Three-dimensional stress analysis by the boundary integral equation method

Acoustic wave scattering by scatterers in layered media is studied using a new method that combines the hybrid ray–mode and boundary integral equation methods. The latter is used to formulate the scattering process and the former is used to provide the Green’s function of the layered environment. Following standard procedures of the boundary integral equation method, integral equations are formulated first for field distributions along the boundaries of the obstacles. By expressing these unknown distributions in terms of appropriate basis functions, the integral equations are then reduced to algebraic equations that are solved numerically. In the integral equations, it is necessary to compute the Green’s function of the layered environment for various arrangements of locations of source and receiver. None of these conventional approaches (rays, modes, and spectral integration) to evaluate the Green’s function are satisfactory for all possible arrangements. Only the newly developed hybrid ray–mode method is best suited for this purpose because it combines rays and modes self-consistently within a single framework and optimizes the advantages of each. Numerical implementation illustrates these aspects.

A quantative comparison between the boundary integral equation (BIE) method and the finite difference (FD) method is presented in which each technique is applied to an elliptic boundary‐value problem (BVP) containing a boundary singularity. Two types of singularity have previously been analysed theoretically, namely those due to a discontinuous boundary potential, which we shall refer to as S1, and those due to a sudden change from the specification of boundary potential flux, an S2 singularity. In this paper the analysis is presented for a third type of boundary singularity, namely an S3 singularity: that arising from a discontinuous boundary flux. Such a condition is frequently encountered in the field of heat transfer where, for example, a system or pipe has a change of lagging material.In general, it is found that the BIE method is superior, with regards to the computational time required to achieve a certain degree of accuracy, over standard FD methods even when there is a boundary singularity. Further, the BIE method determines the solution near the singularity much more accurately than the FD method. The FD method does, however, have advantages for a very restrictive class of problems; for example, when the boundary conditions are of the Dirichlet type and the boundary geometry is rectangular. In this case an optimum relaxation parameter can easily be obtained. A soon as Neumann conditions are prescribed, the BIE is far more efficient than the FD, whatever the boundary geometry.It is concluded that, for fast, accurate solutions of general Laplacian boundary‐value problems, the BIE is appreciably superior to the FD and this is even more pronounced when there is a boundary singularity.

One popular approach to solve the sea surface scattering and propagation in a ducting environment is the parabolic wave equation (PWE) method. An alternative method is the boundary integral equations (BIE) method. The implementation of the BIE in inhomogeneous media (ducting environments) is not straightforward, however, since the Green's function for such a medium is not usually known. In this paper, a closed-form approximation of the Green's function for a two-dimensional (2-D) ducting environment made up of a duct having a linear-square refractive index profile below a medium of constant refractive index, recently published, is used. This paper demonstrates how the BIE method can model the combined effects of surface roughness and medium inhomogeneity. Furthermore, it illustrates its capability of accurately predicting scattering in all directions including backscattering. Then, the PWE combined with the Split-Step Fourier (SSF) method is compared with this method.

A numerical method for the Dirichlet initial boundary value problem for the elastic equation in the exterior and unbounded region of a smooth closed simply connected 2-dimensional domain, is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and a boundary integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the time-depended problem to a sequence of stationary boundary value problems, which are solved by a boundary layer approach resulting to a sequence of boundary integral equations of the first kind. The numerical discretization and solution are obtained by a trigonometrical quadrature method. Numerical results are included.

A hybrid formulation of the 2.5D elastodynamic scattering problem combining the finite-element method and boundary integral equation method is presented and validated. The formulation of the 2.5D boundary integral equation method that is used was presented in detail by Papageorgiou and Pei (1998) and is an extension of the discrete wavenumber boundary integral equation method originally proposed by Kawase (1988) for 2D scattering problems. Modeling of the wave field in the domain of the scatterer is based on the variational principle of virtual displacements, and discretization of the domain is accomplished using the finite-element method.The formulation may be used to study the wave field in models of sedimentary deposits (e.g., valleys) or topography (e.g., canyons or ridges) with a 2D variation in structure but obliquely incident plane waves. The hybrid method exploits the versatility of the finite element method for modeling the scatterer and the effectiveness of the boundary integral equation method for taking care of the radiation condition in the half-space. The advantage of the 2.5D formulation is that it provides the means for calculations of 3D wave fields in scattering problems by requiring a storage comparable to that of the corresponding 2D calculations.

The boundary integral equations are derived in the framework of the analytical five-mode models for propagation of symmetric and skew-symmetric waves in a straight elastic layer of the constant thickness. The forcing problems for fundamental loading cases are solved with the bi-orthogonality conditions employed. By these means, the Green’s matrices are constructed. The derivation of the Somigliana’s identities for the five-mode models is presented. To exemplify application of the method of boundary integral equations, eigenfrequencies of a layer of the finite length are found for two sets of boundary conditions. In the course of analysis, the essential features and advantages of the method are highlighted. The isogeometric analysis at several approximation levels and the standard finite element software are also used to calculate the eigenfrequencies. The results obtained by alternative methods are shown to be in an excellent agreement with each other.

The direct problem of antiplane oscillations of a layer with delamination in the context of the gradient theory of elasticity is considered. The gradient model proposed by Aifantis is taken as a mathematical model. The main attention has been paid to the analysis of mechanical fields at the crack bank and at its tips - stress concentrators. The study is carried out using the method of boundary integral equations (BIE). The BIE for the gradient of displacement field on the crack is constructed. The analysis of the constructed BIE is carried out and the cubic singularity is explicitly revealed. The solution of singular BIE is constructed using approximating Chebyshev polynomials. A study for cracks of small relative length - asymptotic approach is carried out, simple semi-analytical expressions for constructing the crack swap function are obtained, the range of efficiency of the asymptotic approach is obtained. The stress fields in the area of the crack tips are constructed. Numerical results of computational experiments are presented.

Application of the boundary integral equation (BIE) method to transient response analysis of inclusions in a half space

The 2D canonical case of a semi-circular canyon is addressed via two types of boundary-integral equations (BIE) methods which are shown, via Galerkin procedures, to lead to matrix equations with kernels that vanish at two different sets of (what appear as resonant) frequencies. This physically impossible result is interpreted as being the sign that the two BIE methods are generally defective and that the resonances are ‘spurious’. This ‘disease’ is cured by combining two BIE's into one which gives rise to a closed-form solution identical to the exact separation-of-variables solution devoid of resonances.

Three-dimensional shape design optimization is demonstrated using the boundary integral equation method. Design sensitivities are obtained directly by differentiating the boundary integral equations with respect to normals to the surfaces at the design variable points. Coon's patches are employed in order to represent continuously changing design surfaces. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) method which requires first-order derivative information has been adopted for the optimization part of the problems. Two examples, a perforated plate under biaxial stresses and a cantilever beam with end shear loading, are demonstrated for the smoothing of stress peaks and stress levelling.

Boundary integral equation (boundary element) methods have the advantage over other commonly used numerical methods that they do not require values of the unknowns at points within the solution domain to be computed. Further benefits would be obtained if attention could be confined to information at one small part of the boundary, the particular region of interest in a given problem. A local boundary integral equation method based on a Taylor series expansion of the unknown function is developed to do this for two-dimensional potential problems governed by Laplace's equation. Very accurate local values of the function and its derivatives can be obtained. The method should find particular application in the efficient refinement of approximate solutions obtained by other numerical techniques.

Eddy current nondestructive evaluation (NDE) of airframe structures involves the detection of electromagnetic field irregularities due to non-conducting inhomogeneities in an electrically conducting material. Usually, the eddy current NDE problem can be formulated by the boundary integral equations (BIE) and discretized into matrix equations by the method of moments (MoM) or the boundary element method (BEM). The fast multipole method (FMM) is a well-established and effective method for accelerating numerical solutions of the matrix equations. Accelerated by the FMM, the BIE method can now solve large-scale electromagnetic wave propagation and diffusion problems. The traditional BIE method requires O(N) operations to compute the system of equations and another O(N) operations to solve the system using direct solvers, with N being the number of unknowns; in contrast, the BIE method accelerated by the two-level FMM can potentially reduce the operations and memory requirement to O(N). Moreover, several approaches have been proposed for the field calculation in the presence of flaws in three dimensional NDE; however, seldom work has been done in applying efficient methods to seek rapid solution in eddy current NDE simulation. As elaborated in the dissertation, we introduce a fast multipole BIE method for two-dimensional diffusive scalar problem and an efficient BIE method for three-dimensional eddy current NDE. Firstly, we work with the two-dimensional Helmholtz equation with a complex wave number for non-trivial boundary geometry. We describe the FMM acceleration procedure of the BIE method and its features briefly, explaining that the FMM is not only efficient in meshing complicated geometries, accurate for solving singular fields or

Applying the concept of minimum potential energy and the variational method proposed in this paper, one can derive the governing equation and transversality conditions for the critical slip surface of a cohesive land slope described by simplified Janbu's model where both horizontal and vertical inter-slice forces are neglected. The governing equation, transversality conditions and boundary conditions were solved by the boundary integral equation method, which is a one dimensional BIEM, so that the critical slip surface and its associated minimal factor of safety can be determined effectively. By comparison of the results gotten from the boundary integral equation method and other numerical methods, it can be concluded, that, for some simplified cases, by using the boundary integral equation method on slope stability analysis of a cohesive land slope, a more reasonable result can be obtained.

On solving external boundary value problems of the theory of elasticity using the method of boundary integral equations

We study a modification of the qualocation method for boundary integral equations on smooth curves, which allows the same high-order convergence as the original qualocation method but with reduced smoothness assumptions on the exact solution. This ‘tolerant qualocation’ method differs from the original one only in that the exact inner product is used on the righthand side of the qualocation equation, whereas in the original method a specially designed approximate inner product is used on both sides. The modified method achieves exactly the same error estimates as the Petrov-Galerkin method, at greatly reduced cost.