Boundary Integral Equation Methods in the Theory of Thin Plates

Among the boundary value problems of hydrodynamics of viscous fluid, an important from a practical point of view the class of tasks in the region with a free (previously unknown) boundary, which is determined in the process of direct solving, is distinguished. One of the possible approaches to solving such problems in such an area is a method of hydrodynamic potentials, which translates the basic complexity of research and numerical calculations into certain boundary integral equations, which relate only to the boundary of a given region and take into account the boundary conditions directly. This approach allows you to immediately identify unknown values at the boundary without calculating them in the entire area. This advantage distinguishes the method of boundary integral equations from other methods. The aim of the research was to develop a method for numerical solution of the problems of the viscous fluid movement with a previously unknown boundary. The problem of deformation of a viscous elliptical cylinder under the action of forces of surface tension is formulated and solved in the article. The boundary integral equations are used, which are considered together with the kinematic contour condition. An algorithm for the steps of the numerical solution of the system of boundary integral equations for the problem of viscous fluid with a previously unknown boundary was implemented and computational features were analyzed. On the basis of the conducted research the regularities of the deformation process of the viscous elliptical cylinder under the influence of the forces of surface tension, depending on the viscosity of the liquid and the coefficient of surface tension , were established. A convex contour with continuous curvature deforms into a circle, carrying out fading oscillations around the equilibrium position. Zones of stable synchronous oscillations of points of the contour around the equilibrium position are observed, if , and when , ; in both cases, the density of the liquid was taken equal to one. The liquid body reaches the equilibrium position without oscillation, if . Key words: nucleus, contour discretization interval, curvature, value of the Cauchy integral, surface tension forces, elliptic cylinder

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.

We consider the problem of a polygonal plate with free edges. It is a boundary value problem for the biharmonic operator on a polygon with Neumann boundary conditions. Its resolution is studied via boundary integral equations. A variational formulation of the boundary problem obtained by a double-layer potential is given. Finally, we implement the method and give numerical results. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

We propose a new integral equation formulation to characterize and compute transmission eigenvalues for constant refractive index that play an important role in inverse scattering problems for penetrable media. As opposed to the recently developed approach by Cossonnière and Haddar [1,2] which relies on a two by two system of boundary integral equations our analysis is based on only one integral equation in terms of Dirichlet-to-Neumann or Robin-to-Dirichlet operators which results in a noticeable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further we employ the numerical algorithm for analytic non-linear eigenvalue problems that was recently proposed by Beyn [3] for the numerical computation of transmission eigenvalues via this new integral equation.

A dual reciprocity boundary element solution method for the free vibration analysis of fluid-coupled Kirchhoff plates

The coupled structures consist of two- or three-dimensional bounded or unbounded domains of different anisotropic materials. The domains are non-smooth in general, edges and corners can appear. There are different boundary conditions on the exterior boundary and standard coupling conditions on the interfaces. The so called “direct method” is used for the derivation of appropriate boundary integral equations, which is based on a system of boundary integral equations given by the Cal-deron projector. This leads to “local” pseudo-differential operator equations with corresponding STEKLOV-POINCARE operators on the interface. Existence, uniqueness and regularity of the solutions are discussed.

A boundary element modeling for predicting the plasmonic response from a Nanoparticle‐on‐Mirror (NPoM) structure is proposed. The NPoM can be considered as one of the most important platforms in the research on nanophotonics. Instead of using the local response model to describe the optical response of metals, the nonlocal hydrodynamic model is used to account for the dynamics of free electrons in the metallic NP and the metallic mirror beneath. A system of boundary integral equations (BIEs) is derived to describe the interaction of light with the NPoM structure. The BIEs are solved by the boundary element method. To validate the implementation, two exemplary structures are considered: the Nanosphere‐on‐Mirror structure and the Nanocube‐on‐Mirror structure. By contrasting with the results from previous works and the results from a generalized T‐matrix method, both a qualitative and quantitative agreement in the near field and the far field regimes are observed.

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.

Fast and accurate simulation of responses of electromagnetic logging-while-drilling tools in 2D and 3D formations is important for further development of the proactive geosteering service, which is currently based on the 1D layered-model inversion. In search of an optimal method of forward simulation, we have developed boundary integral equations in a 2D formation model with planar boundaries. A peculiarity of the method is that the number of grid nodes required for reaching a certain accuracy, and hence the computation time increases as the transmitter approaches the boundary. To retain efficiency in the case of transmitter-boundary proximity, we use the following modification of the method. The main term of the calculated electromagnetic field, namely, the solution for the two-layered model corresponding to the closest boundary, is calculated by the well-known explicit formulas, whereas the anomalous field is found from the system of integral equations. The modified method is implemented and tested on complex realistic 2D formation models as well as basic models to be used in 2D inversion. Numerical simulations of real propagation and azimuthal resistivity tools show that software based on the modified method can be successfully used for synthetic log simulation in complex formations. However, the proposed method has two shortcomings: poor efficiency when the transmitter is close to the boundary endpoints and limitation to isotropic formations. Both are topics for further research. Currently, the singularity near the endpoints is treated by simply refining the grid, which deteriorates efficiency. Generalization to the anisotropy case would include modification of the basic method and derivation of the closed-form solution for the layered model without the assumption of transverse isotropy.

The Numerical Solution of Integral Equations on the Half-Line

We apply the Laguerre transform with respect to time to a time-dependent boundary-value integral equation encountered in the solution of three-dimensional Dirichlet initial-boundary-value problems for the homogeneous wave equation with homogeneous initial conditions by using the retarded potential of single layer. The obtained system of boundary integral equations is reduced to a sequence of Fredholm integral equations of the first kind that differ solely by the recursively dependent right-hand sides. To find their numerical solution, we use the boundary-element method. We establish an asymptotic estimate of the error of numerical solution and present the results of numerical simulations aimed at finding the solutions of retarded-potential integral equations for model examples.

The propagation of harmonic elastic wave in an infinite three-dimensional matrix containing an interacting low-rigidity disk-shaped inclusion and a crack. The problem is reduced to a system of boundary integral equations for functions that characterize jumps of displacements on the inclusion and crack. The unknown functions are determined by numerical solution of the system of boundary integral equations. For the symmetric problem, graphs are given of the dynamic stress intensity factors in the vicinity of the circular inclusion and the crack on the wavenumber for different distances between them and different compliance parameters of the inclusion.

Acoustic radiation from out-of-plane modes of an annular disk using thin and thick plate theories

In this article the authors introduce and analyze a family of simple quadrature methods for strongly elliptic equations of negative order. The results cover all the classical pseudodifferential equations of this kind, and hence the boundary integral equations appearing in applications. The quadrature methods are based on the trapezoidal approximation of the operator with respect to a uniform mesh, and on $\varepsilon $-collocation. They are proved to have the stability and the convergence with the maximal rate $O(h^{ - \beta } )$ of convergence in general, and $O(h^{ - \beta + 1} )$ for some special values of $\varepsilon $. Here $\beta $ is the order of the operator. The method is applicable to systems of boundary integral equations as well. In this connection the authors have conducted some numerical experiments, which confirm their theoretical results.