Analysis of Bow Wave Breaking by the Slender Body Approximation

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.

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.

A fiilly nonlinear 2-dimensional numerical wave flume, based on the boundary integral equation method, has been developed. Waves are generated by a hinged paddle wave maker at one end of the flume and a sponge type wave absorber at the other end. A fourth order Taylor expansion technique is used for the time stepping of the free surface. Simple monochromatic waves have been generated and very good agreements are found when compared with Stokes wave profiles. The wave flume is used to study deep water wave breaking due to large periodic displacements of the wave paddle. Wave breaking as a result of energy focusing a group of waves of different frequencies and heights, is also studied. For each breaking event, breaking wave parameters such as the wave steepness, breaking height, particle velocities and accelerations are examined in detail. There does not appear to be a definable correlation between the point of breaking, the wave steepness, or the particle velocities. The maximum downward (vertical) and forward (horizontal) accelerations at breaking are found to be independent of the initial conditions, with constant values of g and 1.56g, respectively; and where g is the acceleration due to gravity. It has been proposed (Philips*) that particle accelerations could be used as a criterion for wave breaking. Positive verification of this was first provided in an earlier study by the authors She et af, which used spatially periodic boundary conditions. Both studies support Philips' conjecture, and suggest that the point of breaking is when the maximum vertical and horizontal accelerations are at g & 1.56g. Transactions on the Built Environment vol 40 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509

The principles and development of the boundary integral equation (BIE) method for solving engineering problems are reviewed, with particular emphasis on applications in three-dimensional stress analysis. Its use in problems of this type is illustrated with the aid of three examples, one a classical stress concentration problem, the other two involving cracked components. Compared with a finite-element method, the BIE approach is shown to be capable of being both more accurate and more economical to use in terms of competing resources required and the cost of data preparation. These advantages may, however, be less clear-cut when predictions of stresses and displacements at a large number of points within a component are required, or when the component concerned has a relatively high surface-to-volume ratio.

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation Y. D. Stoynov; Dynamic behavior of cracked magnetoelectroelastic composites at different boundary conditions. AIP Conf. Proc. 2 October 2012; 1487 (1): 256–263. https://doi.org/10.1063/1.4758966 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioAIP Conference Proceedings Search Advanced Search |Citation Search

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.

Mathematical modeling of dynamic problems of continuum mechanics in domains of complicated geometry for various types of boundary conditions usually leads to boundary value problems for hyperbolic systems. The method of boundary integral equations is a convenient tool for solving such problems; it allows one to reduce the original differential boundary value problem in a domain to a system of boundary integral equations on the boundary, thus reducing the dimension of the equations, improving the stability of numerical solution methods, and so on. Now this method is widely used for a large class of problems in mechanics and mathematical physics. The construction of fundamental solutions and kernels of boundary integral equations of the system is a key point of the method of boundary integral equations. The fundamental solutions of classical equations of elliptic and hyperbolic types have been studied quite comprehensively [1, 2]. Less is known about fundamental solutions of systems of partial differential equations. Related papers mainly deal with specific equations of continuum mechanics [3–7]. In the present paper, we consider strictly hyperbolic systems of M equations with second-order derivatives in an (N + 1)-dimensional space. For such systems, we construct the Green matrix and analyze its properties. We also study systems invariant under the orthogonal group.

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.

In this paper the new integral equation method proposed and developed in our previous papers on boundary-value problems is applied to eigenvalue problems. Two formulations by the integral equation expression are performed from the same manner that we adopted on the boundary value problems. As the result, whether the derived integral equation is single or not, the integral kernels do not include the unknown eigen-parameter. Through this notable feature in contrast with the boundary integral equation method, eigenvalue problems on continuous fields can be reduced to the standard eigenvalue problems of matrix equation. Finally, in order to show the applicability and validity of the proposed methodology and procedure, numerical calculations are performed for three illustrative examples, which are eigenvalue problems of Sturm-Liouville type equation, Helmholtz equation and biharmonic type equation. The approximate values of an eigenvalue parameter of each problem demonstrate a sufficient accuracy through the comparison of another values.

Analyzing the 2D fracture problems via the enriched boundary element-free method

In past few years, we have resorted to the methodology of special-purpose computer in order to accelerate the Boundary Integral Equation Method (BIEM). Here we applied a special-purpose computer MDGRAPE-2 to the acceleration of the BIEM for three-dimensional elastostatics. The formulation was developed so that MDGRAPE-2 can evaluate the layer-potentials of the BIEM. The implemented BIEM code using MDGRAPE-2 and the iterative solver GMRES was nearly 70 or more times faster than the naive BIEM code in solving a boundary value problem with 3.9 million unknowns.

Non-stationary boundary value problems of uncoupled thermoelasticity are considered.             A method of boundary integral equations (BIME) in the initial space-time has been developed for solving non-stationary boundary value problems of thermoelasticity under plane deformation. On the basis of the method of generalized functions, generalized solutions of boundary value problems are constructed using the Green's function for the heat equation and the Green's tensor of the Lame equations under the action of non-stationary power and heat sources of various types. Integral representations of the solution of boundary value problems are obtained. These solutions allow, based on known boundary values and initial conditions (displacements, temperature, stresses and heat flux), to determine the thermally stressed state of the medium under the influence of various power and thermal loads. Resolving boundary integral equations are constructed to determine the unknown boundary functions.

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.

This paper presents an effective solution method for predicting acoustic radiation and scattering fields in two dimensions. The difficulty of the fictitious characteristic frequency is overcome by incorporating an auxiliary interior surface that satisfies certain boundary condition into the body surface. This process gives rise to a set of uniquely solvable boundary integral equations. Distributing monopoles with unknown strengths over the body and interior surfaces yields the simple source formulation. The modified boundary integral equations are further transformed to ordinary ones that contain nonsingular kernels only. This implementation allows direct application of standard quadrature formulas over the entire integration domain; that is, the collocation points are exactly the positions at which the integration points are located. Selecting the interior surface is an easy task. Moreover, only a few corresponding interior nodal points are sufficient for the computation. Numerical calculations consist of the acoustic radiation and scattering by acoustically hard elliptic and rectangular cylinders. Comparisons with analytical solutions are made. Numerical results demonstrate the efficiency and accuracy of the current solution method.