Boundary Integral Equation Approach Research Articles

A new meshless regular local boundary integral equation (MRLBIE) approach

A new meshless method based on a regular local integral equation and the moving least-squares approximation is developed. The present method is a truly meshless one as it does not need a ‘finite element or boundary element mesh’, either for purposes of interpolation of the solution variables, or for the integration of the ‘energy’. All integrals can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. No derivatives of the shape functions are needed in constructing the system stiffness matrix for the internal nodes, as well as for those boundary nodes with no essential-boundary-condition-prescribed sections on their local boundaries. Numerical examples presented in the paper show that high rates of convergence with mesh refinement are achievable, and the computational results for the unknown variable and its derivatives are very accurate. No special post-processing procedure is required to compute the derivatives of the unknown variable, as the original result, from the moving least-squares approximation, is smooth enough. Copyright © 1999 John Wiley & Sons, Ltd.

A new meshless regular local boundary integral equation (MRLBIE) approach

A new meshless regular local boundary integral equation (MRLBIE) approach

An inverse boundary value problem for the heat equation: the Neumann condition

We consider the inverse problem to determine the shape of an insulated inclusion within a heat conducting medium from overdetermined Cauchy data of solutions for the heat equation on the accessible exterior boundary of the medium. For the approximate solution of this ill-posed and nonlinear problem we propose a regularized Newton iteration scheme based on a boundary integral equation approach for the initial Neumann boundary value problem for the heat equation. For a foundation of the Newton method we establish the differentiability of the solution to the initial Neumann boundary value problem with respect to the interior boundary curve in the sense of a domain derivative and investigate the injectivity of the linearized mapping. Some numerical examples for the feasibility of the method are presented.

Amplification of waves by an orthotropic basin: sagittal plane motion

Scattering of elastic waves by an orthotropic basin of arbitrary shape embedded in a half-space is investigated for the sagittal plane motion using an indirect boundary integral equation approach. Steady-state results were obtained for incident plane harmonic pseudo P-, S-, and Rayleigh waves. Detailed convergence analysis of the method is presented. Green's functions are evaluated by using adaptive Newton-Cotes or Filon quadratures. Surface ground motion is presented for semicircular and semielliptical basins with different material properties and various angles of incidence. The results show that surface motion strongly depends upon nature of incident wave, geometry and material properties of the basin, and location of the observation point. Comparison with isotropic basin response demonstrates that anisotropy is very important in amplification of surface ground motion.

Circulant preconditioners for ill-conditioned boundary integral equations from potential equations

In this paper, we consider solving potential equations by the boundary integral equation approach. The equations so derived are Fredholm integral equations of the first kind and are known to be ill-conditioned. Their discretized matrices are dense and have condition numbers growing like O(n) where n is the matrix size. We propose to solve the equations by the preconditioned conjugate gradient method with circulant integral operators as preconditioners. These are convolution operators with periodic kernels and hence can be inverted efficiently by using fast Fourier transforms. We prove that the preconditioned systems are well conditioned, and hence the convergence rate of the method is linear. Numerical results for two types of regions are given to illustrate the fast convergence. © 1998 John Wiley & Sons, Ltd.

Circulant preconditioners for ill‐conditioned boundary integral equations from potential equations

In this paper, we consider solving potential equations by the boundary integral equation approach. The equations so derived are Fredholm integral equations of the first kind and are known to be ill-conditioned. Their discretized matrices are dense and have condition numbers growing like O(n) where n is the matrix size. We propose to solve the equations by the preconditioned conjugate gradient method with circulant integral operators as preconditioners. These are convolution operators with periodic kernels and hence can be inverted efficiently by using fast Fourier transforms. We prove that the preconditioned systems are well conditioned, and hence the convergence rate of the method is linear. Numerical results for two types of regions are given to illustrate the fast convergence. © 1998 John Wiley & Sons, Ltd.

On the numerical solution of an inverse boundary value problem for the heat equation

We consider the inverse problem of reconstructing the interior boundary curve of an arbitrary-shaped annulus from overdetermined Cauchy data on the exterior boundary curve. For the approximate solution of this ill-posed and nonlinear problem we propose a regularized Newton method based on a boundary integral equation approach for the initial boundary value problem for the heat equation. A theoretical foundation for this Newton method is given by establishing the differentiability of the initial boundary value problem with respect to the interior boundary curve in the sense of a domain derivative. Numerical examples indicate the feasibility of our method.

The method of analysis of cracks in three-dimensional transversely isotropic media: boundary integral equation approach

The fundamental solutions are obtained for a unit-concentrated displacement discontinuity in a three-dimensional transversely isotropic medium. The method of solution involves reducing the problem to a system of hyper-singular integral equations by application of these fundamental solutions. The near crack border elastic displacement and stress are obtained. Stress intensity factors can be expressed in terms of the displacement discontinuity on the crack surface. Analogy is established between the boundary integral equations for arbitrarily shaped cracks in a transversely isotropic and elastic medium such that once the stress intensity factors in the transversely isotropic medium can be determined directly from that of the isotropic elastic medium. Results for the penny-shaped crack are obtained as an example.

Amplification of SH waves by an orthotropic basin

Scattering of elastic waves by an orthotropic sedimentary basin is investigated for antiplane strain model using an indirect boundary integral equation approach. Both steady state and transient response were obtained for semicircular and semielliptical basins with different material properties. The results indicate that the basin geometry and the impedance contrast between the half-space and the basin have similar effects on the surface ground motion amplification as for the isotropic case. However, the material anisotropy may change significantly the fundamental resonant frequencies of the basin, resulting in different surface displacement amplification patterns. In addition, it was observed that the arrival time of the main disturbance on the surface strongly depends on material anisotropy for different angles of incidence. The results demonstrate that material anisotropy may be very important in explaining surface ground motion amplification for sedimentary basins. © 1998 John Wiley & Sons, Ltd.

On the boundary integral equations approach to a semi-infinite strip investigation

An orthotropic semi-infinite strip under arbitrary boundary conditions is considered. By means of Fourier transforms, boundary integral relations of special type with moving and motionless singularities of the Cauchy type are obtained. These relations lead to a system of singular integral equations corresponding to the various mixed boundary value problem. The power of singularities at the corner points, stresses and stress intensity factors are calculated for different loads and various material properties.

Electromagnetic scattering from 3-D curved dielectric bodies using time-domain integral equations

A boundary integral equation (BIE) approach is developed to calculate transient scattering from dielectric bodies. The treatment is directly in terms of the E and H fields rather than magnetic and electric currents. It employs curvilinear (quadratic) modeling, which allows accurate representation of arbitrarily shaped curved bodies. The treatment is isoparametric with the same quadratic representation of the spatial field variation and with the temporal variation modeled by similar quadratic elements. Integration employs high-order Gaussian quadrature with special treatment of the singular and hypersingular integrals that arise. The treatment is implicit, requiring the solution of a sparse matrix equation at each timestep. This adds only trivially to the cost at each timestep and, by freeing the timestep from the constraint that it be smaller than the smallest nodal spacing, can greatly reduce the number of timesteps that must be employed. Additionally, it produces stable results without resort to the averaging processes proposed elsewhere. Example calculations of scattering from a sphere, a cube, and an almond are presented and compared with earlier published transient results and with results from a frequency domain treatment. Good agreement and improved accuracy is found.

A boundary integral equations approach for the study of the subsonic compressible flow past a cusped airfoil

A boundary integral equations approach for the study of the subsonic compressible flow past a cusped airfoil

HYPERSINGULAR RESIDUALS—A NEW APPROACH FOR ERROR ESTIMATION IN THE BOUNDARY ELEMENT METHOD

SUMMARY This paper presents a new approach for a posteriori 'pointwise' error estimation in the boundary element method. The estimator relies upon evaluation of the residual of hypersingular integral equations, and is therefore intrinsic to the boundary integral equation approach. A methodology is developed for approximating the error on the boundary as well as in the interior of the domain. Extensive computational experiments have been performed for the two-dimensional Laplace equation and the numerical results indicate that the error estimates successfully track the form of the exact error curve. Moreover, a reasonable estimate of the magnitude of the actual error is also predicted.

Moment stress intensity factors for collinear and parallel cracks in Reissner's plate: Boundary integral equation approach

Based on Betti's reciprocal theorem, a displacement discontinuity boundary integral equation is derived for a Reissner plate with line cracks. The boundary element method is used to obtain the moment stress intensity factors for two collinear and parallel cracks. Displayed graphically are the numerical results for the moment stress intensity factors.

Fracture Analysis for a Frictional Fastener Hole with Edge Cracks in an Anisotropic Plate∗

Abstract Stress intensity factors are evaluated for a singly or doubly cracked fastener hole with frictional traction in an anisotropic plate, using a special kernel boundary integral equation (BIE) approach. The integration kernel (Green's function) used in this BIE approach has already taken the presence of the crack (or cracks) into account, thus.avoiding the need for element discretization to model the stress singularity at the crack tip. The Green's function employed is that of an infinite anisotropic plate containing an elliptical hole or crack, and subjected to an arbitrarily positioned point force. Several types of normal and shear traction conditions at the pinhole interface are considered. Numerical results are obtained for various geometrical and loading conditions and are compared with known solutions, where available, for their isotropic counterparts.

An alternative approach for large deflection analysis of axisymmetric plates on elastic foundation

AbstractThe geometric non‐linear behaviour of thin axisymmetric circular plates on an elastic foundation is studied in the paper by using a boundary integral equation approach. A plate analogue procedure is suggested to simplify the solution. With such simplification, the foundation reaction can be arbitrary functions of plate deflection, rotation and curvatures. The boundary integral equation of thin plates and the fundamental solution of a biharmonic equation are employed for both bending and plane stress analysis. A number of numerical examples are given in the paper to show the effectiveness of the proposed method.

An integral equation justification of the boundary conditions of the driven-cavity problem

The driven-cavity problem, a renowned bench-mark problem of computational, incompressible fluid dynamics, is physically unrealistic insofar as the inherent boundary singularities (where the moving lid meets the stationary walls) imply the necessity of an infinite force to drive the flow: this follows from G.I. Taylor's analysis of the so-called scaper problem. Using a boundary integral equation (BIE) formulation employing a suitable Green's function, we investigate herein, in the Strokes approximation, the effect of introducing small “leaks” to replace the singularities, thus rendering the problem physically realizable, The BIE approach used here incorporates functional forms of both the asymptotic far-field and singular near-field solution behaviours, in order to improve the accuracy of the numerical solution. Surprisingly, we find that the introduction of the leaks effects notably the global flow field a distance of the order of 100 leak widths away from the leaks. However, we observe that, as the leak width tends to zero, there is exellent agreement between our results and Taylor's thus justifying the use of the seemingly unrealizble boundary conditions in the driven-cavity problem. We also discover that the far-field, asymptotic, closed-form solution mentioned above is a remarkably accurate representation of the flow even in the near-field. Several streamline plots, over a range of spatial scales, are presented.

On a Transmission Boundary Value Problem for the Time-Harmonic Maxwell Equations without Displacement Currents

This paper considers a transmission boundary-value problem for the time-harmonic Maxwell equations neglecting displacement currents. The usual transmission conditions, which require the continuity of the tangential components of the electric and magnetic fields across boundaries, are slightly modified. For this new problem, it is shown that the uniqueness of the solution depends on the topological properties of the domains under consideration. Finally, existence results are obtained by using a boundary integral equation approach.

Laplace's Equation and the Dirichlet-Neumann Map in Multiply Connected Domains

A variety of problems in materials science and fluid dynamics require the solution of Laplace's equation in multiply connected domains. Integral equation methods are natural candidates for such problems, since they discretize the boundary alone, require no special effort for free boundaries, and achieve superalgebraic convergence rates on sufficiently smooth domains in two space dimensions, regardless of shape. Current integral equation methods for the Dirichlet problem, however, require the solution of M independent problems of dimension N, where M is the number of boundary components and N is the total number of points in the discretization. In this paper, we present a new boundary integral equation approach, valid for both interior and exterior problems, which requires the solution of a single linear system of dimension N + M. We solve this system by making use of an iterative method (GMRES) combined with the fast multipole method for the rapid calculation of the necessary matrix vector products. For a two-dimensional system with 200 components and 100 points on each boundary, we gain a speedup of a factor of 100 from the new analytic formulation and a factor of 50 from the fast multipole method. The resulting scheme brings large scale calculations in extremely complex domains within practical reach.

A unified approach for the linear and nonlinear analysis of plates and shallow shells

The geometric nonlinear behaviour of thin plates and shells is studied in this paper by using a boundary integral equation approach. Two types of plates and shells are considered. One of them is rectangular with movable edges and the other is axisymmetric with various boundary conditions. A plate analog procedure is suggested to simplify the solution. With such simplification, only a knowledge of the boundary integral equation of plates is required in the present study. A number of numerical examples are given in the paper to show the effectiveness of the proposed method.