=1,2). Third, the anisotropic problems can be solved directly without transforming them into isotropic ones so that no inverse transform is required. Finally, the gradient BIEs are independent of the potential BIEs and they can provide variously useful equations. Numerical examples show that a better precision and high computational efficiency can be achieved by the present method.

Fast Galerkin BEM for 3D-potential theory

This paper is concerned with the development of a fast spectral method for solving direct and indirect boundary integral equations in 3D-potential theory. Based on a Galerkin approximation and the Fast Fourier Transform, the proposed method is a generalization of the precorrected-FFT technique to handle not only single-layer potentials but also double-layer potentials and higher-order basis functions. Numerical examples utilizing piecewise linear shape functions are presented to illustrate the performance of the method.

A new method for Stokes problems with circular boundaries using degenerate kernel and Fourier series

AbstractThis study is concerned with the Stokes flow of an incompressible fluid of constant density and viscosity with circular boundaries. To fully capture the circular boundary, the boundary densities in the direct and indirect boundary integral equations (BIEs) are expanded in terms of Fourier series. The kernel functions in either the direct BIE or the indirect BIE are expanded to degenerate kernels by using the separation of field and source points. Consequently, the improper integrals are transformed to series sum and are easily calculated. The linear algebraic system can be established by matching the boundary conditions at the collocation points. Then, the unknown Fourier coefficients can be easily determined. Finally, several examples including circular and eccentric domains are presented to demonstrate the validity of the present method. Five gains were obtained: (1) meshless approach; (2) free of boundary‐layer effect; (3) singularity free; (4) exponential convergence; and (5) well‐posed model. Copyright © 2007 John Wiley & Sons, Ltd.

A minimum-order boundary element method to extract the 3-D inductance and resistance of the interconnects in VLSI

The high frequency resistance and inductance of the 3-D complex interconnect structures can be calculated by solving an eddy current electromagnetic problem. In this paper, a model for charactering such a 3-D eddy current problem is proposed, in which the electromagnetic fields in both the conducting and non-conducting regions are described in terms of the magnetic vector potential, and a set of the indirect boundary integral equations (IBIE) is obtained. The IBIEs can be solved by boundary element method, so this method avoids discretizing the domain of the conductors. As an indirect boundary element method, it is of minimum order. It does not restrict the direction of the current in conductors, and hence it can consider the mutual impedance between two perpendicular conductors. The numerical results can well meet the analytical solution of a 2-D problem. The mutual impedance of two perpendicular conductors is also shown under the different gaps between conductors and different frequencies.

A virtual 3-D fast extractor for interconnect capacitance of multiple dielectrics

As features in VLSI reach submicron sizes, the parasitic interconnect must be calculated even more quickly and accurately. In this paper, we present a virtual 3-D extractor of multiple dielectrics that is very fast. In the indirect boundary integral equations from classical potential theory, we replaced the plane charge distribution on the surfaces of conductors with metal mesh charge distribution to simplify the 3-D structure. We also adopted the multipole acceleration with improved non-uniform cube partitioning to further simplify the computational complexity. Numerical results show that the computational complexity of our algorithm is about O( n).

Dynamic modelling of the flat 2-D crack by a semi-analytic BIEM scheme

We present an efficient numerical method for solving indirect boundary integral equations that describe the dynamics of a flat two-dimensional (2-D) crack in all modes of fracture. The method is based on a piecewise-constant interpolation, both in space and time, of the slip-rate function, by which the original equation is reduced to a discrete convolution, in space and time, of the slip-rate and a set of analytically obtained coefficients. If the time-step interval is set sufficiently small with respect to the spatial grid size, the discrete equations decouple and can be solved explicitly. This semi-analytic scheme can be extended to the calculation of the wave field off the crack plane. A necessary condition for the numerical stability of this scheme is investigated by way of an exhaustive set of trial runs for a kinematic problem. For the case investigated, our scheme is very stable for a fairly wide range of control parameters in modes III and I, whereas, in mode II, it is unstable except for some limited ranges of the parameters. The use of Peirce and Siebrits' e-scheme in time collocation is found helpful in stabilizing the numerical calculation. Our scheme also allows for variable time steps. Copyright © 2001 John Wiley & Sons, Ltd.

Equivalent boundary integral equations for plane elasticity

Indirect and direct boundary integral equations equivalent to the original boundary value problem of differential equation of plane elasticity are established rigorously. The unnecessity or deficiency of some customary boundary integral equations is indicated by examples and numerical comparison.

Boundary integral equation method for the diffraction of elastic waves using simplified green's functions

AbstractAn alternate formulation of the ‘substructure deletion method’ suggested by Dasgupta in 19791 has been successfully implemented. The idea is to utilize simple Green's functions developed for a surface problem to replace the more complicated Green's functions required for embedded problems while still being able to generate an accurate solution. Since the exterior medium is usually represented by a continuum model, the interior medium in the present approach will also be represented by a continuum model rather than a finite element model as suggested originally, thereby eliminating the incompatibility between the solutions of the interior and exterior media. Detailed studies of the method's accuracy and limitations were performed using two‐dimensional examples in wave scattering of canyons and alluvial valleys, problems which are more suitable for this method than the embedded foundation problem. The results obtained indicate that the alternate formulation gives accurate results only when the vertical dimension of the scattering object is not too large; if the aspect ratio (vertical over lateral) exceeds a certain limit, the results will not approach the known results given by boundary integral equation solutions or indirect boundary integral equations no matter what the refinement of the model may be. The greatest advantage of the present method is that the task of calculating Green's functions is reduced significantly; computational time using this new formulation is approximately five times less than for conventional boundary integral equation methods.

Boundary element analysis method for 3-D transient eddy current problems based on the second order vector potential formulation

The authors consider a boundary-element method (BEM) for 3-D transient eddy current problems formulated in terms of the second-order vector potential (SOVP). Indirect boundary integral equations are derived using the time-dependent fundamental solution and fictitious source densities. In the numerical implementation, a time-stepping convolution approach is adopted to eliminate the need for volume discretization and domain integration. Numerical results of a benchmark problem demonstrate the validity and accuracy of the technique. The proposed scheme, compared with an alternative volume-based recurring initial condition approach, not only provides considerable reduction in the manpower requirements for modeling, but also permits a more precise solution, especially for 3-D problems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>