Direct Boundary Integral Equation Method Research Articles

Stress analysis of composites with interfacial bridged cracks and damaged zones

Stress analysis of composite structures with bridged cracks and damaged zones is performed with the direct boundary integral equations method in axisymmetric formulation. It is assumed that: a) the crack faces interact in some zones behind of the cracks edges (bridged zones); b) these zones width can be comparable to the whole crack size; c) adjacent parts of the structure can be connected by mechanical ligaments (including interfaces ahead of cracks fronts) and the relative displacements of these parts can occur under the loading (these zones consider as damaged zones). The boundary element software to solve axisymmetric problems of elasticity with interfacial bridged cracks and damaged zones was developed. The results for different properties of composite materials and bonds are presented. Applications the bridged crack approach to modeling of cracks self-healing is remarked.

Nyström method for BEM of the heat equation with moving boundaries

A direct boundary integral equation method for the heat equation based on Nystrom discretization is proposed and analyzed. For problems with moving geometries, a weakly and strongly singular Green’s integral equation is formulated. Here the hypersingular integral operator, i.e., the normal trace of the double-layer potential, must be understood as a Hadamard finite part integral. The thermal layer potentials are regarded as generalized Abel integral operators in time and discretized with a singularity-corrected trapezoidal rule. The spatial discretization is a standard quadrature rule for smooth surface integrals. The discretized systems lead to an explicit time stepping scheme and is effective for solving the Dirichlet and Neumann boundary value problems based on both the weakly and/or strongly singular integral equations.

Stress concentration analysis in technical structures with boundary integral equations

The direct boundary integral equation (DBIE) method in the multi-domain formulation is used for three-dimensional stress concentration problems solution. The computational results of several new stress concentration problems are presented and analyzed. The following problems have been studied: flat plates with oblique, conical and round-edges holes under in-plane tensile loading for different angles of holes inclination. The stress concentration problem for an aviation engine turbine disk with cooling hole has also been considered.

Analysis of interaction of bridged cracks and weak interfaces

Structures with interfacial bridged cracks and weak interfaces are analyzed with the direct boundary integral equations method. Within the framework of this approach it is assumed that: a) the crack faces interact in some zones behind the crack tips (bridged zones); b) the sizes of these zones can be comparable to the whole crack length; c) different materials can be connected by mechanical ligaments and relative displacements of initially adjacent materials may occur under the loading (non-ideal or weak interfaces). The relations for the bridged crack opening and for the relative displacements at weak interface layer are defined by the transverse separation and tangential slipping to an interface surface. Several new problems for the interface bridged cracks, weak interface and interaction of bridged cracks with ideal or weak materials interface were solved and analyzed.

Boundary element modelling of dynamic behavior of piecewise homogeneous anisotropic elastic solids

A traditional direct boundary integral equations method is applied to solve three-dimensional dynamic problems of piecewise homogeneous linear elastic solids. The materials of homogeneous parts are considered to be generally anisotropic. The technique used to solve the boundary integral equations is based on the boundary element method applied together with the Radau IIA convolution quadrature method. A numerical example of suddenly loaded 3D prismatic rod consisting of two subdomains with different anisotropic elastic properties is presented to verify the accuracy of the proposed formulation.

Structures with bridged cracks and weak interfaces

Structures with interfacial bridged cracks and weak interfaces are analyzed. In the frames of this approach is assumed that: a) the crack surfaces interact in some zones behind the crack tips (bridged zones); b) size of these zones can be comparable to the whole crack length; and c) the interface between different materials can be connected by mechanical ligaments and a relative displacement of initially adjacent materials may occur (non-ideal or weak interface). Several new problems for interface bridged cracks and interaction of bridged cracks with material interface were solved and analyzed by the direct boundary integral equations method.

Direct local boundary integral equation method for numerical solution of extended Fisher–Kolmogorov equation

In this paper, the local boundary integral equation (LBIE) method based on generalized moving least squares (GMLS) is proposed for solving extended Fisher---Kolmogorov (EFK) equation. Since the local weak forms are directly approximated from nodal data using a GMLS approximation, this method is called direct local boundary integral equation (DLBIE) method. In addition to DLBIE method, three other meshless local boundary integral equation methods such as LBIE method based on moving least squares, LBIE method based on radial point interpolation, and LBIE method based on moving Kriging are applied to find the numerical solution of EFK equation. A comparison between these methods is done from the perspective of accuracy and computational efficiency. The computational efficiency is the most significant advantage of the DLBIE method in comparison with the other local boundary integral equation methods. DLBIE shifts the numerical integrations over low-degree polynomials instead of over complicated shape functions and this reduces the computational costs, significantly. The main purpose of this paper is to show that the local boundary integral equation methods can be used for solving the fourth-order non-linear partial differential equations especially EFK equation. The numerical results confirm the good efficiency of the proposed methods for solving our model.

Influence of Poroelasticity on the 3D Seismic Response of Complex Geological Media

AbstractElastic wave propagation in 3D poroelastic geological media with localized heterogeneities, such as an elastic inclusion and a canyon is investigated to visualize the modification of local site responses under consideration of water saturated geomaterial. The extended computational environment herein developed is a direct Boundary Integral Equation Method (BIEM), based on the frequency-dependent fundamental solution of the governing equation in poro-visco elastodynamics. Bardet’s model is introduced in the analysis as the computationally efficient viscoelastic isomorphism to Biot’s equations of dynamic poroelasticity, thus replacing the two-phase material by a complex valued single-phase one. The potential of Bardet’s analogue is illustrated for low frequency vibrations and all simulation results demonstrate the dependency of wave field developed along the free surface on the properties of the soil material.

Scattering of a plane harmonic SH wave by multiple layered inclusions

Scattering of a plane harmonic SH wave by an arbitrary number of layered inclusions in a half-space is investigated by using a direct boundary integral equation method. The inclusions of arbitrary shape and placement are embedded within an elastic half-space. The effects of multiple scattering, the geometry, and the impedance contrast of the materials for layered inclusions and pipes are considered in detail.

Three-Dimensional Nonlinear Force-Free Field Reconstruction of Solar Active Region 11158 by Direct Boundary Integral Equation

A 3-D coronal magnetic field is reconstructed for NOAA 11158 on Feb 14, 2011. A GPU-accelerated direct boundary integral equation (DBIE) method is implemented. This is about 1000 times faster than the original DBIE used on solar NLFFF modeling. Using the SDO/HMI vector magnetogram as the bottom boundary condition, the reconstructed magnetic field lines are compared with the projected EUV loop structures from different views three-dimensionally by SDO/AIA and STEREO A/B spacecraft simultaneously for the first time. They show very good agreement so that the topological configurations of the magnetic fields can be analyzed, thus its role in the flare process of the active region can be better understood. A quantitative comparison with some stereoscopically reconstructed coronal loops shows that the present averaged misalignment angles are at the same order as the state-of-the-art results obtained with reconstructed coronal loops as prescribed conditions and better than other NLFFF methods. It is found that the observed coronal loop structures can be grouped into bundles of closed and open loops with some central bright coronal loops around the polarity inversion line (PIL). The reconstructed highly-shearing magnetic field lines agree very well with the low-lying S-shaped filament channel along PIL. They are in a pivot position to all other surrounding coronal structures, and a group of electric current lines co-aligned with the central bright EUV loops overlying the filament channel is also obtained. This central lower-lying magnetic field loop system must have played a key role in powering the flare. It should be noted that while a strand-like coronal feature along PIL may be related to the filament, one cannot simply attribute all the coronal bright features along PIL to manifestation of the filament without any stereoscopically information. It shows that DBIE is rigorous and effective.

A direct traction boundary integral equation method for three-dimension crack problems in infinite and finite domains

This paper presents a direct traction boundary integral equation method (TBIEM) for three-dimensional crack problems. The TBIEM is based on the traction boundary integral equation (TBIE). The TBIE is collocated on both the external boundary and one of the crack surfaces. The displacements and tractions are used as unknowns on the external boundary and the relative crack opening displacements (CODs) are introduced as unknowns on the crack surface. In our implementation, all the surfaces of the considered structure are discretized into discontinuous elements to satisfy the continuity requirement for the existence of finite-part integrals, and special crack-front elements are constructed to capture the crack-tip behavior. To calculate the finite-part integrals, an adaptive singular integral technique is proposed. The stress intensity factors (SIFs) are computed through a modified COD extrapolation method. Numerical examples of SIFs computation are presented to demonstrate the accuracy and efficiency of our method.

AN INTERPOLATING BOUNDARY ELEMENT-FREE METHOD WITH NONSINGULAR WEIGHT FUNCTION FOR TWO-DIMENSIONAL POTENTIAL PROBLEMS

In this paper, an improved interpolating moving least-squares (IIMLS) method with nonsingular weight function is presented. The shape function of the IIMLS method satisfies the property of Kronecker δ function. The IIMLS method can overcome the difficulties caused by the singularity of the weight function in the interpolating moving least-squares (IMLS) method presented by Lancaster and Salkauskas. By combining the boundary integral equation (BIE) method with the IIMLS method, an improved interpolating boundary element-free (IIBEF) method is presented for two-dimensional potential problems. The IIBEF method is a direct meshless boundary integral equation method in which the basic unknown quantities are the real solutions to the nodal variables, and the boundary conditions can be applied directly and easily. Thus, it gives greater computational precision. Some numerical examples are presented to demonstrate the IIMLS and IIBEF methods.

Integral Equation Method for the First and Second Problems of the Stokes System

Using the integral equation method we study solutions of boundary value problems for the Stokes system in Sobolev space H1(G) in a bounded Lipschitz domain G with connected boundary. A solution of the second problem with the boundary condition \(\partial {\bf u}/\partial {\bf n} -p{\bf n}={\bf g}\) is studied both by the indirect and the direct boundary integral equation method. It is shown that we can obtain a solution of the corresponding integral equation using the successive approximation method. Nevertheless, the integral equation is not uniquely solvable. To overcome this problem we modify this integral equation. We obtain a uniquely solvable integral equation on the boundary of the domain. If the second problem for the Stokes system is solvable then the solution of the modified integral equation is a solution of the original integral equation. Moreover, the modified integral equation has a form f + Sf = g, where S is a contractive operator. So, the modified integral equation can be solved by the successive approximation. Then we study the first problem for the Stokes system by the direct integral equation method. We obtain an integral equation with an unknown \({\bf g}=\partial {\bf u}/\partial {\bf n} -p{\bf n}\). But this integral equation is not uniquely solvable. We construct another uniquely solvable integral equation such that the solution of the new eqution is a solution of the original integral equation provided the first problem has a solution. Moreover, the new integral equation has a form \({\bf g}+\tilde S{\bf g}={\bf f}\), where \(\tilde S\) is a contractive operator, and we can solve it by the successive approximation.

Boundary element analysis of structures with bridged interfacial cracks

The direct boundary integral equations method has been applied to analyze stresses in a fracture process zone (a crack bridged zone) and to calculate stress intensity factors module for structures with bridged interfacial cracks under mechanical loading. Bridged zones at interfacial cracks are considered as parts of these cracks with assumption that surfaces of interfacial cracks are connected by distributed spring-like bonds with given bond deformation law. For numerical analysis of piecewise structures with bridged interfacial cracks the multi-domain formulation of the boundary elements method is used. The stress intensity factors module evaluation is performed on the basis of displacements and stresses computed at nodal points of special quadratic boundary elements adjoined to a crack tip. The comparative study between the results obtained by the boundary elements method and the results obtained previously by the singular integral---differential equations method is performed and the validity of the presented numerical formulation is demonstrated. The new problem for a bridged circumferential crack between a cylindrical inclusion and a matrix in plate of finite size is also solved. Stresses distributions along the bridged zone and the stress intensity factors modulus dependencies versus the bridged zone length and bonds stiffness are presented and discussed for this problem.

Nonlinear force-free magnetic field extrapolation in AR 11158 by GPU-DBIE

AbstractA GPU-based acceleration for the direct boundary integral equation method (GPU-DBIE) to extrapolate solar coronal magnetic fields is developed, which is about 1000 times faster than the original DBIE. The 3-d coronal magnetic field is reconstructed for NOAA 11158 on 14-Feb-2011 with the SDO/HMI vector magnetogram as bottom boundary condition. The extrapolated results agree well with the projected SDO/AIA, EUV loops and the STEREO EUV sideviews, which verifies the correctness of our GPU-DBIE method. It is also found that the group of bright EUV loops along magnetic neutral lines agree well with current lines, which may have played an important role in the flare process of the active region.

A direct boundary integral equation method for the numerical construction of harmonic functions in three-dimensional layered domains containing a cavity

We consider boundary value problems for the Laplace equation in three-dimensional multilayer domains composed of an infinite strip layer of finite height and a half-space containing a bounded cavity. The unknown (harmonic) function satisfies the Neumann boundary condition on the exterior boundary of the strip layer (i.e. at the bottom of the first layer), the Dirichlet, Neumann or Robin boundary condition on the boundary surface of the cavity and the corresponding transmission (matching) conditions on the interface layer boundary. We reduce this boundary value problem to a boundary integral equation over the boundary surface of the cavity by constructing Green's matrix for the corresponding transmission problem in the domain consisting of the infinite layer and the half-space (not with the cavity). This direct integral equation approach leads, for any of the above boundary conditions, to boundary integral equations with a weak singularity on the cavity. The numerical solution of this equation is realized by Wienert's [Die Numerische approximation von Randintegraloperatoren für die Helmholtzgleichung im R 3, Ph.D. thesis, University of Göttingen, Germany, 1990] method. The reduction of the problem, originally set in an unbounded three-dimensional region, to a boundary integral equation over the boundary of a bounded domain, is computationally advantageous. Numerical results are included for various boundary conditions on the boundary of the cavity, and compared against a recent indirect approach [R. Chapko, B.T. Johansson, and O. Protsyuk, On an indirect integral equation approach for stationary heat transfer in semi-infinite layered domains in R 3 with cavities, J. Numer. Appl. Math. (Kyiv) 105 (2011), pp. 4–18], and the results obtained show the efficiency and accuracy of the proposed method. In particular, exponential convergence is obtained for smooth cavities.

The effect of impedance contrast upon surface motion due to scattering of plane harmonic P, SV, and Rayleigh waves by a randomly corrugated elastic inclusion

Scattering of a plane harmonic P, SV, or Rayleigh wave by a corrugated elastic inclusion completely embedded in a two-dimensional isotropic half space is investigated by using a direct boundary integral equation method. The corrugated scatterer is generated by superimposing a random perturbation of arbitrary amplitude to a smooth elliptical shape. The probability density function for the randomly corrugated inclusions is assumed to be of uniform or normal types. The displacement fields are evaluated along the half-space surface for a range of impedance contrasts, frequencies, five incident waves for circular- and elliptic-based rough inclusions. Subsequently, the roughness factor is introduced in terms of the surface response for the rough and the corresponding smooth inclusions. The results clearly show that this factor strongly depends upon the impedance contrast of the materials, the basic inclusion geometry, the type of incident wave, and the frequency. The peak values of the roughness factor are observed for the near-grazing SV incidence. Furthermore, the impedance contrast of the materials has a non-uniform effect upon the surface motion. While the smallest roughness factor is consistently observed for the minimal impedance contrast the peak values of the same factor may take place for different non-minimal impedance contrasts. For the range of frequencies considered here, the type of the random interface corrugation (random-uniform vs. random-normal) has a minimal effect upon the roughness factor.

Maurice Jaswon and boundary element methods

Abstract In the direct boundary integral equation method, boundary-value problems are reduced to integral equations by an application of Green's theorem to the unknown function and a fundamental solution (Green's function). Discretization of the integral equation then leads to a boundary element method. This approach was pioneered by Jaswon and his students in the early 1960s. Jaswon's work is reviewed together with his influence on later workers.

Convergence of the Neumann Series in BEM for the Neumann Problem of the Stokes System

A weak solution of the Neumann problem for the Stokes system in Sobolev space is studied in a bounded Lipschitz domain with connected boundary. A solution is looked for in the form of a hydrodynamical single layer potential. It leads to an integral equation on the boundary of the domain. Necessary and sufficient conditions for the solvability of the problem are given. Moreover, it is shown that we can obtain a solution of this integral equation using the successive approximation method. Then the consequences for the direct boundary integral equation method are treated. A solution of the Neumann problem for the Stokes system is the sum of the hydrodynamical single layer potential corresponding to the boundary condition and the hydrodynamical double layer potential corresponding to the trace of the velocity part of the solution. Using boundary behavior of potentials we get an integral equation on the boundary of the domain where the trace of the velocity part of the solution is unknown. It is shown that we can obtain a solution of this integral equation using the successive approximation method.

A coupled BEM and FEM for the interior transmission problem in acoustics

The interior transmission problem (ITP) is a boundary value problem arising in inverse scattering theory, and it has important applications in qualitative methods. In this paper, we propose a coupled boundary element method (BEM) and a finite element method (FEM) for the ITP in two dimensions. The coupling procedure is realized by applying the direct boundary integral equation method to define the so-called Dirichlet-to-Neumann (DtN) mappings. We show the existence of the solution to the ITP for the anisotropic medium. Numerical results are provided to illustrate the accuracy of the coupling method.