Integral Equation Method for linear Water Waves

The Boundary Integral Equation Method (BIEM) is extended to solve three-dimensional, linearized, transient water wave problems. Only the radiation problems concerning long waves generated by an upthrust block of the sea bed are considered in this paper. Due to the linearization, free-surface boundary conditions are applied on the still water level rather than the actual location of the free surface. Thus the problem becomes a fixed boundary problem. A radiation boundary condition is used to truncate the computational domain. Reasonable agreement between present numerical results and previous analytical solutions is observed.

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.

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.

New boundary integral equation method which was previously proposed have been examined in detail and extended to the computer‐aided design (CAD) of asymmetrical dielectric slab waveguide bends. Many numerical results of symmetric waveguide discontinuities and asymmetric waveguide bends confirm the validity of the new method. They show that all numerical results satisfy the energy conservation law within an accuracy of 1% and indicate that the new integral equation method is very suitable for the basic theory or a solver of CAD software for various open dielectric waveguide circuits.

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.

Dispersive shallow water waves over a porous sea bed

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 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.

This paper compares the accuracy of the integral equation method and the multiple scales method in solving the amplitude equation of multi degree of freedom nonlinear vibration systems. We consider three examples: a two-degree-of-freedom stainless-steel beam system controlled by a saturation controller, a three-degree-of-freedom rotating compressor blade model and a four-degree-of-freedom horizontally supported Jeffcott-rotor system controlled by a PPF controller. The amplitude equations are obtained by applying the integral equation method and the method of multiple scales. The stability analysis is achieved based on the Floquet theory together with Routh–Hurwitz criterion. Furthermore, we modified the iterative procedure of the integral equation method to make the analytic approximate solution more accurate. Finally, the analyses show that in most cases, the analytical solutions obtained by the integral equation method are more excellent agreement with the numerical solutions than the most commonly used method of multiple scales. Therefore, the integral equation method is worth popularizing to obtain the approximate analytical solutions of the multi-degree-of-freedom nonlinear vibration system.

Ship hydrodynamics in a constant waterway have been well studied. With a coordinate system fixed on the moving ship, the boundary value problem (BVP) is usually treated as a steady one. However, the hydrodynamics of a ship moving in confined waterways with abrupt changes in width or depth are complex due to their unsteady nature. The studies on such unsteady problems are insufficient. In the engineering practice, the hydrodynamic unsteadiness can be fully reflected by a scenario when a ship enters a lock. Prior studies have predominantly focused on predicting the ships' hydrodynamic forces without considering the unsteady terms on the free-surface boundary conditions. Obviously, such steady or quasi-steady methods overlooked the crucial unsteady phenomena on free water surface. To address this gap, the present study introduces a novel three-level difference scheme to discretize the free surface condition, preserving unsteady terms while maintaining temporal continuity of cells on the free surface. With the implementation of such fully unsteady BVP, we observed some interesting unsteady free surface motions, which were not well documented in the existing literature. To verify our new observations, as well as to validate the numerical method proposed in this study, two physical model tests were designed and conducted in a towing tank: a submerged ellipsoid enters a deep lock at relatively high speeds, and a box enters a shallow and narrow lock at very low speeds. The discussion is highlighted on the unsteady waves in front of the moving bodies, as well as the unsteady resistance induced by such waves.

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.

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.

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.

A boundary element method for analysing non-linear phenomena on gas-liquid two-phase free surface at middle and higher Reynolds numbers has been developed. Employing this method we analyze two kinds of non-linear phenomena on gas-liquid two-phase free surface (l) Solitary wave on free falling thin liquid film. (2) Surface tension drives convective flows in open-boat type crystal-growth techniques in microgravity environment. INTRODUCTION Although some new schemes of discretizing convective terms are developed as QUICK and QUICK ER in finite difference methods[FDM] and finite element methods[FEM], but it is very difficult to avoid numerical diffusion introduced by convective discretizations, especially, at high Reynolds number. In [1], they developed a boundary element iterative scheme(BEIS) to calculate convective terms and compared the advantages of using upwind or central difference schemes and BEIS. As shown as numerical results, BEIS of the convective terms was the most accurate of the two approaches. However, computation was complex. The main objective of this research is to develop a boundary element method for analysing some non-linear phenomena on gas-liquid two-phase free surface at middle and higher Reynolds numbers. On the basis of the boundary element formulation with the pressure penalty function, the divergence theorem is applied to the non-linear convective volume integral of the boundary element formulation. Consequently, velocity gradients are eliminated, the complete formulation is written in terms of velocity. This avoids the difficulty of convective discretizations, and provides considerable reducing storage and computational requirement while improving accuracy. In order to calculate free surface, we construct shooting iterative method of free surface on the basis of normal stress balance condition Project supported by the National Natural Science Foundation of China Transactions on Modelling and Simulation vol 6, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 168 Free and Moving Boundary Problems similar to [2] [3]. The method is applied to analyze two kinds of free surface problems. (1) Solitary wave on free falling thin liquid film. This problem was researched by Wasden and Kheshgi etc. with finite difference method [4] and finite element method [5]. However, in finite difference method[4], one applied an adjustment procedure for wave velocity in order to obtain reasonable flow fields, in finite element methods[5] numerical solution was limited to low flow rates. Moreover, the upwind schemes of discretizing convective momentum terms common to FDM and FEM introduce numerical diffusion or oscillatory behavior of the wave peak region, more importantly, the upwind scheme lacks sensitivity to cross stream diffusion and source terms which are especially important in the case of a thin film. Therefore it is valuable to research a non-traditional numerical method such as BEM. (2) Surface tension drives convective flows in open-boat type crystal growth techniques in microgravity environment. The analytical or numerical study of free surface problem in crystal-growth techniques is very difficult. We shall restrict ourselves to model a problem of open-boat type crystal growth technology, and we only study the melt-gas free boundary, assume the melt-crystal interface shape is flat. The free surface problem of open-boat type is simplified as the free surface problem in open rectangular cavity. The left vertical boundary of the cavity can be interpreted as the crucible wall. The right vertical boundary of the cavity can be interpreted as the melt-crystal interface. The temperatures equal #/ and 0,., respectively. Some important numerical results are presented in paper. BASIC EQUATIONS We consider a system of steady incompressible viscous liquid in a open rectangular cavity with differentially heated lateral walls (the temperature difference is A0*). There is a gradient of surface tension on the free surface since the distribution of temperature is non-homogeneous. The gradient of surface tension drives the convection of liquid. On the basis of the basic law of fluid mechanics and Boussinesq approximation, the governing equations of these problems can be described by a tensor notation and a non-dimensional form as follow: continuity equation «,-„• = o (i) momentum equations energy equation V,-^,=0 (3) where, #,„% is the Kronecker delta symbol, m = x for solitary wave, m = y for thermocapillary convection. The lengths, velocities are scaled with respect to H, Transactions on Modelling and Simulation vol 6, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X